Large eddy simulation of turbulent front ... - Semantic Scholar

Large eddy simulation of turbulent front propagation with dynamic subgrid models Hong G. Ima) and Thomas S. Lund Center for Turbulence Research, Stanford University, Stanford, California 94305

Joel H. Ferziger Department of Mechanical Engineering, Stanford University, Stanford, California 94305

~Received 12 May 1997; accepted 22 August 1997! Dynamic models for large eddy simulation of the G-equation of turbulent premixed combustion are proposed and tested in forced homogeneous isotropic turbulence. The basic idea is to represent the ‘‘filtered propagation term’’ as ‘‘propagation of the filtered front at higher speed,’’ where the enhanced filtered-front speed is modeled. The validity of the linear relation between the turbulent flame speed and turbulence intensity is examined through the use of filtered direct numerical simulation ~DNS! data. These tests show a range of scalings from linear to cubic depending on the ratio of the turbulence intensity to flame speed as well as the filter type. Filtered DNS data are also used to evaluate the proposed dynamic model for the turbulent flame speed. It is found that the model is very sensitive to the manner in which the subgrid-scale kinetic energy is estimated. It is also found that accurate predictions of the turbulent flame speed can be obtained provided a good estimate of the subgrid-scale kinetic energy is used. Simulations are also run using the new dynamic model and the results are shown to compare well with DNS results. © 1997 American Institute of Physics. @S1070-6631~97!03512-5#

I. INTRODUCTION

Direct numerical simulation of turbulent reacting flows places extreme demands on computational resources. At the present time, simulations can be performed only for greatly simplified reaction systems and very low Reynolds numbers. The cost of direct simulation of flows at higher Reynolds number including multiple species and numerous chemical reactions will exceed available computational resources far into the future. Thus there is a clear need to develop large eddy simulation ~LES! for reacting flows. Unfortunately, this task is complicated by the fact that combustion occurs at the smallest scales of the flow. Capturing the large-scale behavior without resolving the small-scale details is extremely difficult in combustion problems. Thus LES modeling of turbulent combustion encounters difficulties not present in modeling momentum transport, in which the main effect of the small scales is to provide dissipation. The difficulty is more pronounced in premixed combustion, where detailed chemistry plays an essential role in determining the flame speed ~or overall burning rate!; in nonpremixed combustion infinite rate chemistry can be assumed to a first approximation, eliminating the small scale features. One of the practically relevant and better understood types of turbulent premixed combustion is the laminar flamelet regime,1 in which the characteristic chemical time is much shorter than the characteristic flow time. Under this condition, combustion can be represented as the propagation of laminar flamelets, corrugated by turbulent eddies. It has been suggested2 that such a propagating front may be captured by defining the front as a level contour of a continuous function G, whose governing equation is a!

Present address: Combustion Research Facility, Sandia National Laboratories, Livermore, California 94551.

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Phys. Fluids 9 (12), December 1997

]G ]G 1u j 5S L u ¹G u . ]t ]x j

~1!

In this equation, all information about the flame structure is carried by the flame speed S L . This provides a convenient opening for large eddy simulation; the flame structure need not be modeled. Since the flame retains its laminar structure, if necessary, explicit expressions for S L as a function of the flow variables may be taken from asymptotic studies3 or computations. In LES, only filtered flame fronts are resolved. These fronts can be viewed as flame brushes that propagate at a speed, ¯ S , higher than the laminar flame speed. The problem is closed if one can provide an explicit expression for ¯ S as a function of available quantities. Several previous studies have attempted to derive relationships between ¯ S and the turbulence intensity u 8 .4–7 However, the existing theoretical and empirical results for ¯ S (u 8 ) do not agree with one another, and thus the functional form of ¯ S (u 8 ) remains an open question. Even if the question is resolved, there will be a constant or function to be determined and a model will be needed for u 8 . In this study, we present an attempt at LES using the dynamic modeling approach similar to that which has been successfully applied to a variety of turbulent flows.8 The more primitive formulation was derived earlier,9 and is currently modified to incorporate the effect of subgrid transport. Unlike previous LES approaches for the G-equation,10 the model constants are computed dynamically as a part of the calculation procedure rather than being prescribed. Also, a diffusive term is included in the model, in contrast to the procedure used in some other models.11 Dynamic modeling of turbulent flow has been shown to exhibit correct behavior, for example, in the near-wall region of boundary layers,

