Colloquium FLUID DYNAMICS 2007 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 1

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1 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 1. CRITICAL PROBLEMS OF BOUSSINESQ-TYPE TURBULENCE MODEL APPLICATION IN INTERNAL COMBUSTION ENGINES Olřich Vítek, Miloš Polášek, Karel Kozel, Jan Macek Czech Technical University in Prague, Josef Božek Research Center, Prague Abstract The presente paper eals with iscussion of weak points when Boussinesq-type turbulence moels are applie for moelling phenomena in ICE. The following are the important problematic factors turbulence momentum transport, issipation equation, moelling of SI engine combustion. Examples are presente. The contribution is just stating some facts, no general conclusions how to solve the mentione problematic cases are presente. 1 Introuction Nowaays it is a usual routine to apply two-equation turbulence moels for moelling of compressible transient turbulent flow in internal combustion engine (ICE). No special turbulence moels for ICE are evelope. Instea, classical moels (evelope for the nees of external aeroynamics) are applie (the moel constants might be slightly ifferent). The exception might be a three-equation turbulence moel of the company AVL Graz, the moel is labelle k ζ f an is applie in their CFD coe FIRE [1]. There is really a large amount of papers where results of such simulations are presente incluing comparison with experimental ata. What is missing (or it is unknown to the authors) is that no comprehensive comparison of measure Reynols stresses with preicte/calculate ones is carrie out (the authors re aware of the fact that it is not easy to measure Reynols stress tensor components in ICE uner typical operating conitions). It is a critical question as correlations of fluctuating terms of Favre average mathematical moel (equations 1, 2 an 3) are the ominant ones in ICE. If there is not goo corresponence with momentum turbulent transport, internal energy (temperature) turbulent transport cannot be truste either. This woul have significant consequences for all the calculations base on chemical kinetics as these computations are very sensitive to local temperature. When working on [14], these problems arisen again. The presente paper main target is to stress some problems of turbulence moelling base on Boussinesq hypothesis for the case of ICE. Ing., Ph.D., Olrich.Vitek@fs.cvut.cz Doc. Ing., Ph.D., Milos.Polasek@fs.cvut.cz Prof. RNDr., DrSc., Karel.Kozel@fs.cvut.cz Prof. Ing., DrSc., Jan.Macek@fs.cvut.cz

2 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 2 2 Theoretical Backgroun In this section, a brief escription of the applie mathematical moel is presente. A more etaile escription of the whole mathematical moel cen be foun in [14,19]. The mathematical moel is base on a Favre-average integral Navier-Stokes equation set written for arbitrary movable control volume (Avance Multizone Eulerian Moel) equations (1), (2) an (3) plus aitional relations (4) - (13). Favre ecomposition is efine through equations (4)-(9). ρ V = ρ ( w j w Gj) n j S (1) ρ w i V = ρ w i ( w j w Gj) n j S + ρg i V (2) + [ ( ) ] p δ ij + τ ij + ρ w i w j n j S ρẽ V = ρẽ ( w j w Gj) n j S + + ρg j w j V (3) [ ( ) ] p δ ij + τ ij + ρ w i w j w i n j S + + ( τ ij w i 1 ) 2 ρ w j w i w i n j S [ ( ) ] q j + ρ w j u + p δ ij w i n j S Q V V ρ = ρ + ρ (4) p = p + p (5) w i = w i + w i (6) u = ũ + u (7) T = T + T (8) ρe ρẽ = ρũ + ρ w i w i 2 + ρw i w i + ρψ (9) 2

