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1 AD- A ,Ii i~11 1 ~ A(GARD-AR-291 DVIORY GROUP FOR AEROSPACE RESEARCH & DEVELOPMENT FRtJC ANCELLE 922CO NEUILLY SUJR SEINE FRANCE AGARD ADVISORY REPORT 291 -; "TechniCal Status Review Appraisal of the Suitability of Turbulence Models in Flow Calculations (Revue Technique - L'Evaluation de IApplicabilitc des Mod~les de Turbulence dans Ic C a lcul des Ecoulrments) +CWI/, NORTH ATLANTIC TREATY ORGANIZATION Votmfa wyafj

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3 AGARD-AR-291 ADVISORY GROUP FOR AEROSPACE RESEARCH & DEVELOPMENT 7 RUE ANCELLE NEUILLY SUR SEINE FRANCE :..,., - AGARD ADVISORY REPORT 291. Technical Status Review Appraisal of the Suitability of Turbulence Models in Flow Calculations /! j (Revue Technique - L'Evaluation de lapplicabilitdes Modles de Turbulence dans le Calcul des Ecoulements) Om This Advisory Report was prepared at the request of 0 the Fluid Dynamics Panel of AGARD at a Technical Status Review -- held in Friedrichshafen, Germany, 26th April I - North Atlantic Treaty Organization Organisation du Trait* de I'Atlantique Nord Nil inim.ll lmmm m J I m

4 The Mission of AGARD According to its Charter, the mission of AGARD is to bring together the leading personalities of the NATO nations in the fields of science and technology relating to aerospace for the following purposes: - Recommending effective ways for the member nations to use their research and development capabilities for the common benefit of the NATO community; - Providing scientific and technical advice and assistance to the Military Committee in the field of aerospace research and development (with particular regard to its military application); - Continuously stimulating advances in the aerospace sciences relevant to strengthening the common defence posture; - Improving the co-operation among member nations in aerospace research and development; - Exchange of scientific and technical information; - Providing assistance to member nations for the purpose of increasing their scientific and technical potential: - Rendering scientific and technical assistance, as requested, to other NATO bodies and to member nations in connection with research and development problems in the aerospace field. The highest authority within AGARD is the National Delegates Board consisting of officially appointed senior representatives from each member nation. The mission of AGARD is carried out through the Panels which are composed of experts appointed by the National Delegates, the Consultant and Exchange Programme and the Aerospace Applications Studies Programme. The results of AGARD work are reported to the member nations and the NATO Authorities through the AGARD series of publications of which this is one. Participation in AGARD activities is by invitation only and is normally limited to citizens of the NATO nations. The content of this publication has been reproduced directly from material supplied by AGARD or the authors. Published July 1991 Copyright 0 AGARD 1991 All Rights Reserved ISBN I Printed by Spe ialised Printing Serves Limited 40 Chigwl Lane, Loqhion, Esex IGIO 37Z

5 Recent Publications of the Fluid Dynamics Panel AGAIU)OGRAPHS (AG) Experimental Techniques in the Field of Low Density Aerodynamics AGARD AG-318 8(E), April 1991 Techniques Experlusentales Likes A I'Akrodynarnique i Basse Denaiti AGARD AG-3 18 (FR). April 1990 A Survey of Measurements and Measuring Techniques in Rapidly Distorted Compressible Turbulent Boundary Layers AGARD AG-315. May 1989 Reynolds Number Effects in Tnsmonic Flows AGARD AG-303. December 1988 Three Dimensional Grid Generation for Complex Configurations - Recent Progress AGARD AG-309, March 1988 REPORTS (R) Aircraft Dynamics at High Angles of Attack.~ Experiments and Modelling AGARD R-776, Special Course Notes, March 1991 Inverse Methods in Airfoil Design for Aeronautical and Turbomachinery Applications AGARD R-780, Special Course Notes. November 1990 Aerodynamics of Rotorcraft AGARD R-78 1, Special Course Notes, November 1990 Three-Dimensional Supersonic/Hypersonic Flows Including Separation AOARD R-764. Special Course Notes. January 1990 Advances in Cryogenic Wind Tunnel Technology AGARD R-774. Special Course Notes, November 1989 ADVISORY REPORTS (AR) Rotary-Balance Testing for Aircraft Dynamics AGARD AR-265. Report of WG 11. December 1990 calculation of 3D Separated Turbulent Flows in Boundary Layer Limit AGARD AR-255. Report of WG 10, May 1990 Adaptive Wind Tunnel Walls: Technology and Applications AGARD AR-269. Report of WG 12, April 1990 Drag Prediction and Analysis from Computational Fluid Dynamics: State of the Art AGARD AR-256. Technical Status Review. June 1989 CONFERENCE PROCEEDINGS (CP) Vortex Flow Aerodynamics AGARD CP-494, July 1991 Misaile Aerodynamics AGARD CP-493 October 1990 Aemyncs o6.ba Aircraft Controls and of Grond Effects Computational Methods for Aerodynamic Design (Inverse) and Optimization AGARD CP-463, March 1990 Applications of Meshs Generation to Complex 3-D Configuratioms AGARI) CP-464, March 1990

6 Fluid Dynamics of Three-Dimensional Turbulent Shear Flows and Transition AGARD CP-438. April 1989 Validation of Computational Fluid Dynamics AGARD CP-437, December 1988 Aerodynamic Data Accuracy and Quality: Rcquirements and Capabilities in Wind Tunnel Testing AGARD CP-429. July 1988 Aerodynamics of Hypersonic Lifting Vehicles AGARD CP-428. November 1987 Aerodynamic and Related Hydrodynamic Studies Using Water Facilities AGARD CP-413, June 1987 Applications of Computational Fluid Dynamics in Aeronautics AGARD CP-412, November 1986 Store Airframe Aerodynamics AGARD CP-389. August 1986 Unsteady Aerodynamics - Fundamentals and Applications to Aircraft Dynamics AGARD CP-386, November 1985 Aerodynamics and Acoustics of Propellers AGARD CP-366. February 1985 Improvement of Aerodynamic Performance through Boundary Layer Control and High Lift Systems AGARD CP-365, August 1984 Wind Tunnels and Testing Techniques AGARD CP-348, February 1984 Aerodynamics of Vortical Type Flows in Three Dimensions AGARD CP-342, July 1983 Missile Aerodynamics AGARD CP-336, February 1983 Prediction of Aerodynamic Loads on Rotorcraft AGARD CP-334. September 1982 Wall Interference in Wind Tunnels AGARD CP-335. September 1982 Fluid Dynamics of Jets with Applications to V/STOL AGARD CP-308. January 1982 Aerodynamics of Power Plant Installation AGARD CP-30 1, September 1981 Computation of Viscous-Inviscid Interactions AGARD CP February 1981 Subsonic/Transonic Configuration Aerodynamics AGARD CP-285, September 1980 Turbulent Boundary Layers Experiments, Theory and Modelling AGARD CP-27 1, January 1980 Aerodynamic Characteristics of Controls AGARD CP-262, September 1979 High Angle of Attack Aerodynamics AGARD CP-247, January 1979 Dynamic Stability Parameters AGARD CP-235, November 1978 * Unsteady Aerodynamics AGARD CP-227. February 1978 * LamANw-Turbulent Transition AGARD CP-224, October 1977 i.1 1

7 Preface The past decade has seen a rapidly accelerating development of computational methods, following the ever increasing computing power offered by modem technology. The new tool has found immediate and wide application in fluid dynamics initiating a new era in this field by opening unlimited. as we see it at the present time, horizons in research and development with corresponding technical implications for the aerospace industry. In retrospect it appears, that the fluid dynamicist of the sixties or even the seventies, could have hardly imagined that solving problems of considerable complexity as those handled by today's computers, could have been possible within the elapsed short period of time. During the first part of the past decade the ability of the developed computer codes to produce relatively economically a great amount of information regarding flow characteristics around bodies of complex geometries created a sense of euphoria mainly among users of computational techniques. However, it was soon recognized that in order to produce valid answers to questions related to intricate flow cases by employing CFD methods, more is required than simply developing computer capabilities matching the complexities of the problem. Validation procedures, initiated as early as 1968 with the first Stanford conference, have been gradually introduced as an indispensable part of developing valid computer codes. Equally important, it has been also widely recognized, that suitable modelling of physical processes in flows, such as turbulence transport, constitutes a crucial factor for the development of successful prediction codes. This cannot be achieved without adequate understanding of the underlying physical aspects of these phenomena. The Fluid Dynamics Panel of AGARD has shown a keen interest in computational fluid dynamics in general and for turbulence modelling in particular, following very closely the developments in these areas, as related to the arising needs, especially in the aerospace industry. As a result, several activities in the form of symposia, specialists' meetings, working groups etc. have been proposed and organized by the panel during the last decade. In this context the FDP has organized a Technical Status Review activity on the "Appraisal of the Suitability of Turbulence Models in Flow Calculations" which took place in Friedrichshafen, Germany on April 26, 1990, resulting in the publication of the present Advisory Report. The general scientific scope within which the present TSR activity was placed, has been defined as follows: i) To carry out an in depth appraisal of the suitability of existing turbulence models for use in flow calculations. For this purpose, available information from the existing theoretical and experimental work on the subject should be reviewed and critically evaluated. In this process, the underlying physical concepts for each particular turbulence model should be considered as they apply to different flow cases, therefore establishing the range of validity of each model for flow calculations. Existing uncertainties, discrepancies, inadequacies and failures in reported results, rising from inherent limitations in the employed model and/or its inappropriate application, should be pointed out and discussed. ii) To evaluate the potential of existing knowledge regarding turbulence models, in covering present and future needs in the field of flow calculations. iii) Based on the above studies, to issue guidelines for future theoretical and experimental work and propose a number of key experiments and flow calculations cases to be conducted in the future. Since the capability of a Technical Status Review activity to fulfil such an ambitious scope is rather limited, a subcommittee has been initiated and operates, in order to follow developments on the subject and suggest to the panel future appropriate actions to be taken. As the chairman of the Technical Status Review I would like to express my gratitude to each and all members of the committee responsible for organizing and implementing this activity. Their interest and enthusiasm has guaranteed the participation of well-known members of the scientific community working on the subject. I would also like to express my appreciation to the executive of the Fluid Dynamics Panel, Dr WGoodrich for the interest he has shown and the assistance extended to the committee in carrying this effort to its final goal of publishing the present Advisory Report. D.D. Papailiou Chairman of the TSR committee v -

8 Prefface La derire decennie a &ta marquee par le developpement fulgurant des method~es de calcul grimc a laccroisaement de la puissance des ordinateurs offert par les technologies modemnes. Ce nouvel outi a trouve des applications diverses et inuuddiatsa en dynamique des fluides, ouii da permis d'ouvnir un noveau domnained'intirkt avec, apparemosent, des horizons iilmiuis deci serche et diveloppenaesn syant des implications techniques importantes pour l'induatrie aerospatiale. Rhtrospectivement, ii eat certain que le sp~cialiste en dynamnique des fluides des annees soixante ou soixante (fix aurait eu du ins] h imaginer que la rdsoution de probibim aussi complexes que cetix qui sont traites par les ordinateurs modernes serait acquise dans un laps de temps si court. Pendant [a premseire parie de la dernatre decennie, la capacite des codes machine ivohu6s pour generer, a faible codit relatif, o volume considerable de donnees sur lee caracteristiques des ecoulements autour de corps de geometries conmpleses a creei 1'euphorie, principalement chez les utilisateurs des techniques de calcul. Cependant, ii a 6tW vite rcconnu que pour obtenir des reponses valables aux questions touchant des cas d'ecoulements complexes avec des methodes d'aerodynarnique numerique, ii ne suffit pas seulenient de developper lee capacit~s de l'ordinateur pour qu'il corresponde aux complexites du problem Les procedures de validation, dont lee premieres datent de 1968, epoquc de Ia premiere confertence de Stanford, se sont revelees peu h pcu indispensable au developpemnent de codes machine valables. 11 est egalement important de noter qu'il a e&6 largement rcconnu que Ia modelisation adequate de certains proccasus physiques dans les icoulements, tel que le transport de la turbulence, constitue un facteur critique pour Ie developpement de codes de prediction perfonnants. Ccci ne peut se faire sans l'acquisition de connaissances appropries des aspects physiques snusjacents de ces phenomenmes. Le Panel AGARD de [a dynamique des fluides s'intrresse vivement a l'aerodynanuque numerique en general et a Ia modelisation de la turbulence en particulier. HI suit de tri% pres lee travaux en cours dana em domaines, dana la mesure oo ils correspondent aux besoins qui se font sentir, en particulier dana l'industrie acrospatiale. Par consequent, diffirentes activitis telles quc symposia, reunions de spicialistes. groupes de travail etc.. ont ete propoaces et organises par Ie Panel au cours de Ia derme~re decennie. Dans ce cadre, le FDP a dielopped une activite de revue technique de ['&atdel'art intitulee26valuation de I'applicabiliti de modee de turbulence dana le calcul des ecoulements" le 26 avril 1990 h Friedrichshafen en Allemagne. ILe present rapport consultatif est le fruit de certe reunion. Le cadre scientifique general de cc rapport est difini comme suit: i) Faire one evaluation detailee de l'applicabilite des inodiles de turbulence existants pour Ic calcul des ecoulements. A certe fin, lea informations rfsultant des travaux thenriques et experimentaux doivent itre revus et evalues de fan critique. Dans ce proc~di, lee concepts physiques de base pour chaquc modede turbulaice doivn ttre conaiddnfs par rapport A differents cas d'ecoulements. Lea incertitudes, divergences, irsuffisances et ichecs dont dl eat fait mention dana lee resultats. et qui proviennent des limitations propree du modele employe et/ou de son application inopportune doivent etre tignalees et diacutees. ii) Evaluer le potentiel des connaissancee actuellee en cc qui concemne lee modeles de turbulence pour couvrir lee besoins actuels et ftzturs dans le domaine du calcul dee ecoulements. iii) Sur ia base dle ces etudes, etablir des directives pour de futurs travaux th6oriques et experimentaux et proposer on certain nomnbre d'expedriences des et deema calcul d'ecoulements a lancer A l'avenir. Un programme si ambiticux diborde du cadre d'un tel xamen et un sous-comiti a donc cit6 cr&c 11Ia pour mandlat de suivre les developpements dana ce domaine et de recommander des actions futures A prendrhe par le Panel. En tant quc Prisident de cette revue technique de Nt de I'artjc tiens aexprimer ma reconnaissance envers tout et chacun des membres du comite responsable dc l'organisation et de Is misc en oeuvre de cette activud. Grfice a leer motivation et a leer enthousiasme, nous avons pu compter sur la participation de plusieurs membres eminents de la conmmnaute scientifiquc qui travaillent sur ce sujet. Je tiens enfin h remercier l'administrateur du Panel de Ia Dynamnique des Fluides. le Dr W Goodrich. pour l'intirit qu'il a manifesti pour cc project et pour r'aide qu'il a bien svulu apporrer Au comite pour la concretisation de em efforts sous Is forme du present rapport consultatif. D.D. Papailiou vi

9 Fluid Dynamics Panel Chaman: Dr WJ.McCroskey Deputy Chatirman:- Professor Ir.W Slooff Senior Staff Scientist National Aerospace Laboratory NLR US Army Aero Flightdynamics Directorate Anthony Fokkerweg 2 Mail Stop N-258-1I 1059 CM Amsterdams NASA Ames Research Center The Netherlands Moffett Field. CA United States PROGRAMME COMMITTEE Prof. D. Papailiou (Chairman) Prof. Dr T Ytrehus Dept. of Mechanical Engineering Division of Applied Mechanics University of Patras The University of Trondhseim Kato Kostritsi The Norwegian Inst. of Technology Patras Greece N-7034 Trondheim - NTHI Norvody Prof. R.. Decuypere Ecole Royale Militaire Prof. J. Jimenez Chaire de Mcanique Apptiqu~c Escuela Tecnica Superior de Avenue de la Renaissance Ingenieros Aeronauticos B Brussels, Belgium Departansento de Mecanica de Fluidos Plaza del Cardenal Cisneros 3 M. l'ing. G~n~ral B. Monnerie Madrid. Spain Directeur Adjoint de W~rodynamnique D.Kya pour les Applications DTU.Sayna ONERA UA B.P. 72 Havactlik ye Uzay San. A.S Chitillon Cedex. France P.K. 18 Kavalidere 1)6690 Ankara. Turkey Dr. WSchmidt Messerschmitt-Bdlkow-Blohm GmbH Prof. A.D. Young Director. Air Vehicle Engineering 71) Gilbert Road Military Airplane Division Cambridge CB4 3PD Muinchen 80, Germany United Kingdom Prof. M. Onorato Dr S. Lekoudis Dipartinlento di Ingegneria Director (Acting) Mechanic8 Div. Aeronautics e Spaziale Code 1132 I'olitecnico di Torino Office of Naval Research C. so Dues degli Abruzzi 24 Arlington. VA )129 Torino, Italy United States Ir A. Elsenaar National Aerospace Laboratory NLR Anthony Fokkerweg, CM Amsterdam, The Netherlands It PANEL EXECUTIVE j. Dr WGoodrich Monl fron Etupe: Mail from US mid Canada: AGARD-OTAN AGARD-NATO Attn: FDP Executive Attn: FDP Executive 7 me Ancelle APONewYorkO9777 F Neuilly-sur-Seinc France Tel: 33 (1) Telex: F Telefax: 33 (1)

10 Contents Page Recent Publications of the Fluid Dynamics Panel Preface Preface Fluid Dynamics Panel and Programme Committee iii v vi vii Reference Turbulence Modelling: Survey of Activities in Belgium and the Netherlands, an Appraisal of the Status and a View on the Prospects by B. van den Berg I Calculation of Turbulent Compressible Flows 2 by I Cousteix Some Current Approaches in Turbulence Modelling 3 byw.rodi Turbulence Models for Natural Convection Flows along a Vertical Heated Plane 4 by D.D. Papailiou Turbulent Flow Modelling in Spain. Overview and Developments 5 by C. Dopazo, R. Aliod and L. Valifio Computational Turbulence Studies in Turkey 6 by I. Kaynak Appraisal of the Suitability of Turbulence Models in Flow Calculations: 7 A UK View on Turbulence Models for Turbulent Shear Flow Calculations by G.M. Lilley Collaborative Testing of Turbulence Models 8 by P. Bradshaw vifii

11 ,I 2 I-1 TURBULENCR MODELLING: SURVEY OF ACTIVITIES IN BELGIUM AND THE NETHERLANDS, APPRAISAL OF THE STATUS AND A VIEW ON THE PROSPECTS AN B. van den Berg National Aerospace Laboratory NIX P.O. Box BN Amsterdam The Netherlands SUMKARY 2- SURVEY OF LOCAL ACTIVITIES Turbulence research proceeding presently at various At NIX in the Netherlands the activities in 3D places in the Netherlands and Belgium is briefly turbulent boundary layer research have been taken reviewed. Subsequently some experimental results up again in an extensive measurement program of obtained in turbulent boundary layers, as occur on mean flow and turbulence properties in the shear airplane wings, are considered in relation to the layers of a swept wing (Fig. 1). This is a European usual turbulence model assumptions. The status of collaborative project with companion measurements turbulence modelling is found not to be satisfac- in the French F2 wind tunnel (Van den Berg, 1988), tory. To support the development of semi-empirical Further work at NLR concerns 2D turbulent shear models of acceptable accuracy, a more extensive layers, namely turbulence measurements in wakes at base of reliable turbulence data would be desir- strong adverse pressure gradients, as occur on able. wings with flaps. k constant in mixing length relation I mixing length p static pressure q turbulent kinetic energy u,v,w fluctuating velocities U,V mean velocities, in x- and y-direction x,y coordinates, approximately parallel and normal to the mean flow direction 6 boundary layer thickness 6* displacement thickness W pressure gradient parameter r shear stress Subscripts e at boundary layer edge tr at transition s at shock W at wall r... I. INTRODUCTION The need for reasonably accurate turbulence models is becoming more and more pressing due to the progress made with the development of efficient solution procedures for the Reynolds-averaged Navier- Fig. 1 Garteur swept ling model for 3D turbulent Stokes equations. Though turbulent flow research is shear layer measurements in the NLR 3 x 2 being done at many places for quite some time now, wind tunnel progress has been slow. The field of research is wide and comprisea very different flow types, e.g. turbulent boundary layers along airplanes or ships, atmospheric boundary layers, open water flows, internal flows, flows with chemical reactions, etc. A sumary of the research work going on presently At Delft University three research groups are acin Belgium and the Netherlands will be given here. tive in turbulence research. At the Faculty of The mein aim of the paper, however, is to review Aerospace Engineering an experimental investigation concisely the state of the art in general and to of the flow in the vicinity of an airfoil trailing discuss the prospects for accurate turbulence edge is near completion (Absil & PasschLer, 1990). models. Pitot, Laser-Doppler and hot-wire measurements are carried out on a fine measurement grid to resolve After the simary of local work in section 2, the the sudden local changes in the flow near the traistate of the art in turbulence modelling will be ling edge (Fig. 2). At the seame Faculty recently exemplified for a few simple flows in section 3. On measurements ae" been initiated to obtain turthe basis of the results of the brief analysis some conclusions will be drawn. Finally in section 4 the bulence data downstreem of a Iminar separation bubble, as well as in turbulent wakes at lar author's view on the status and prospects of turbulence modelling will be expressed. adverse pressure gradients to supplont the LA date.

12 1-2 v y tmm) 30 '. stratified flows are investigated (Uittenbogsard & Baron, 1989). The application of turbulence models for the prediction of flows with suspended par- 10.>' titles is considered at Shell Laboratories in 0 -Amsterdam (Roekaerts, 1989). 10 L... % In Belgium turbulence research work also is going on at several places. At VKI, Rhode-SL-Gdn&se, both.20 the Aerospace Department and the Environmental and Applied Fluid Dynamics Department are active in the.....field. One research topic is the 3D turbulent flow in ducts of different shapes for Mach numbers up to Mean vlody, UIU ref one. These flows are being computed with the parab- 30 olized Reynolds-averaged Navier-Stokes equations, 201 using different turbulence models. Navier-Stokes ulmpm).solutions for 2D airfoils at low Reynolds numbers Y(M have also been obtained (Alsalihi. 1989) and extensons of the work employing special low.reynolds- 0. number turbulence models are in progress..10 Further work at the Aerospace Department of VKI y(mm) "20 comprises the development of a one-equation model for the prediction of the transition from laminar X 10. to turbulent flow (Deyle & Grundmann, 1990). Similar work, using modifications of a two-equation turbulence model, is going on at Ghent University. N=a skess, uide 30 Figure 3 shows the turbulence intensity development 20 in the transition region of a boundary layer as measured and computed with a two-equation model and taking into account the turbulent flow intermitten- 10 cy (Vancouilli & Dick, 1988) 'W~t C-c,bn: Vmcl wd... & MWk Dic Fig. 2 Laser-Doppler measurement results at 0.5% chord behind an airfoil trailing edge in.i43 the Delft University 1.8 x2.25 m2 wind tucflel r At the Fluid Dynamics Division of the Faculty of 10 i Mechanical Engineering work on numerical simulation of turbulent flows is going on. modelling the small-scale eddies. The large eddy simulations have 5 * been done primarily for atmospheric boundary layers (Van Haren & Nieuwstadt, 1987), but extensions to \7"" *. " m x engineering flows are in progress. Experimentally, x coherent structures are being investigated in a turbulent boundary layer in a water channel. A Y (mm} project on turbulence manipulation by surface ribleta is carried out in cooperation with Eindhoven Fig. 3 Turbulence intensity in a boundary layer University, Faculty of Physics (Pulles et al, transition region. Comparison between expe- 1989). Though the direct objective is here viscous riment and calculations at Ghent University drag reduction, an important byproduct may be a based on a turbulence model with an allobetter comprehension of turbulent flow structu' wance for Intermittency near the surface. A new project at gindhoven University concerns measurements on the decay of swirl in a pipe to support turbulence moel At the Environmental and Applied Fluid Dynamics development for these flows. Department of VKI non-linear algebraic Reynolds stress models have been tested with encouraging At the Faculty of Physics, Delft University, natur- results for the prediction of the turbulent flow al convection boundary layers ale S a heated verti- over a back-ward facing step (Benocci & Skovaard, cal plate and in a square cavity with a heated 1989). An other direction of research concerns vertical side wall are investigated using different droplet transport in turbulent flows using difturbulence models (see e.g. Lankhorst, Honkes & ferent turbulence models. Further. large-eddy simu- Hoogedoorn, 1988). For these calculations a spe- lations of turbulent flows are being carried out, cial-purpose processor has bean developed to solve first in a 2D channel (Pinelli & Bennoci. 1989), efficiently the Naviar-Stokes equations: the Delft Navier Stokes Processor. Turbulence modelling for ship stern flows, using the parabolizod Reynolds-averaged Navier-Stokes while extensions to 3D channels and free-shear layers are under development. Numerical simulations of the large eddies in turbulent flown is alsn research topic of Louvain University, Finally, at the Freo University of Brussels extensive moasureequations, is studied at the Dutch maritime re- ment, have been carried out in the shear layers of search institute KARIN (Hookstra, 1987). At the an oscillating NACA 0012 airfoil (Dn Inyk at &I. Delft Hydraulics Laboratory turbulence models for 1988). Figure 4 shas typical results obtained with i ti.....