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© 1997 American Institute of Physics

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without the need for additional modification. The LES models for the G-equation suggested in this study have these features, allowing the possibility of application to practical combustion systems. II. SOME REMARKS ON THE G-EQUATION

Before we proceed with LES modeling, some numerical issues related to the G-equation should be pointed out. If one wishes to solve Eq. ~1! in the constant-speed limit, i.e., S L 5S 0L , numerical difficulties arise due to the formation of cusps as the front propagates. Cusps are a natural consequence of constant-speed propagating fronts in much the same way as shocks are a characteristic feature of the Burgers equation; they are discontinuities in the derivatives of the solution and make numerical treatment very difficult. To overcome the numerical difficulties associated with cusps, previous studies introduced various types of diffusive terms.2 These terms are not entirely ad hoc, however; they can be shown to represent the effect of thermal relaxation under transverse heat diffusion in the preheat zone of a wrinkled front.3 Using the asymptotic relation for S L , Eq. ~1! can be written as

]G ]G 1u j 5S 0L u ¹G u 1D¹ 2 G, ]t ]x j

~2!

where only the leading term has been kept; this is a reasonable approximation provided the flame thickness is sufficiently small compared with the hydrodynamic scale. In the above relation, D5S 0L L is the Markstein diffusivity, where the Markstein length, L, is typically normalized by the flame thickness l F . Since l F 5 a /S 0L 5 ~ 1/S 0L !~ n / Pr ! ,

~3!

we find D5 ~ n / Pr ! M a,

~4!

where a and n are the thermal diffusivity and kinematic viscosity, Pr the Prandtl number and M a5L/l F the Markstein number, which is ideally a property of the mixture4 and typically of O(1) quantity in practical flames.12–14 We solved Eq. ~2! with various values of D in order to examine the effect of the Markstein diffusivity on the overall burning rate, or the turbulent flame speed. Figure 1 shows the DNS result for the overall flame speed using the passive G-equation in isotropic turbulence, plotted as a function of D/ n 5M a/ Pr. It shows that the overall flame speed depends significantly on the magnitude of the diffusion term in the G-equation. Furthermore, although the numerics could not resolve the solutions in the constant-speed limit (M a→0), it is clearly seen that the flame speed increases sharply as M a decreases. Indeed, we suspect that the flame speed is unbounded in the limit M a→0 as should be the case if cusps are present. This is not surprising; it demonstrates that one must be careful about choosing the diffusive term, especially when comparing the computed flame speed with experiments. Accurate estimation of the Markstein number is mandatory for such comparisons. For this reason, we maintain a constant value of the Markstein number in this study so that Phys. Fluids, Vol. 9, No. 12, December 1997

FIG. 1. Turbulent propagation speed of the G-equation as a function of Markstein diffusivity, M a/ Pr, for u 8 /S 0L 52.

the properties of the laminar flame do not interfere with the subgrid scale modeling which is the main objective of the present study. In this study, the LES models are validated by comparing with DNS results based on Eq. ~2! with 643 resolution. We note that results with 1283 resolution led to the same conclusions; further study at higher Reynolds numbers is needed to achieve the scale similarity that will allow proper assessment of the accuracy of dynamic modeling. Most of the LES were performed at 323 resolution using the filtered DNS fields as initial conditions. III. SUBGRID-SCALE MODELS FOR THE G -EQUATION

We now describe the dynamic subgrid-scale models for the passive G-equation. As in other turbulent flows, we assume scale invariance of the G-equation in the inertial range of turbulence; this has been shown to exist by Yakhot15 and ¯ , and the test filter Gˆ Pocheau.7 We define the grid filter G respectively as

E E

¯f ~ x! 5

¯ ~ x,x8 ! dx8 , f ~ x8 ! G

ˆf ~ x! 5

f ~ x8 ! Gˆ ~ x,x8 ! dx8 ,

~5!