3 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 3 pv = mr T (10) ũ = ũ 0 + c v ( T T ) T ref ( T T ref ) (11) ( τ ij = µ( T wi ) + w ) ( j + λ S 2 ) x j x i 3 µ( T wk ) δ ij (12) x k q i = λ( T ) T x i (13) Boussinesq hypothesis (14) is applie for moelling of momentum turbulent transport term algebraic, one- an two-equation turbulence moels are use, the excact moel efinition can be foun in [14, 21]. The turbulence moelling theory is base mainly on [21], the remaining fluctuating correlations are moel by means of equations (15)-(16). S ij 1 2 ( wi + w ) j x j x i τ R ij ρw i w j = 2µ T ( S ij 1 ) w k δ ij 2 3 x k 3 ρk δ ij (14a) (14b) ( q turb,i ρ w i u + p δ ji w j = µ T cv ( T ) + r ) P r T T x i (15) ( τ ij w i + 1 ) ( 2 ρw j w i w i = µ + µ ) T k (16) σ k x j From numerical point of view, the finite volume cell-centere metho is implemente. The numerical solution is performe by means of explicit multi-stage Runge-Kutta metho (e.g. [5, 13]). As far as moelling of combustion is concerne, the Level Set approach is applie equations (17)-(18). Arbitrary function ( G) transport is use to moel flame front propagation in the space omain more etaile escription can be foun in [10,11]. Mass balance of each consiere component (air, fuel an burnt gas) is written in equation set (19), iffusion across the flame front is neglecte. Source term in energy equation ue to combustion process is efine in Equation (20). The combustion in each finite volume is moelle by means of constant rate-of-heat-release (ROHR) epening only on the amount of combustible mixture in a finite volume an combustion uration being function of size of the finite volume only. The ROHR is labelle as x function in presente equations. Combustion itself is starte when prescribe level of the arbitrary function G is reache in the finite volume. ρ G V = ρ G (w j w Gj) n j S + ρs 0 G T ( ) G G 1 2 x j x j G x i n i S (17)

4 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 4 ρ pal V = ρ vz V = s 0 T s 0 L w w i = ( ) w n = 1 + C 1 (18a) s 0 L = f(p, T, λ) s 0 L 2 (18b) 3 k (18c) ρ pal V V ρ pal ( w j w Gj) n j S lib 1 x x ρ pal V V ρ vz ( w j w Gj) n j S L lib T 1 x ρ pal V V ρ vyf V = ρ vyf ( w j w Gj) n j S + (1 + L T ) lib 1 x ρ pal V V Q V V = H lib u 1 x x x x (19a) (19b) (19c) The mathematical moel was applie to create a Fortran coe AMEM3D [15] which eals with moelling of in-cyliner phenomena uring compression stroke an expansion stroke of spark ignite (SI) 4-stroke ICE. (20) 3 Compute Cases A very simple engine geometry was selecte to carry out parametric stuies to test the mathematical moel of ICE. Engine combustion chamber consists of a flat piston an a flat cyliner hea it has the shape of a cyliner. The bore is 85 mm, the stroke 92 mm an the compression ratio is 8. Heat transfer via coole walls was enable an specific heat capacity was a function of temperature. Algebraic, one- an two-equation turbulence moels (base on Boussinesq hypothesis) were use. Gasoline is use as a fuel. The experimental ata [4, 8] were use for comparison with ata compute by means of [15]. All the results are presente in [14], the most important results were publishe in [16 20]. 4 Result Discussion One of the encountere problems was that Reynols stresses were too small (figures 1-3) when compare with measurement [4] regarless of applie turbulence moel, initial