13 II 1-3 a rotating single hot-wire in the partially separa- -2- ced trailing edge region. Work on the validation of turbulence models has been initiated. 0(8/C) 10-2 Tie C) AW Wcal Reduced freq a) scae: - 40% CP 0 O.2-10 b) oc taxlu mi4ajure scwe- 4% SCO-.4% CPCRIT b) \0 0- A4 1/l / Fig. 4 Some hot-wire measurement results near an oscillating airfoil in the Brussel Free University I x 2 m2 wind tunnel Large adverse pressure gradients normally occur over the rear part of the airfoil, especially on the upper surface. In figure 5 typical order of magnitudes for the boundary layer pressure gradient parameter,. - (S*/r.) (dp/dx), are indicated. Ac- curate pree;f.tions of the turbulent boundary layer development in these adverse pressure gradient regions are a further requirement. On the upper surface moreover a small region with very large pressure gradients often exists due to the presence of a shock wave in the flow. The interaction of the shock with the boundary layer can have a large effect on the viscous flow development and this is an example of a complex region where turbulence modelling requires special attention. It is clear from the foregoing that interesting contributions are being made at various places in Belgium and the Netherlands. One comment may be that, so far work on modelling is being done, it concerns in general earlier applications of existing turbulence models rather than the development of new models or new concepts. Further the survey of work suggests that the various activities seem to progress sometimes rather independently, notwithstanding the small scale of the countries considered. This can be at least partly attributed to a specific interest in a particular type of rurbulent flow. Since this review is related to an AGARD activity, the type of turbulent flow on aircraft configurations will be discussed in more detail in the following section. 3. STATUS FOR SOME SIMPLE AIRFOIL P11W1 F Fig. 5 Typical pressure distribution on a modern airfoil section and associated levels of the B.L. pressure gradient parameter, n= (6;/T w )(dp/dx) Attached turbulent boundary layers are often tom- puted using algebraic eddy-viscosity or mixinglength models. These models contain at least three empirical constants. In the wall region of the In reviews of turbulence modelling attention Is often focussed on complex turbulent flows and the A -0.4y deficiencies of turbulence models in these circumstances. However, turbulent shear flows on prac- 0.l0, tical airplanes are for the most part simple boun- Jt/8 dory layer flows. An accurate prediction of these 0.06_ 73 simple turbulent flows is a prerequisite for satis- 1.9 / factory calculations of the whole flow field. For 05"X.0 this reason it was regarded useful to discuss here 5 / - - turbulence models for simple shear flows, and to refer to companion papers In this AGARD report for x a discussion of more complex turbulent flows To support the discussion a typical pressure distribution on a modern transonic airfoil has been O.02 plotted in figure 5. In the first place it may be noted that along a considerable part of the airfoil - L upper surface the surface pressure gradient is close to zero. The boundary layer development along A that part will generally differ little from that along a flat plate at constant pressure. As flow velocities are high, local viscous losses will be Fig. 6 Mixing lengths deduced from turbulent shear comparatively large, Consequently, any turbulence stress measuremeats in equilibrium boundamodel should at least yield accurate predictions ry layers at various pressure gradlits for constent-preosure turbulent boundary layers. (Sest et &l, 1979)

14 1-4 boundary layer outside the viscous sublayer the bulent boundary layers at favourable pressure graturbulent shear stress, r, is normally expressed in dients and in three-dimensional boundary layers the mixing length relation r- 12p(aU/y) 2 with (Spalart, 1986, 1989). The obtained solutions of I - k.y. In this relation is y the distance to the the time-dependent three-dimensional Navier-Stokes wall and k one of the empirical constants, the so- equations also seem to indicate that the log law called Von Karman constant. Evidence Is gradually velocity variation in the wall region is preserved accumulating, however, that the value of k is not a and that the mixing-length relation is altered. constant, but depends on the surface pressure gradient. Mixing length variations, deduced from tur- On the whole, substantial evidence has been accumubulence measurements in equilibrium boundary layers lated that the value of k in the mixing length at different non-dimensionalized pressure gradients relation might not be constant. Yet in the majority., have been plotted in figure 6 (East et al, of algebraic mixing-length or eddy-viscosity tut- 1979). Accoiding to these measurements the value of bulence models now in use, k is assumed constant. k varies widely and differs generally substantially The value of k is known to have a comparatively from the usually assumed value for the constant, large effect on the calculated boundary layer devek lopment. Referring to figure 5, typical adverse pressure gradients on airfoils, x - 2 or more, The measurement data shown were obtained a decade would lead according to figure 7 to a k-increment ago, but the point that k varies with the pressure of 20 % or more. The effect of neglecting the varigradient has been made even earlier, amongst others ation of k in the calculations is likely, thereby a group at Cambridge University (Galbraith, fore, to be far from negligible. Sjolander & Head, 1977). One of their plots has been reproduced in figure 7, completed with the An other assumed empirical constant in many models data from figure 6. The turbulent shear stresses is the dimensionless eddy viscosity, u*/u.6*, or were not directly measured in the cases considered mixing length, 1/6, in the outer region of the initially e. Cambridge. Instead, the stresses were turbulent boundary layer. However, turbulence hisderived f'om the equations of motion and the measu- tory effects are known from several experiments to red mea,, velocity profiles, which were represented be very important in this region and to affect the by a rrofile family assuming a log law velocity value of the "constant" substantially. To ilvariation in the wall region: u - (in y* + A)/k,, lusrrate the severity of history effects, the with k - constant. It is easy to see that, if the mixing length development downstream of a laminarshear stress variation in the wall region is not turbulent transition region has been shown in figunegligible, the latter implies that k in 1 - k.y re 8. The data were obtained at Delft University in can not be constant and must vary according to k - a boundary layer at a small adverse pressure gradik.(r/r.). To a first approximation the shear stress ent (Van Oudheusden, 1985). The value of k is convariatior near the wall may be written as r/r. - i + stant here within the measureeent scatter, but the x(y/6*). This suggests that if k. - constant, the mixing length ratio 1/6 in the outer region is seen average value of k in the wall region will vary to vary considerably. The ratio 1/6 appears to be globally as k - ku(l+c.m). This relation with k - twice as large at the first measurement station 0.4 and a suitable value of C appears to than the ratio at the last station, where the represent the found variation of k with x well. mixing length has relaxed to a more normal level. This means also, however, that conclusions about k The last station is situated at nearly 80 boundary based on the presumption that k - constant, i.e. layer thicknesses behind the transition region. The that the log law velocity variation is preserved in relaxation of the transitional flow with high turpressure gradient flows, are not convincing. bulent shear stresses to the lower stresses in equilibrium conditions appears to take a con- This objection does not hold, however, for the data siderable streamwise distance. of figure 7 based on turbulent shear stress measurements,,hich support the claimed variation of k 0.4 wii. pressure gradient. Further support has come rccently from direct numerical simulations of tur- _9. (i-i =0.41 y k. 3 J /0.4 ttal " Ond&hk /" ari iia C M I.... Fig. 8 Mixing leagthe deduced from measuremmnt in an advorao-praure-radiot turbulent Fig. 7 Effect of pressure gradient on value o4 k boundary layer developing downstream of a in t- k. y for equilibrium boundary layers transition region (van Oudheuden, 1965)

15 ]-5 8- Y - - o x xo, S o o 2 -- o - / " / A oj o. U 01 rg U. -T e U 2 Fig. 9 Boundary layer properties downstream of a shockwave B.L. interaction region. Comparison with data at saw Hi and R in sore near-equilibrium conditions (Delery et al, 1986) On transonic airfoils, shock boundary layer inter- gy. q - 1/2(u 2 +v 0 -+w7), reads for two-dimensional actions may lead to substantial changes in the mean flows: turbulence properties, which also relax downstream only slowly. The turbulence intensities and shear U L + V L n - (J'0) a + diffusionstresses measured downstream of a shock at a dis- ax ay a Tu dissipation tance of about 50 boundary layer thicknesses are compared in figure 9 with the corresponding data in turbulence turbutence a turbulent boundary layer in more near-equilibrium history production conditions (Delery & Marvin, 1986). Evidently turbulent stresses are still very high at this large In the usual thin shear layer approximation, the distance behind the shock. As indicated in figure spatial derivative in the approximate mean flow 5, the boundary layer thickness on the airfoil direction, a/ax - O(L-'), is supposed to be much upper surface at the position of the shock, say at smaller than the crisswise derivative, a/dy - about mid-chord, is typically of the order of 1 % 0(6-1). as L >> 6 in thin shear layers. In that case chord. Consequently the adverse pressure gradient turbulence history, as represented by the leftregion behind the shock up to the airfoil trailing hand side terms of the above equation, would be edge is likely to be dominated by turbulence his- negligibly small compared to the production term. tory effects of the shock boundary layer interac- The large turbulence history effects found in extion. perent in thin shear layers must come, therefore. from other terms in the turbulence transport equa- It is clear that turbulence history effects are not tions, most likely the dissipation term responding taken into account at all in simple algebraic eddy- slowly to flow condition c anges. Note that this is viscosity and mixing-length models. However, also one of the terms relying heavily on semi-empirical more advanced models often fail to predict the modtlling and therefore not necessarily predicting correct order of magnitude of turbulence history a history effect of the right mount. effects, even if they are based on turbulence transport equations, which they usually are. The tur- Finally the effect of three-dimenalonality of the bulence transport equations can be derived by suit- mean flow on the turbulence properties will be able manipulations from the Navier-Stokes equa- briefly discussed. As turbulence is inherently tions. The equation for the turbulent kinetic ener. three-dimensional one might assume that the genera- 0"-O! - t B.L. twista*igm 1.04!,- S " 'mu Fig. 10 Mixing lengths deduced from turbulence measurements In a thre-dimeinsonal boundary layer (2lsenaer et al, 1975) I:t

16 1-6i lization of turbulence properties in two-dimen- different flows without taking the time-dependence sional shear flows to three-dimensional flows would of the flow into account, or without simulating at be straightforward. Measurements indicate, however, least the large eddies in time-dependent calculathat this is not true and that the three-dimen- tions. sionality of the mean flow tends to reduce turbulence activity in general and the turbulent shear If a universal turbulence model ioes not exists, stresses especially. As an example the mixing turbulence models with a limited validity, restriclength values derived from turbulence measurements ted to certain turbulent flow conditions, must be in a two-dimensional boundary layer developing into accepted. Then turbulence modelling comes down to a three-dimensional one are shown in figure 10 approximating for certain flow conditions the (Elsenaar & Boelsma, 1975). The mixing length ratio Reynolds stress terms by a convenient mathematical 1/6 in the outer region of the boundary layer is model on the basis of available data, usually from seen to drop substantially, when the amount of experiments. In this view turbulence modelling is twist of the boundary layer velocity profiles be- much more a semi-empirical art of engineering level comes significant. Turbulence history effects may than a fundamental scientific problem. Note that play a role again, but it is likely that in addi- even if universal validity can not be attained, it tion the velocity profile twist hampers the develo- remains useful to attempt to stretch the range of pment of turbulent eddies and so leads to a reduc- validity of semi-empirical turbulence models, since tion of turbulence activity (Bradshaw & Pontikos, zonal models valid in parts of the flow may be 1985). Few turbulence models available up to now expected to pose problems at the zone boundaries. predict such a decrease of turbulent shear stress magnitude with mean-flow hree-dimensionality. The The foundation for any development of a semi-emnon-alignment of the shear stress and the velocity pirical model should be a reliable and extensive gradient, i.e. the non-isotropic eddy viscosity data base. The data must come from experiments or observed experimentally, is neither taken into from solutions of the full time-dependent Navieraccount in most existing turbulence models. Stokes equations. Although the latter source of data is promising for the future, most turbulence The observations made in this section indicate that data presently still come from experiment. A data even for simple boundary layer flows, available base of sufficient extent and reliability is not turbulence models will often not meet the accuracy yet available, even not for simple turbulent flows, requirements, certainly not those made normally by as he examples discussed in this paper indicate. aerospace industry, i.e. prediction accuracies of the order of I % or better. Uncertainties exist even in seemingly well-established data, such as ACKNOWLEDGEMENT the constant in the mixing length relation for the wall region. Acceptable predictions of the tur- Thanks are due to those who informed the author bulence development are still more remote for com- about the turbulence research going on at their plex turbulent flow regions, like shock boundary places, and in particular to Dr. J.A. Essers, who layer interactions. Due to turbulence history, the collected the information from Belgium. downstream effects of such a complex flow region may remain appreciable for quite some distance. On the whole, the position that adequate turbulence R models for accurate predictions are available seems still remote. i. Asbil, L.H.J. and Passchier, D.M., "LDA measurements in the highly asymmetric trailing edge flow of a NLR 7702 airfoiln. Fifth Inter- 4. GENERAL VIEW ON TURBULENCE MODELLING national Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon. It looks sensible to start with a discussion of the origin of the turbulence modelling problem. In the 2. Alsalihi, Z., "Compressible Navierfirst place it is important to note that the equa- Stokes solutions over low Reynolds number airtions which describe turbulent flows are actually foils'. Proc. Int. Conf. on Low Reynolds Number well-known: the time-dependent Navier-Stokes equa- Airfoils (Springer). tions. Consequently one can not state that there is any fundamental problem to be solved in turbulence. 3. Benocci, C. and Skovgaard, H., "Predi- There is only a practical problem, as it is nearly ction of turbulent flow over a backward facing impossible to solve in reality the full time-depen- step". Sixth Conf. on Num. Methods in Laminar dent Navier-Stokes equations for most turbulent and Turbulent Flows, Swansea. flows. Numerical solutions can not be obtained in general because of the fine computational grid in 4. Berg, B. van den, "A European collaboraspace and time needed. To circumvent the problem tive investigation of the three-dimensional the time-averaged or Reynolds-averaged Navier- turbulent shear layers of a swept wing'. ACARD Stokes equations are solved as a rule, but these Conf. Proc. No contain extra unknowns, the apparent stresses due to turbulence or Reynolds stresses. This means that 5. Bradshaw, P., "The turbulence structure the need for turbulence modelling only occurs be- of equilibrium boundary layers". NPL Aero cause the full equations are not being solved. Report It should be noted that there is no reason why it 6. Delery, J. and Marvin, J.G., 'Shockwould be possible to obtain accurate solutions wave boundary layer interactions". AGARDograph using a reduced set of equations. Actually in the No author's opinion it seems unlikely that solving the full time-dependent Navier-Stokes equations can be 7. Deyle, H. and Grundmann, R., *Transition really circumvented. This statement amounts to model for prediction and calculation of bounstating that it is unlikely that a truly universal dary layers". VKI Aerospace Report. turbulence model exists. The evidence about turbulence available up to now does not support univ- 8. East, L.F.. Sawyer, W.G. and Nash. C.R., ersal properties in various flows. Large eddies "An investigation of the structure of equiplay an important role in turbulent flows and the librium turbulent boundary layers". RAE Report large eddy properties are known to differ essen- SR tially for different flow conditions. It is not clear how the large eddies can be predicted for 9. Galbraith, R.A.M., SJolander, S. and Read, S I i

17 mi-7 M.R., 1977 "Mixing length in the wall region flow". VKI Techn. Note 171. of turbulent boundary layers". Aeron. Quart Pulles, C., Krishna Prasad, K. and Nieuwstadt, F.T.M., "Turbulence measurements over 10. Glowacki, N.J. and Chi, S.W., 'Effect of longitudinal micro-grooved surfaces". Appl. pressure gradient on mixing length for equi- Scient. Research librium turbulent boundary layers". AIM Paper Roekaerts, D., Sixth Conf. on Num. 11. Haren, L. van and Nieuwstadt. F.T.M., Methods in Laminar and Turbulent FlowsSwansea. "The behavior of passive and buoyant plumes in 18. Ruyk, J. de, Hazarika, B. and Hirsch, Ch a convective boundary layer, as simulated with - "Transition and turbulence structure in the a large eddy model". Journ. Appl. Meteor. 28, boundary layers of an oscillating airfoil" Free Univ. Brussel VUB Report STR Hoekstra, M., "Computation of steady 19. Spalart, P.R "Numerical study of sinkviscous flow near a ship's stern". Notes on flow boundary layers". Journ. Fluid Mech. 172, Num. Fluid Mech. 17, 45 (Vieweg) Lankhorst, A.M., Henkes, R.A.W.M. and 20. Spalart, P.R., "Theoretical and numeri- Hoogedoorn, C.J., "Natural convection in cal study of a three-dimensional turbulent cavities at high Rayleigh numbers, computations and validation". Proc. of Second UK Nat, Conf. boundary layer". Journ. Fluid Medh. 2Q, 319. on Heat Transfer. 21. Uittenbogaard, R.E. and Baron, F "A 14. proposal: Extension of the q 2 - t model for Oudheusden, B.W. van, "Experimental stably stratified flows with transport of ininvestigation of transition and the development ternal wave energy". Seventh Symp. Turb. Shear of turbulence in a boundary layer flow in an Flows, Stanford. adverse pressure gradient". Delft Univ., Dept. Aerospace Eng., Master Thesis. 22. Vancoilli, G. and Dick, E., "A turbulence model for the numerical simulation of 15, Pinelli, A. and Benocci, C., "Large eddy the transition zone in a boundary layer". simulation of fully turbulent plane channel Journ. Eng. Fluid Mech. 1. I.-- "1

18 2-i CALCULATION OF TURBULENT COMPRESSIBLE FLOWS J. COUSTEIX ONERA/CERT Complexe scientifique de Rangueil 2 avenue E. Belin Toulouse Cidex FRANCE SU~MMARY s ij 6u Ia 1 av fij = 2 U 1 sij + This paper discusses the use and the suitability 3 of turbulence models for calculating compres where < - > denotes a statistical average. sible flows in Aerodynamics. As the compressible form of turbulence models is generally The main question is to choose how to decom extended from a basic incompressible form, the pose < p ui> and <p ui uj> in equations (I) and emphasis is placed on the pertinence of these (2). The decomposition of <fij> is less important extensions and on the peculiarities of corn- because the viscous stresses are negligible in pressible flows, turbulent flow. A discussion of this problem has been done by CHASSAING, INTRODUCTION The pressure and the density are decomposed The calculation of compressible turbulent flows as : in Aerodynamics has been performed by using P = <P> + P <P> = 0 more or less empirical methods, These methods P = < P > + '; < P,> = 0 include correlations techniques and integral but many possibilities can be used to decommethods for calculating boundary layer flows, pose the velocity. Such methods are still in use today but is is clear that more detailed or more accurate me- If we use the decomposition thods are needed. Ui=<u>+u"i ; <u"i>=0 we have : In the seventies a great hope has been placed <pui>=p><ui>+<p'u"i> in the development of transport equation mo- <puiuj>=<p><ui>.<uj>+<p'u-i><uj> dels to represent the effects of turbulence on + <p' uj > < ui> + <p < u"i > <uj the mean flow. After a period of enthusiasm + < p' u"i u" > where these techniques enabled us to make a decisive step towards a real improvement in Compared with the incompressible case, many the calculation methods of turbulent flows, it additional terms appear and the formulation of appears that the progress is now much slower, equations is very complicated. Most of the works use a mass-weighted velo- c city ui. This type of average has been introdu- ced by FAVRE, 1958, in turbulence studies. We have : This paper is trying to state where we are with the turbulence models, what is being done and what could or should be done to improve these turbulence models, ti= ui u=i+u'i, <pu'i>=0 2. PFECLIARITMS OF TURBULFNT P rompripm9ibly FLOWS and < P Ut> =<p>ui 2.1 Avarkmed aumtions < p ui uj> = < p> Zi nj + < p u'i U' j > Deflnifion Qf averaes For practical applications, the calculation of The average equations ar turbulent flows is approached by using statisti = 0 (4) cal equtions which are at a;i at a; at -x i 1 ax( at ax 2x aui at u> q~l,- (2)- wit : F, i < 2I sj - 3 s j at ~ j -- ax i 8X i It

19 _2 Equations (4) and (5) have an "usual" form in Kinetic energy equations the sense that they look like their incompressible counterpart. The mean value of kinetic energy is decomposed as The concept of average stream surface has the same meaning as in incompressible flow. In L <puiui>=.l pui Ui + I addition, <pu'iu'i>= the pk equations + pk for the Reynolds (8) stresses - < p' u'i u) > are obtained in a natural way and not too many additional terms are in- The first part K corresponds to the averaged troduced. motion and the second part k to the fluctua- The danger is that the equations are too close tions. to the incompressible case and it is tempting to apply them an "incompressible" closure,which The corresponding equations is are not always justified. D- P aui F _U 1 + pu. aui It is worth mentioning that many other de- Pt= k <puxiu k>composition are possible and this problem is (9a) not closed. +-a[ui (-Pgik + Fik - <pu'iu'k> Nevertheless A the mass-weighted averages are axk used in the rest of this paper., u'i. au' > Equations with mass-weighted averages Dt ax axi + <Pu'iu'k >-U + 9b)i To simplify the writing of the averaged equa- alk P i (9b) tions, <p> is written p. For example, we have + <u'ip'> ik + <fiku'i>- <pu'kuiui > but when p is combined u= with u a random func- 1 tion inside a sign <>, not apply the :with same convention does <pu'iu'j> * <P> <u'iu'j> fik = 2 - au/ Sk JU+ auk When no confusion is expected, the sign (~) is 3 ax; 2\axk x 1 ) omitted Ui-- i Fik = < fik > h = h The dissipation rate of the instantaneous kine- Continuity equation tic energy is p = fik -.i Its averaged value is ap+ PUk = (6) decomposed as : at axk <q> + pb with ogmentum equation ap - lu>) (7 P'-D- -(-P8k+fi u.iuk) ()4 ax i axk D t with 1 014V=Fkau, Fik=2 at >au', sk- 8k 0 is the dissipation rate of the kinetic energy of Sik = aui +auk the averaged motion which appears in eq. 9a. 2ik + ax, axi x <p'> is the netic dissipation energy rate of the of averaged the fluctuating ki- motion which At high REYNOLDS number, the viscous stress appears in eq. 9b. The exchange of energy bet- Fik is negligible compared with the turbulent ween K and k is represented by the work of stress - P < u'i U'k >. Near a wall, this is no Ion- the REYNOLDS stresses - <pu'iu'k > U i ger true. Even if the fluctuations of viscosity axk are neglected, the expression of Fik is not simple because eaue u < U a <pui> In the k-equation, < ana',u'> aa the compressibility appears a, aexplicity through two terms :<p' -> and axl C1

20 This does not mean however that 2.2 Variations of density and temnera- P axi *u e the compressibility cannot influence other terms. REYNOLDS stress equations We define Tij as The flows under consideration are characterized by a high Mach number and very often by a heat transfer at walls. Therefore heat is produced by direct. dissipation and transferred by the turbulent fluctuations. Tij = < p u'i u'j > / p These phenomena imply a non uniform averaged temperature and density, which influence The REYNOLDS stress equation is the velocity field. D +(10) In a boundary layer developing on an adiabatic P-=p Pij-p Dij+pij+pCij+----(p aixk Jijk) (10) wall, the large amount of dissipation near the wall leads to a large static temperature in this P Pij p Ti k i U " - P T.k au region. Then the kinematic viscosity is larger axk axk than near the external edge of the boundary layer and the local REYNOLDS number is p Dij < ufik.'.> + < ufj 1 > smaller. Compared with an incompressible axk axk boundary layer, the viscous sublayer is thicker. )+---q/> S ui,. + u' 1 The variation of density in itself does not imply P1ij = < p' a modification of the turbulence structure. L x axj xi example, the study by BROWN and ROSHKO of a low-speed mixing layer with a mixing of gases = <P*u'i> ap <pu 1 > ap with different densities showed that the P axj P axi spreading rate of the layer is not affected by the variation of density. On the contrary, the P Jijk <pu'ku'iu'j > + <fiku'> + <fjku'i> spreading rate of a mixing layer of air is significantly reduced in supersonic flow. This means - <u'i p' > hjk - <u 1 P' > Ik that there is a genuine compressibility effect in this case. It is not clear however if this is due The interpretation of this equation is nearly to an effect on the turbulence structure. At the same as in incompressible flow. PPij is the least partly, the reduction of the spreading rate production term ; pdij is the destruction term can be attributed to an effect of compressibility (p Di is the dissipation of k) ; p 4ij is the on the stability properties of the flow which are at the origin of the large scale structures. velocity-pressure correlation PJijk is the PAPAMOSCHOU-ROSHKO studied ten configuradiffusion due to interactions between velocity tions of free shear layers obtained by using the fluctuations, due to viscosity and due to flow of various gases (N2, Ar, He) at various interactions between pressure and velocity. MACH numbers (between 0.2 and 4). These authors introduced a convective MACH number The compressibility appears explicitly only which is defined in a coordinate system moving through the term Ci, but, once again, the mo- with the convection velocity of the dominant delied form of the other terms can be influen- waves and structures of the shear layers. The ced by compressibility. For example, the mo- theoretical analysis is performed by studying delling of Oij is based on the POISSON equation the stability of a compressible inviscid vortex for p which is obtained by taking the diver- sheet. PAPAMOSCHOU-ROSHKO showed that the gence of the momentum equation. In this growth rates of the various free shear layers equation, many additional terms appear due to fall nearly onto a single curve. This result indithe compressibility of the flow cates that the shear layer question is more related to a stability problem than to a turbua 2 lence problem. - A.-,.- 2 a (<P>uU=)*2 a, (Pu U) Deeampsition of the fluetuatine + (p., U ) field In three mod" * p u' ge'm - <p n', U'.>) (11) Another feature of compreusible flows is that axtaxi, all the flow characteristics we fluctuating : ve- 2p 2 locity, temperature, density and pressure. t- a a ' F) KOVASZNAY showed that tnese fluctuadons cam - ~ -.-- be exptessed as a fuactiom o three bas modes

21 2-4 (see FAVRE et al.) : fluctuations of vorticity, possible vorticity generation interactions are entropy and pressure (acoustic fluctuations) vorticity-entropy, vorticity-acoustic pressure, and when the level of fluctuations is low, the entropy-acoustic pressure). equations describing the evolution of vorticity and pressure are separated. (The correlation From experimental data, MORKOVIN, 1961, coefficients between the various modes are not showed that the acoustic mode and the entropy necessarily low). mode are negligible in boundary layers with usual rates of heat transfer and Me < 5. When the fluctuations are no longer low, a second order theory predicts various possible According to MORKOVIN, these flows are such interactions between modes. In supersonic as flows, a strong interaction is the vorticity-vorticity interaction which is at the origin of the << I ; 1i <<I aerodynamic noise. P These flows are also characterized by the pres- Using the state law and assuming that the sure fluctuations (which are isentropic) which velocity fluctuations u'/u are not too large, we can be transmitted over long distances as have MACH waves. The loss of turbulent energy by sound radiation is low but the radiated energy pi Tu' can have a strong effect on the laminar-turbu- p-- T -- ) M 2 u lent transition. In supersonic and hypersonic wind tunnels, the transition on a flat plate or The following relationships are deduced on a cone is strongly dependent on the noise generated by the turbulent boundary layers <P2/2 =<.2>1 developing on the test section walls. The <-1) M 2 UI2a) ( transition location is correlated with the test p T 1 section size because the noise affecting the r, T,= <11T'> = -I transition depends on the distance between the (< u ' 2 > < T' 2 >)I /2 (12b) model and the tunnel walls. The role played by the pressure fluctuations in This is the so-called strong REYNOLDS analogy. the turbulence modelling can be very different In fact, the basic hypotheses presented above in compressible or incompressible flows. For are not very well founded and improvements example, the influence of compressibility on of the analysis have been proposed by the <V axi ul' > term appears through the GAVIGLIO, Nevertheless, practical results such as formulae (12) can give reasonable orders of magnitude. For example, in a boun- POISSON equation for the pressure which con- dary layer on an adiabatic wall in supersonic talus much more terms in a compressible flow. flo, ra.'j is of order The averaged pressure gradient can also modify the processes of turbulence generation or The use of the strong REYNOLDS analogy should be done with care, in particular when the flow destruction in supersonic flows. The pressure undergoes rapid variations. gradient can be very strong (through a shock wave or an expansion fan) and the interaction It should also be noticed that some of the forwith the term <pus> in the turbulent kinetic mulae deduced from the strong REYNOLDS p analogy are not galilean invariant ; this is the energy equation can be significant. This also case of the formula (12a) because M 2 /U is not means that the wall curvature is an important galilean INVARIANT. parameter because it implies the existence of a normal pressure gradient and an effect on the BRADSHAW, 1977, associated the validity of turbulence, the MORKOVIN hypothesis with low values of p>l2. For boundary layers with an external 2.4 The MORKOVIN hvnothnesi MACH number lower than 5, the condition is Let us go back to the decomposition of the tur- fulfilled as <p2>1/2 is smaller than 0.1. bulent field into three modes : vorticity, en- P tropy and acoustic pressure. At low MACH BRADSHAW noticed that at higher MACH numnumbers in an isothermal flow, only the vor- bers, the total temperature fluctuations are no ticity made remains. In a compressible flow, i longer negligible but when the wall is cooled, the voiticity genteratli by interactions bet- the level of temperature and density fluctuaween modes is structure is unaffected negligible, by the compresibility ttr (the number. increases At these only higher slowly MACH with the numbers, MACH the pressure fluctuations increase and the turbo-