ˆ , is larger than that of the where the width of the test filter, D grid filter, D. For the homogeneous flows considered here, the filter kernels depend only on the distance u x2x 8 u . Then, applying the grid filter to Eq. ~2!, we obtain ¯ ]G ] ¯! 1 u G ~¯ ]t ]x j j 52

¯ ] ] 2G ¯ ! 1S 0L u ¹G u 1D uG . ~ u j G2 ¯ ]x j ] x 2j

~6!

¯ ), and the filHere both the subgrid scalar flux, (u j G2 ¯ u jG tered modulus term, u ¹G u , need to be modeled. In the preIm, Lund, Ferziger

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vious study,9 both these terms were modeled by a single ¯ u . This model, however, suffiltered propagation term, ¯ S u ¹G fered from numerical instability because there is a tendency to form surfaces containing cusps with very slight rounding if no subgrid dissipation was provided ~the cusps remain rounded to a radius of approximately Markstein length, which is unchanged while grid size increases!. In this study, the subgrid flux term is modeled by an eddy diffusivity model analogous to the Smagorinsky model, i.e., ¯ 52 a t g k5 ¯ u kG 2 ¯ u kG

¯ ]G , ]xk

¯u , a t 5C G D 2 u S

~7!

¯ u is the magnitude of the strain rate tensor: where u S ¯ u 5 ~ 2S ¯ S ¯ 1/2 uS ij ij! ,

S

D

¯ ¯ ¯ 51 ] u i 1 ] u j . S ij 2 ]x j ]xi

~8!

Similarly, at the test filter level, we obtain C d C 52 aˆ ] G , G k 5u k G 2uC k G t ]xk

~9!

aˆ t 5C G Dˆ 2 u SR u .

~10!

can be used to determine the constant C G . Using Eqs. ~7!, ~9!, ~10! with the least-squares method,16 we obtain F iHi , H jH j

~11!

¯ C d ¯u ] G , R u ]G 2uS ˆ /D ! 2 u S Hk5~ D ]xk ]xk

~12!

C G D 2 52

so the model constant C G for the scalar transport can be determined dynamically. Next we consider the modeling of the propagation term, S 0L u ¹G u . This term can be written as ~13!

which defines v , a quantity analogous to the subgrid stress term in the Navier–Stokes equations. Similarly, the propagation term at the test-filter level becomes d 0 C u 1S 0 @ ud C C S 0L u ¹G u 5S 0L u ¹G L ¹G u 2 u ¹G u # 5S L u ¹G u 1V.

~14!

It is then easy to derive a second analog of the Germano identity d ¯ u 2 u ¹G C u#, V2 vˆ 5S 0L @ u ¹G

~15!

where the right-hand side can be computed from the gridlevel solution. 3828

Phys. Fluids, Vol. 9, No. 12, December 1997

C u 1V5SC u ¹G C u, S 0L u ¹G

~16!

such that the unknowns v and V are substituted with ¯ S and SC , ¯ C respectively. It is then necessary to model S and S as functions of known quantities, such as the turbulence intensity. Theoretical studies suggest the functional form ¯ S /S 0L 511C S ~ q/S 0L ! p ,

SC /S 0L 511C S ~ Q/S 0L ! p ,

~17!

