Modelng - J. P. Chabard and D. Laurence TURBULENT FLOW MODELING J. P. Chabard Thermal Transfer and Aerodynamcs Branch, EDF Research & Development, France D. Laurence Natonal Hydraulcs Laboratory, EDF Research & Development, France Keywords: turbulent flow, turbulence modelng, energy cascade, length-scales, Reynold's number, Naver-Stokes equatons Contents 1. Introducton 2. The Reynolds Averaged Naver-Stokes Equatons 3. Energy Cascade and Length-Scales n a Turbulent Flow 4. Turbulence Modelng 5. Examples of Industral Applcatons 6. Concluson and Future Developments Acknowledgements Glossary Bblography Bographcal Sketches Summary After a hstorcal ntroducton, ths artcle descrbes the state of the art and the trals for new developments n the feld of turbulent flow modelng. O. Reynolds showed that when a nondmensonal parameter called the Reynolds number becomes hgh, the flow becomes turbulent. Ths means that perturbatons can be amplfed and can generate unsteady eddes. Consequently, a turbulent flow s usually unsteady, three-dmensonal, and contans eddes of very dfferent szes. Due to ther large Reynolds numbers, most real lfe flows are turbulent. A frst approach for smulatng turbulent flows s to compute only the averaged moton. Incompressble turbulent flows obey the Naver- Stokes equatons. When the averaged moton s concerned, these equatons are averaged usng the Reynolds averagng. Then the Reynolds averaged Naver-Stokes equatons are obtaned. Unfortunately, due to the nonlnear advectve term of the Naver-Stokes equatons, new unknowns are obtaned the exact Reynolds stresses equatons can be derved for the Reynolds stresses, but agan, they nvolve hgher order statstcal correlatons that are unknown. Consequently, the Reynolds averaged Naver-Stokes equatons have to be modeled; ths s called the turbulence closure problem. Before ntroducng the dfferent closures used n practce, ths artcle ntroduces the Kolmogorov theory, whch explans how turbulence s possble,.e., how turbulent knetc energy s produced, and how t s dsspated, explanng why varous szes of eddes exst n a turbulent flow. Then the dfferent closures are presented, frst the one based on the Boussnesq eddy vscosty concept (the mxng length model and the k-ε model) and then Reynolds stress transport equatons models. In parallel, recent approaches based on the drect resoluton of the unsteady Naver-Stokes equatons are Encyclopeda of Lfe Support Systems (EOLSS) 186
Modelng - J. P. Chabard and D. Laurence ntroduced drect numercal smulaton f all the szes of eddes are computed, or large eddy smulaton f only the largest eddes are computed, the effect of the smallest eddes beng ntroduced through modelng. As a concluson, the trals for developments n the frst decade of the twenty-frst century are presented n the felds of both the Reynolds averaged Naver-Stokes equatons, and drect numercal smulaton. 1. Introducton Real flows are beautful. They are always changng, wth eddes randomly formng At the end of the ffteenth century, ust magne Leonardo da Vnc (1452 1519), bent over a brdge and gazng at the rver, fascnated by these eddes, and tryng to catch ther beauty by drawng on hs notebook. Real flows are beautful because they are turbulent. A turbulent flow s characterzed by ts unsteadness and the presence n the flow of eddes of varous szes. Thus, a turbulent flow s usually three-dmensonal. Most flows n nature (flow n atmosphere, flow n rvers) and n ndustral devces (ppng systems, power plants) are turbulent. After Leonardo s pctures, there was a wat untl the nneteenth century for a descrpton of the turbulent moton. The frst qualtatve descrpton of turbulent flows was publshed by Hagen n 1854. He obtaned flow vsualzatons by usng glass ppes and partcle laden flows. He exhbted two regmes of flow: a regular regme wth smooth partcle paths (lamnar flow), and a regme where these paths become chaotc wth fluctuatons (turbulent flow). Barré de Sant Venant (1797 1886) was the frst to lnk the stresses n a turbulent flow to the ntensty of the eddes created by the flow. Then, Boussnesq (1842 1929), n hs essa sur la theore des eaux courantes (theory of fresh waters) presented at the Academy of Scences of Pars n 1877, demonstrated the necessty of workng on averaged quanttes for the study of turbulent flows. He also ntroduced the eddy vscosty