22 lence structure can be affected (pressure-vor- 2.S Comniressibillitv transformations 2 ticity and pressure-entropy interactions can generate vorticity fluctuations). The idea that compressible flows behave like BRADSHAW also noticed that in free mixing incompressible flows led many authors to look for transformations which reduce the study of layers, the level of velocity fluctuations compressible boundary layers to that of an <u' 2 >t/2/u can reach 0.3 so that the density equivalent incompressible boundary layer. fluctuations are larger. This implies that the limit of validity of the MORKOVIN hypothesis In fact, there is no method which enables us to (<p,2>t/21p < 0.1) is limited to external MACH transform exactly the equations of a compresnumber less than 1.5. This is in rough agree- sible boundary layer into incompressible ment with experimental data but as already equations. said, it is not clear if the effect of Mach number on the spreading rate of the mixing layer is due Among the various problems, we can cite the to an alteration of the turbulence structure or presence of large temperature gradients norto an effect on the stability of the flow (which mal to the wall, the dissipation effects, the is at the origin of the large structures). fluctuating density terms,... which have no counterpart in incompressible flow. The 'incompressible" behaviour of boundary layers in supersonic flow can be illustrated by The idea of compressibility transformation is examining shear stress profiles (figure 1). however used for specific purposes, for MAISE and McDONALD. 1968, determined the example the study of the law of the wall or the evolution of the shear stress in a flat plate construction of a skin friction law. boundary layer at Me = 0 and Me = 5 for a REYNOLDS number R9 = 10 4 and an adiabatic 2.6 Transition and low REYNOLDS numwall. The comparison of the profiles ber effects - <pu'v'>/w shows that the effect of compressibility is small. Similar results have been ob- To study this problem, BUSHNELL et al., 1975, tained by SANDBORN up to Me = 7. In the same characterized the boundary layer with the paway, quantities like <pu 2 >ftw are not affected rameter 8 based on the thickness 8 by compressibility (SANDBORN, 1974). 8 (,m a/pw)i /2 MAISE and McDONALD also showed that the mixing length distribution is nearly independent + VW of MACH number. This means that the where the index "max" refers to the maximum turbulent shear stress is expressed as shear T in the boundary layer. <puv> p 2(13) S(y)2 Qualitatively, the importance of low REYNOLDS numbers are given in a (8 +, Rx) plane (figure 4). where US has the same evolution of y/5 as in incompressible flow (figure 2). This formula This diagram shows that the effects of low can be used to calculate equilibrium or near REYNOLDS numbers can be important even if equilibrium boundary layers, but no shock PUx wave-boundary layer interaction for example. the REYNOLDS number R 1 = I'e is large. Many features of supersonic boundary layers BUSHNELL et al. observed that the level of the are close to the incompressible case but some mixing length in the outer region can be doueffects of compressibility on turbulence struc- bled or more when P is of order 100 (figure ture can be noticed. For example, the intermit- 5a). However this increase is a function of distency function defined as the fraction of time tance downstream of transition (compare fithat the flow is turbulent is sharper in super- gures 5a and 5b) ; it takes about a distance of sonic flow ; this means that the region with in to wash out the low REYNOLDS number termittent turbulence is narrower in superso- effects. nic flow (figure 3). The transition onset is greatly affected by The entrainment coefficient is also affected by compressibility effects : influence of MACH the MACH number. Compared to an equivalent number, influence of wall to boundary layer boundary layer in incompressible flow, the edge temperatures ratio. ARNAL used the elentrainment coefficient of a boundary layer on transition criterion to evaluate these effects. an adiabatic wall is approximately doubled at Let us recollect the principle of this technique. Me : 5. The stability properties of laminar boundary layers are determinded by solving the ORR- SOMMERFELD equations. These solutions indicate whether small perturbations are stable or unstable (the perturbations we waves charac- 2-5

23 2-6 terized by their frequency and wave length). Another result of these solutions is the amplification by the PECLET number rate of the unstable waves. Then, it is Te = RtP possible to calculate the total amplification rate A/A0 of the most amplified waves. The transi- For air, the PRANDTL number is close to unity tion criterion introduced by VAN INGEN, 1956, so that the thermal field is fully turbulent for and SMITH-GAMBERONI, 1956, tells that the same range of REYNOLDS numbers as the transition occurs when A/A0 reaches a critical velocity field. level en. The factor n is an empirical input which characterizes the quality of the external Then the turbulent heat fluxes are often analyflow : when the external flow is noisy, the va- zed by using the turbulent PRANDTL number Pt lue of n is small (in noisy supersonic or hyper- defined as : sonic wind tunnels, n = 2-4) ; in a clean envi- <puv (P14 ronment, n is of order 8-10 (BUSHNELL et al., - 1 Ipv u (14) 1988) cly ay The calculated effects of MACH number (adiabatic wall) and of the wall to edge tempe- In certain analyses, the value Pt = I is taken. rature ratios are shown in figures 6 et 7. Let This is the so-called REYNOLDS analogy (which us notice that these results are at least in qualitative agreement with experimental data. is different from the strong REYNOLDS ana- logy). For a boundary layer developing on an adiabatic wall, an increase in MACH number 'tabilizes For flat plate boundary layers in air, the turbuthe transition (the transition REYNOLDS num- lent PRANDTL number is of order 't = bet is larger) except in the range 2 < M < 3.5, with a tendency to increase near the wall and where the opposite effect is observed (figure to decrease near the outer edge. 6). Figure 7 shows that, for a given MACH number, a cooling of the wall increases the No systematic effects of MACH number, low transition REYNOLDS number compared to the REYNOLDS number or blowing have been obcase of an adiabatic wall at the same MACH served (BUSHNELL et al, 1977). number. It is also noted that the beneficial effect of wall cooling is less pronounced as the The data of BLACKWELL et al. show a decrease MACH number increases, in ', when the pressure gradient is positive (see LAUNDER, 1976). 2.7 Turbulent heat flux The value of the PRANDTL number can be influenced by boundary conditions. For example, The diffusion of heat in a turbulent boundary on a rough wall compared with a smooth wall. layer is due to the molecular diffusivity and to the increase in heat flux is less than the inthe transport by turbulence. The corresponding fluxes are crease in the skin friction. at On the other hand, the value of the PRANDTL - y and Qt = <v'h> number depends on the type of flow. In free flows, 't is significantly different from unity in When the flow is fully turbulent, the ratio Qt/Qt the central part of the flow. For round jets, P, is of order 0.7. For wakes, values of order 0.5 is large and the thermal transfer is mainly due have been measured. This means that calculato turbulence. ting a compressible turbulent flow with a turbulent As already seen, the correlation between ve- PRANDTL number is a simple solution but not the best. (figure 8). locity and temperature fluctuations is good. The coefficient I <vt'> I/(<v'2><T2>)l/2 can be of The analogy between the fluctuations of velothe order 0.6. Then, the order of magnitude of city and temperature has been extensively QdQ[ is: studied by FULACHIER and ANTONIA. They found that there is a good analogy for the Q1 u f <T'2>1/2 energy-containing pprt of the spectra of the v A' -T temperature fluctuations and of the total velocity fluctuations ; this result has been obtained where, is the PRANDTL number ; AT is a cha- for different types of flows. racteristic temperature difference within the These authors also showed that the spectral boundary layer ; u is a characteristic velocity of distribution of the PRANDTL number is not at turbulence ; I s a characteristic length scale of uion The an l nmeit at turbulence. all unifom. The analogy between velocity and temperature fluctuations is analyzed by using the parameter B: Thus the thermal turbulence is characterized 1

24 Conference. The integral method gives right 1 / 2 y trends in the range of parameters investigated S(Me<5,0.2<Tw/Tad< ). 2-7 ay where q' is the fluctuation of the total velocity. FULACHIER and ANTONIA observed that B varies from flow to flow but is nearly constant within a given flow. In addition, the spectral distribution of B is nearly uniform (except for high values of frequency). CEBECI-SMITH also presented comparisons of experimental skin friction coefficients for adia- batic flat plate boundary layers by using results obtained with their method. The calculations have been performed with the CEBECI- SMITH mixing length model. In the range 0.4 < Me < < Re < , figure I I shows that the calculations reproduce the experiments very well. Another application of the integral method 2.8 Other nroblems proposed by COUSTEIX et al. is shown on figure 12. In the experiments performed by As already seen, the compressibility of the flow HASTINGS-SAWYER, the MACH number is adds nreiy complexities as compared to the in- nearly constant (Me = 4) and the wall is adiacompressible flow. The list of problems discus- batic. Good results are obtained on boundary sed in this section is not complete and many layer thickness and skin friction. other effects could be cited. The following application concerns a flat plate The hypersonic vehicles often have a blunt boundary layer with heat transfer (COLEMAN shape so that a bow shock wave exists in front et al.). The calculations have been first perforof them. Then, the streamlines which cross this med in turbulent flow with a mixing length shock wave do not have the same entropy scheme (figure 13). It seems that the quality of jump and the boundary layer is fed with va- results is poor as the MACH number increases. riable entropy streamlines ; the associated va- Calculations have also been performed by riations of free stream characteristics normal to ARNAL by taking into acount transition effects. the wall can be large. The influence on the tur- In the transition region, the shear stress is calbulence structure is not known. culated by : Instability like GORTLER vortices can develop -<pu'v> = pud in supersonic or hypertsonic flow. This has been ay observed in the flow on a compression ramp for example. The eddy viscosity is given by a mixing length The shock scheme. wave-boundary In incompressible layer interaction flow, the intermitis tency function Z is described according to figure obviously a problem of prime importance in 14 (0 is the momentum thickness calculated at transonic, supersonic and hypersonic flow. This the current point and 0T is the value determitopic great has details. been reviewed by DELERY, 1988, in ned e at ttetasto the transition point). on) In ncmrsil compressible flow, the intermittency function i is expressed by the same function but 0 - I is replaced by 3. EXAMPLES OF CALCULATION OF 0 TURBULENT COMPRESSIBLE FLOWS ( - 1)/( M 2 ) to take into account the 3.1 Flat plate boundary layers lengthening of the transition region at high speeds. In the application shown on figure 15, the transition point is prescribed according to the experiments. Qualitatively, well behaved computational results are obtained but the le- vel of heat fluxes is slightly overestimated in the turbulent region. An integral method has been used (COUSTEIX et al, 1974) to determine the effects of MACH number and wall temperature on the skin friction of the flat plate boundary layer. The integral method is based on the solution of the global equations of continuity, momentum and energy. The closure relationships are obtained from self-similarity solutions calculated with a mixing length scheme. A FIn Smerical results with the VAN DRIEST II results, These latter results are in good agreement with the experimental data and are recommended as references at the STANFORD 3.2 Boundary layer with pressure ea. 32Bnd the experiments of CLUTTER-KAUPS, the boundary layer is studied along a body of revolution with different conditions of velocity, pressure gradient and wall temperature. In the example presented, the MACH number is -~ 1

25 2-8 around 2.5, the pressure gradient is slightly by JONES-MUSONGE, Figure 21 shows the negative and the ratio Tw/Tad is around 0.6. application of this model to the experimental data obtained by TAVOULARIS and CORRSIN in The calculations have been performed using a nearly homogeneous shear flow with a linear three methods : the integral method proposed temperature gradient. The results represented by COUSTEIX et al., a mixing length scheme and in figure 26 are the turbulent PRANDTL numthe k-e JONES-LAUNDER model. This case does ber and the ratio of heat fluxes. Here again, this not pose special difficulties and all the methods case illustrates the value of a model with heat are in good agreement with experiments flux transport equations. (figure 16). JONES-MUSONGE applied their model with the In the experiments of LEWIS et al., 1972, the same sucess to a thermal mixing layer downsboundary layer is studied along the inner wall tream of a turbulence grid and to a slightly of a cylinder and a centerbody placed along the heated plane jet in stagnant surrounds. axis generates the pressure gradient. The wall is adiabatic. The calculations shown in figure 17 have been performed by CEBECI with his 3.5 Shock wave-boundary layer interacmixing length model. The calculated results re- tions produce the effects of adverse and favorable pressure gradient very well. BENAY et al., 1987, performed a critical study of various turbulence models applied to the calculation of shock wave-boundary layer inte- 3.3 Boundary layer with variable wall raction in transonic flow. This study has been temperature performed by using the boundary layer equations solved in the inverse mode (the displa- Extensive studies of boundary layers with cement thickness distribution is introduced as pressure gradient, heat fluxes and blowing and a datum in this calculation method). The ausuction have been performed by MORETTI- thors verified the validity of this approach by KAYS, MOFFAT-KAYS. The case presented in fi- comparison with NAVIER-STOKES solutions. gure 18 deals with a negative pressure gra- They tested mixing length models (from dient and a variable wall temperature. The cal- MICHEL, models with ALBER transport and BALDWIN-LOMAX) equations (the JOHNSONculations and by CEBECI-SMITH follow the experi- KING model which includes an equation for the mental data remarkably well.kigmdlwihicueanqatofrte maximum shear stress, the k-e JONES-LAUNDER model, the Algebraic Stress Model which is 3.4 Calculations with heat flux transport obtained from the RODI proposal applied to the eations HANJALIC-LAUNDER three equation model). In a general way, the authors concluded that the FINSON and WU, 1979, developed a REYNOLDS models with transport equations behave better stress transport equation model which also in- than the other models. The best results are cludes transport equations for the turbulent obtained with the Algebraic Stress Model heat fluxes. This model has been applied with (figures 22b-c-d). It is noticed that the mean the boundary layer approximation. FINSON-WU velocity profiles are well calculated with this used their model to calculate boundary layers model whereas the turbulence characteristics on rough wall. To do this, they added rough- are not. However the experiment reveals that ness functions in the momentum equation, in the flow is not strictly steady and the unsteathe turbulent kinetic energy equation and in diness can interact with turbulence ; on the the dissipation equation. other hand, the experimental data ar not analysed by taking into account this unsteadiness. Figures 19 and 20 show the results of their calculation at low speed and comparisons with Another example is provided by calculations measurements by HEALZER et al.. The calcula- performed at NASA with NAVIER-STOKES tions reproduce the increase in the skin friction equations (see MARVIN-COAKLEY). The expecoefficient and in the STANTON number rimental configuration is depicted in figure (St = ipw/peue(hie-hw)). It is interesting to 23a. The results obtained with three models notice that the increase in the heat flux is less are compared with the experimental data : the than the increase in skin friction. This means CEBECI-SMITH model, the BALDWIN-LOMAX that the REYNOLDS analogy is not preserved, and a q-m model which has been proposed by This case illustrates the interest in using a COAKLEY, 1983 ; this model has been modified model with heat flux transport equations. to take into account compressibility corrections and finally a heat transfer correction is inclu- In incompressible flow, several models have ded (in the eddy viscosity, the length scale bebeen proposed for calculating a scalar field comes f = min(2.4y, q/w) in order to reduce the with scalar flux equations. The scalar can be heat transfer in the region of reattachment). temperature. Such a model has been developed The results are given in figures 23b and 23c L

26 2-9 her which is assumed essentially to be a consfor two angles of the corner :0 = 15 and tant. This hypothesis influences directly the e = 38. A reasonable agreement is obtained calculation of the wall heat fluxes. In many siwith the three models for 0 = 15* ; for the case tuations, the value of the PRANDTL number is 0 = 380, the results obtained with the CEBECI- not the value determined in a flat plate boun- SMITH model have not been given because of dary layer. Therefore, it is certainly valuable to the difficulties in computing the displacement try to develop transport equations for the turthickness distribution. It should also be noticed bulent heat fluxes. This work is often perforthat the overshoot of the heat flux near the med when the temperature can be considered reattachment is not predicted by any model. as a passive scalar which is a first approach to the more general problem of compressible flow. Useful information can be gained from 3.6 Calculation of the free shear layer the studies of the mixing of non reactive gases and from the study of homogeneous compres- The free shear layer is a flow where the inade- sed turbulence (REYNOLDS, 1987). quacy of "incompressible" models has been attributed to compressibility terms. Indeed the The question of including compressibility cormodels extended from the incompressible case rections in the transport equations is not solwithout compressibility effects predict practi- ved because these terms have been used for cally no effect of the MACH number on the rate flows such as the free shear layer or shock of expansion of the free shear layer whereas wave-boundary layer interaction. It seems that the experimental results indicate a decrease in the inclusion of such terms has often been bethe expansion as the MACH number increases. neficial but it is not clear if these compressibi- The results shown in figure 24 are concerned lity terms are completely justified or if they with a shear layer with a zero velocity on one mask other problems. In the case of the shock side and a non zero velocity Ue on the other wave-boundary layer interaction, models are side. The thickness 8 is defined as the distance available which give reasonable agreement on between the points where the velocity is pressure distribution for example, but none of 4-T U, and '1--.9Ue. The computed results them give the viscous parameters with the rehave been produced by BONNET for the quired accuracy. 81 STANFORD Conference. To calculate this flow, BONNET tried to include compressibility A third important problem is the near wall terms in the modelling of the pressure-velocity treatment (which is not specific of the comcorrelation term. He argued that the pressure pressible flows). In most of the applications, a equation in a steady compressible two-dimen- simple model is used (for example a mixing sional thin shear flow suggests that compres- length or a one-equation model) but efforts are sibility affects mainly the return-to-isotropy devoted to develop more general models valid term. Accordingly, the modelled form of this in the fully turbulent region and in the near term is multiplied by a compressibility depen- wall region (LAUNDER-TSELEPIDALIS, 1988). dent factor. Improvements of the same qua.ity Indeed the near wall model is very important have been obtained by VANDROMME and by because not only it influences the prediction of DUSSAUGE-QUINE who also introduced com- the skin friction for example, but also it has an pressibility effects. These compressibility cor- important effect on the numerical behaviour of rections lead to improved results (as compared the model. The numerical properties of the with experimental data) but it is not sure if the models are rather rarely studied, although they effects of MACH number are attributable to are of great practical relevance ; indeed it is modifications of the turbulence structure or to not very useful to have a well physically founa problem of stability which modifies the large ded model which leads to untractable numeristructures of the shear layer in which case it is cal difficulties. not justified to accuse the turbulence model. The evaluation of turbulence models is based on comparisons with experimental data but it Cappears that accurate data are not very numerous in compressible flows. Surprisingly good The calculations of classical compressible tur- data are available for complex flows like bulent boundary layers not too far from equili- shock-wave/boundary layer interactions but brium have often been approached with rather for simpler boundary layer flows the data are simple models extended in a straightforward limited. Good and detailed data are needed to manner from the incompressible case. For evaluate the effects of MACH number, wall these caea, this approach is justified even at temperature, pressure gradient. For this latter MACH numbers as high as 10 except perhaps case (effect of pressure gradient), simple expefor the calculation of wall heat flux where some rimonts are difficult because the generation of uncertainty is still present. Indeed, in most of strearwise pressure gradient induces the prethe calculations, the turbulent heat flux is sence of a pressure gradient normal to the wall. evaluated by using a turbulent PRANDTL num- Now, when boundary layer codes are used the I:.

27 2-It) presence of normal pressure gradient is not ac- BUSHNELL D.M., CARY A.M., HOLLEY B.B. counted for. Then it could be instructive to de- "Mixing length in low REYNOLDS number comvelop second order boundary layer codes to pressible turbulent boundary layers" AIAA analyze the validity of turbulence models in Journal Vol. 13 NO 8 (1975) such cases. BUSHNELL D.M., CARY A.M., HARRIS J.E. Another direction which is being worked on to "Calculation methods for compressible turbuimprove turbulence models is the use of full lent boundary layers" VKI Lecture Series LS86 numerical simulations. In incompressible flow, (1976) NASA SP-422 (1977) the results of these simulations have led to improvements (for example for the study of rota- BUSHNELL D.M., MALIK M.R., HARVEY tion effects) and it is believed that the numeri- W.D. "Transition prediction in external flows cal simulations are useful to generate complete via linear stability theory" IUTAM Symp. sets of data for basic flows and the analysis of Transonicum III, G6ttingen (may 1988) these data is certainly a precious guide to improve turbulence models. CEBECI T., SMITH A.M.O. "Analysis of turbulent boundary layers" Academic Press (1974) It is also worth mentionning the problem of laminar-turbulent transition. The accurate CHASSAING P. "Une alternative A ia formulaprediction of the transition location and of the tion des Equations du mouvement d'un fluide A transition length are obviously very important masse volumique variable" Journal de parameters for the description of the flow but Mdcanique Thdorique et Appliqu~e Vol. 4 n' 3 the influence on the characteristics of the (1985) downstream turbulent flow is not very well known. CHAMBERS A.J., ANTONIA D.A., FULACHIER L. "Turbulent Prandtl number and spectral characteristics of a turbulent REERENCEN mixing layer" Int. J. Heat Mass Transfer, Vol. 28 n' 8 (1985) ARNAL D. "Laminar-turbulent transition problems in supersonic and hypersonic flows" CLUTTER D.W., KAUPS K. "Wind tunnel in- AGARD FDP VKI Special Course on vestigations of turbulent boundary layers on Aerothermodynamics of Hypersonic Vehicules axially symmetric bodies at supersonic speeds" (30 May-3 June 1988) DOUGLAS Aircraft Division Rep. NO LB BALDWIN B.S., LOMAX H. "Thin layer ap- (1964) proximation and algebraic model for separated COAKLEY T.J. "Turbulence modelling methods turbulent flows" AIAA Paper NO for the compressible NAVIER-STOKES equa- (1978) tions" AIAA , DANVERS, MA (JULY 1983) BENAY R., COET M.C., DELERY J. "Validation de modilies de turbulence appliqu6s A l'interaction onde de choc-couche limite trans- COLEMAN G.T., ELESTROM G.M., STOLLERY JL. "Turbulent boundary layers at supersonic sonique" La Rech. Airosp. N* (1987) and hypersonic speeds" AGARD CP 93 (1971) BLACKWELL B.F., KAYS W.M., MOFFAT RJ COUSTEIX J., HOUDEVILLE R., MICHEL R. Rept. HMT-16, Mech. Eng. Dept., Stanford "Couches limites turbulentes avec transfert de University (1972) chaleur" La Rech. Atrosp. NO (1974) BONNET J.P. "Comparison of computation with DELERY J. "Shock/shock and shock experiment" AFOSR-HTTM STANFORD wave/boundary layer interactions in hyperso- Conference on Complex Turbulent Flows, Ed. SJ. nic flows" AGARD FDP VIK Special Course on KLINE, BJ. CAN'WELL, G.M LILLEY Aerothermodynamics of Hypersonic Vehicles (i988) BRADSHAW P. "Compressible turbulent shear layers" Annual Review of Fluid Mechanics, DUSSAUGE J.P., QUINE C. "A second order (1977) closure for supersonic turbulent flows - Application to the supersonic mixing" BROWN G.L., ROSHKO A. "On density effects Workshop on "the Physics of Compressible and large structures in turbulent mixing Turbulent Mixing" Princeton. NJ. (1988) To be layers" J.F.M. Vol. 64 part 4 (1974) published in "Lecture notes in engineering" BUSHNELL D.M., WATSON R.D, HOLLEY Spnger Verlag B.B. "MACH and REYNOLDS number effects on turbulent skin friction reduction by injection" J. Spacecraft Vol. 12 NO8 (1975)

28 2-li FAVRE A., KOVASZNAY L.S.G., DUMAS R., MAISE G., McDONALD M. "Mixing length and GAVIGLIO J., COANTIC M. "La turbulence en kinematic eddy viscosity in a compressible m6canique des fluides" Ed. GAUTHIER- boundary layer" AIAA J. 6, 73 (1968) VILLARS (1976) MARVIN J.G., COAKLEY T.J. "Turbulence FINSON M.L., WU P.K.S. "At. of rough modelling of hypersonic flows" The Second wall turbulent heating with apphiwtion to Joint EUROPE/US. Short Course in Hypersonics, blunted flight vehicles" AIAA Paper Colorado Springs (1989) (1979) MEIER H.U., ROTTA J.C. "Temperature dis- FULACHIER L., ANTONIA R.A. "Spectral re- tributions in supersonic turbulent boundary lationships between velocity and temperature layers" AIAA J. 9, 2149 (1971) fluctuations in turbulent shear flows" Phys. Fluids 26(8) pp (1983) MICHEL R. "Couches limites - Frottement et transfert de chaleur" Cours de l'ecole Nationale GAVIGLIO J. "Reynolds analogies and expe- Supdrieure de l'a~ronautique (1967) rimental study of heat transfer in the supersonic boundary layer" Int. J. Heat Mass MOFFAT R.J., KAYS W.M. "The turbulent Transfer Vol. 30 N' 5, pp (1987) boundary layer on a porous plate ; experimental heat transfer with uniform blowing and HANJALIC K., LAUNDER B.E. "A Reynolds suction" Int. 1. Heat Mass Transfer II (1968) stress model of turbulence and its application to thin shear flows" J.F.M. Vol. 52, Part 4 MORETTI B.M., KAYS W.M. "Heat transfer to (1972) a turbulent boundary layer with varying free stream velocity and varying surface tempera- HANJALIC K., LAUNDER B.E. "Contribution ture : an experimental study" Int. J. Heat Mass towards a Reynolds stress closure for low Transfer 8 (1965) Reynolds number turbulence" J.F.M. Vol. 74 n' 3 (1976) MORKOVIN M.V. "Effects of compressibility on turbulent flows" C....- n. CNRS N' 108, HASTINGS R.L., SAWYER W.G. "Turbulent Mdcanique de lia -'rbulence, Ed. CNRS (1961) boundary layers on a large flat plate at M = 4" RAE TR P,, IAMOSCHOU D., ROSHKO A. "The compresible turbulent shear layer : an experimen- HEALZER J.M., MOFFAT R.J., KAYS W.M. tal study" J.F.M. Vol. 197, pp (1988) "The turbulent boundary layer on a rough porous plate : experimental heat transfer with REYNOLDS 4A.C. "Fundamentals of turbulence uniform blowing' Thermosciences Division, for turbulence modelling and simulation" Dept. of Mech. Eng., STANFORD University AGARD Report n* 755 (1987) Report n* HMT-18 (1974) RODi W. "The prediction of free boundary JONES W.P., LAUNDER B.E. "The prediction layers by use of a two-equation model of turof laminarization with a two-equation model of bulence" Ph. D. Thesis, University of LONDON turbulence" Int. I. of Heat Mass Transfer Vol. (1972) 15 (1972) SANDBORN V.A. "A review of turbulence JONES W.P., MUSONGE P. "Closure of the measurements in compressible flow" NASA TM Reynolds stress and scalar flux equations" X-62, 337 (1974) Phys. Fluids 31 (12) pp (1988) SMITH A.M.O., GAMBERONI N. "Transition, LAUNDER B.E. "Heat and mass transport" in pressure gradient and stability theory" "Turbulence" Ed. P. BRADSHAW - Topics in DOUGLAS Aircraft Co. Report ES26 388, EL Applied Physics - Vol. 12 Springer Verlag, SEGUNDO, CALIFORNIA (1956) BERLIN-HEIDELBERG-NEW YORK (1976) TAVOULARIS S. CORRSIN S. "Experiments in LAUNDER B.E., TSELEPIDAKIS D.P. nearly homogeneous turbulent shear flow with "Contribution to the second-moment modelling a uniform mean temperature gradient" J.F.M. of sublayer turbulent transport" Zoran Zaric Vol. 104 (1981) Memorial, BELGRAD (1988) VAN DRIEST E.R. "Turbulent boundary layer LEWIS J.E, GRAN R.L., KUBOTA T. "An ex- in compressible fluids" J. Aeron.Sci. 18, 145 periment in the adiabatic compressible turbu- (1951) lent boundary layer in adverse and favorable pressure gradients" J. Fluid Mech. 51 (1972)