5 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 5 Figure 1: Comparison of corresponing Reynols stresses (XY component) at three ifferent space locations measurement ata accoring to [4] (left figure) an compute ata (right figure) for the moel 1280x35-konv3-k-τ-VN=5 at engine spee n M = 1000 min 1 ; ata plotte as a function of crank angle (360 egca correspons with engine top ea center (TDC)). conitions or other parameters (numerical solution preciseness, mesh coarseness, etc.). Consiering the fact that the engine geometry is very simple, an hence the flow structures are relatively simple as well, it was a negative surprise. On the other han, it was proven (e.g. [14, 19]) that Boussinesq-type turbulence moels applie for the case of ICE have many of the qualitative properties that are known from experience. When comparing preicte an measure integral length scale (Figure 4), the measurement [8] confirms the assumption that turbulent properties are more or less homogeneous, especially near TDC. The theoretical results basically preicts the same. However, the experimental ata suggest that the value of integral length scale is almost symmetrical with respect to TDC. This was not proven by calculations. This is the main qualitative ifference preicte integral length scale is not symmetrical with respect to TDC. If the measure integral length scale is applie for the case of one-equation turbulence moel, there is no improvement of preicte Reynols stress tensor components. Even if algebraic turbulence moel is use an very high value of turbulence viscosity is prescribe, the results were not satisfying either. The AMEM3D coe was verifie against the commercial CFD coe [2] to make sure that there is no funamental problem within the evelope algorithm. The main result of this comparison [20] was that the results of both CFD coes are comparable. It is well know that turbulence properties ecrease uring compression stroke (unless they are increase by means of combustion chamber shape) an that turbulence generate uring intake stroke is umpe at early compression stroke. Moreover, if there are significant ifferences of properties at the en of intake stroke, there will be much smaller at the en of compression stroke. Generally speaking, this empirical knowlege can be confirme by computations. Even if ifferent turbulence moels are applie, the results satisfy that (Figure 5, right sub-figure) uner the assumption that there is significant swirling motion The authors are aware of the fact that the exact efinition of the turbulence integral length scale is not the same as the value of the length scale which is use in applie turbulence moels. However, the authors believe that these two length scales are comparable an their qualitative behavior shoul be similar. The length scale calculate by applie turbulence moels is reference as preicte integral length scale, even if this is not a precise terminology.

6 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 6 Figure 2: of corresponing Reynols stresses (XZ component) at three ifferent space locations measurement ata accoring to [4] (left figure) an compute ata (right figure) for the moel 1280x35-konv3-k-τ-VN=5 at engine spee n M = 1000 min 1 ; ata plotte as a function of crank angle (360 egca correspons with engine top ea center (TDC)). Figure 3: of corresponing Reynols stresses (YZ component) at three ifferent space locations measurement ata accoring to [4] (left figure) an compute ata (right figure) for the moel 1280x35-konv3-k-τ-VN=5 at engine spee n M = 1000 min 1 ; ata plotte as a function of crank angle (360 egca correspons with engine top ea center (TDC)). Figure 4: Comparison of integral length scale measurement ata accoring to [8] (left figure) an compute ata (right figure) for the moel 1280x35-konv3-k-τ-VN=5 at engine spee n M = 1000 min 1 ; ata plotte as a function of crank angle (360 egca correspons with engine top ea center (TDC)).

7 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 7 Figure 5: Influence of applie turbulence moel for the moel 720x25-konv3 at engine spee n M = 1000 min 1 in-cyliner average specific turbulence kinetic energy for the case of initial swirl number VN=0 (left figure) an initial value of swirl number VN=5 (right figure). at the beginning of compression stroke. On the other han, if there is no initial swirl an ifferent turbulence moels are applie, the results are significantly ifferent (especially RNG k ɛ moel). The ifferences among consiere turbulence moels become more an more significant. The biggest ifferences occur at the TDC (Figure 5, left sub-figure). More etaile analysis in [14] suggests that the iscusse phenomenon is mainly ue to non-linearity of issipation equation, especially its sink term. It was confirme that if the consiere process is ominate by prouction of turbulence specific energy k (k increases in time), the issipation is not ajuste quickly enough the ifferences are even more significant uring the computation. However, if the process is ominate by issipation of k (k is ecrease in time), all moels preict almost the same results. Regaring application of simple combustion moel base on the Level Set approach, there are two issues. Firstly, the moel shoul be suitable for SI engine combustion moelling as the mixture is alreay well mixe an turbulent transport is the ominant phenomenon this statement is vali if temperatures are high enough which is the case when engine loa is high. Taking into account this fact, it is a bit surprising that the shape of ROHR is relatively ifferent when compare with experimentally measure one (Figure 6). Moreover, the ROHR shape cannot be change by means of tuning the moel constants. However, the preicte shape of ROHR seems to be in a goo corresponence with moelle processes. The combustion itself is a significant source of turbulence which causes that turbulent flame spee is increase. This leas to even faster flame front propagation which causes that more energy is release per unit time. Another important fact is that as the combustion procees in time, the unburnt mixture is compresse (ensity is higher). These two phenomena clearly must cause that the ROHR is more significant at the en of combustion process. The only fact which may change the shape of calculate ROHR is the chemical issociation (not taken into account in presente results; more etails can be foun in [22]) it can work as a capacitor which stores the energy at very high temperatures an once the temperature ecreases, the energy is release back. There seems to be no other reason that combustion spee shoul be ecrease at the en of combustion process if turbulent transport is the ominant factor. Seconly, it was proven that combustion moel constants shoul be tune in such a way that they compensate for the influence of both applie turbulence moel an mesh coarseness. Especially the latter phenomenon is very