¯i )(u i 2u ¯i ) # 1/2 where q5 @ (u i 2u and Q5 @ (u i 2uC i ) 1/2 3(u i 2uC i ) # are the square roots of the subgrid kinetic energy associated with the grid-filtered and test-filtered volume. Previous theoretical studies4,7 predicted quadratic (p52) and linear ( p51) behavior, respectively, in the weak and strong turbulence limits, while p54/3 has also been suggested based on the stochastic nature of turbulence.6 In the next section, this functional relation will be examined using an a priori test based on DNS data. Combining Eqs. ~15!, ~16! and ~17!, we obtain ~18!

from which the unknown constant C S is determined. In general, C S is a function of space and time. For the present study of homogeneous isotropic turbulence, we assume C S to be uniform and a function of time only, in a manner analogous to that for the Smagorinsky model constant.8 Then, when Eq. ~18! is volume averaged, it becomes p ¯ u & # 5S 0 p @ ^ ud ¯ u & 2 ^ u ¹G C u & 2 ^ qd C u& #, u ¹G ¹G C S @ ^ Q p u ¹G L ~19!

where

¯ u 1S 0 @ u ¹G u 2 u ¹G ¯ u # 5S 0 u ¹G ¯ u1v, S 0L u ¹G u 5S 0L u ¹G L L

¯ u 1 v 5S ¯u ¹G ¯ u, S 0L u ¹G

p ¯ u # 5S 0 p @ ud ¯ u 2 u ¹G C u 2qd C u#, u ¹G ¹G C S @ Q p u ¹G L

A generalization of Germano’s identity, d ¯ C G C ¯ F k 5G k 2 gˆ k 5u k G 2u k ,

Now the enhanced propagation terms, v and V, need to be modeled. We adopt the notion of representing the ‘‘filtered propagation term’’ as ‘‘propagation of the filtered front at higher speed,’’ by writing

where the brackets denotes a volume average. We remark that, although Eq. ~19! is obtained from the mathematical identity ~15!, it can be physically interpreted as the conservation of the overall burning rate, recalling that ^ u ¹G u & represents the total front area within the volume.2 In the Appendix, some alternative forms of Eq. ~19! are derived using physical considerations, leading to slightly different forms. Our numerical tests in homogeneous isotropic turbulence showed that these modifications produce negligible differences in the predictions of the model constants and averaged flame speed. This needs to be reevaluated for the case for which the model constant, C S , is no longer uniform. Equation ~19! is closed only if the subgrid scale kinetic energies, q and Q, are available. This requires an additional model for the subgrid scale kinetic energy as a function of the large scale quantities. As will be seen later, unlike the case of Smagorinsky’s model, the model for the subgrid kinetic energy is crucial to accurate prediction of the flame speed. We shall consider the following three models: ¯ u , deduced from dimensional reasoning ~1! Model 1: q;D u S ¯ similar to that used in Smagorinsky’s model. Since S can be computed, this model is applicable with any numerical method. Unfortunately, this model overpredicts Im, Lund, Ferziger


the turbulent flame speed when the turbulence is not in the inertial subrange, as will be shown later. ~2! Spectral curve fit, which can be used with spectral methods. The turbulence energy spectrum is described by an algebraic power relation E ~ k ! 5C k k 2m ,

~ Model 2A! :

which is appropriate in an inertial subrange, or an exponential ~ Model 2B! :

E ~ k ! 5C k exp~ 2mk ! ,

which may be more appropriate in the dissipative range. The two unknowns C k and m of either model can be ¯) and E(kˆ ). The subgrid kinetic determined from E(k energy can then be computed as q ~¯ k !5

E

`

¯ k

E ~ k ! dk.

~20!

~3! Since the spectral curve fit has limited application, we suggest another model similar to that of Bardina et al.17 The subgrid kinetic energies are estimated by ~ Model 3! :