29 2-12 VANDROMME D. "Contribution A la modlijsation et A la pr6diction d'dcoulements turbulents & masse volumnique variable" Doctoral d'etat _ Universit6 de LILLE (1983).8C VAN INGEN J.L. "A suggested semi-empirical.8low SPEED 1 method for the calculation of the boundary0 layer transition region" Univ. of Technology,.60 Dept. of Aero. Eng., Rept. UTH-34. DELFT (1956) 0 FIGURES 0 MAISIE & MCDONALD M OWEN AND HORSTMAN, M "BEST ESTIMATE" OF SUPERSONIC 03 LAOERMAN AND DEMETRIADES, M o SHEAR STRESS OISTRfBUTIC"*:.'A, L L IAf <M_ < N Y/ 6.8 Fgure3 :Effect of Mach number on the intermittency function from SANDBORN. Figure taken from BUSHNELL et al., M- I "FULLY DEVELOPED" X ILEDANOFF, Tr TURBULENT FLOW INCOMPRESSIBLEM3 AZ WT. ZORIC, INCOMPRESSIBLEM6 SIMILARITY REYNOLD CS Tw/TT -'0. 7 ~ 00 REYNOLDS 0.2 A via EiueI Effect of Mach number on the shear stress distribution in a flat plate boundary layer Figure taken from MAR VIN-COAKLEY i C3 re... Fguare 4 : Importance of low Reynolds number C03 effects at high Mach number.08 From BUSHNELL et at., J ' C0 KLEBANOFF FLAT V6 n16 PLATE M"-0 Rea E3 BRADSHAW AND FERRIS.0-MODERATE ADVERSE PRESSURE.12? GRADIENT M - 0 Re DISTANCE FROM WALL, y/3 06 a Figure 2 Effect of Mach number on the mixing , j+ 1tI le~ length distribution from MAISE and McDonald. Flue i Variation of the outer level of the Figure taken from MfAR VIN-COAKLEY mixing length downstream of natural transition on plates, cones and cylinders From BUSHNELL etal

30 NOZZLE WALL CASE TYPICAL " TRANSITION"\ M< - M>l TYPICAL MEASURING Rx' REGO STATION.14 t MeO.9 [A JU6)m 9 T ,, 4 1o 5 Taw Eivure : Effects of low boundary layer Reynolds number on the outer level of the mixing length for flows on nozzle walls without 7 laminarization-retransition 1 From BUSHNELL et al.. p p p Figure 7 Theoretical effect of wall cooling on the flat plate transition Reynolds number as calculated by the en method (n=9) -6 From ARNAL, RX Rxto is the transition Reynolds number for n= 10 Tw/Taw = I (adiabatic wall). Rxto depends on the value fo the Mach number P t , , 0, ,00,0 0,2 0,4 0, 0's 1,0 2 Fia... 8a 1.2-,,o M e A -! ~0,6 Prandtl number in a boundary layer From MEIER-ROTTA Figur 6 Application of the en method to the,4. boundary layer transition on a sharp cone with an adiabatic wall 02 From ARNAL, O's 1.0 I',s Figff,.k : Prandtl number in a plane jet From FULACHIER-ANTONIA i. :

31 , St " 0., C/CO 0,4, 0,6" ,00 0,10 17 :igna..8c Prandtl number in a mixing layer From CHAMBERS et al. 0,0 0,0 0,2 0,4 0,6 0,8 1,0 a a a a 0 a wt o.6, eure 10 Effect of wall temperature on the 0.6. skin friction of a flat plate boundary layer (Me=5) 0,4- Tw = wall temperature - Tad = adiabatic wall 0.2i temperature, Free flight at m.r = Cf = a ox/h 11 From VAN DRIEST 11 applied to KARMAN- SCH(ONHERR equation :iue8 Prandtl number in a homogeneous 13 Integral method shear flow From TAVOULARI$-CORRSIN 1,2-0,, '--. 0,4" 0 1, 2 3 0, Figr I Comparison of calculated and Aft experimental local skin friction values for adiabatic flat plates :igua-2 Effect of Mach number on the skin Calculations from CEBECI-SMITH friction of a flat plate boundary layer (insulated wall) Free flight at m RB = Cf 0 = From VAN DRIEST i applied to KARMAN- SCHONHERR equation 1 Integral method

32 r-, Me I IPle= Pa 3.8( (Mm) [ Cf 10-3 ~ ~ o razor blade 0 Xb lance Xi)50 Eivure12 : Calculation of the experiments of HASTINGS-SAWYER with an integral method I " Z.. I Ch A qi MeS5 5.1i0-4 ~~e 2.10-'transitionI Exp. Coleman et at. +00 ME-9 V ' 4 Fiure 1 Calculations of COLEMAN et al experiments with a mixing length scheme

33 Fimn.1 :Intermittency function used by ARNAL in incompressible flow Experiments Colema., Elfsrom and Stoityry * 1.t6 0. Calculatons (WlCml T' K 10 i 20 I TTi MS ' 2~ o ,XIm) 'I 0 Fivure, : Calculations of COLEMAN et al experiments taking into account the transition region.o., From ARNAL, L) oo o I A OfLEW.S ET &.,q? () is l 110 -F- igre1i : Calculations of LEWIS et al experiments by CEBECI-SMITH 404 (k*wjn- 2 ) X(in) Eig : Calculation of CLUTrER-KAUPS experiments - integral method mixing length k - i model (JONES-LAUNDER) 2%'

34 DATA OF MORETTI -KAYS (1965)12 U, 000 '1 a i 'z p%% 6Pb &ap _ P 00 q:? 4 0, 12 L 5 6 -<'T > St 2 - t2 X 10 0 ) iaure 21 Scalar turbulence in a nearly X 00 ~ homogeneous shear flow with a linear mean Figure 18 Boundary layer with variable wall temperature temperature gradient -4 Experiment TAVOULARIS-CORRSIN... Experiments MORETI-KAYS Calculations: JONES-MUSONGE Calculations CEBECI-SMITH MESUREMENTS z.47f9i 0 U_ 32 It/$.C t! 0 ISO z CALCULATIONS U. 1R0UGH WAL- 20 U ~ 4242 ft / Figure 19 Calculations of the skin friction 5 coefficient on a roughened flat plate _ By FINSON-WV 005In 0U. 3 t " lay rshock a e Calculations of a boundary 190 IS ae/sokwv interaction with an inverse 2 CACW.A42 ~ boundary layer method -MC 0 = 1.36 I, CACLAIN The experimental displacement thickness U. distribution is used as an input of the method From BENA Y et al. -SMOOTH WALL fgr20 Calculations of (he heat transfer coefficient on a roughened flat plate By FINSON-WI)

35 2-18 y (mm) Michel et al. 20X ,2 125,5 154, i Alber 0... Baldwin-Lomax Y (mm) BadiLoa X-~ 20IX = ,4 328, ,6,:::.~.~z. 11.-~ ).H "~.* _ J nn-ng [kae...asm 0.2B a X - X o Y(mii Y (l [ X =38, , ,2 experimental pressure distribution 10" Johnsn-Kig ::: El : Compa rison- bteen g ulate and expeimenl bonda ilye velocity profiles jii-

36 Y (mm) Michel et al. 20 X 38, , ,2 115 (a) 0 EXPc 10 KLL Alber5 10 0_ oo(i Y (mm) Baldwin-Lomax "_'" -, 20 X = , ,4 328, L -- Figure 23b Comparison between calculated and 10; ' experimental data on a compression corner flow 20 -o=ois* - Pressure and heat transfer go. EXP 10 "s., // o.:.l70 -* q-,bc 0 kl , Y (mm) (k, el model 2 0 X = ,2 125, / " 30 10?:b) 1:Es~~* i --u'v" ISM Ib) bewe i t5 Figure 22d : Comparison between calculated and experimental boundary layer shear stress 10 profiles M-g.20 - r 1r,. -/ I ///// " 38'. Hj;rn. 2 Comparison between calculated and experimental data on a compression corner flow Fiure 23a :. Study of a compression corner flow -, = 38 - Pressure and beat transfer From MARVIN-CAOKLEY distributions!x

37 4t df 0, Figure 24 Calculation of the free shear layer expansion rate 0 experimental data Calculations (J.P. BONNET)

38 3-1 SOME CURRENT APPROACHES IN TURBULENCE MODELLING W. Rodi Institute for Hydromechanics University of Kartwuhe 7500 Karlauhe, F.R. Germany SUMMARY fields as well as for simulating transport and history eftects and the anisotropy of turbulence, and they automatically The paper reviews some recent work in the area of modelling account for certain extra effects on turbulence such as due turbulence in near-wall regions and by Reynolds-stress-equation models. Various low-reynolds-number versions of the k-c to streamline curvature, rotation, buoyancy and flow dilatation. RSE models have recently been subjected to model and their damping functions are examined with the aid of fairly wide testing and are presently built into commercial results from direct numerical simulations and they are compared CFD codes. However, the main testing and application has with respect to their performance in calculating boundary layers under adverse and favourable pressure gradients. A two-layer been restricted to fairly basic versions of RSE models which do not satisfy certain theoretical requirements like the model is presented in which near-wall regions are resolved with a realisability conditions and compatability with the limiting one-equation model and the core region with the standard k-e 2D state of turbulence near walls. Another active area of model Various applications of this model are shown. The ability research is the development of more refined models which of the various models to simulate laminar-turbulent transitioni in boundary layers is discussed. Recent applications of a fairly do satisfy these requirements. Further, RSE models were applied so far mainly in connection with wall functions, but simple, standard Reynolds-stress-equation model to two complex recently research has focused also on near-wall RSE flows of practical interest are presented. Finally, the ppper reports modelling. on some recent proposals for improved Reynolds-sr-,r-equation models and provides an outlook on possible fit,rc turbulence- Much of the recent turbulence modelling activities described model developments, briefly under (ii) and (iii) was prompted by the increase in computing power. This also triggered another development in turbulence research, as direct numerical simulations are now 1. INTRODUCTION possible, at least for moderate Reynolds numbers. The direct simulations performed to-date already provide a wealth of data In conventional turbulence w"alling, whose task it is to provide that allow the checking of each and every detail of the model models for calculating the turbulent stresses appearing in the Reynolds-averaged e-uadons, three main approaches have proposals and may form the basis for devising improved models. Hence, turbulence-model development and testing benefits emerged: increasingly from the direct numerical simulation data, but since these are so far restricted to fairly low Reynolds numbers, the (i) Fairly sirnpte eddy-viscosity models, which are either greatest impact of the direct simulations was so far on near-wall entirely algebraic and of the mixing-lenth-model type modelling. (Cebeci-Smith [1], Baldwin-Lomax [2]) or employ an ordinary differential equation for the maxhnim shear stress The present paper describes some recent developments in areas (Johnson and King [3]) are used extensively for calculating where research is particularly active, namely the areas of nearte flow over airfoils and turbomachinery blades at all wall modelling and the application of basic RSE models to speeds. The usefulness of the various models depends on complex flows. Further, trends in the development of more the degree of flow separation. While the entirely algebraic refined Reynolds-stress closures are briefly discussed. Of models cannot simulate transport and history effects, the necessity, only some areas of recent turbulerc-modelling work Johnson-King model can account for such effects and can be covered, and the ppe focuses on incompressible flow therefore appears to perform best in the presence of separation regions. However, this model was designed for and on the modelling of Reynolds stresses (rather than turbulent heat and mass fluxes). external s,,odynamic flows with relatively small separation zones and is not so suitable for other situations with 2. NEAR-WAIL MODELLING masvely separated flow. The simple eddy-viscosity models listed in the Introduction under (i) The second masin class of turbulence models in piecal use (i) resolve the viscous sublayer nea walls by relying on the van today are the k-a-type models employing two differential Driest damping model for the eddy viscosity. The application of equations for calc i the eddy viscosity, one for the velocity scale and the other for the length scale of the these models is restricted mainly to external aerodynamic flows without massive separation. Dictated by the limitations of turbulent fluctuations. k-e-type models ae widely used for computer resources, most practical calculations with mote practical calculations and we built into many commercial complex turbulence models such as k-a- and Reynolds-stressgeneral-purpose CFD codes. They can handle situations equation models were carried out so far by bridgig the rather with massive separation; yet there as no guarantee for good thn, viscosity-affected near-wall layer by wall functions. The accuracy. The use of an isotropic eddy viscosity, a ofen gradietsiin this Isr wa thereby not resolved as establishing a close link between the tobalens sure and the first point away ftom C wall wa placed outside the the strain rate, is probably too simple for highly non- viscous aublayer. As described in ine detail in [4] for the k-. isomrpic: situations prviin In os with complex sluai model, the wall funitons relating the velocity and turbulence fields. Most preac" k-mel calcullations are still catle queaulles at the firs grid point mainly toth dietio velocity lean out with wel bealms Wtdgite viscous sublayer, but hevlon the assmtion ofalpiua elctsiabto reenl alo low-reynoldanumer versions resolvingl slis andoflcleuibumftruen.thnaspinsw sublyew be mer popular, and these versiona moa be certainly not gerally valid, for example not whe strong used whesever the flow isatrnsitional. Considera secondary flows extend into the ublayer [51 and als not in mmcli aivihy ima been pvi on in the uea of ne-wall paaed flows The wall fuistions am of eourse particularly modelon&unuu for sepaain d rencutgio n Due to ccon him., in 00os pam W the neesto tan wall fnction Oi) The third class of models in use an Reyon-stmss- is now much, and cosiderable acityw recenly equit (R!) mdek also know ma lela di1w1sed 1 towass tesa; ad dewioplag lowsaoemes. They do no employ the lsemlor e ty mpo modls for me )Cmoltlag viny nean nwmentio ais In but solve model-trusapeet equations fior the Individlual the folowinig, om ofthe. evle will be decrbfr k-e- Reynod nmese They we bow sulsed fur comsplex smain type models. Is should be mn ioein thil contex thst for 6.

39 3-2 sungly compreisble aerodynamic flows, there is some vend in the oppoim direction. namely to use specially developed wall functions [6]. For various transonic and supersonic flows, d Mavin[7 repons superir k- model predictions with these wall.. deduced from f ions over the use ofalow-reversion. experiments lawjgrkgm s 1 - rom direct simulotio l5)... ft 0~ Various nw-wall versions of the widely used k-e turbulence * f - - mdlhave been Virpoed, and the pre 1994 models have been reviewed by Patel et a. [8]. These versions employ the eddy- - visosityconcept and detemine the eddy viscosity vt from Mk[ l V, =.- 2 (1) 18] 9._-LB fi- function C N111 in Cc /. SMII} whe!i is a damping funcdion and k and e are determined from I 292v1k from l.] the foll6wing equations: 0.1 fro 8k k ia1ato 0r0vi - jtv+l- Ok J --C (2) Fig. 1: ftr-function i ow-re k-e~models 81 r0.2ie In some model versions, t is equal to the actual dissipation e, while inoetherit is " =e-d, whereddepends ontheversion consdered and is non-j-i ong.y in the viscosity-affeted region. Equation (3) for the varishle e contans the additional damping funtons f and v f2 and in smie versios an extra tem E. The versions published before 1984, their damping functions and the 0 extra terns involved have been discussedin detail by Patel 8] y Here, three mote recent versions [9, 10, 11] are included and Fi.2Dsubto ofcladnenchnlfrt treissues are adrssd namey the f -funcon.omparison 2 direc smlton [YSJdot nd~= with direct simulatio dta, and the performnce for bondary sml, [ layers under adverse pointed and favourable out already in [8], the damping presstute gradients. of the turbulent As was When cit is chosen as 0.09 and the actual dissipation rate is momenum transfer ner wallsisdue to both viscos effectsand taken for ' in (I), there follows the fis-distibution shown in the reducton of velocity fluctatons normal to the wall by the i g. I from Gilberts direct simultio which is in surprising Eis3re-reflect mechanism. The secon d mechanism is, to first accor with the distibtion deduced by Pateaei a. [8] from separate the two viscosity-dependent effects they ar f..-functio usually although both they modelled are p by the y in pointed crvy out by na Chapman h wal whr tbhvsa - and Kuhn [17]. The data also follow /y a correlated only for the viscosity efects. Thi must be considered a pragmatic approch which is physically not veay sound [12], this trnd and Myong an Kaag [91 functon in their.hatve.atull k- moel which exhiits tis behavior used af.- whie but ra prper delneaion of the two processes is pssble ony in the fb.-funedon used in the rather poplar model of Lam and the bhreaiouear context of e a nodrmalmfluctuating Reynolds-stress-equation c onntand model where the idmprin Breinche an is usedin [d18 goes (1) which to goesto ss at the wall. On the other hand, wen oat the wall, fcn behaves as awe simlated direty by Use model~bhvoro 01 h ~mlfutail n n t apn fp - y. The models of Nagano and Hishida, NH [ 10], and Shih and Mansor, SM [11], rom ti high-re ssio-n uatd o model of Gibs and Launder dependent fs-functions follow are this also approach inluded. Their in Fig. strictly. y+- SM [13] applied to a local-equaldbium shear layer follows that t he dent erined teir function with the aid of the direct smution shear.rean is related to the velocity adieu by -uv = reslts of Kim t al [19. Itris thereforenolt ssrising t their with Ai/ft jy wh illow-re _. models se -uv vt from ( T i). yb fc=ua.92 insnrk if the formula =,vtoau/ from the fttfunction deduced from agrees direct coely simlation, with the albeit f 1 -curve a diffent in Hg. one I whichwas [15]. Very Gibo-Lander model is applicabe so the low-re reaion. This close to the wall areement is alo.onde, bu t this is not obviou relationsip derived by under [ 14] Is omared in i f. 1 with o m ig. i because, as was mntioned already, as different the f t-stroion dets in (8] from experimental data, taldng definitions have been used. It should be noted here that SM used over the ba fo r the izk-data from (14]. This possi so * C suggests that the near-wall damping expssed byf is mostly eniployed Only an in addiional (I) but pres solved s hfo equation term (3) for e. in the k-c Also, qution they du h o the r of the normal flucatonse, which is mainly (2) which is effective v ar walls. oy controlled by no-viscous wall effects. It threfore appears reasonable to coelat re eriell f with a 114 parameter Iscomare involving infix she wall th firm After Fig. all the I beae above discussion sw a men00 t it must oned be alrmei said ad that d the ie exact distace, e.. y, a wa doe In lie moe rcent pirponals [10, distribution of fp ve,. ner sh wall ha lintle effect o the It d]. be aloul noted, however, that y defined f in the usul predictionsec In I egi fi tre dominlte ovi way via the fritlotve o vel t oity tble only for attached turbulent o ne Thet Is bore out by a ompafriso of kinetic flows. i. 1 als includes he fl -disre lbutio deduced by enercy and eddy-viscosity proih obtained with varos model [15] (i.hss direct simulatio of channel flow. irs, v oe fr canmnlaflown FIa. 3.dTheRey ntone 1,er ofthe * however, sugeosided his results which for showtlht the mdu thfi ftcit reprduced lt yonstant In I. 2 f velocity) w e and lo dicnmlatio (3 bu aed res on e (3)afor aion C. vaile Aso me fo outide the - e gio. These resulct do however is [19]. his cv w from walls 3 that ver new the wail wa ian the fici t on yud ssuae l ohnensl fo rlaehr ones. This isbme oui t b he pain th k- flw.fg loicue h itiuinddcdb m n dyvsolypoie bandwt aiu oe Giler o [31frm canelflw.ph, isdiec Z erullfo canelflw n ig 3 M Rynld mrb.rofth LIIJ I II 1

40 3-3 distribution and also gives too low eddy viscosity at intermediate wall distances. while the Lam-Brembote [1] model yields too 0- large eddy viscosities towards the centre of the channel. C1 MK CH LS * Samuel and Joubeit 0] k 3 a[ C 2 s" 1.5 2S Is a) Adverse-pressure-gradient situation 1 of Samuel and Joubert (30] ios /NR Chien Jones & Lunder 122 & Bremhorst -Loin Shih Nagono & Mansourilil & Hisido ONS (19 - L -,\. 2L *Badfl Nowyaran and Ramjee 1311 " 10 ~~~~0.0o5 2' -- ' O o b) Favourable-pressure-gradient situation y + of Bdri Narayarman and Ramjee (311 Fig. 4: Friction coefficient in boundary layers with su'camwise pressure gradient, from [ ; Fig. 3: Near-wall kinetic energy (k) and eddy-viscosity (vt) NR = Norris-Reynolds (241 one-equation model, distributions in low Re channel flow, from [20] LB = Lam-Brenhorst 118], LS = Launder-Sharma (36], CH Chien (211, NH = Nagano-Hishida 1101, MK = Myong-Kasagi [9] low-re-k-c models, Although corect near-wall and low-reynolds-number behaviour 2L = two-layer model, of a model is an important feature, it does not necessarily guarantee good performance in engineering calculations. In these, At the end of this section it should be emphasized once more that one of the most important parameters to be calculated is the it is of course desirable for a model to have the correct near-wall friction coefficient cf, and Figs. 4a and b show respectively how behaviour, but also that the details of this behaviour very near the various models perform in calculating this coefficient in boundary wall have in fact little influence on the overall model performan e layers under adverse and favourable pressure gradients. The and in particular on the mean-flow quantities of engineering calculations shown were carried out by Fujisawa [25, 261. It interest. The judgement of the perfrmance of the model for should perhaps be mentioned first that all models succeed in engineering calculations can only be based on test calculations for predicting correctly the cf behaviour under um pressure-gradient as wide a variety of flows as possible. Further substantiation on conditions. Fig. 4a shows that all low-re versions of the k-e this point will be made in the section on transition modelling. model overpredict the friction coefficient under adverse pressuregradient conditions. This behaviour was traced by Rodi and 2.2 TwoLAl 1cls Scheuerer (231 to the high-re-fom of the E-equation (and in particular to the value of the coefficient ctl in this equation) Low-Re k-c models have the undesirable feature of requiring employed which leads to a too steep increase of the turbulent very high numerical resolution near the wall, and it was shown length scale under adverse pressure-gradient conditions. Fig. 4a also includes a prediction with the Norris-Reynolds [24] oneabove that they perform rather poorly in adverse-pressure- gradient boundary layers. Fur ther damping fuactions in thes equation model, which only solves an equation for k but uses the usual linear length-scale prescnption. It can be seen that with this models were develosd for attached boundary layers and are not always well behaved in sepaed flows. Hence, in order so saw length-scale specification the friction-coefficient distribution in g points and thesfeow computing time and also to introduce the boundary layers with adverse M gradient cas be predicted fairly well established length-scale distribution very near the wal correctly. The situation i quite diffe ent for boundary layers into model, ie one teout mend in practical calculations is to use under favourable pressure gradient, for which predictions with the k-e model only away from the wall and to reaolve the neamvarious models are shbwn in Fi Norris-Reynods ose-equation model (witho any modification) performs rather wall viscosity-affected layer with a simpler model involving a eno-,cale Prescription. poorly while the Launder-Shtama (LS) and LamBremhorst (LB) versions of the k-e model do fairly well. On the other hand, the Chien (CH). Myong and Kasagl (M) and N gano and Hishida At the University of Karurhe, a two-layer model is under (NH) models also overpredict cf consiably- From these ezusi oln whch uestnam walls thm oae-eqution model doe to calculations it appears that the low-re finctions in the latter N0rl and Reynolds [241 because this has beet found to per00m models are not very suitable for favourable pressure-gradient well in advese-paussure-gradent boundary layes (231. I conditions. the edd viscosty fromt ai