concept, whch s used n a large number of turbulence models. He lnked the eddy vscosty to the ntensty of the mean fluctuatons produced n a turbulent flow. Osborne Reynolds (1842 1912) exhbted a nondmensonal parameter whch s related to the transton between the lamnar and the turbulent flow regme. Ths parameter s ρvd VD called the Reynolds number, Re = μ = ν, where V s the velocty, D a length μ scale, ρ the densty, μ the dynamc vscosty of the flud, and ν = the knematc ρ vscosty of the flud). However, t was Lord Kelvn (1824 1907) who ntroduced the denomnaton of turbulence. At the begnnng of the twenteth century, t was the German school of Göttngen Unversty, created by Ludwg Prandtl (1875 1953), whch generated advances n turbulent flow modelng, and especally n the feld of turbulent near-wall flows. Prandtl ntroduced the frst turbulence model the mxng length model, whch s well suted for the computaton of turbulent near-wall flows. Blasus (1883 1970) produced expermental results of turbulent channel flows and Von Karman (1881 1963) worked on turbulent wakes behnd obstacles and on turbulent boundary layers. Encyclopeda of Lfe Support Systems (EOLSS) 187
Modelng - J. P. Chabard and D. Laurence Kolmogorov (1903 1987) ntroduced n 1942 the concept of the turbulent energy cascade from large (sze L t ) to small (sze λ 0 ) eddes. He demonstrated that theses eddes of dfferent length scale also have very dfferent velocty scales, and that the rato of the L t 3/4 largest to the smallest scale s Re λ =, explanng the coexstence n a turbulent 0 flow of eddes of very dfferent length scales. The development of powerful tools for scentfc computng (ncludng both computers and accurate numercal methods) enabled advances n turbulence modelng n the 1970s. Launder and Spaldng ntroduced n 1972 the so-called k-ε model, based on the Boussnesq eddy vscosty concept, whch s stll commonly used for smulatng turbulent flows. In 1975, Launder, Reece and Rod proposed the frst Reynolds stress transport model, removng the eddy vscosty assumpton, and openng the way for more accurate smulatons of turbulent flows. The mathematcal presentaton of turbulence modelng s summarzed n the frst secton of ths artcle, focusng on ncompressble flows,.e., flows wth constant densty. In such a flow, moton of flud partcles s descrbed by the so-called Naver-Stokes equatons. These equatons cannot be solved exactly for a turbulent flow, because tme scale and length scale of the smallest eddes n a turbulent flow are very tny, and ther exact smulaton would requre too much memory and tme, even on the most advanced computers. Followng the Boussnesq dea, t s necessary to ntroduce equatons on averaged velocty and pressure. Ths averagng was ntroduced by Reynolds and led to Reynolds averaged Naver-Stokes equatons. Unfortunately, these equatons contan new unknowns comng from the averagng of the nonlnear advectve terms the Reynolds stresses related to the fluctuatng velocty components. Exact equatons can be obtaned for the Reynolds stresses, but they nvolve agan hgher order unknown statstcal correlatons. A turbulence model usng these Reynolds stress transport equatons s called a second order turbulence model. Secton 2 of the artcle focuses on the energy transfer n turbulent flows, based on the Kolmogorov model for homogeneous turbulence. Frst the knetc turbulent energy equaton s establshed. It demonstrates that a turbulent flow s possble because the mean flow produces turbulent energy through the nteracton of large eddes wth mean flow stran. The Kolmogorov model explans that ths energy s cascadng from large eddes towards small eddes (Kolmogorov length scale) where t s dsspated by vscosty. Secton 3 presents varous turbulence models. The smplest one s the mxng length model. A more advanced and predctve model s the k-ε model. Ths commonly used for real lfe flow smulaton. More accurate models based on the Reynolds stress transports equatons are also presented. In parallel, the way of drectly solvng the Naver-Stokes equatons s ntroduced. It s very demandng n term of computer power, and s thus lmted to smple flows. Nevertheless, ths approach s very useful for producng rch numercal experments. Encyclopeda of Lfe Support Systems (EOLSS) 188