8 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 8 Figure 6: Comparison of compute ROHR (Level Set approach) with typical experimental curve of ROHR (Vibe exponent in the range from 1.5 to 2.0) ifferential form of ROHR (left figure) an integral form of ROHR (right figure). Figure 7: Influence of mesh on ROHR for the moel konv3-k-τ-heat-real_gas- G_function at engine spee n M = 5000 min 1 the same value of combustion moel calibrating constants applie for all meshes (left figure) an ifferent value of calibrating constants for each mesh to keep the same flame front propagation spee (right figure). significant (Figure 7, left sub-figure). This correspons well with the alreay mentione facts concerning turbulence flame spee an issipation equation the finer mesh causes that velocity graients are smeare less. This causes higher source term of k, issipation moel is not fast enough to take some correction action. 5 Conclusion The presente contribution eals with significant problems of Boussinesq-type turbulence moel application in ICE. It shoul be stresse that two-equation turbulence moels are routinely use when performing multi-imensional simulations in ICEs. The presente ata an encountere problems where obtaine when working on [14]. The paper main target was to stress the important problems of turbulence moelling in ICE. It seems that Reynols stress quantitative preiction is important weak point of the consiere approach. If turbulent momentum transport cannot be truste, temperature turbulent transport is most likely wrong as well. This conclusion might cast oubts over chemical-kinetics-base simulations for which the proper temperature preiction is very

9 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 9 important. Moreover, it seems that ifferent turbulence moels vary significantly in moeling of issipation. This causes that the results might be very ifferent. On the other han, it shoul be mentione that two-equation turbulence moels have such properties that qualitative influence of many phenomena is in a goo corresponence with experimental knowlege (etaile escription is presente in [14, 19]). Simple combustion moel base on assumption that turbulent transport is the ominant term (which shoul be vali for SI engine uner high engine loa) has many positive properties (more etails can be foun in [14]). However, the shape of ROHR an significant influence of applie mesh are the most important weak points. Many hopes are put in the Large Ey Simulation (LES) techniques (e.g. [7]). Some promising results have been obtaine using this approach e.g. [3, 6, 9, 12]. However, the authors are a bit sceptic about it. The LES approach shoul prove first that progress can be mae for proper moelling of momentum an energy transport in ICE. After that, aitional effort for eveloping avance moels for other phenomena (combustion, pollutant prouction, cycle variability, etc.) might be important. Acknowlegement This work has been supporte within the MŠM project Z "Komplexní ynamické systémy v termoynamice, mechanice tekutin a těles". This help is gratefully appreciate. References [1] Fire 8.2 for Winows [CD-ROM]. AVL List GmbH, [2] Fluent 6.1 Documentation [CD-ROM]. Fluent Inc., February [3] Celik, I., Yavuz, I., an Smirnov, A. Large Ey Simulations of In-cyliner Turbulence for Internal Combustion Engines: a Review. International Journal of Engine Research, Vol. 2(2): , ISSN [4] Dimopoulos, P. an Boulouchos, K. Reynols Stress Components in the Flow Fiel of a Motore Reciprocating Engine. SAE Technical Paper Series, March Paper [5] Dvořák, R. an Kozel, K. Matematické moelování v aeroynamice. Lecture notes of CTU. CTU in Prague, Prague, ISBN (In Czech). [6] Haworth, D. C. Large Ey Simulations of In-cyliner Flows. Oil an Gas Sci. Technol. Rev., IFP, Vol. 54(2): , [7] Lesieur, M., Métais, O., an Comte, P. Large-Ey Simulations of Turbulence. Cambrige University Press, 40 West 20th Street, New York, NY , USA, ISBN