41 3-4 Vt fuc, kl with f, I -ep( F, Ry friction coefficient. The two-layer-model prediction included in (4) Fig. 4b for the favorable-pesr-gradient case shows fairly good agreement with the date. It should be mentioned, however, that this good performance could be achieved only by making the involving the damping fumetion fs In this, the parameter A + is parameter A+ in the damping function fp (4) a function of the usually given a value of 25 exgept r accelering and tansitional pressure gradient as recommended by Crwford and Kays [271: bounda layen as descrbed below. The distribution of k is by solving the k-equaion (2). The dissipation rate e A2(30.175p.1),p * - P (6) appearing in this is determined from dx S (5) A constant value of A+ = 25 was adopted for zero and adverse- YkL/v pressure-gradient boundary layers and is also used for the separated flow calcuatos reported below. which also involves a viscosity influence. The length scale L is assumed proportional to the wall distance y in the near-wall layer Cordes [28] tested the two-layer model for the flow over a to which the application of this model is restricted. The model is backward-facing step as studied experimentally in [29]. Fig. 5 matched with the high-re k-e model at a location where either the compares his calculations obtained with this model and with the damping function fl, or the ratio of turbulent to laminar viscosity, vrv, has a prescribed value (e.g. f± = 0.95 or vt/v = 36) ensuring standard k-e model employing wall functions with measirements. The two-layer model predicts the rett chment length in much that the matching takes place in a region where viscosity effects better agreement with the experiments and also produces a small are small, second corner eddy which is absent in the calculation with the standard k-e model. Also, the velocity and shear-stress The model was tested by Fujisawa et al. [26] for various distributions improve and are in generally good accord with the boundary-layer flows. For a boundary layer with zero pressure data. Fig. 6 shows similar teat calculations performed by Btilule gradient, the mean-flow behaviour was predicted in good [32] for the flow over a T-configuration as studied experimentally agreement with experimental data and so was the shear-stress by Jaroch and Fernholz [33, 34]. Although these authors point distribution, but the predicted near-wall peak in k is too low out the basically thre-dimensional nature of the flow in their (similar to that in Fig. 3 for the Jones-Launder model), experiment, a two-dimensional calculation was performed for the Predictions of the friction coefficient for boundary layers with flow in the symmetry plane. It can be seen from Fig. 6 that again adverse and favourable pressure gradients are included in Figs. the two-layer model predicts the reattachment length in much 4a and b, respectively. The first figure shows that the two-layer better agreement with measurements than the standard k-e model model is only slightly better than the k-e model under adverse- employing wall functions. On the other hand, the velocity profiles pressure-gradient conditions; her it is again the e-equation used indicate that the separation zone is predicted too thin by both in the outer part of the boundary layer which yields too high a models. This then leads to a shift in the shear-stress distribution length scale and in turn also too high a turbulent shear stress and towards the wall. It can further he seen that the shear-stress level ~~~~~~x R~k R2ly ' ~ p a) Calculated streamlines in separation region XIH 20 X/H 4.0 XIH t 50 X/H 12.0 U W O." 1 0' lf t 1 1 I' t I lot lie It 11 It ' e it, is' W i' I 1t W I it' 1W -clv I b) Velocity (LO/ and sha stes (5) profiles Fig. 5: Flow over backwssd-fwing step with aspet ratio of Calcukioer with M.-ayer model (291, -- Calculation with stadard k-e model [24], 0 experiments 1291

42 3-5 X, 1 XR, 2L XR -P MR a) Flow configuration b) calculated streamlines in separation region I- -- X/HF = XjHF X,'F XJHF X/HF =27.75 X/HF = r - 0. ~-0.01 Z , - r - -T - - t Y/HF Y/HF Y/HF Y/HF Y/H r Y/HF c) Velocity (U) and shear stress (v) profiles Fig. 6: Flow over T-configuradon; - Calculatios with 2-layer model [32], --- Calculations with standard k-c model [321, 0 experiments [33] is underpredicted, and it is not entirely clear at present whether Shanrma [36] model, which is a successor to the Jones-Launder the higher shear stress in the experiments is associated with the [221 model, was found to be less sensitive, but there is still some basically three-dimensional nature of the flow. However, for practical purposes the overall features of the flow are fairly influence of the al-value on the transition location. For boundary layers with zero pressure gradient, the usual value of the structure reasonably predicted by the two-layer model. Further testing of parameter, al = 0.3, was found to give the correct transition the two-layer model is in progress, also in three-dimensional location. The models of Chien [21] and Myong and Kasagi [9] situations, were found to be insensitive to the value of al, but transition was predicted by these models to occur considerably too early when 2.3 Transitio Mcompared with the data of Abu Ghannam [37] for Tu = 1.9%. For the prediction of laminar-turbulent transition occurring for example on airfoils and turbine blades, the use of a low- For transition calculations with the two-layer model described in Reynolds-number model is an absolute necessity. With k-c-type 2.2, the parameter A + occurring in the damping function in models natural transition occurring at low free-stream turbulence equation (4) is made a function of the boundary-layer state. In levels (say below 1%) via Tollmien-Schlichting waves cannot be laminar boundary layers, a large value is chosen for A + (here simulated but requires artifical triggering. At higher free-stream 300) so that a small vt results, and for fully turbulent boundary turbulence levels, which are common in turbomachinery flows, layers A + is made a function of the pressure gradient according to transition occurs in a bypass situation due to the turbulent free- (6), with a value of 25 for zero and adverse pressure gradients, as stream disturbances. This mechanism can be simulated by k-e- was described above. In the region of laminar to turbulent type models which mimick the diffusion of free-stream turbulence transition, A + is assumed to vary between these two limiting into the laminar boundary layer. However, in order to start the values accor ling to the following fornula boundary-layer calculations near the leading edge. initial k- and c- 3 profiles must be specified in the laminar boundary layer, and A %A ' o (1-sn '(300-At) R R~ (9) some low-re-versions of the k-e model have been found to be Rtr sensitive to this specification. Usually, the kinetic-energy distribution lrelated o the velocity distributio by which is applied whenever the local momentum thickness Reynolds number Reg is larger than the critical transition k/ke - (U/Ue) 2 (7) Reynolds number Retr but is smaller than twice this number. For the critical Reynolds number, the following empirical dependence where "e" relates to values in the free swean The e-disibution is on free-stream turbulence vel and pressure gradient due to Abu expressed by the following equilibrium approximation: Ghannam and Shaw [381 is taken: e - alk au/ry (8) + m (F(A,) - '(A') -T) 6.91 (10) where at is an empirical structure parameter (- - A) for which Rodi and Scbeuerer [351 proposed a correlation with the free- Suzam turbulence level To for use in the Lam-Bremhor's model AI A4 ', < 0- : Fujisawa mdel version [25] examined to the coeflficient the sensitivi I i d heam of various to the low-re prescription k-e F(A,) 1.T a> A, = -of9 qp-u.a of the initial e-pmofle. He found that the Lam-Bramhorat model is most sensitive, which was the reaso for Rodh and Scheuere to intoduce the above mentioned function for al. The Launder- It where e is the momentum thickness.

43 3-6 3a 3 (b) C H,L B H H B LS 22 L27~ LS Xet et = C 0*1 00'. 0. xm) 0 05if s (M) Xim) a) Zero pressure gradient, Tu = 1.9% b) Favourable pressure gradient, Tu 3.5 % c) Adverse pressure gradient, Tu =1.8 % Fig. 7: Shape factor H in transitional boundary layersfrom [25. 26]. 9 experiments [37], key to model see Fig St a.blair and Werle [391 Fujisawa [25, 26] has tested the ability of the Lam-Bremhorst (LB) model, the Launder-Sharma (LS) model and the two-layer (2L) model described above for calculating the transition of Tu =7.51 boundary layers. For the LS model he used a constant value of al =0.3 in (8). while for the LB-model calculations he used the Tu- 0 - dependent correlation of Rodi and Scheuerer [35]. For boundary layers with zero pressure gradient as studied experimentally by Abu Ghannam [37], Fujisawa found that at Tu = 1.2%. the LB Tu=6.21. model produced no transition while the IS model predicted 0 transition correctly. When the turbulence level is increased, both models lead to transition but at intermediate Tu-levels the LB model produces late transition while the LS model predicts the L 2 Tu =S*I correct location. At higher Tu-levels (3.3% in the calculation 0 example) both models yield roughly the same results and the I*LS correct transition behaviour. For the intermediate turbulence leveli 1 the results for the shape factor are given in Fig. 7a. Similar j L8 Tu z 3/ conclusions can also be drawns from heat transfer calculations. Fig. ga presents the variation of the Stanton number predicted0 with various models for boundary layers with zero pressure is gradient and various free-streamn turbulence levels and provides X(M) comparisons with the data of Blair and Wede [39]. At the lowest a) Zero pressure gradient turbulence level (1.3%), the LB model predicts transition too late and the LS rrwdel slightly too early while the two-layer model yields very goou agreement with the data. When the turbulence... = iio level is raised to 2.5%. transition moves forwatrd and is predicted fairly well by all three models. For the higher turbulence levels (6.2% and 7.5%) transition occurred before the first measurement St station in the unheated-wall pars very close to the leading edge. This very early transition is predicted by all the aiodels. Fig. 7b compares prdce shape factors with the measurements D2 of Abu Ghannam[37 orboundary layer with mild favouablepressure grdient and Tu -3.5%.In ths case, both the LEand.. she LS model yield a satisfactory prediction of transitton. 5 However, for a strongly aceelerated boundary layer, the distribution of the for Stantor number is given which in Fig. gb. th to 1.5 X 2.0 situation is differentl Here the strong acceleration at modest freestream turbulence level (2.1%) causes a delay in the onset of b) Favourable prssure gradient transition and also transition to take place over a aie distac (k -VIUe -duejdx = 0.75 x 10-6), Tu =2.1 % thn thezeo-re radieut case (see Fig. 8a). BohM h LB and the LS Ioe Oredict transition to oceur too early and too Fig. 8: Stanton number St in tsitional boundary layers abrupdy and only the two-layer model AppMr to dio justice toth from (25, 261, key to modiels see Fig. 4 observed transition proessa. Finally. in 1.g 7c the shape factor variation for a situation with avrepesr rdetadt 1.8% is shown. Her, all three s dofairl god job, but 3. REYNOWDS-STRESS-EQUATIObl MODBLLING die two-layer model pict * gender transiton whsichs is closer to die observed behaviour. Due so increased computing power and improved numerical Pro the tat calculadions carried out so far, it appears that the techniques, models employing tansport equations for the individual Reynolds stresses uiu~j have recently undergone L.Aunder-Sharma model is somewhat superior to the Lam- extensive tsting and as now applied also, to relatively cmlex greinbos model with regard to transition simulation and it has flow situations involving recrulation and to 3D flows the advantpe that Iis considerably less sensitive so the Cosiderabe experiencelasnow availa"l with these models, and posseibeditia pofle latthe Isinar"ounary layer. The two- this has been summarilsed in various review articles, e.g. [42, layan whc Modelea &andon corlation 43]. So far, all the aplieatons reporte for practically relevant apersas ver ois...,apresently tsted under flows have been obtained with relatively smle vesin Of unsteady conditions for bonaylayers on turbine blades stesmuto mdl ee~ om hn1 years ago 144. ubete o passing wakes gen e 'by the preceding blade row 13). The model equation for ssiuj used most Benn,(and also 141. in the qiplemo eitanles gvnbelow) is the Vtlwn

44 I ~~~i'so-called rapid part of the pressure-stain term is simulated with Ce a. a., - the analogous isotropation of production model. Near walls, the a. I -all k a1 dsmpng ofte normal fluctaornsdueo the Pe=encOf the Wall rate of amci difnion P,*,...dct is accounted by an extra surface correction to the pressure-strain clwq. 1 model e.g. the one due to Gibson and Launder [13]. Further. the,-..~ i~ tnear-wall region is either bridged by using wall functions or is -a' Uj4.ii - Olk) -21c, ri( Pq q. - T 1 resolved with the aid of simpler near-wall models (two-layer Sapproach discussed abowe). 3-7 * trassmx--strain (11) 3.1 A jwlzzk which, together with a model equation for determining the dissipation rate E, is model 2 of Launder, Reece and Rodi 144]. Local isotropy of the turbulence was assumed to model the dissipation rate of the individual stress components uuj and the diffusive transport of Zi7-j- is simulated by a simple gradient diffusion model T7he return to isotropy put of the pressure-strain mechanism is simulated with Rom'as linear model assuming this part to be proportional to the anisotropy of the turbulence. The In their rie parson second-moment-closure modelling, Laundler [42)1 a1 echle 43] provide a niumber of calculation examples where the use of a Reynolds-stress-equaiion model leads to markedly improved predictions compared with those obtained with the k-c model. Among the examples are the flow through an aisymmetric annular diffsor involving two bends with strong curvature effects, rotating channel flow, strongly swirling flows with separation (coaxial swirling jets in a pipe), 5/K , a) Geometry and velocity field predicted with RSE model 3 U 5-1l 24 xa I I - a2reqik-- 4X UC 0/C.5.0 A8.21A xD M rid RE QUICK2tSQUC k2rsehywi k-r3a32-rutbri ) Raiao pofe oenua-ine velocity j Fig. 9: 3D flow in combuator model, lines are calculations (461. o experiments (451

45 3-8 flow in a plenum chamber and flow around a square-sectioned U- bend. Reynolds-stress-equation models are now used vortex shedding was obtained, and a sequence of streamlines covering approximately one period is shown in Fig. 10. Values increasingly for solving more complex flows, which will be of th dimensionles shedding frequencies (Strouhal number S) demonstrated in die following by providing two calculation and time-averged drag coefficients c6 are compared in Table I examples with experimental values. The k-e model yields too low values for both Strouhal number and drag coefficient, while the two- The first example concerns the very complex three-dimensional layer RSE model produces too high values; the RSE model with flow in a combustor, which was studied experimentally under wall functions yields results in closest agreement with the isothermal conditions by Koutmos (451. A swirling jet with measurements, at least as far the parameters St and CD are relatively small flow rate enters the combustor at the left and dilution jets from two rows of holes (introducing 83% of the total flow rate) enter radially. Lin [46] carried out calculations for this Table I Glol= parametm for vortex-shedding flow past squre cylinder [471 flow using both the basic RSE and the k-e model and also both the hybrid central/upwind differencing scheme and the more St cj accurate QUICK scheme. Because of periodicity, the calculations could be restricted to a 600 segment as shown in Fig. 9a, and this - 2 layer k-s model figure also gives an indication of the boundary-fined grid used and the very complex flow evolving in this situation. Fig. 9b RSE model with wall functions compares calculated and measured distributions of the centre-line velocity. The fairly large negative velocity value indicated by one 2 layer RSE model measurement point at x/d is not reproduced by the k-s model; with the RSE model such large negative velocities can be Experiments obtained, but only when the numerically more accurate QUICK scheme is used. Fig. 9c shows the development of the axial velocity profiles. Altogether the agreement of both models with concerned. Fig. 10b displays the distribution of the time-averaged experiments is only modest, but the flow behaviour predicted by velocity along the centre-line. It is clear from this figure that the the RSE model is somewhat closer to reality. k-e model produces too long a separation region. This indicates that the model generates not enough mixing and hene not enough The next example has a simple geometry but the flow is very momentum exchange due to the unsteady vortex shedding which complex due to its unsteady nature involving periodic vortex is too weak in these calculations. The RSE model calculations shedding. As the latter motion is not turbulence, it cannot be yield too short a separation zone and hence this model seems to simulated by turbulence models and the flow cannot be calculated generate too much vortex-shedding motion. This finding is by solving time-averaged equations but only by resolving the supported by the distribution of the total (periodic plus turbulent) periodic shedding motion in an unsteady calculation. For this fluctuating energy along the centre-line shown in Fig. l0c. The purpose, equations for ensemble-averaged quantities need to be RSE calculations agree fairly well with the measurements, but the solved, in these ensemble-averaged stresses appear which need to turbulent contribution of the total energy (not shown here) is be simulated by a turbulence model. For the flow around a square underprexicted so that the periodic conibution must be too large. cylinder at Re = 22000, Franke [47] performed calculations with On the other hand, the k-s model grossly underpredicts the the k-s model and the basic RSE model, in both cases with wall fluctuation level behind the cylinder. In front of the cylinder, functions as well as with the one-equation Norris-Reynolds model [24] for resolving the viscous sublayer. In the calculation however, the k-e model leads to unnaturally high turbulent fluctuations. This problem with the k-s model in stagnation flows using the k-e model and wall functions, a steady solution resulted is now well known and can be traced to the fact that the turbulent and no vortex shedding was obtained. In the calculations with energy production in this flow is due to normal stresses (not the other three model variants, unsteady motion with periodic shear stresses) and these cannot be simulated correctly with an a) Streamlines calculated with RSE model and wall functions for phases /T 120, 5/20, 9/20, 13/20, Z O S.OS " '"' ,. * b) Distribution of time-averaged velocity md total kinetic energy of fluctuations (penodic + turtulent) along cntre line. Calculations [47]:-- 2 layer k-e model, - RSE model with wall functions, 2layer RSEmodel;experments: o Lyn [481 Re-22O., x Duntoetal. [49J Re Fig. 10: Vortex-shedding flow pat a squens cyilnder caluladona [471 we for Re

46 isotropic ed-viscosity model, leading to excessive energy is now available, and the most recently developed ones show the Pouto. T e does not arise when an RSE model is correct near-wall behaviour as demonstrated by comparison with used. Overall, the RSE model leads to significantly improved direct simulation dat. However, the various model versions were predictions for voertex-shedding flows c with the k-e tested mainly for simple boundary layers and their predictive model predictions because it can account for the anisotropy, ability. inludinthat for transition, has yetso be established for a history and trasport effects which are important in these flows. wider nmge of C s, especially for the newly proposed models. As the various damping functions in the low-re k-e models were 3.2 Advan-, a designed for use in boundary layers, the performance of these models in separated flows is not clear and has to be examined before the models can also be used with some confidence for The model approximations that have entered the Reynolds-stressequation (11) are fairly simple and do not satisfy certain these flows. For both boundary layers and 21) separated flows, encouraging results wer obtainedwith a two-layer model, which theoretical constrains such as the realisability conditions due to Schumann [501, and they are also not compatible with the limiting employs the k-e model only in the outer part of the flow but resolves the near-wall region with a one-equation model. This two-dinensional behaviour of turbulence very near walls. Hence, considerable effort has gone into the development of improved model also needs further testing, in particular for three- dimensional situations. Here the question arises whether the models. Launder [42] has recently given a summary on these developments Here only the main trends ae briefly outlined, computer resources available now and in the near future are sufficient for resolving the viscous sublayer with such a model also in 3D situation. Most important is the modelling of the pressure-strain term in the Reynolds-stress-equation, and hence most research effort has gone into this zmodelling, with the return to isotropy part Ij,t, Reynolds-stress-equation models will be used more and more for practical calculations in place of the k-s model, particularly for and the rapid part,j2 being treated separately. Rotta's linear situations with complex strain fields and special effects on model for % 1 (given in equation 11) has the incorrect behaviour turbulence like due to streamline curvature, rotation etc.. Slowly, that o1j does not vanish in 2D turbulence as it should. This was the more refined of the RSE models will come to be used, but remedied by introducing more elaborate forms involving second considerable testing is still necessary before these models are and thin invariants of the anistropy tensor a = ( ik q), see e.g. [42]. The isotropation of production model or the rapid ready for practical application; die same is true also for near-wall RSE models. The imporance of direct numerical simulation data pan given in equation (11) is also incompatible with 21 for tea ing the turblence models and asbasis fortie development turbulence and shows an incorrect response in homogeneous of new model assumptions will increase dramatically. However, shear layers to the variation of the ratio of production to as the Reynolds numbers possible in direct numerical simulations dissipation of kinetic energy, Pk/e. Also for this par, more will go up only slowly, the resulting data will be restricted to elaborate models have been proposed [42, 52], involving fairly low Reynolds numbers and the main impact will remain in quadratic and cubic terms. The more refined pressure-strain the area of near-wall modelling. For high Reynolds numbers, models were tested for relatively simple shear layers and were perhaps results from lge-eddy simulations can be exploited in a found superior to the model given in (11), but further testing of similar way, but these simulations am of course themselves the somewhat complex models is certainly necessary before their dependent on model assumptions entering the subgrid-scale general validity can be established, model, and sometimes they may rather be used directly as a predictive tool; in complex flow sitution where the structume of The assumption of local isotropy leading to the dissipation model turbulence is impotant or 3D time-dependent calculations have to in (11) is also not compatible with 2D turbulence. Hence, more be carried out anyway, lage-eddy or very-larpe-eddy simulations elaborate reaton for the dissipation rate of the individual scres are likely to become the methods to be used ma few years'time components near walls were proposed, e.g. in [531. More [54]. However, in many application areas, conventional elaborate diffusion models have also been suggested in the turbulence models will still be used and needed for many years to lterature, but in his review paper Launder [421 argues that in general their use is not really warranted, if only because the come. weaknesses of model predictions can seldom be traced to weaknesses o the diffusion model. 5. ACKNOWLEDGEMENTS The work reported here was suppported partly by the Deutsche In the practical applications of standaid RSE models such as the Forschungsgemeinschafs. The author is grateful to J. Cordes, one given by eq. (11), the near-wall layer was not resolved ad R. Franke and D. Lyn for providing unpublished calculation the damping ofthe normal fluetuations was accounted for by wall results and to R. Zschernitz for the efficient typing of the corectos to the pressure-stan model. The new, more complex iuilusclip-. pressue-strain models just discussed, which are consistent with the lmiting 2D behaviour of turbuleim at ihe wall, should help to 6. REFERENCES smuate this damping effect. However, experience gained with these models so far [42] indicates that the use of these m odes. Cebeel, T. and Smith A.M.O.. 'Analysis of Turbulent alone seems not sufficient and that wal-correction terms, Boundary Lays". Academic Ptan, New Yoek, although reduced In mnanitude, may still be necessary. In some 2. Baldwin. W.S. md Lomax, H., "Thin-layer approximation aid nea-wall RSE models which appeared in the literature, viscositydependent functions were introduced to simulate the damping ebrans odel for separated turbulent flos, AIAA Paper S effect, which appears to be mainly a pressure-reflection effect and 3. Johnson, D.A. and King, LS.. "A mathematically simple is therefore not really affected by viscosity. However, some tutulence cloue model for attached and se ed turbulent influne of viscosity may have to be brought in. Work on near- boundary layers", AIAA J., 23,1995. pp wall RSE modeig is in vsn o s 4 L ausi der, B.E. asid M D.B., -rte tamieridc Stanford (e.n. at U University MST / NASA hster, Ames), A o but State it is fair Univerty to say that and at of tt pp. 2W9289. t flow-, Coup. leth. in Appl. Me. ek d n PsntRSE mdd odel is available.. lacovides, H. Lemle. 3., 'The nesecal simlon of flow aid heat tusafer in tubes in outhogumnl-mde otin", 4. CONCLUSIONS AND OUTLOOK Pe. t1967. t y T Teflom, Stie. Simple tuhulence models involving no differential equations or 6. Vinla,..., Rubesin, M.W. and Hosmuu, C.C., "O the ise only an orinsy differeatial equsdon for the maximum shear of wall teeie as boundary uanditios for two-dimeiuionl, stres will continue to be used in nony practical calculatios, separd oampdline bowm. AIAA qr S-013O especially for external aerodynaemic flows, and they will be 7. Marvin,.0., "Prgresn and challengs in modeling turbulent refined furthr. The k-e model will also continue for sme time to aerodynamie flows", in Pre. it. Symu o Basavefn play a key role in engineering calculations, and most of these Tlblie b It r bgandl Memnr ew. Im& Elevier calculations will be cared out with wall funcdons for many years Publshers, 1I so come. Howver, near-wall models will be used increasingly to 8. Paid, V.C, Rd, W. an Schewer, G.. Ir"flace models replace the wall functions, at least in incompressible flows. A fr mar-wal ad low-reyiuldg-mmber flows: A mvew, AIAA fairly lrge variety of different low-reynolds-number k-e models L pp. 13(

47 9..HK n aai.," ejraht h 32. Dfldhle. P. "Nimesae Deredig der oadusnm Strfluwa Irovata urbuenc of -c modl h -boale che flow", SMEIrkJouraL eris U p~ 6-72Was Other clue seoaclie chose Plante it eloser zr Amtrilrat6 Trompane in der Syimetreclmas". Slhiadel an 10. Na= o Y. and Hisdda, h.. "improved form of the k-c model QM Wl1ytomedinlk e Udiersis* Kmlutte fo alturbulent shear flows". ASMdE J. Fluids Hng.. 10", 33. Isroda, M., "Nin kuldadie Detradmuing der Methiode diskrewe 1967, pp Wirsid abla Mded lrein mtine isd~u Shbh, T.IL said Mansour, N.N.. *Modelling of near-wall Abifieblase auf der Bai experrimenteller Elkeun *s. tutbubence" Pruc. nlot Symp. on Engineer n Turbulence Prtdb~eVU1 tet , 19C?. Modelinug uid Measurements. ed. W. Rodi, Elsevier Publism 34. luoch M.P. and Femnboly. ILH., -Mh diece-dimeaunasl 199. character of a nominally two-dimensional segmaed tuntruleti 12. Halic, K.. "Practical predictiona by two-equation said othrer show flow". J. Fluid MedL pp fast methods'. Proc. Zac Memorial he. Seminar on Wall 35. Rod[, W. anid Sdreumite, G., "Calculation of laminar-turtulort Turbulence, Dubrovnik, Yugoslavia, Hemisphere Publishing boruidary -layer transition an narbine blades", Proc. AGARD- Corp PP65th Symp., DBan, Norway. May I9M. 13. Gibson. M. sand Launder,.iBY. "Ground effects on pressure 36. Larunder, D.E said Sharma DiL. "Application of dhe energyflucuatios in the atmospheuic bounidary layer', J. Fluid Mect. dfissipation model of turbulbence to die calcuation of flow near a , pp swrimhig disit".lemtrin Hen aidmasstrmsle, , pp. 14. Launder. B.F., 1.0w-Reynolds nmbser tutbulence near walls" UMIST Manchester, Dept. of Mchl. Hog. Rope. TFDiW Abu Glaussam. D.I., "Boundary-layer ttansition in relssion to aadley,u blades", PILD. fiest Unilverity of LverpooL 15. Glbert N., "Numsneclie Simulation der Transition von der UVer~pOOL U.K lamlrraron in die trarbulente Kanalatrmung-. Report DFVLR Abu Uhannam, Di. and Shaw, R.. 'Natural transition of PD 88-55, DLR Gouingen. F.M. Germany bounaray layers - the cectsof turbulence,.rsuegsi aid 16. RodI. W.. -ft prediction of fire turbulent boundary layy flow blueory". L. Medi. Eng. Sd , pp. 213,-228. use of a two-equation model of turbulence", Ph.D. University of London, is, 39. ilir M.P. said Weue, MJ.. -"ftr influence of free-streani tuarbulaee on die too paree gradlmfully trbulent bounarey 17. Capan. DXR and Kuhn, G.D.. '"lre limiting beharviour of layet", U1MC Rep. RS0.143M tunbulence near a wall". J. Fluid Mech.. 170, 1986, pp Dlair. M.F. and Werle. Mi., "Combined influence of free turbulence ad favourable jam radiefia on bomdary-layer 18. LAMn. C.G. aid Demhbors LA. "Modified form of the k transition said beam trnfr, TRC Rept. R , model for predicting wall turbulence". ASME J. Fluids Eng, ,1961. pp Roth. W.. LU, X aid Scihiuong. B., "rrasitional bouidary deel. e oh.ne c~.i. P. i w low Renod numur eej li Numerical aid Physical Aspects of Aerodynamc lowln deo ed pp. number". Fluid66 Beach. Caifornia, Jauaruy hlktjl odeflornef~wd Moulece orarinceand 42.Launder. D.E.. 'Second-moment closure: present... and effet of ' *aronntubulece CM AnnualResearchlBricka futute?'. Irk J. Heat aid Fluid Flow, 10,19I9, pp Cattl fr~abulence Research, NASA Araes Research Ceunte 43. Laidaibrre MA. Secorid-mumest loowse for complex flows-. said Stanfordi Universlly 199,. pp Proc. bit. Form on Matheatical MoDdelfhof Processes tn 21. Chron, T.-Y.. "Preictions of close arid bouidary-layer flm Energy Systems, tnt Centre for Hea d Mass Transfer. with a low-reynolds-tssber turbulence model", AIAA J.. 20, SW*vo. Muchi , pp Launder. BY_, Ramc. G.C. said Rodi. W.. "Progress In dt 22. Jones. W.P. said Launder, D.E.. 'Mla calculation of low- developmnt of a Reynolda-stress turbulence closure", J. Fluid Renls-ubrphenomea wih a two-equation model of Me&t. it pp turbuence. 1ne.J. Heat Mass Transfer, 16, pp Kournros, F.. Ph.D. Thess University of London, Unx CA., Ph.D. Thesis Faculty of Technology, University of 23. Rodh, W. and Sceresr. 0.. "Scrudinizing die k-c model under Manihster, adveseers gradient condidons". J. Fluids Eng. I, 478e.6,pve omtctln Wp f.47 Pt R.piaecaoodn Norris, LIt, said Reynolds, W.C., '"lurulest channrel flow wit 48. LMn D., private oosinicaleu, a moving wavy boundary, Rope. No. FM-tO, Stanford 49. Durao. D.F.0.. Heltm, M.V. and Pereira, J.C.F.. Unbversity, Dept. Macht. Ea keoaa orut umd IaIDONrun = 25. Fujiaaa N.. "COLflo of transitional bounasy-layeea with 2834 a refned low-reyneldeiraamber version of a k-c model of ourbulene", Proc. lot. Symp. on Engineering Turtbulence 50. Schuannrsa U., "Realisablt of Reymlds-suess tublec Modelfti aid Measumentsea W. Rodl, Bevier Piduidress models. Physics of Plaids, A0,1977, pp &. 51. Lumley. IL. VCo I o frtbulia t owi trauldl boundary layers with a two-layer model of Acaduele P=4u New York.p.1376 turbulence. Proe. 3rd IoL limp. on Trasport Phenoamena aid Dynamics oflldg Madistery, Ibenirk, April 1990 a2 A04 PM. aid Lumley, JIL. "Merelig of I u-~ A W tam in Reyusib mins Mud bawfl epolon. Rapt. FDA- 27. Crawfod,MEend Keys, MW,STAN 5 -a irogram hor Sibley School of Uedh. an Aesopee IBaf. Comnell I umonft~osmlod- ad eoana University, 198& boss Wsa-lae fl, NASA CR-2742, Limier, BI. sand Reysreldt. W.C.. "Asyppttc: groai 26. ads L., pelve omuiofelm stro.-dlmltulon ran in atlem flw, physics of Fluds, 29. Drive, DX said Seegmller. HIL, "Fmarsof a revci 24, 1963, pp ISSt ttoler drear Ina ia divergent dimel flow". ALAA J L ireynolds, W.C.. "lb potetia and jimlsfie of Anns Mad 1IM. pp laspe-idy jelatmaowf, Filabim w at Cshm Wd 30. SoomI. AX sand Jaoshe, P.M., "A boudy kae deveumoa Turbule? Iot Twboloso as Coss Roar., Coroena Ii aivs promine gin.. FlidO Maci. tivelty, u 22-4k 1909.m Ine p eido by asprnge in 31. Dod N Z. hrmpb ftm lk S hha.,v d o MooeW i In a two eusilboudary layer flow", J. Fluid Meat., X.,1969, pp Naq d 14j 4