Modelng - J. P. Chabard and D. Laurence Secton 4 gves examples of turbulent flow smulatons n varous confguratons, and usng dfferent turbulence models. As a concluson, trals for future developments are presented. 2. The Reynolds Averaged Naver-Stokes Equatons 2.1 Introducton The modelng of turbulent flows s ntroduced here for ncompressble flows, whch are representatve of a large number of flows encountered n practce: ar flows n the atmosphere, rver flows, water flows n ppes, etc. Usual ncompressble vscous flows are represented by the Naver-Stokes equatons. If V = (u 1, u 2, u 3 ) denotes the velocty vector, p the pressure, ρ the densty and g = (g 1, g 2, g 3 ) the gravty acceleraton, these equatons can be wrtten as: u u 1 σ + u = + g t x ρ x u = 0 x In the prevous equatons and n all of ths artcle, the Ensten notaton s used, whch u 3 u means summng on repeated ndces,.e., u = u x x σ = (σ ) stands for the stress tensor. Usual flows follow the Newtonan consttutve law: u u σ pδ 2μS pδ 2μ = + = + + x x Where μ s the dynamc vscosty of the flud. Then the Naver-Stokes equatons become: 2 u u 1 ρ u + u = + g t x ρ x xk xk u = 0 x = 1 (1) (2) μ where ν = s the knematc vscosty of the flud (unt: m 2 /s). ρ The Naver-Stokes equatons reman applcable when a flow becomes turbulent. Startng from a steady lamnar soluton and ncreasng the velocty leads to amplfcaton of small perturbatons through the nonlnear advectve term. These perturbatons ultmately occupy the whole doman and are almost of comparable magntude to that of the mean flow. Turbulent eddes wth a wde varety of scales Encyclopeda of Lfe Support Systems (EOLSS) 189
Modelng - J. P. Chabard and D. Laurence appear. Ther evoluton s stll gven by the Naver-Stokes equatons, but an ncreasng number of degrees of freedom s requred to represent the smaller detals of the turbulent flow properly. The dffculty of modelng turbulent flows arses from the fact that the fluctuatons assocated wth these perturbatons have a sze whch s often too small to be captured by measurng devces or computatonal meshes. The Naver-Stokes equatons can be seen as the averaged moton of a large ensemble (called flud partcle ) of ndvdual molecules. In the same way, the equatons of the mean velocty can be derved by averagng over a large ensemble of eddes or flow realzatons. The resultng equatons, however, wll requre hypotheses, and result n model equatons that are far from unversal. The man reason s that whle the scale of a flud partcle s several orders of magntude larger than that of the ndvdual molecules, the scales of the mean and turbulent motons, on the other hand, are often of comparable magntude and nteract strongly. 2.2 Reynolds Experment (1986) Osborne Reynolds (1842 1912) was the frst to quantfy the transton between a regular lamnar flow and a turbulent flow. The Reynolds experment was very smple a water tank dschargng through a glass ppe of dameter D. The flow s vsualzed by usng dye whch s ntroduced through a small tube nto the large ppe. The bulk velocty (flow rate dvded by secton area) n the large ppe s V. For small values of V the dye draws a straght lne because the molecular dffuson s slow compared wth the advecton by the mean flow. The flow s sad to be lamnar. By gradually ncreasng V, the dye streak starts to oscllate; ths frst appearance of unsteadness s called transton. Further ncrease n V results n a chaotc pattern, and dye s rapdly dspersed n the entre cross secton of the tube. The flow s sad to be turbulent. The same observatons can be made f the vscosty (or the dameter of the ppe) s changed, whle V s kept constant. That s, the transton from lamnar to turbulent flow regmes s only characterzed by the Reynolds number defned as: VD Re = (3) ν In a crcular ppe, transton occurs for a crtcal Reynolds number of Re c = 2500. If D = 10 cm, the flow becomes turbulent above a velocty of only 2.5 cm/s. Flows n nature or ndustral systems are thus very turbulent. 2.3 Mean and Fluctuatng Components of the Flow Assume that some parameter F s measured by some adequate devce; repeat N tmes the same experment ; N measured values of F are obtaned: F, ( = 1,...,N). If N s large enough, then the mean value of F s approxmated by: F1 + + FN F = N Encyclopeda of Lfe Support Systems (EOLSS) 190