¯i¯ q 2 5u u i 2u% i u% i ,

where the ‘‘double-bar’’ denotes applying the same filter twice. Similary, at the test-filter level we can estimate the unresolved kinetic energy, Q 2 , as the difference between the test-filtered ( -ˆ ) and the twice-test-filtered field, namely, ˆ¯ ˆ¯ ˆ ¯ ˆ ¯ Q 2 5uC i uC i 2u i ui We note that, although Model 3 requires some additional filtering operations, both q and Q can be readily computed as long as the grid-filtered field ( ¯ u ) is known. Strictly speaking, the cutoff filter cannot be applied in this model since ¯ u 5u% , but one can define the second filter as one with a more severe cutoff and proceed in the manner described above. In the following section, a priori tests are performed by applying these models ~referred to as Models 1, 2A, 2B and 3! for the propagation term, combined with the Smagorinsky-type subgrid transport model. It is noted that we used the Gaussian filter, G (k)5exp(2k2D/24), for Model 3 since it is consistent with the principle of the model. On the other hand, it is more logical to use cutoff filter ~in Fourier space! with Model 2 in which the exact values of ¯) and E(kˆ ) are important. Model 1 is not restricted to E(k any particular filter type, so the results are presented for both cases. Although the cutoff filter is a more obvious choice with the spectral method employed in this study, the Gaussian filter may be more practical with finite difference technique. The effects of filter type are discussed in the next section. IV. A PRIORI TESTS OF THE MODELS

The subgrid models proposed in Sec. III are tested by post-processing DNS results. We performed DNS based on Eq. ~2! with forced incompressible homogeneous isotropic Phys. Fluids, Vol. 9, No. 12, December 1997

FIG. 2. Energy spectra obtained from the DNS, 5 large-eddy turnover times after the g-field was initialized. Turbulence energy: —; scalar energy (g 8 2 /2) for u 8 /S 0L 50.5: •••; scalar energy for u 8 /S 0L 52: –•–.

turbulence. The turbulence is forced at the lowest wave numbers in order to hold the total kinetic energy approximately constant. The numerical method is pseudo-spectral in space with second order Runge–Kutta time integration.18 A developed flow field is used as the initial condition and the G-field is initialized as a linear function G5x. To make the G-field homogeneous, we define g5G2x and solve for g. Following initialization of the g field, the simulation is run for 5 large-scale eddy turn over times, at which point the g field is fully developed. The Reynolds number based on the Taylor microscale is about 76, and M a/ Pr54 so D54 n . Two cases were computed; u 8 /S 0L 50.5 and 2, which was accomplished by adjusting S 0L ; the turbulence field was the same in both cases. Figure 2 shows the DNS results for the turbulence and scalar energy (g 8 2 ) spectra for both values of u 8 /S 0L . The slope 25/3 is also shown. It is seen that the turbulence spectrum has a slope 25/3 over a very limited range of k due to the low Reynolds number of the flow. Since Smagorinsky-type transport models have been studied extensively in previous LES work, the main emphasis in this work is on the modeling of the propagation term. We first examine the functional relation of ¯ S /S 0L and q/S 0L , Eq. ~17!. These quantities are computed from the DNS spectrum by using q5u i u i 2 ¯ u i¯ ui,

~21!

¯ ¯ u&. S /S 0L 5 ^ u ¹G u & / ^ u ¹G

~22!

Both the cutoff and Gaussian filters are used to obtain the ¯. filtered field, ¯ u i , and G ¯ Figure 3 shows S /S 0L 21 as a function of q/S 0L for various filter sizes; increased abscissa corresponds to increased filter size. The logarithmic plot shows the value of the exponent p of Eq. ~17! for different turbulence intensities and filter types. It is seen that the slope of the curve varies with turbulence intensity for both filter types. The variation in Im, Lund, Ferziger

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FIG. 3. Enhanced filtered flame speed ¯ S /S 0L vs normalized subgrid kinetic 0 energy q/S L . Cutoff filter: —; Gaussian filter: •••; u 8 /S 0L 50.5: h; u 8 /S 0L 52: j.