48 4-1 Introduction: TURBULENCE MODELS FOR NATURAL CONVECTION FLOWS ALONG A VERTICAL HEATED PLANE by D.D.PosIiliou Laboratory of Applied Therrmodynamics, Dept. of Mechanical Engineering. Univerity of Patras Kato Kostrilsi. Patras 26500, (recec The present work has been prepared for presentation in the Technical Status Review on "Appraisal of the Suitability of Turbulence Models in Flow Calculations", organized by the Fluid Dynamics Panel of AGARD (Friedrichshafen, April 26, 1990). Its objective is to review the state of development of turbulent models currently employed in the computation of turbulent free convection flows along heated plane surfaces. In this context some experimental results recently obtained in the Laboratory of Applied Thermodynamics, University of Patras, are also compared with corresponding computations in which existing turbulence models were used. Although free convection flows such as plumes, buoyant jets, thermals and those originated from heated surfaces, find wide application in various fields of science and technology including aeronautics, the understanding of turbulence transport processes in these cases is lagging behind as compared to other flows. This is generally due to the currently existing lack of sufficient knowledge regarding the effects of buoyancy on the turbulence structure of these flows and also its participation and role in the corresponding transport processes. The reasons causing this inadequacy of necessary information will be discussed in the following. As a consequence, in computing buoyancy induced flows, turbulent transport models are adopted from forced convection or ordinary flows, mostly in modified forms Ḃy reviewing the existing experimental evidence and computational methods currently in use for predicting the simple case of free convection turbulent flow along heated planes, it becomes possible to appreciate the difficulties presented in developing turbulence models for buoyancy driven flows. It also becomes possible to identify the sources of existing problems and limitations and finally, to suggest a course for future research activities in this particular area. Most of the reported in the literature experimental and computational work on turbulent free convection, refers to Grashof or Rayleigh numbers corresponding to fully developed turbulent flow conditions. However, apart from the fact that both transitional and non-fully developed turbulence states are important on their own merits, their closer investigation might substantially help in improving free convection computational schemes. The study of these states of flow development could for instance result to the selection of more representative initial conditions and/or a more effective method for introducing turbulence action in the computations. More important, conditions prevailing in these states is known to affect the physical processes occurring in the following state of fully developed turbulence. The corbined eperimental and computational effort in the region of Grashof numbers between 10 and 10, presented in this work aims at exploring some of the above mentior-.,: fesmibilities. Turbulence Models for Free Convection Flow Along a Vertical Heated Plane: The problem of turbulent free convection along a vertical heated wall, has been the subject of investigation for the past several decades. Numerous experiments have been reported in the literature (1-14), the majority of which has been limited to measurements of overall and local heat transfer coefficients and mean temperature distributions in the turbulent boundary layer. This is caused by existing inherent difficulties in measuring velocity in buoyant flows, especially near a heated wall where a combination of very low velocities and very steep temperature gradients dominate. The lack of sufficient experimental information, which exist in abundance for most of the ordinary flows, appears to be mainly responsible for the inadequate understanding of * buoyancy dominated free convection turbulent transport phenomena. As a consequence, this prevents the development of convincing theoretical arguments on which the modelling of turbulence transport terms will be based. At the present,almoet all the theoretical efforts on the subject are based on analogies "borrowed" from forced convection flows. The turbulence models which are presently employed in free convection boundary layer computations fall into three categories namely, algebraic (15, 16), standard x-a (17, 18) and low Reynolds number x-s (19-26). Algebraic models are using the concept of eddy diffusivity in forms either directly transferred from forced convection or modified for free convection. The latter two categories employ two equation s-s models describing turbulent kinetic energy and dissipation dynamics. In the case of standard x-s modelling the use of wall functions is necessary to avoid the steep gradients prevailing near the wall and to provide boundary conditions for the x and a differential equations. Sines no wall functions exist for turbulent free convection boundary layer,log"ithmic wall functions, appropriate for forced convection boundary layer with small presure gradionts, are comonly used in computations (27). In the case of low-reynolds s-. models, the diminishing presence of turbulenme near the wall is modelled by the introduction of a number of functions in the differential equations for x and a (27). It should be noticed that, here again, low-reynolds nu models used in free convootion,with the exception of the model developed by To and Humphrey (26), have been originally developed for forced convection boundary layers. 4

49 9 4-2 Henken and Hongendoorn (28) have conducted a systematic comparison of turbulence models of the three categories, by employing a selected number of them in the computation of a free convection turbulenj boundary layer along a vertical plate. Their calculations covered the range 10 < Gr, < 10 corresponding to a fully developed flow. According to the methodology followed by the authors, calculations started at Gr. - 10, where laminar similarity velocity and temperatufe profiles were employed. The selected turbulence models were introduced at Gr, - 2*10. An amount of turbulence kinetic energy was also introduced at the edge of the boundary layer when x-s models were employed. Their calculation. ended at Gr The conducted comparison included heat transfer rates at the wall and also, velocity and temperature profiles. A sensitivity study was also carried, regarding the choice of the wall function for the standard x-e model as well as the influence of the parameters of the low-reynolds number x-e models on the obtained results. The conclusions reached by the investigators are summarized in the following: The calculated heat transfer rates by the selected algebraic turbulence model of Cebeci -Khattab (15) are too low. This model is also producing laminar-like velocity profiles due to serious underestimation of turbulence viscosity. The standard x-e model calculates too high values of wall-heat transfer which, as the sensitivity studies indicated, depend on the choice of the wall functions. The low-reynolds number models of am and Bremhorst (21), Chien (23) and Jones and Launder (20) perform best up to Gr Beyond this value the first two models predict heat transfer rates which are higher than the experimental, while the model of Jones and Launder gives more accurate results. However, the latter model exhibits a late transition to turbulence as shown in fig. 1. It should be noted, that all low-reynolds number models produce velocity and temperature distributions which fall above the experimental at the outer region of the boundary layer. Finally, the authors suggest that an accurate low-reynolds number x-e model for turbulent free convection boundary layer computations can be constructed by replacing wall functions by zero wall conditions for x and a and by adding certain functions of the Chien's model to the standard x-e model equations. Regarding the computations conducted by using the low-reynolds number x-e models, difficulties can arise in achieving turbulent transition. According to the authors, the solution of the low-reynolds number models is not unique. Assuming that the solution for large Gr, is independent of the starting profile, both laminar and turbulent solutions exist for large Gr., if homogeneous boundary conditions for x and e are applied at the wall and the outer edge of the boundary layer. Similar problems regarding transition to turbulence and uniqueness of solution, have been reported to exist in other configurations of free convection computations (29). An interesting work offering some physical insight in the structure of a fully developed turbulent natural convection boundary layer forming along a heated vertical surface, has been reported by George and Capp (30). For the case under consideration the authors correctly recognized the necessity of treating the flow in two parts namely, an inner part next to the wall in which viscous and conduction effects are dominant while mean convection of momentum and heat are negligible and an outer part, in which the opposite conditions prevail. They employed momentum and energy equations for these parts in which, according to the above stated arguments, only appropriate for each part terms were retained. By using classical scaling methods of analysis, were able to reach some useful conclusions regarding the structure of the layer and to obtain relations for the velocity and temperature distribution as well as heat transfer rates. According to their analysis the inner part is characterized by constant heat flux and consists of two sublayers and a buffer layer between them. More specifically, next to the wall there exist conductive and viscous sublayers with linear velocity and temperature distributions, their relative thickness depending on the Prandtl number. The outer part of the constant heat flux inner layer is occupied by a buoyant sublayer characterized by velocity and temperature distributions varying as the cube root and the inverse cube root of the distance from the wall respectively, that is: T-T, K A -(r K -Y_-.B(Pr) B (1) where, K,, K2, are universal constants, A(Pr), B(Pr) universal functions of the Prandtl number, U,, T,, inner velocity and temperature scales and n,, inner length scale which, for constant temperature wall, na"[ (2) 2 9A (T.-T.) These theoretical predictions are well supported experimentally as will be further discussed in the following. Heat transfer and friction laws have also been derived in this work which, as in the case of the velocity and temperature profiles, contain as yet unspecified functions of the Prandtl number. The authors point out in their concluding remarks, that additional experimental and theoretical work is needed to supply missing information regarding the physics of the problem and also, to make possible the calculation of the unknown Prandtl functions mentioned above. This is important in developing new computational models, since the velocity and temperature distributions of the experimentally confirmed buoyant sublayer could be utilised as inner boundary conditions, therefore avoiding the difficulties presented in modelling turbulence in the inner part of the boundary layer. Finally, the work of liehere, Noffatt and Shwind (14) should be mentioned, in which extensive measurements of heat transfer oseffiok 1 e and teerature profiles in air ar Presented, for Orambof numers up to 2*10.Regarding the measured temperature

50 4-3 distributions, they found that in the outer region, beyond the buoyant mublayer, they obey a logarithmic relation of the form: ST.-- - o. 8 In f) (3) where 6, is the thermal boundary layer thickness, a, dy(4) TT The preceded survey on the suitability of the turbulence transport models, presently employed in computations of natural convection flows along vertical heated planes, leads to the conclusion that, those models are not capable of accurately predicting heat transfer rates and/or velocity and temperature distributions, within a sufficiently extended range of Grashof and Prandtl numbers. It in therefore evident that, an will be discussed in the following, further experimental and theoretical work is needed. From the experimental point of view, os already mentioned, an abundance of mean temperature distribution and wall heat transfer measurements exist in the literature although for a restricted range of Grashof and Prandtl numbers. Also, the statistical characteristics of the turbulent thermal field have been measured to a certain extend (3,4,9,10,31-34). However, measurements of the mean and statistical character of the corremoondinq turbulent velocity field are very limited (35-38), for reasons already explained. Especially, the present lao._ dires&_.measurements of momentum and heat turbulent transport terms of the form u'u, and uw, impose severe limitations in the understanding of these processes in the different parts of the boundary layer. It is evident, that without such knowledge, the development of turbulent models offering adequately accurate predictions cannot be expected. Recent attempts to conduct velocity measurements by using laser velocimetry, appear promising in providing such information in the near future (31). Further experimentation is also needed, in order to establish firmly the mathematical form of mean temperature and velocity distribution in the various parts of the turbulent boundary layer as well as the skin friction and wall heat transfer. In this direction the already discussed findings and suggestions contained in reference (30), might form the basis for further investigation. It should be added that future experiments regarding both mean and statistical turbulent flow characteristics should be planned to he conducted in an extended range of Prandtl and Crashof numbers. In buoyancy generated flows where the velocity field is initiated and sustained by the corresponding thermal field, a strong interaction exists between the two fields. Although this interdependence probably dominates the turbulence dynamics as well as the transport processes in these flown, it has not been adequately investigated. Addressing this question will certainly enhance the understanding of the physical aspects of turbulent transport mechanisms in buoyant flown, resulting to improved modelling of the corresponding terms in the conservation equations. It could also provide physically founded relations between them, to replace the empirical Prandtl number dependent formulas currently in use. The conducted survey has shown the limitations of the presently employed turbulence models in free convection computations, strongly indicating that suitable models for these flown cannot be developed by modifying existing ones corresponding to ordinary or forced convection flows. This task demands adequate understanding of the buoyancy influenced turbulence structure and related transport mechanisms not presently existing. To supply the needed information the widely accepted procedure of comparing carefully selected experiments, conducted along the discussed directions, with corresponding computations should be adapted. However, the method of interplay between experiment and computation cannot produce the sought knowledge if confined to comparing only overall quantities, as the trend has been in the past few years. This methodology should be used as a tool for a detailed investigation of the structure of the various parts of the turbulent velocity and temperature fields, in which different physical processes prevail. In the case of a heated vertical plane these parts have been identified to correspond to an inner layer consisting of a viscous, conduction and buoyant sublayers and an outer layer, as already mentioned. The described approach is also expected to help in selecting appropriate boundary conditions in order to avoid a number of computational difficulties as for instance, handling the low turbulence region near the wall. Experiments in the non-fully developed turbulent states The history of flow development towards a fully developed turbulence state is generally known to affect both the structure of turbulence and the occurring in this state physical processes. In turbulent free convection, studying turbulence in its developing stages and ccqparing experimental results with computational predictions, is expected to allow a better understanding of turbulence transport processes and especially the influence of boyancy on them. It will also assist in recognising existing shortcomings in the presently employed turbulence models and in acquiring useful information for the development of suitable ones for free convection flow. In the conducted preliminary investigation reported in the following the experimental results obtained in a non-fully developed turbulent freq convection thermal layer formed along a heated vertical plane are compared with corresponding computational predictions. The experimental apparatus, consisting of a stainless steel plate of dimenaics lo00xsooxl2 me heated by ton uniformly spaced resistance elements, has been described in refernce (34). By controlling the powr delivered to each element, a uniform temperature over the surface of the plate we achieved. The obtained measuremnts covered both mean and statistical

51 4-4 characteristics of the turbulent temperature field by using a hot-wire anemometer with the sensor operating as a resistance thermometer. Heat transfer rates were also estimated from the obtained temperature distributions and from wall temperature and heat flux measurements by using Newton's law. In the present work only mean temperature distributions and heat transfer rates are reported. As already discussed, Henkes and Hoongendoorn (28) have computed heat transfer rates by using different turbulence models as shown in figure I in which, corresponding measurements obtained in the present experiments, are added. It can be seen in this figure that the majority of the turbulence Models fail dramatically to predict heat transfer rates in the mentioned range of Grashof numbers. As a next step, the form of the inner and outer parts of the Measured mean temperature distributions were examined as shown in figures 2 and 3. The measured distributions corresponding to the inner part (fig. 2) exhibit a gradual development with increasing Grashof number towards the theoretically predicted inverse cube root form, characteristic of th- buoyant sublayer in a fully developed state. Furthermore, computationa were carried for two Grashof numbers namely, Gr *10 and Gr, - 9.7*10 and for three different turbulence models as shown in figures 4 and 5. These, were compared with measured in the present experiments temperature distributions at the same Grashof numbers, corresponding to non-fully and fully developed flows. It is evident that none of the computed profiles agrees with the experiment and that in the case of the higher Grashof number only the model of Jones and Launder predicts the inverse cube root form in a small part of the buoyant sublayer. Equally significant none of the employed turbulence models appears capable to produce computed distributions following the trend exhibited by the measured profiles, in the range of Grashof numbers corresponding to non-fully developed flow conditions. The comparison of the form of the outer part of the measured temperature profiles with the logarithmic expression suggested in (14) leads to similar remarks as those made for the inner part (fig. 3). Although not as dramatic, there is again a gradual approach to the logarithmic form with increasing Grashof numbers. In conclusion, the present state of development of turbulence models employed in free convection flow computations appears unsatisfactory. Existing efforts, mainly attempting to use turbulence Models "borrowed" from ordinary or forced convection flows in modified forms, have produced only limited results. To develop turbulence models suitable for this category of complicated flows it is suggested, that research efforts should be directed towards understanding the effects of buoyancy on turbulence transport processes and the mechanism of the existing strong interaction between the flow field and the sustaining it thermal field. The known methodology of interplay between experiment and computation consists a most valuable approach to the problem. However, this should not be used only for validation purposes, but as a tool for the study of the turbulence structure and transport processes in the various parts consisting the velocity and temperature fields associated with these flows. Finally, the reported experiments suggest, that studying the development of turbulence towards a fully developed state offers valuable information assisting in the clarification of a number of questions discussed in this work. References l.- E.R.G. Eckert and T.W. Jackson "Analysis of a Turbulent Boundary Layer on a Flat Plate" NACA TN 1015, C.Y. Warner and V.S. Arpaci "An Experimental Investigation of Turbulent Natural Convection in Air at Law Pressure Along a Vertical Heated Flat Plate" Int. J. of Heat and Mass Transfer, Vol 11, pp , G.C. Vliet and C.K. Liu "An Experimental Study of Turbulent Natural Convection Boundary Layers" J. Heat Transfer, Vol. 91, p. 511, R. Cheeaewright "Turbulent Natural Convection from a Vertical Plane Surface" J. Beat Transfer, Vol. 90, pp 1-8, F.J. Bayley "An Analysis of Turbulent Free Convection Heat Transfer" Proc. Inst. Mch. Eagre, Vol. 169 (20), p. 361, H.S. Kutateladge, A.G. Kirdyaskin and P.V.Ivakin, "Turbulent Natural Convection on a Vertical Plate and in a Vertical Layer" Int. J. Heat and Mass Transfer, Vol. 15, pp , J. Countanceau "Convection Naturelle Turbulence sur une Plaque Verticals Isotherm, Transition, Echange de Chaleur at Frottement Parietal Lois de Repartition de Vitesse at de Temperature" lnt. J. Heat and Mass Transfer, Vol. 12, pp , J.M. Piau "La Convection Naturelle en Regime Turbulent, Aux Grandes Humbles de Grashof", C.R. Bebd. Seanc. Aced. Sci. Paris, Ser. A294, pp , R.R. Smith, "Characteristics of Turbulence in Free Convection Flow Past a Vertical Plate" Ph. D. Thesis, University of London, D.D. Papailiou, P.S. Lykoudis "Turbulent Free Convection Flow" Int. J. Heat Mass Transfer, Vol. 17, pp , II.- K. Not* and R. Mateumoto "Turbulent Heat Transfer by Natural Convection Along an Isothermal Vertical Flat Surface", J. Heat Transfer, Vol. 97, pp , J.W. Elder "Turbulent Free Convection in a Vertical Slot*, J.Fluid Nech. Vol 23 (1), pp , Z.H. Qureshi and B. Gabbart "Transition and Transport in a Buoyancy Driven Flow in Water Adjacent to a Vertical Uniform Flux Surface" Int. J. Heat Mass Transfer, Vol 21, pp , D.L. Sibers, R.F. Moffatt and R.G. Schwind -Experimental, Variable Properties Natural Convection From a Large, Vertical, Flat Surface*, Transactions of ASNE, Vol. 107, pp , IS.- T. Coei and A Khattab "Prediction of Turbulent Free Convective Heat Transfer From a Vertical Plat PLate", J. Heat Transfer, Vol. 97, pp , f. Nason and R.A. Seban "Numerical Predictions For Turbulent Free Convetion from Vertical Surfaves", Int. J. eat and Nass Transfer, Vol. 17, ppi , 1974.

52 O.A. Plumb and L.A. Kennedy *Application of a x-e Turbulence Model to Natural Convection froma Vertical Isothermal Surface', J. Neat Transfer, Vol. 99, pp 79-85, 18.- S.J. Lin and S.W. Churchill " Turbulent Free Convection from a vertical, Isothermal Plate", Humer. Heat Transfer, Vol 1, pp , S. Hassid and M. Poreh "A Turbulent Energy Dissipation Model Reduction", for Flows J. with Fluids Drag Eng., Vol 100, pp , W.P. Jones and B.N. Launder "The Prediction of Laminarization with a Two-Equation Model of Turbulence", Int. J. Heat and Mass Transfer, Vol. 15, pp , C.K.G. Lam and K. Bremhorst "A Modified form of the x-e Model for Predicting Wall Turbulence", J. Fluids Engng, Vol 103, pp , K.Y. Chien, "Predictions of Channel and Boundary Layer Flows with a Low-Reynolds number Two-Equation Model of Turbulence" AIAA , K.Y. Chien, I Predictions of Channel and Boundary Layer Flows with a Low-Reynolds number Turbulence Model", AIAA J., Vol.20, pp 33-38, W.C. Reynolds, "Computation of Turbulent Flows", Ann. Rev. Fluid Mech., Vol.8, pp , G.H. Hoffman, "Improved Form of the Low Reynolds Number x-e Turbulence model", Physics of Fluids, Vol.18, pp , W.M. To and J.A.C. Humphrey, "Numerical Simulation of Buyoant Turbulent Flow-l.Free Convection Along a Heated Vertical Flat Plate', Int. J. Heat and Mass Transfer, Vol. 29, pp , V.C. Patel, W. Rodi and G. Scheuerer "Turbulence Models for Near-Wall and Low-Reynolds number Flows a Review', AIAA J., Vol. 23, pp , R.A.W.M. Henkes and C.J. Hoogendoorn "Comparison of Turbulence Models for the Natural Convection Boundary Layer Along a Heated Vertical Plate", Int. J. Heat and Mass Transfer, Vol.32, pp , H. Ozoe, A. Mouri, M. Ohmuro, S.W. Churchill and N. Lior "Numerical Calculations of Laminar and Turbulent Natural Convection in Water in Rectangular Channels Heated and Cooled Isothermally on the Oposing Vertical Walls" Int. J. Heat and Mass Transfer Vol.28, pp , W.K. George and S.P. Capp "A theory for natural convection turbulent Boundary Layers Next to Heated Vertical surfaces-, Int. J. Heat and Mass Transfer,Vol.22, pp , M. Niyamoto, H. Kajino, J. Karima and I. Takanami "Development of Turbulence Characteristics in a Vertical Free Convection Boundary Layer', Proc. 7th Int. Heat Transfer Conf., Vol.2, pp , D.D. Papailiou "Statistical Characteristics of a Turbulent Free-Convection Flow in the Absence and Presence of a Magnetic Field*, Int. J. Heat and Mass Transfer, Vol. 23, pp , K. Kitamura, M. Koike, I. Fukuoka and T.S. Saito "Large Eddy Structure and Heat Transfer of Turbulent Natural Convection Along a Vertical Flat Plate ", Int. J. of Heat and Mass Transfer, Vol. 28, pp , T.J. Tsirikoglou and D.D. Papailiou "The Structure of the Turbulent Temperature Field Above Heated Planes". Published in the " Advances in Turbulence ", Editors: G. Comte-Bellot and J. Mathieu, Springer-Verlag, pp , R. Cheesewrite and B. Ierokipiotis "Velocity Measurements in a Natural Convection Boundary Layer", Queen Mary Colege, Faculty of Engineering Research, Report EP 5022, R.Cheesewrite and E. Ierokipiotis "Velocity measurements in a Turbulent Natural Convection Boundary Layer" Proc. 7th Int. Heat Transfer Conf., Vol. 2, pp , Munich, F.R.G., C.J. Hoogendoorn and H. Euser, "Velocity Profiles in the Turbulent Free Convection Boundary Layer", Proc. 6th Int. Heat Transfer Conf., Vol. 2, pp , M. Hishida, Y. Nagamo, T. Tsuji and I. Kaneko, "Turbulent Boundary Layer of Natural Convection Along a Vertical Flat Plate*, Trans. Japan Soc. Mech. Engrs, Vol. 47, pp ,

53 Of*s Is 'W*.,4wt asf v - WC. ft h 10 J.*1ww. L~ I.e~ * Lw KP - Se Gr. l Fig. 1. Computed and measured wall heat transfer for air. Henkes and Hoogendoorn ref. (28), kahg Linear + 0.6A 0 0 a A 00a Gr.=3.71xl0 Smiho 0 **' * Gr,.O.97 Present Experiments Gr,-0.84 AAAAA Gr,0O o0 Gr=O Gr, Fig. 2 Temperature profiles In boundary layer along a ri~~bg turbullenit natural convection setc urface -inner part.

54 op A 9=0.08Ln(Y/,gt) *****Gr,=0.97x O00 0 Gr,= AA Gr.= Grx= Gr.=0.27 Fig , Y/dt Temprature profiles in a developing turbulent natural convection bouary layer along a vertical surface -outer part e rs 1.C05u angr-0ully Presente Expromn. U~i- Smit

55 '4 4-8 I Gr=371x Smith00 *****Gr 1=0.97x 1010 Present Experiments -Inverse Cube Root Cebeci -Smith &ha*" Jones -Launder 0.0 O- K-9 model a & 0.4' 0.'0 FIg. 5 Compan of computed and measured temperature proffles. Fur~ developed flow.