Modelng - J. P. Chabard and D. Laurence The rgorous mean value would be obtaned wth N tendng toward nfnty. The mean value obtaned by averagng over N experments s called an ensemble average. It s more convenent to average over tme n a sngle experment but ths tme should be suffcent so that a large number of dfferent eddes have gone past the sensor. 2.3.1 Ergodcty Hypothess The ergodcty hypothess assumes that the average defned by the followng tme averagng s dentcal to the ensemble average: 1 T F = lm F( x, t) dt T T 0 In practce, T should be large compared wth the ntegral tme-scale defned further. Ths procedure only makes sense f the mean flow s consdered to be at steady state (constant flow-rate for nstance). When the mean flow can be consdered homogeneous (.e., statstcal values are constant n space), the mean defned by tme averagng s the same as the space averaged value, on a space volume over whch the flow s homogeneous: F = 1 lm F( x, y, z) dx dy dz Ω Ω Ω The sze of ths doman should be much larger than typcal eddy scales, or more precsely the ntegral length-scale defned further. 2.3.2 Probablty Densty Functons In the mathematcal sense, probabltes allow the defnton of the averagng operator ndependently from the prevous hypotheses. The probablty that a random parameter F should happen to fall nsde a certan nterval s: Probablty F [ F, F + df] = P( F ) df ( ) 0 0 0 Ths defnes the probablty densty functon P(F). Naturally: + PF ( ) 0 and PFdF ( ) = 1 The knowledge of P(F) allows the constructon of any statstcal quantty related to F: F ' s the fluctuaton, defned by F' = F F, and has a mean of zero. The probablty densty functon allows a rgorous mathematcal approach. It s useful for demonstratng that the averagng operator commutes wth space or tme, and for obtanng the followng propertes for the averagng operator: Encyclopeda of Lfe Support Systems (EOLSS) 191
Modelng - J. P. Chabard and D. Laurence ( af + bg) = af + bg ( a and b are constants) F F F F = and = x x t t ( ) ( ) F = F and FG. = FG. F' = F F and F' = 0 ( FG) but. FG. (4) 2.3.3 Mean Velocty Equatons Startng from the Naver-Stokes equatons, Reynolds ntroduced for a turbulent flow the decomposton of the velocty and pressure nto mean and fluctuatng components: u = u + u ' p = p + p' and then appled the averagng operator. Gven relatons n Equaton (4), the lnear terms n the Naver-Stokes equatons gve rse to dentcal terms n the mean velocty equatons. Only the nonlnear advecton term requres specal attenton. Usng the ncompressblty condton ths term becomes: ( u ' ) ( ' )( ' ) + u u + u u + u ( u + u' ) = x x u u + u' u + uu' + u' u' = x Applyng the averagng operator, t s noted that: uu s already a mean quantty, so averagng t agan has no effect. Moreover: ( ) u' u = u' u = 0 u =0, but on the other hand u' u' 0 Ths term s carred over to the rght hand sde, leavng on the left hand sde only the advecton by the mean velocty. The averaged Naver-Stokes equatons, called Reynolds averaged Naver-Stokes equatons (or Reynolds equatons), are thus: 2 u 1 ' ' u p u u u + u = +ν + g t x ρ x xk x k x u = 0 x (5) The Reynolds stress tensor s defned by R = u' u',.e.: Encyclopeda of Lfe Support Systems (EOLSS) 192
Modelng - J. P. Chabard and D. Laurence u' u' u' u' u' u' 1 1 1 2 1 3 = u' 1u' 2 u' 2 u' 2 u' 2 u' 3 u' 1u' 3 u' 2 u' 3 u' 3 u' 3 R (6) Now the Naver-Stokes equatons can be wrtten n a vector form, wth σ as the stress tensor, then: V 1 + ( V. grad ) V = dv( σ ) + g t ρ dv( V ) = 0 thus the Reynolds equatons are wrtten as follows: V 1 + ( V. grad ) V = dv( σ ρ R ) + g (7) t ρ dv( V) = 0 Ths clearly demonstrates that R, whch s the mean of the exteror product of the velocty fluctuaton vector: R= V' V', appears as a new stress tensor n the equatons. 2.3.4 Necessty of a Closure Hypothess To obtan the mean velocty by solvng Equaton (7), an evaluaton of R and subsequently of the fluctuatng moton, s needed. The equaton of the fluctuatng velocty s obtaned by subtractng the Reynolds equaton from the Naver-Stokes equaton, then multplyng by the fluctuaton and averagng: u' ' ' ' ' u u u u u u = = = u ' + u ' = t t t t t t Leadng to: 2 u' ' 1 ' ' ' ' ' u p u u u u u + u = +ν u' + u' + (8) t x ρ x xk x k x x x u ' x = 0 or dv( V ') = 0 (9) In a compact notaton, from the momentum equaton, ths gves: V = f( V, p, V' V ') t A transport equaton can be obtaned for the Reynolds stresses: V' V' = f( V, V' V', V' p', V' V' V') (10) t Ths new set of equatons depends on new unknowns, for nstance V' V' V'. Agan, an equaton can be obtaned for ths thrd order tensor: V' V' V' = f( V, V' V', V' p',, V' V' V' V') (11) t Encyclopeda of Lfe Support Systems (EOLSS) 193