10 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 10 [8] Michar, M., Lengyel, I., an Grosjean, N. Measurements of Turbulence Length Scales in a Motore Reciprocating Engine. In: 2n International Symposium on Experimental an Computational Aerothermoynamics of Internal Flows Proceeings, Vol. 2: , July [9] Naitoh, K., Itoh, T., an Takagi, Y. Large Ey Simulations of Premixe Flame in Engine Base on the Multilevel Formulation an the Renormalization Group Theory. SAE Technical Paper Series, February Paper [10] Peters, N. Turbulent Combustion. The Press Synicate of the University of Cambrige, The Pitt Builing, Trumpington Street, Cambrige, ISBN [11] Sethian, F. A. Level Set Methos. Cambrige Monographs on Applie Computational Mathematics. Cambrige University Press, Cambrige, U.K., [12] Smirnov, A., Yavuz, I., an Celik, I. Diesel Combustion an LES of In-Cyliner Turbulence for IC-Engines. In: ASME Fall Technical Conference on In-Cyliner Flows an Combustion Processes, Ann Arbor, Michigan, pages , Paper 99-ICE-247. [13] Swanson, R. C. an Turkel, E. Multistage Schemes with Multigri for Euler an Navier-Stokes Equations. NASA Technical Paper 3631, NASA, 800 Elkrige Laning Roa, Linthicum Heights, MD , August [14] Vítek, O. Simulace ějů v pístovém motoru s vlivem turbulence. Ph.D. Thesis, CTU in Prague, Department of Automotive Engineering, (In Czech). [15] Vítek, O. an Polášek, M. AMEM3D: CFD Program for Moelling of In-cyliner Phenomena, Program Library Ú 220. [16] Vítek, O., Polášek, M., Kozel, K., an Macek, J. First Approach to Combustion Moeling in an SI Engine. In: Topical Problems of Flui Mechanics 2004, Prague, pages 15 16, February ISBN [17] Vítek, O., Polášek, M., Kozel, K., an Macek, J. Premixe Turbulent Combustion Moeling Using Level Set Approach. GAMM 2005 Abstracts, Vol. 1:182, March [18] Vítek, O., Polášek, M., Kozel, K., an Macek, J. U-RANS Moel of Internal Combustion Engine. Proceeings: Symposium on Hybri RANS-LES Methos, page 17, July [19] Vítek, O., Polášek, M., Kozel, K., an Macek, J. Linear Turbulence Moel Valiation for the Case of 3-D In-cyliner Phenomena Moeling. Journal of Mile European Construction an Design of Cars (MECCA), Voloume V. (04/ /2007):24, ISSN

11 Institute of Thermomechanics AS CR, Prague. October 24-26, 2007 page 11 [20] Vítek, O., Polášek, M., an Mareš, B. Prouění ve spalovacím motoru během vysoktlaké části oběhu. In: Sborník přenášek KoKa 2004, Brno, pages , September (In Czech), ISBN [21] Wilcox, D. C. Turbulence Moeling for CFD. DCW Inustries, Inc., 5354 Palm Drive, La Canaa, California 91011, 2. eition, March ISBN [22] Zacharias, F. Mollier-I,S-Diagramme für Verbrennungsgase in er Datenverarbeitung. Motortechnischezeitschrift (MTZ), 31(7): , 1970.