56 5-1 TURBULENT FLOW MODELLING IN SPAIN. OVERVIEW AND DEVELOPMENTS. C. Dopazo, R. Aliod and L. Valio Fluid Mechanics Group, School of Engineering University of Zaragoza M. Luna, 3, Zaragoza Spain ABSTRACT Reynolds averaging techniques consider the contribution to a fluctuating variable coming from the whole A brief overview of ongoing flow modelling activities in spectrum of length and/or time scales25. The closure Spain in the fields of turbulent boundary layers, two- problem can be more severe and difficult to solve than phase flows and turbulent reactive flows is presented. the equivalent one in LES for the velocity field or of the same order to complexity for a reactive scalar in a Turbulent shear flows laden with small solid particles turbulent flow. Moment and probability density function and the turbulent isothermal mixing of two chemical (pdf) equations have been obtained after Reynoldsspecies undergoing a second order irreversible chemical averaging the corresponding local-instantaneous reaction are discussed in more detail. An eulerian consevation equations. Most modelling activity has t approach is followed in the former, employing a phase revolved around moment-equationsl.' 9. Pdf transport indicator function conditioning technique for the equations condense the information equivalent to an continuous phase and a Boltzmann type velocity infinite hierarchy of moment equations 9,1025.It has been distribution function for the dispersed phase. The latter shown that the pdf formalism is the natural one for is treated via a probability density function (pdf) turbulent reactive flows' 0 and that pdf models can be formalism where the joint statistics of concentrations proposed based on second order closure models 29. and concentration-gradients is investigated. Predictions are compared with available experimental results in both An overview of ongoing activities on turbulent flow cases. modelling in Spain is presented in Section 2. Modelling work on turbulent boundary layers, two-phase flows and Some concluding remarks on the specific needs on turbulent reactive flows is briefly described. More modelling or turbulent two-phase and reacting flows are details with emphasis on closure models are provided in listed. Section 3 for two selected cases: Turbulent shear flows laden with small solid particles and turbulent isothermal 1. INTRODUCTION mixing of an acid and a base undergoing a second order irreversible chemical reaction; these cases exemplify As in most fields of scientific research, the ultimate goal two areas in which significant modelling efforts are in the investigation of turbulent flows is to develop a required. In the former even the basic equations and quantitative tractable predictive theory. This objective is methodologies to be used as a starting point are being hampered by the existence of a wide spectrum of length questioned. The latter is adequate to illustrate the and time scales, ranging from the characteristic sizes and difference between what seems to be the natural pdf frequencics of the large turbulent structures to the approach and what most moment modellers in the field Kolmogorov length and time microscales. This prevents are doing. Section 4 contains some concluding remarks direct numerical simulation (DNS) of turbulent flows in on the two selected cases. most cases of practical interest. DNS 13, 16,31, 36 of some idealized and limit cases can provide useful information 2. OVERVIEW on the asymptotic behavior of turbulent fields. The following ongoing turbulent flow modelling Existing computers set a limit on the length and time activities in Spain have been brought to the attention of scales that can be numerically resolved. Large eddy the authors: simulation (LES) methodology 2 ' filters the contribution to a fluctuating variable coming from large structures 2.1 Turbulent Boundary Layers averaging the effects of small scales below a given threshold. The latter introduce unknown terms in the Asymnotic Expansion Techniques 726 resulting equations. Models or closure hypothesis must be proposed in order to proceed fi.ther to the numerical Turbulence modelling is being conducted via the simulation. In some instances, dumping the small scale asymptotic techniques originated within the framework contributions into a collective pool may yield good of "periodic homogeneization theory". The flow results. This can be the case of the turbulence modelling variables are asymptotically expanded and the model where most energy production and energy transfer can, thus, be formally justified. The averaged partial processes are dominated by large/intermediate size differential equations have a structure analogous to that eddies, while the overall small-scale action upon the of the compressible Navier-Stokes equations. flow can be simulated and quantified as a diffusive-like irreversible mechanism. However, if a process is tied to The starting point is the so called MPP turbulence the small fluctuating scales the closure difficulties might models 20, valid for incompressible render LES strategies flows of useless. ideal or This is the case of weakly viscous fluids. These models have been chemically reacting turbulent flows. generalised in a first attempt 7 to inviscid compressible

57 5-2 isentropic flows. No closure hypothesis are required at where the flux vectors F and G can be expressed as this level The Reynolds stress tensor is obtained from F = F I + FV, G = G I + GV The superscripts I and V the solution of a non-linear differential equation. The denote inviscid and viscous contributions, respectively, gradient of the inverse lagrangian coordinate associated and to the mean velocity field, the average turbulent kinetic 2 energy and the average helicity appear as parameters in =u FI = p11 + P that equation. The unclosed terms can then be tabulated U = F G+ 1 P I and it is to be expected that the corresponding table has a P L universal charater26. These models do not take into e uh jpvh account the flow behavior near solid walls. On the other hand, mechanisms for turbulence generation are not considered and models are typically non-stationary. 0 l - I As a first step, a reformulation of asymptotic models for FY R R Cyr moderately high Reynolds number and moderate Mach u V v l number compressible isentropic flows has been achieved. Non-standard boundary conditions adding 0 new non-linearities to the problem and based on the well f -V12 known logarithmic law of the wall have been introduced v 12 to take into account the existence of boundary layers.the methodology followed to numerically solve G- "22 the a large problem number is described of numerical in detail results in Reference are presented. 26 where u1t 2 - v *t22 + r (2) In a second step, new models describing the turbulent Here R. and Pr denote the Reynolds and Prandt motion of the fluid over the whole domain, including the numbers respectively, u and v the two components of the regions near solid walls, have been formulated. The velocity vector and p, e, p, h and T are the dimensionless flow domain is decomposed into a turbulent boundary density, specific total energy, pressure, specific total layer and an inviscid outer zone. Small parameters are enthalpy and temperature of the fluid. The Sutherland introduced and asymptotic expansions for the flow relation is used to take into account the dependence of variables are hypothesized for the different zones. the molecular viscosity coefficient, IL, on temperature Formal manipulation of the equations, identifying the and Mach number. The non-dimensional components of coefficients of the various powers of the small the stress tensor, "e11, e12 and t22 are assumed to parameters, lead to averaged variable equations for the follow the Navier-Poisson relation for lamin flows. inner and outer zones. The original initial and boundary The equation of state complete the problem conditions are supplemented with those obtained from specification. "matching" the inner and outer variables. The use of these models allows, among other things, to rigorously For turbulent flows IL is replaced by "gip + It". W is the justify, from the asymptotic point of view, the laws of laminar viscosity. For the "eddy viscosity coefficient", the wall. lit, the Baldwin-Lomax algebraic model 4 is used. The variables in Equ. (2) stand then for average values. The numerical solution rest upon semi-implicit time Cebeci's modifications avoid the calculation of the discretization schemes and finite element type approximation for the space variables. A least square boundary layer edge. type formulation is utilized followed by an algorithm of In this two-layer model the turbulent viscosity is given the Buckley-Le Nir conjugate gradient 6. Finally, the by, numerical problem is reduced to the solution of a "cascade" of linear problems, some of them analogous in A for Y 5 Y Co nature to the Stokes problem, which can be solved using, 1" for y > yco (3) for example, the Glowinski-Pironneau method, and others of the Poisson type. where Itti and Iito stand for inner and outer values of t, y is the normal distance to the wall and yo is the cross- These techniques are being developed within the over value, the minimum y for which the inner and framework of the HERMES Project of the European outer values of g±t are equal. Space Agency (ESA) in order to predict the thermal behavior of this space aircraft during its reentry into the Van-Driest formulation is followed for pdi, earths's atmosphere gebraic Model of Turbulence' pd = Ci p 12 w (4) The two-dimensional (2D) compressible laminar where Ci is a constant, and Navier-Stokes equations are considered in a nondimensional form, Il=k y [1 - exp (- y+/a+)] (5) au -i + W, 0 w is the mean vorticity module given by, )iij

58 2 Should the particle cloud viscosity be ignored no macroscopic mechanism exists to set a limit to strong gradients. A discrete system of particles has been (6) analysed in order to find a strong-gradients limiting mechanism for the continuum model. The inviscid flow and Y+ Y yu/n, with u. being the friction velocity and v around an ensemble of spherical particles has been the kinematic viscosity, investigated. None of the microscopic terms can prevent Clauser formulation is used to express tro, the development of strong gradients in the particle concentration. The consideration of the cloud viscosity p4to=k Cop Fw Fk(y) (7) seems thus to be neccesary in order to describe small wavelengths. K is Clauser constant, CCp is an additional constant and On the other band, the functional dependence of the =min m (yn2x w~ Fa F, Cwk /Fmax) (8) particle cloud pressure is important and has significant Y 2 imacroscopic consequences upon the flow developmenti 5. Fmx is the maximum value of The specification of the particle cloud viscosity and F(y) = y [I- exp (- y+/a+)] (9) pressure are therefore open topics. The viscosity data measured by Schtlrge1 32 for glass spherical particles of and ymax is the value of y at which F(y) = Fmax. FK(y) is various diameters have been analysed. The viscosity the Klebanoff intermittency factor, increases considerably when the fluidised bed concentration approaches the maximum packing FK(y) = [I- 5.5 (CK y/ymax)] (10) concentration; on the other hand, it has been observed that a dimensionless viscosity can be obtained by scaling Udif is the difference between the maximum and the experimental data with the particle density, the minimum values of the velocity for a profile. The fluidization velocity and a characteristic length which is standard values are assigned to the constants. a function of the particle diameter and Reynolds number. A universal curve of the dimensionless The solution is obtained by advancing the unsteady non- viscosity as a function of the volume concentration can dimensional form of the equations to steady state by means of a hybrid timestepping scheme. The convergence is accelerated by the use of local timestepping and the incorporation of implicit residual averaging. means of a Spatial Galerkin discretization finite element is accomplished method assuming by a general unstructured grid of linear three-noded triangular elements. In the code, this is achieved by the equivalent process of computing the element contributions from a loop over all the element sides in the mesh. To ensure complete vectorisation of this procedure, the element sides are automatically coloured by the mesh generator'. The resulting scheme is stabilised by the application of an artificial dissipation operator which is formed as a blended combination of second and fourth differences, 5-3 then be plotted. More experiments are however necessary in order to ascertain the suspension rheological behavior. At moderately high loads the particles interact mainly with wt the h surrounding urudn fluid li and n the h stresses tesso of the h cloud lu are mostly due to particle/fluid interactions. On other hand, the for high cloud concentrations the frequency of collisions between particles increases and the pressure of the particle cloud should substantially grow, preventing the cloud from attaining concentrations above that of maximum packing. Neither experimental data nor theoretical developments exist to model the cloud pressure at high concentration. However, the correct modelling of the pressure is essential, as this is an important mechanism limiting the maximum value of particle concentration. 2.2 TIbaeflw Euleran Formulation of Average Eguations Disosed Phase as a Continuumls In the theoretical derivation of two-phase flow A monodispersed cloud of rigid spheres syspended in an equations, extensive use is made of conditioned incompressible isothermal viscous gas is considered. No averaging techniquess.l1. Spatial, temporal, or ensemble mass transfer between the phases is included. The average and double/mixed average operators are dynamic continuous behavior of th partiles is velocity, characterized by frequently found in the literature. pressure Two consecutive and concentration fields, averages are also encountered in turbulent two-phase For a distribution of particle sizes, as many velocity, flow modeling. Nevertheless the existing formalisms * pressure and concentration fields are required as the are not quite satisfatory, specially in dealing with number i be tted, of size turbulent flows. families Volume or time averaging produce filtering of small-scale turbulent structures or large The particl/particl and particle/fluid interactions are fluctuating frequencies. To avoid that filtering effect, simulated by the introduction of a "pressure" and a dispersed elements must be much smaller than "viscosity" of the particle cloud and an inter-phase force Kolmogorov scales and the number of elements inside taking ito account the momentum transfer between the the probe vohirne large enough to get stable values. Plums. ithose situations rarely occur in practice and are impossible to satisfy in dilute flows, Ensemble averages

59 5-4 avoid these problems but the interaction terms have no An axisymmetric turbulent jet configuration has been straightforward meaning and must be interpreted by chosen to validate the model for both single phase and comparison with other averages. Applying two two-phase flows. A classical finite volume four-node consecutive averages lead to equations with a large formula space-marching fully-implicit scheme has been number of terms with no direct physical meaning and used. Validation has been conducted by comparing the include high order fluctuation correlations, creating model predictions with available experimental data some serious closure problems. Preliminary predictions indicate a reasonably good Following the usual conditioning technique based on a agreement 2 ". phase-indicator function 3 and properly defining a joint 2.3 Turbulent Reactine Flows 34 probability density function to describe the dispersed phase dynamics, some of the ideas pointed out by Herczynski and Pienkowska1 4 are generalized, The mixing of dynamically passive inert and reactive scalars convected by a field of nearly homogeneous Theorems and associated properties are given allowing turbulence is being investigated. The transport equation a general, rigorous and systematic treatment of the for the joint probability density function (pdf) of scalars continuous phase of a dispersed two-phase flow system. The conditioned mean value of any variable derivative and their gradients is used. Exact treatment is possible of the processes of the straining by mean velocity can be expressed as the derivative of the variable gradients, the chemical reaction and the dissipation of conditioned mean value, plus an extra interface source scalar fluctuations. Closure approximations are required term, which will introduce in the equations information for the non-closed terms describing the straining of about the action of the second phase.the formalism scalar gradients by fluctuating velocity gradients and the employed, shares some features in common with those dissipation of scalar/scalar-gradient and of scalarpreviously developed. The difficulties mentioned above gradient correlations by molecular diffusion. are however overcome. Using those results conditioned average continuity, momentum turbulent kinetic energy A cumulant expansion technique17 3 is used to treat the and turbulence dissipation rate equations for the turbulent straining term. The limit of large Kubo or continuous phase can be obtained. The main Reynolds number ("strong perturbation limit") is characteristics of these equations are: identified. Furthermore, the "small time limit" allows to truncate the cumulant function operator series expansion - General applicability. In principle no restrictions are at the second-order level. This small time pdf behavior placed on the sizes of dispersed elements or their is all that is required in order to derive a scalar-gradient concentrations, increment random equation. For isotropic turbulent velocity gradients the straining is inversely proportional - Continuous phase variables appear statistically to the square of the Kolmogorov time micro-scale. The averaged. Therefore no spatial or temporal filtering is scalar gradients display a tendency to grow and to performed. become isotropic. - Continuous phase equations are obtained in a singlestep averaging operation. A binomial sampling mode1 33, 34 is used to close the molecular dissipation terms. This model has features in - Continuous phase equations are formally indentical to common with the LMSE closure approximation 9.'o and those in classical turbulent single-phase flows, with with Langevin models 29. This closure model leads to a similar terms and the same order of fluctuation qualitatively reasonable relaxation process and produces correlations. correct asymptotic results 3. - A few additional interaction terms, with direct physical A stochastic scalar-gradient straining model is proposed interpretation and mathematical properties which help in terms of products of two random vectors. This in modeling them, enter the continuous phase equations. simulation also implies a model for the turbulent velocity gradient and, therefore, for the turbulent strain General eulerian equations for the moments of the rate tensor and for the fluctuating vorticity. Only the particle velocity are next obtained. This is done from a consequences of this model on scalar gradients have been "Boltzmann type" probability density function transport explored until now. equation which is assumed to correctly represent the particle cloud dynamics 322. Dispersed phase average A Monte Carlo simulation 2 7 of the molecular dissipation continuity, momentum and fluctuating kinetic energy processes is performed considering non-interacting equations are then generated. Neither a particle cloud particles. During a time step, all the particles are pressure nor viscosity appear in the average equations. affected by a LMSE process, while only a fraction Particle fluctuating velocity correlations and phase acquire new values via a binomial or gaussian sampling interaction terms are unknowns. technique. Scalar mixing frequencies can be calculated as a part of this joint pdf formulation. The ratio of A k-e turbulence model is used for the continuous phase. scalar-gradient to scalar characteristic mixing The dispersed phase is modelled via a gradient transport frequencies is assumed to be a constant greater than approximation where the particle turbulent diffusivity is one23" rephrased in terms of a root mean square particle fluctuating velocity and, in the case of a dilute dispersed The previous formulation is then applied to study the Phase, a particle relaxation length. Clownr hypothesis evolution I atclr of an h inert rdcino scalar grid h generated octain. turbulence. are also proposed for the phase interaction terms. In particular the prediction of the concentrstion

60 variance and the characteristic mixing frequency derivatives. The following relation holds", displays an excellent agreement with the experimental data for the evolution of dilute CINa injected at the grid (I = - into a main water flow 5. The flatness factor of the " (11) concentration gradient grows to values significantly * stands for any continuous phase variable and, I for the greater than 3. The latter can be taken as an indication of derivative with respect to X (time or space). ( ) is meant the internal intermittency of the scalar field, for statistical average. The last term in equ. (11) The reaction between sodium hydroxide diluted in t represents the phase interaction. Should the characteristic length of the dispersed elements be much main flow and methyl formiate injected as a dilute smaller that the mean overall flow characteristic length, solution at the grid generating the turbulence has been this interaction term can be expressed as 3 investigated in detail by Bennani et al.5. Predicted means and T7U variances experimentally agree determined very well with segregation experimental coefficient data. d d*ij=- o Xcixs + F_ O stabilizes at about -0.7, and it is acurately predicted. The vd * P (12) coefficient of skewness of the injected species is positive while that of the species diluted in the main flow are negative, both increasing downstream. Large values of where otd is the "particle concentration", the overbar the flatness factors of both species clearly suggest the followed by a star denotes statistical average conditioned spatial segregation of reactants. The computation of to the center of the dispersed elements being positioned characteristic mixing frequencies for both species shows? * how moderately fast chemical reaction (Damkdhler at point x at time t, V is the dispersed element number of order unity) can significantly modify the average volume, the surface integral extends over the time and length scales of the scalar fields. This is a call of interface, Sp, r 4 ;j is a residue usually much smaller than attention on some extended practices in turbulent the first term on the right side of equ. (12) and combustion modeling which overlook this fact. The. coefficients of skewness and the flatness factors of the f-v n if X =t magnitude of the concentration gradients increase n. ink if X = x k (13) downstream reaching high positive values. This is a clear sign of the small scale intermittency. The vs and n are the interfae velocity and the Unit Vector predicted pdf of the concentration gradient magnitude vsrandicar the interface elit ate ut vt for both reactants display a log-normal behavior. Th Perpendicular to the interface element located at x, dissipation of the injected scalar fluctuations, is pointing towards the continuous phase, respectively. The significantly faster than that of the reactant diluted in the subscript k designates the k component of the main flow, due most probably to the initial structure of corresponding vector. the injected field that favors the action of the stretching Continuous Phase Conditioned Euuations and mechanisms and enhances molecular effects. Mo1ina Information on the statistics of length scales defined as The instantaneous conservation equations multiplied by I the ratio of concentration and concentration gradients anaed afterain equ (12)iyied he could be obtained from the formulation. The sensitivity and averaged, after making use of Equ. (12) yield the of the strain-rate/scalar-dissipation correlation continuity, momentum, turbulent kinetic energy and coefficient to changes in the Schmidt or Prandtl numbers dissipation rate averaged equations. could also be evaluated. The same formalism could probably be extended to investigate the fluctuating A k-c model is used to represent the turbulence. The vorticity dynamics in nearly homogeneous shear flows following expresions are used for the unclosed terms, with constant mean velocity gradients. 3. puesentatin OF TWO SELECT(D CASES J 3 The following two cases have been selected for a more (14) detail description: - a 1, VTTk 3.1 Eulerian Formulation of Two-Phase flow Average (k " (15) Fluid Phas~e Condii gin Technioue 3_ b _i ~average A phase indicator function, I, is defined such that 5.1, where a is the void fraction, U and u are the mean and fluctuating velocity, respectively, k and k are the and the instantaneous-fluctuating turbulent kinetic energy. rspecvely, 8ij is the Kronec r delta, t I if (x,t) is inside the continuous phase p' is the fluctuating pressure, ok is set equal to one and 10 otherwise v T = C(P,)k2 /e is the turbulent viscosity. P ande stand for the turbulent k ic eegy production and In the process of averaging the Navier-Stokes equations dissipation rat especively. conditioned upon the presence of the fluid phase the indicator function appears as a factor of time and space

61 5-6 'The weakest point in the model is the e -transport dad 2 d equation, which is replaced by - * - "+ (ai + au )=( VT + + V-(ad Vij + (ad V") 1 (20) + a -(Ce2 P - kaj(c~ C.e ) +Il 2 1 e e (16) ad_** d - = vd(ddj (adkd% k (21) where the constants o E, Cc and C 2are assigned the (21) values corresponding to single-phase flows and the phase interaction term is designated by Ip Based on dimensional considerations and order of magnitude estimates the following model is proposed, where v is the dispersed phase fluctuating velocity and vd is the particle turbulent diffusivity, which for a dilute dispersed phase is given by, V d = Cpkd/a. od has been set equal to one. I,=±a-EF -C. (k-kd) The phase interaction terms to be modelled in the P k e 3 dispersed phase equations are identical to those entering 21 the fluid phase equations and closure assumptions have +Ce K (Ui Vi) + k (1- e)j (17) already been proposed. A detailed derivation of the equations and a justification where p d and kd are the dispersed phase density and of the models are presented in References 2 and 3. fluctuating kinetic energy, respectively, a is the inverse of the particle relaxation time, K = (R/q) + a TL,with Numerical Results R and Tl being the particle radius and Kolmogorov An axisymmetric turbulent jet configuration has been length microscale respectively and TL=0.5 (k/a), V is the chosen to validate this model for both single phase and mean dispersed phase velocity and 0 = / L/('CLthroughout + /a). two-phase flows. The pressure is assumed constant the flow field. Ce 3 and Ce4 are both tentatively taken equal to one. A classical finite volume, four node formula, space The phase interaction terms in the momentum and marching, fully implicit scheme has been used. turbulent kinetic energy equations are modelled as, Due to this particular methodology applied to a 2! aparabolic shear flow, the radial momentum equation is IUi = - a (U i - Vi) (18) not solved. An additional closure assumption is required for Wr = Ur - Vr (the radial relative velocity) namely I ( -0) -! a (k - k d) W v d aad (19)(1)Ca ad (22) Should mass transfer at the interface be negligible and with Ca = 0.5. Justification for this hypothesis is for solid particles the lu, Ik and le include all the provided in Reference 3. significant phenomena taking place at the solid/fluid interface. The standard values for the single-phase k - E model coefficients are used: A rigorous presentation of the equations and closure assumptions is contained in Reference 3. CD = 009 ( G), Ce DL d Phase Averagc Euations and losure s Ce2 =1.92(1 -Q0 3 5 G), k LO, a= L3 A Boltzmann type equation is assumed to accurately G is the deceleration parameter, modifying the describe the particle cloud dynamics The generalised traditional CD and C. values, introduced by Rodi3O in transport equation for the particle velocity distribution 2 function can be multiplied by any function of the order to get the proper single phase jet spreading rate. velocity field and integrated over the phase-space in order to generate moment equations. For clouds of solid Based on a few initial runs, the preliminary values for non-diffuive particles undergoing elastic collisions and the additional two-phase model coefficients an: allowing neither diagregation nor agglomeration, average continuity, momentum and fluctuating kinetic Cp =0.I,Od = LO, Cc = LO, Cc =LO equations are easily obtained 2 3. 'lbe unknown tem are then modelled as,

62 ii Model dissipation rate be used, the closure problem associated with the turbulent transport can also be avoided 2s. Here Available experimental data for turbulent jets of air, some preliminary ideas on the use of the joint pdf of laden with solid particles or liquid droplets in highly scalar fields and their gradients are presented 3 4. dilute concentrations, were scrutinized. Restrictions setting the range of applicability for the model, are not 3.2.1Bai gton fully satisfied by most of the published works. Furthermore, only a few publications supply direct The isothermal turbulent mixing of two scalar fields measurement of each individual phase, and on measured undergoing a second order chemical reaction obeys the turbulent parameters and boundary conditions required local instantaneous equation, for proper model calibration. Nevertheless, the experimentaldata from References 12 and 24 have been L C, = Sa (C) (23) selected to test the model predictions. Three basic cases have been documented: where Q 2RLW ] k ill RMIO L= +va + v D DVV2 2 J~ii (1) 0(10) (24) (1) (1) 0(10) 0(100) is the convection/diffusion operator. vj (x,t) is the local instantaneous velocity field at point x and time t. which can be decomposed into a mean field, Uj (x,t), and where Ob stands for the initial particle mass loading fluctuation, uj (x,t). D is the molecular diffusivity ratio. assumed identical for the two speceis. C = (Ca), for Comparison between model predictions and a= 1,2, stands for the concentrations of the two chemical experiments are shown in Figures I to 6. Comments are species. C (x,t) can also be decomposed into a mean, self-explanatory. r and x represent the radial and axial <Ca(x,t)>, and a fluctuating field, ca (x,t). For a coordinates, respectively. x = 0 is located at the nozzle second order irreversible chemical reaction the exit. The nozzle diameter is D. In order to make clear chemical source term is expressed as the dispersed phase influence upon the flow, single phase So(C) = - Kc C 1 C2, for a=l,2, where Kc is the chemical jet measurements and predictions are included in each rate constant. figure. Single phase jet data were taken from References 24 and 35. Let the gradient Ca be defined as Caj = WCaC 0 xi. Its evolution foliows the equation, Figures I to 3 relate to case 1, the best documented one. Predicted and measured mean velocity, turbulence av. asa intensity and turbulent shear stress profiles are LCJ Ca+ presented for both phases. The agreement is reasonable. a=t* (25) Figures 4 and 5 refer to case 2. The values of the variables at the centerline are slightly underpredicted. Summation over repeated latin and greek indexes is The turbulent axial velocity is accurately estimated. implied. A transport equation for the joint pdf of the Centerline velocity values have been shown to be very scalar and scalar-gradient fields can readily be sensitive to the value of C. derived 34, Figure 6 belongs to case 3. Centerline particle and fluid ap a a as(n S P+ velocity are correctly predicted. However, the Sa(fP] + 7x-- Pi computed profile for the dispersed phase seems to be x, sharper than what the few experimental points available suggests. D)\j r 3.2 TruWm knlmizingz andcmialrealima 34 ara ar~ (0 ci~' It seems now clear that neither large-eddy simulation 2 ( ac (L0ES) nor diect numerical simulation (DNS) can be 2D 'a _ J_ successfuly applied a present to solve general turbulent 1a ajr-a i\dx reacting flow poblemsu. Some investigators advocate the pressuva pdf method in order to evaluate the equation9, o is by far the m omethodology to (K U (26) cope with no-linears ro M ese terms ar close in the pdf fomulation. Should a wherepdrd X = =P(r, X;t)drd X is the probability usport equation for the joint pdf of velocity, kinetic ofhjitvnr C(X)<rS r d enei diutdm rate, conentraio and scalar of the joint evet r. 5 < a + d r a and NJr tt