Modelng - J. P. Chabard and D. Laurence Bblography Baldwn B. S. and Lomax H. (1978). Thn layer approxmaton and algebrac models for separated flows, AIAA Paper No. 78 257. [A paper presentng mprovements of the mxng length turbulence model.] Cebec T. and Smth A. M. O. (1974). Analyss of turbulent boundary layers. Appled Mathematcs and Mechancs 15. [A paper related to the mxng length turbulence model.] Encyclopeda of Lfe Support Systems (EOLSS) 222
Modelng - J. P. Chabard and D. Laurence Daly B. J. and Harlow F. H. (1970). Transport equatons n turbulence. Physcs of Fluds 13, 2634 2649. [A paper on the modelng of trple correlatons n Reynolds transport equatons turbulence models.] Drest E. R. van (1956). On turbulent flow near a wall. Journal of Aeronautcal Scences 23, 1007. [A paper on the smulaton of wall boundary layers usng mxng length models.] Gbson M. M. and Launder B. E. (1978). Ground effects of pressure fluctuatons n the atmospherc boundary layer. Journal of Flud Mechancs 86, 491 511. [A paper on the modelng of pressure-stran correlatons n Reynolds transport equatons turbulence models: wall echo terms.] Hanalc K. and Launder B. E. (1972). A Reynolds-stress model of turbulence and ts applcaton to thn shear flows. Journal of Flud Mechancs 52, 609 638. [A reference paper presentng the frst Reynolds stress transport equatons model.] Launder B. E., Reece G. J., and Rod W. (1975). Progress n the development of a Reynolds stress turbulence closure. Journal of Flud Mechancs 68, 537 566. [A paper presentng the basc lnear Reynolds stress transport equatons model.] Launder B. E. and Spaldng D. B. (1972). Mathematcal Models of Turbulence. London: Academc Press. [Includes a reference paper presentng the frst k- model.] Patel V. C., Rod W., and Scheuerer G. (1985). Turbulence models for near wall and low Reynolds number flows: a revew. AIAA Journal 23(9), 1308 1319. [A very good revew on low Reynolds number k- turbulence models.] Prandtl L. and Weghardt K. E. G. (1945). Uber ene neue Formelsystem für de ausgebldete Turbulenz. Nachrchten Akademe Wssenshaft Göttngen, Math.-Phys. Klasse, p. 16 [A reference paper on the mxng length model.] Rod W. (1993). Turbulence Models and Ther Applcaton n Hydraulcs a State of the Art Revew, Thrd Edton, IAHR Monograph Seres, Delft. [A concse and nformatve textbook contanng many example of cvl engneerng and hydraulcs applcatons of turbulent flows.] Rotta J. C. (1951). Statstsche Theore nchthomogener Turbulenz. Zetschrft für Physk. 129, 547 572. [A paper on the modelng of pressure-stran correlatons n Reynolds transport equatons turbulence models: slow term.] Bographcal Sketches Jean-Paul Chabard graduated Master of Flud Mechancs, Ecole Central des Arts et Manufactures. Snce October 1997 he has been Drector of the Thermal Transfer and Aerodynamcs Branch of the EDF (Electrcté de France), Research and Development Dvson. Formerly he was Drector of the Laboratore Natonal d Hydraulque (LNH) (1994 1997), and n the Research Dvson of EDF was responsble for expermental and numercal studes n flud mechancs and hydraulcs as well as for the development of general purpose Computatonal Flud Dynamcs computer software (LNH devotes resources to studes n flud mechancs and hydraulcs on behalf of French and foregn organzatons Martme Sectons of Publc Works Contractors, Rver Basn Agences Prvate Industres). Chabard lectures at the French Hgh Engneerng Schools (Ecole Natonale des Ponts et Chaussées and the Ecole Centrale des Arts et Manufactures), teachng Appled Flud Mechancs and Numercal Turbulence Computaton Methods. He s a member of the IAHR and Charman of ts Dvson II: Appled Hydraulcs. Domnque Laurence graduated from the Ecole Natonale des Ponts et Chaussées n 1980. He s a Senor Research Engneer, Expert at EDF n Turbulence Modelng and n Computatonal Flud Dynamcs. He teaches three courses n Flud Mechancs at ENPC: Appled Flud Mechancs; Mathematcal Soluton Methods; and Heat Transfer, Stratfed and Compressble Flows. He s an Edtoral Board Member of the Journal of Heat and Flud Flow and revewer for the IAHR and the ASME. Encyclopeda of Lfe Support Systems (EOLSS) 223