63 q 5-8 X Ca/(x,t) < Xi + d X. Eq,. (26) bas approximate P(, X;t). Astochasticpale n is defined been restricted to the case of statistically homogeneous by its concentration and concentration-gradient turbulence and scalar fields. (v I rxz ) stands fo {~1 (a) (0'1. the expectation of the variable V conditional upon C (xt) = rand VC(xt) = Z IStlraining All the ters on the left hand side (LHS) of Equ. (26) Equ. (28) can be formally transformed i,-.,o a finiteare closed and physically represent the accumulation of increment equation by using the definition of the P probability, the probability transport in concentration derivative and a Taylor series expansion. After a few and concentration-gradient spaces due to chemical algebraic manipulations, the following stochastic model reaction and due to molecular mixing of concentrations, (equivalent to a closure assumption) is proposed 34, respectively. The terms on the right hand side (RHS) of t I2 V ()() (n() Equ. (26) are not closed. The phenomenon of turblence A C (t) = (8t)(x. ln n) (n) (n) straining of scalar-gradients (similar to the vortex 60v stretching mechanism of turbulence 25 ) is adscribed to (n) J (a) the first term on the RHS. The last two terms represent + z kn) the 8.4 cross-dissipation C(. of scalars/scalar-gradients (t) and the k (29) dissipation of scalar-gradient/scalar-gradient correlations due to molecular mixing, respectively. In The LHS is the change experienced during a time step, order to use Equ. (26) the three terms on its RHS must &, by the ith component of the gradient of the scalar a be approximated. for the stochastic particle n at time t. x, y and z are constants with analytically deermined values, Turbulence Straining of Scalar-Gradier.. x= , y= , z = Since a sequential Monte Carlo (MC) simulation 27 of TI and 4 are statistically independent random vectors Equ. (26) will be conducted hereafter, the effect on P of with zero mean and unity correlation matrix. the terms on the RHS can be separately treated M In the limit of large turbulent Reynolds number and of small time, Kubo technique1 7 leads to the following A recently developed binomial sampling mode 3 is representation of the first term on the RHS of Equ. (26), used to close the last two terms on the RHS of Equ. (26). Although the last term on the LHS of Equ. (26) is closed, it is also treated in this manner. It has been shown that this model produces a qualitatively reasonable evolution (27) TX [Xajand the correct asymptotically gaussian relaxation for large times. The subscript I on the LHS of Equ. (27) f' icates that hemi all acion the P variation is due only to the first tenii n the RHS of The change of scalar and scalar-gradient in a time step, Equ. (26). It is pertinent to remark that Equ. (27) implies only mathematical approximations but no due to chemical reaction me exactly given by, closure hypothesis in the classical sense. 4nl) -(t) n) For statistically homogeneous and isotropic fluctuating (30) velocity-gradient fields and incompressible flows Equ. (27) finally becomes, c(a) as LC((t) W ap) ~ A C ((80) _;C _ C t t ( ) XaiXpj-Xcj [oi) (31) Numril Reslt 2 p The mixing and chemical reaction of an acid and an ' j(28) alkali diluted in water in a field of grid generated turbulence have been investigated where by Bennani <e> is the et al. average The turbulent kinetic energy acid is injected at the grid and the base is convected by dissipation rate 25 and v is the kinematic viscosity, the main flow. These experiment has been predicted using the previously explained pdf methodoly. The evolution of concentration means and variances is presented in Figure 7 showing a good agreement with Due to the high dimensionality of P. Equ. (26) is the experimental data. The segregation coefficient in simulated via a sequential MC technique 27. Every Figure 8 also shows a reasonable agreement with the physical process represented by the various terms in measuted constant experimental value of The Equ. (26) is then considered to act separately and skewness and flatness factors for the concentration and sequentially upon N stochastic particles which concentration-gradients ae pesented in Figures 9 and. A,>

64 5-9 10; no experimentas data are available. The large values - The conceptually different framework of the pdf of the flatness factors of the concentrations are a clear methodology makes it appear somehow unnecessarily indication of the spatial segregation of the reactants, cumbersome to some investigators. while those of the concentration-gradients might also reflect the internal intermittency of he scalar fields. The - Moreover the inclusion of the velocity and/or evolution of the characteristic turbulent frequencies for dissipation rates or gradients into the formulation leads both reactants is plotted in Figure 11. Due probably to to a pdf with a large number of independent variables. the difference in the initial condition of the acid and the Non-standard Monte Carlo simulations must be used to base the frequencies for both reactants evolve in a obtain numerical solutions. different way. This is not taken into account in present combustion models. - Since reaction takes place at the molecular level, LES or vortex dynamics techniques are also confronted with 4. CONCLUDING REMARKS closure problems identical to those present in moment formulations. An overview of turbulent flow modelling activities in Spain have been presented. Two selected examples have - DNS of simplified chemical kinetics in simple flows been described in detail: turbulent two-phase flows and can be of some help as a guidance for the asymptotic turbulent reactive flows. While there seems to be a behavior of models. consensus that a moderately significant progress in the last two decades has been achieved in the field of - Most available experimental data pertain to the field of turbulence models using moment equations, there are turbulent combustion. Experiments on irreversible not reasons to be so optimistic in the two cases described- bimolecular chemical reactions in incompressible grid- The following comments seem then pertinent: generated turbulence where the interactions among turbulent straining, molecular mixing and chemical 4.1 Turbulent Two-Phase Flows reaction can best be evaluated are badly needed in order to validate model predictions. - No agreement has been reached on the local instantaneous basic equations to be used as a starting In summary, in the foreseable future significant effort point, in particular for the dispersed phase. While some will be required to model turbulent two-phase and investigators treat the dispersed phase as a continuum, reactive flows. A judicious combination of physical defining a particle cloud pressure and viscosity, others insight, theoretical developments, analysis via LES prefer to describe the particle cloud dynamics in terms and/or DNS of simplified flows and examination of of moments of a Boltzmann type equation. reliable experimental data is the best support for successful modelling endeavors. - The type of average employed to generate moment equations also leads to significantly different transport ACKNOWLEIGEMENTS equation structure. The authors wish to express their gratitude to Professors - Classical k - e or Reynolds stress models can be used Femnundez-Cara and Chac6n and Drs. Abbas and for the fluid phase. However, the unknown terms in the Hernmndez for supplying excellent summaries of their dispersed phase equations and the phase interaction modelling activities. Thanks are due to Ms. P. Ezquerra terms deserve special and careful consideration. for patiently and carefully typing the manuscript. - LES or vortex dynamics techniques can help to gain significant knowledge on peculiar phenomena REFERES influencing the particle dynamics. 1. Abbas, A.S., CASA Technical Report, DNS of some simplified two-phase flow systems can provide rational basis to the asymptotic behavior of the 2. Aliod, R. and Dopazo, C., "Modelling and proposed models. Numerical Computation of Turbulent Axisymmetric Jets Containing Dilute - Reliable experimental data for simple flow Suspensions of Solid Particles", Sixth Symposium on configurations to validate model predictions are scarce. Turbulent Shear Flows, Toulouse (France), 4.2 Turbulent Reactive Flows Septemlxi' Aliod, R. "Formulation and Numerical Treatment of * - The non-linearites in the chemical source terms Mathematical Models for Turbulent Two-Phase introduce important difficulties in moment Flows" (in Spanish), PhD Dissertation, University formulations. Even for some idealized situations of Zaragoza, February (diffusion controlled limit, premixed systems, etc.) existing models are not as accurate as desirable. 4. Baldwin, B.S. and Lomax, H., "Thin Approximation and Algebraic Model for Separated Turbulent -The pdf formalism overcomes te closure problem Flows", AIAA 16th AerspaceSciences Meeting, associated with the chemical source terms. The closure Huntsville, Alabama, January 1978, pp of the molecular diffusion effects has proved however to be avery difficuhtone. }

65 Bennani, A., Gence, J.M. and Mathieu, J. "The 20. Maclaughlin, D., Papanicolau, G and Pironneau, Influence of a Grid-Generated Turbulence on the Development of Chemical Reactions", AIChE J. 0., SIAM J. Appl. Math. 45,1985, pp , 7, 1985, pp Mansour, N.N., Moin, P. Reynolds, W.C. and 6. Buckley and Le Nir, Math. Programm. 27, , Ferziger, J.H., "Improved Methods for large Eddy Simulations of Turbulence", in Turbulent pp Chac6n, T., PhD Dissertation, Univ. P. and M. Curie, Paris VI, Shear Flows I, F. Durst et al. eds., Springer Verlag, Berlin, 1979, pp Marble, F.E., 'Dynamics of a Gas Containing Small Solid Particles", Proceedings 5th AGARD 8. Delhaye, M. and Achard, J.L "On the Averaging Combustion and Propulsion Symp., Pergamon, New Operators Introduced in Two -Phase Flows York Modeling", OECD/NEA Specialists Meeting on Transient Two-Phase Flow, Toronto (Canada), 23. Meyers, R.E. and O'Brien, E.E., "The Joint PDF of a Scalar and its Gradient at a Point in a Turbulent Fluid",Combust. Sci. Technol. 26, 1981, pp Dopazo, C. and O'Brien, E.E., "An Approach to the 134. Autoignition of a Turbulent Mixture", Acta Astronautica 1, 1974, pp Modaress, D., Than, H. and Elgobashi, S.E., 'Two Component LDA Measurements in a Two-Phase 10. Dopazo, C., "Non-Isothermal Turbulent Reactive Turbulent Jet", AIAA J. 22, 5, 1983, pp Flows: Stochastic Approaches", PhD Dissertation, State University of New York, Stony Brook, June 25. Monin, A.S. and Yaglom, A.M., "Statistical Fluid Mechanics", Vol. 2, MIT Press, Cambridge, Dopazo, C., "On Conditioned Averages for 26. Orteg6n, F., PhD. Dissertation, Univ. P. and M. Intermittent Turbulent Flows", J. Fluid Mechanics Curie, Paris VI, , 3, 1977, pp Pope, S.B., "A Monte Carlo Method for the PDF 12. Elgobashi, S.E., Abou-Arab, T., Rizk, M. and Equations of Turbulent Reactive Flow", Mostafa, A.,"A Two-Equation Turbulence Model Combustion Sci. Technology 25, 1981, pp for Two-Phase Jets", Proc. 4th Symp.Turbul. Shear 28. Pope, S.B., "Computations of Turbulent Flows, Karlsruhe (W. Germany), September 1983, pp Combustion: Progress Lecture, 23rd. Symp. and (Intemat.) Challenges", on Plenary 13. Eswaran, V. and Pope, S.B., "Direct Numerical Combustion, Orleans (France), July Simulations of the Turbulent Mixing of a Passive 29. Pope, S.B., "PDF Methods for Turbulent Reactive Scalar",Phys. Fluids 31, 1988, p Flows", Prog. Energy Combust. Sci. 11, 1985, pp. 14. Herczynski, R. and Pienkowska, J., "Towards Statistical Theory of Suspensions", Ann. Rev. Fluid Mech. 12, 1980, pp Rodi, W., "The Prediction of Free Turbulent Boundary Layers by Use of a Two-Equation 15. Hemandez, J.A., "Analytical and Numerical Study Model of Turbulence, Ph D Dissertation, mpel ollee d ist, of Instabilities in Two-Phase Flows", (in Imperial College of London, U.K.,1972. Spanish), PhD Dissertation, Universidad 31. Rogallo, R.S. and Moin, P. "Numerical Simulation Polit~cnica de Madrid, of Turbulent Flows", Ann. Rev. Fluid Mech. 16, 16. Kida, S. and Murakami, Y., "Statistics of Velocity Gradients in Turbulence at Moderate Reynolds 1984, pp Schagerl, K., "Rheological Behvior of Fluidized Numbers", Fluid Dynamics Res., 4, 1989, pp.347- Systems", in "Fluidization", Academic Press, 370. London, Kubo, R., "Stochastic Liouville Equations", J. 33. Valiflo, L and Dopazo, C., "A Binomial Sampling Mathematical Physics 4, 2, 1963, pp Model for Scalar Turbulent Mixing", to appear 18. Launder, B.E., Reece, G.J. and Rodi, WJ., Phys. Fluids A, "Progress in the Development of a Reynolds 34. Valifio, L, "Computation of Homogeneous Stress Turbulent Closure", I. Fluid Mechanics 68, Turbulent Flows with Chemical Reaction: 1975, pp Monte Carlo Numerical Simulation of Velocities, Concentrations and Concentration-Gradients", 19. Lumley, ii., "Computational Modeling of (in Spanih), PhD Distation, University of Turbulent Flows", Adv. Appl. Mech., 18, Zaragoza, October , pp

66 35. Wygnanski,. and Fiedler, H.,"Some Measurements in the Self-Preserving Jet", J. Fluid Mechanics,38, 3, 1969, pp Yeung, P.K. and Pope, S.B., "Lagrangian Statistics Case from Direct Numerical Simulations of Isotropic 2.0- Fluid, single phase ft. Turbulence", J. Fluid Mechanics 207, 1989, pp , 2 Fluid, two phase fi A,,/UUx,,s.0h._. V x/ Ux,os Ph. 1.0" 2.0"0 Case Is Fluid, single phase fltow , Fluid 2phase flow 3 a Solid, 2 phase flow N r/x 0.5-~ 1/Dp~ =1 Dpsop5 JA Figure 3 urbulent Shear Stress profile at lf/6d=2:o * ote of.odarress et a]. (24 ). Case r/x Figure I IMean velocity profiles St X/D=20 Date of Modarress et 6). (24) case I _ U =/Ux,c,s.p. V 2.0- / Ux,OA,. 2 ".a 0 0 ase (2; EMa L2~annkiF d9(3%l Fluid, ML - /2 1 < ' M et)24 f, I Fluid, single piase flow (,I. 0 Fluid t (24 flo 1 A 2 & Fluid, 2 phase flow Fluid, two p she flow 3 a Solid. 2 pese flow [3 * SOlid. two p5sse flow,) 1.0-p50 & ~0.5- illi r/x Figure 2 [Turbultnce intensift profils at xld Dote of ModefTele st el. (24) for 2 phi flow. Case I Wigantskl and Fiedler (35) foe single phese J0t -- GNP r/x Figure 4 IMean VeiOCit' profiles at X/D=20 I Dats of Moderress et S. (24 )Cte 2

67 5-12 r * Fluid, two phase flow 3 4 Solid, two phase flow Z 0.j,4.00 Dimensionless time 39.0 In 0 0 : V r/x '000 Figure 5 ITurbuleace intensity profiles at x/d z Ds',a of Madarress at a). (241 for 2 ph. flow. Case Dimensionless time 39.0 li'5gnanski and Fiedler (35 ) for single phase jet Figure 7. Evolution of means and variances of two reactive scalars. Comparison with Benanni et al 5 experimental data (Kc =47.5 V/mol s). U X/ UX..,S$ Case Fluid, single phase flow 2.0- & 2 Fluid, two phase flow * 3 Solid, two phase flow 1.5- in0 DP 2700j, SE 1.00 CP.~ r/x F~~gure rolisat~ Dimensionless time 39.0 ~l Date of Eighobashi St ai. I'12) Cass 3 Figure 8. Evolution of the segrgation coefficient. we amounateexperimental value 5 is 0.7. (K=4. Jrnol s).

68 C: C4 4) 4.00 Dimensionless time e 4.00 Dimensionless time A9, 4.00 Dimensionless time 39.0 Figure 9. Evolution of the skewness coefficients and fatness (Kc = 47.5 factor /tool of s). two reactive scalars Figure 11I. Evolution of the characteristic turbulent fr-equencies Kc= 47.5 /tool of two s). reactive scalars "d2 4.~0 Diesolsstm49 0 C4) LU0 40 i ~.s0 V,, - -.*0 -, 4.00 Dimensionless time Figure 19. Evolution of the skewness coefficients and fltess factor of two reactive scalar graeients (downstream direction derivative) (K 47.5 s). 7/sol

69 6-1 COMdPUTATIONAL TURBUL3NCZ STUDIES IN TURKIEY tnver Kaynak TUSAJAerospace Industries (TAI), Inc. P.K. I , Kavakhdere Ankara, TURKEY SURMAXY RESULTS In this paper, applications of various turbu- 1) Caleaisao a LowSpeed lnaite Sweo-Wing Flow lence models to different flow problems that have recently been carried out in Turkey are The Berg and Elsenaar' incompressible turbupresented. Navier-Stokes, boundary layer and lent boundary l-ayer experiment under infinite vorticity-stream function methods are used to swept wing conditions was chosen to compare solve two- or three-dimensional steady/unsteady the performances of the zero-equation algebraic flow problems. Examples are given in low- model of the Cebeci and Smith$ and the halfspeed and transonic flow regimes for axisymmetrc bodies, airfoils, rigid ripples and Jet flows. KnL aixn tended to three-dimensions. The Different turbulence models are used such a Johnson-King model which includes the nonalgebraic, half-equation and h - i models. It equilibrium effects in a developing turbulent has been demonstrated that improved accura- boundary layer was found to significantly imcies can be obtained by using the so-called half- prove the predictive quality of a direct boundequation (nonequilibrium) turbulence model for ary layer method. The improvement was espesome three-dimensional configurations. Suitabil- cially visible in the computations with increased ity of different turbulence models is explored for three-dimensionality of the mean flow, larger ina variety of flow cases such as dynamic stall, jets- tegral parameters, and decreasing eddy-viscosity in-croesflow and oscillatory boundary layers. and shear stress magnitudes in the streamwise direction; all in better agreement with the ex- INTRODUCTION periment than simple mixing-length methods. The Johnson-Kinq model accounts for the convection and diffusion effects by solving an ordi- Turbulence modeling and grid generation seem nary differential equation (o.d.e.) governing the to be the most important issues for computa- streamwise development of the maximum shear tional fluid dynamics today. Although the algo- stres derived from the turbulent kinetic energy rithms have reached to a relative state of matu- (t.k.e.) equation. The o.d.e. was originally derity in some areas, turbulence modeling remains rived in two-dimensions. For three-dimensions, as a major difficulty for many flow problems. this o.d.e. was simply interpreted as an equa- Recognizing this fact, in parallel to the efforts tion governing the development of the maximum going on in the world, there have been some ef- shear stress on a local streamline. This offers torts in Turkey to contribute to our understand- a lot of simplicity and economy in three dimening of turbulence. The activities are generally aions. concentrated on the computational sld3 of the turbulence modeling. Two of the major centers for incre!5sd activity in turbulence are the In the Berg-Elsenaar2 experiment, an adverse TUSAS Aer)space Industries (TAI), Inc. and pressure gradient is applied on a 35" swept flat the Middle East Technical University (METU). plate. In the leading edge, the turbulent bound- In this paper, firstly the following model prob- ary layer is very nearly two dimensional. Downlems that have been studied at TAI are consid- stream it becomes three-dimensional due to the ered: calculation of a low-speed infinite swept sweep angle and the adverse pressure gradient wing flow using the boundary-layer equations, (Fig. 1). The difference between the wall flow calculation inltratsonc of an axisymmetric bump proble angle (#,), and the flow angle at the boundary- tran nic flow using tisy the e Tin-Layer T hin-layer Npvoblem Navier- layer fa t I edge t (a.) ow that s r is m (A di ) e in i seen n. N to r increase m a u - Stokes (TLNS) equations, calculation of the dy- fast In the downstream direction. Near measurnamic stall o a NACA0012 airfoil using the Ing station 9 the wall flow angle exceeds 4, = 5, TLNS equations, and numerical simulation of which means that the flow i parallel to the plate the jets-n-croesflow problem using the TLNS leading edge showing separation. equations. In all these problems, zero- or halfequation models have been used. The main Calculations were done on a grid with dimentheme of the research conducted at TAI has sions 6o x 10 x 8o in streamwise, spanwise and been the extension of the half-equation turbu- normal directions respectively, using the Van lence model of the Johnson and King, Into three- Dalsem and Steger' time-relaxation algorithm dimensions. This model is particulary attrac- in the direct mode. On the IBM 3090 Model tive because in the past, much improved accu- 160E scalar computer Installed at TAI, the CPU racles have been obtained for two-dimensional time for the Johnson-King model was only 20% flow problems at a little extra coot. Three- higher than the Cebeci-Sm~tb model and about dimensional extention of this model was success- the same number of iterations were required for fully tested for the first two of the problems convergence. Figure 2 shows the angle 06 beenumerated above. Secondly, two more exam- tween the wail shea strew vector and the exple are given related with the research done at ternal streamline of the boundary layer. The METU: computation of the unsteady turbulent baseline Cebeci-Smith model departs from the flow over rough surfaces using a vorticity-stream experiment after the fifth station. The interal function formulation with a k-t model, and ex- method of Coustei' doen't fare better as It de. tension of the law-of-the-wall for conduits hay- parts after the third station. The NLR methods Ing sharp comers. In the following lines, each of lies very close to the Cebecl-Smith model. On thens model problems will be presented: the other hand, the present method goes as far.4

70 6-2 as the seventh station, than nearing the separa- the type of viscous-inviscid interactions that can tion falls to go any further, develop on airfoil sections at transonic conditions, foot transonic and the model wind tunnel was tested and in 6- by-6 Ames foot 2-by-2 su- In fact, it is known that an indirect procedure personic wind tunnel. A finited ieece grid is needed to handle the separation, but in this was braically generated to simulatete t ar instance the success of the turbulence model Is wbualp rfig. 7). t has dimensions x x 41 in measured by how far downstream it can go be- the streamwise, circumferential and normal difore diverging from the experiment. As for the retresecire nta anpnrmal i integral parameters, Fig. 3 shows the stream- rections respectively. Since the experiment is wise displacement thickness for the various mod- axisymmetric only o planes were used in the cirels compared with the experiment. The nonequi- cumferential direction in a 90 deg. pie. The prelibrium model yields thicker displacement thick- liminary runs were done at TAI's IBM E nesses than the baseline$ (and other) models. A computer using the ARC3D' flow solver. The turbulent eddy viscosity profile (at Station Statn 7) computations were also repeated at the NASAte Ames Research Center using the Cray Y-MP and nonequilibrium in i mode mowel g. ene Icuts down the outer por- Cray-2 supercomputers. i36 The fine grid size was x 6 x 4s. The Mach number in the experition of the eddy viscosity significantly in a better ment was and the Reynolds number was agreement with the experiment. Finally, Figs. 5 1&6 X 106/m. and 6 show the streamwise and crosflow velocity profiles inside the boundary layer. Again, the Results of the axial bump simulation are shown nonequilibrium model gives closer results to the in Figs In Fig. 8, coefficients of presexperiment than the baseline model. For further sure for the equilibrium models of Cebeci and details, the reader is referred to Ref. 7 on this Smith' (C-S) and Baldwin and Lomax" (B-L) subject. and the nonequilibrium model of Johnson and King, (J-K) are compared against the exper- 2) Truene Modeling Studies for Trasnic Separated imental data. As seen, the equilibrium models Flows predict shocks too aft of the experimental one in result of the rapid rise in the shear stress The aerodynamic characteristics of aircraft in and lesser boundary layer growth. The presthe transonic regime are very sensitive to the sure plateau that forms because of the shock viscous effects, and the selection of the turbu- induced separation is just nonexistent. On the lence model is no less important than the splec- other hand, using the J-K model correctly pretion of the numerical algorithm. It is well known dicts the exact shock location as well as the presthat turbulence is slow to respond to changes in sure plateau after the shock. Figure 9 compares the mean strain field. The so-called equilibrium the displacement thicknesses around the trailing models that are widely used in Navier-Stokes edge and again demonstrates the ability of the computations fail to model this aspect of the tur- J.1-K model to produce a more accurate viscous bulence and exaggerate the turbulent boundary layer Frowth. Figure 10 shows the streamwise layer's ability to produce the turbulent Reynolds variation of the maximum shear stress for the shear stresses in regions of adverse pressure gra- experiment and computations. The C-S model dient. As a consequence, too little momentum predicts a rapid rise in the shear stress typical loss within the boundary layer is predicted in the of that mode, whereas the J-K model predicts region of the shock wave and along the aft part of a smoother rise. Also, the C-S model displays a the aerodynamic body. This in turn causes the faster decay rate for the shear stress than for the equilibrium models to predict the shock waves J-K model. In Fig. 11, the boundary layer mean too far aft of the experiment, velocity profiles are given at the trailing edge of the bump. The J-K model is clearly superior Recognizing the shortcomings of the equilib- over the equilibrium models inside the boundary rlum turbulence models, a collaborative research layer. The axial bump experiment was a model project between the TAI and NASA/Awes Re- problem to check the performance of the new search Center was started. The purpose of the turbulence model. Work is underway to test the project was to explore the possibilities of the ex- new model for a truly three-dimensional problem tention of the Johnson-King model into three- such as a finite wing. dimensions for use in the Navier-Stoke flow solvers. Under this project, a computer program 3) CAicuistie of te Dynamic Stall was written for the Navier-Stoke flow solvers in general curvilinear coordinates using the stream- In parallel to the efforts going on in the steady wise Integration of the governing o.d.e. for three-dimensional low-speed or transonics area nonequilibrlum effects. The streamwise integra- as described above, unsteady motion of airfoils tion idea was provoked by the fact that the t.k.e. including dynamic stall Is also investigated. The equation upon which the present o.d.e. was de- purpose of this research is to asess the imrived describes the time rate of change of the portance of the turbulence model for the dyturbulent kinetic energy ewking a ftw pmjide. namic stall phenomenon and explore the suitability of the J-K model for this type of flow. In this study, the NASA-Ames ARC3D0 finite- Majority of the numerical simulations encoundifference Thin-Layer Navier-Stokes flow solver tered in the literaure seems to be conducted for was used. The preliminary calculations using laminar flows." For turbulent flows, a computhe newly written turbulence model were done tational study was recently made by Rumesy on the axsymmetric bump flow experiment of Anderson12 for unsteady airfoil motion. In their Bachalo and Johnson.' In this experiment, a thin study, significant differences between the B-L walled cylinder (16.2 outer diam.) with an ax- model and J-K model were found which showed I ymmetric circular bump attached 61 cm. from a need for further studies on this line. The the cylinder leading edge. The bump has a thick- llght and deep dynamic stall regimes are connewo of 1.9 cm. and a chord length of 20.8 cm. sidered In the present research. Especially, the The model was specifically designed to simulate light stall is probably the one that most war-