slope is more prominent for the use of the cutoff filter ~the solid curves!, in which p varies from approximately 1 to 3. For the Gaussian filters ~the dotted curves!, both curves give about p52. It should be noted that, although the results are plotted against the subgrid scale energy, the low energy part of the curve corresponds to cases in which the filter lies in the dissipation range where one would expect different behavior. It may well be that ¯ S is a function variables other than q.19 The reason for the nonlinear behavior being more prominent for the stronger turbulence case can be found from Fig. 2; for stronger turbulence (u 8 /S 0L 52) the fluctuations in the G-function are controlled more by the turbulence than by the propagation, so there is more scalar energy at low wave numbers. Similar behavior has been observed in another recent study.20 Physically, as shown schematically in Fig. 4, propagation diminishes front corrugation, a process called ‘‘kinematic restoration’’ by Peters,21 but it also produces small-scale wrinkles at the trough, which leads to cusp formation in the constant-speed limit. Therefore, the effect of propagation is to transfer energy from low to high wave numbers. Consequently, the higher u 8 /S 0L , i.e., the less sig-

FIG. 4. Schematic depicting the mechanism of kinematic restoration by the flame propagation. 3830


FIG. 5. A priori tests of the ratio q/Q with the cutoff filter; DNS result: —; Model 1: –•–; Model 2A: ---; Model 2B: ••• .

nificant the propagation, the more the nonlinearity in Fig. 3, because less energy is present at high wave numbers. With the Gaussian filter, however, this distinction is less pronounced because the filter covers a broader range of the spectrum, thereby resulting in less sensitive behavior of p with u 8 /S 0L . In the present study, since treating p as variable complicates the dynamic modeling procedure, we prescribe the values of p as obtained from Fig. 3 in the calculation; for the cutoff filter, p51 and 3 respectively for weak and strong turbulence, and p52 for the Gaussian filter. Next we use the DNS data to check the accuracy of the models for the effective propagation defined in Sec. III. First, the 643 DNS field is used to compute the actual flame speed, taken to be the volume average of u ¹G u . The DNS field is then filtered with D of twice the DNS grid size using the cutoff or Gaussian filters. This grid-filtered field is then used to compute the filtered flame speed; the ratio ¯ u&5¯ S /S 0L is thus the target value for the model. ^ u ¹G u & / ^ u ¹G Subsequently, using only the filtered information, Eqs. ~19! and ~17! are solved for ¯ S /S 0L using each of the three models for the subgrid kinetic energy. In doing so, a second ‘‘test ˆ 52D. The procefiltering’’ of the DNS data is made with D dure is repeated at each time step of the DNS. As discussed in the previous section, Models 1 and 2 are tested with the cutoff filter, and Models 1 and 3 for the Gaussian filter. We first note that, since Eq. ~19! is an extrapolation procedure, it is expected that an accurate value of the ratio q/Q is important to good prediction of ¯ S /S 0L . First shown in Fig. 5 are the results with the cutoff filter. It is seen that Model 1, the extended Smagorinsky’s model, gives poor prediction of the target value. Model 2A improves the results somewhat, and the best result is obtained with Model 2B. This may be due to the low Reynolds number of the turbulent flow field studied; such that Model 2B best approximates the dissipation part of the turbulence spectrum tested. The poorer results with Models 1 and 2A may improve if the turbulence possesses a sufficiently wide inertial range. Figure 6 shows a similar comparison of Models 1 and 3 with the Gaussian Im, Lund, Ferziger


FIG. 6. A priori tests of the ratio q/Q with the Gaussian filter; DNS result: —; Model 1: –•–; Model 3: –••–.

filters. As in Fig. 5, Model 1 gives a poorer prediction of q/Q and the improvement provided by Model 3 is comparable with that by Model 2A in Fig. 5. Since Model 2 cannot be used with the Gaussian filters, these results suggest that the modeling of subgrid kinetic energy may be of more serious concern with the finite difference method. Figures 7 and 8 show ¯ S /S 0L computed from various subgrid kinetic energy models at various times during the DNS, for u 8 /S 0L 50.5 and 2, respectively. Here the cutoff filter and prescribed values for p are used. Figures 9 and 10 give similar results with the Gaussian filter. In both cases, it is clear that the behavior of the various models exactly follows that of Figs. 5 and 6, Model 1 being the poorest and Model 2B the best. These results indicate that prediction of turbulent flame speed using LES of the G-equation depends strongly on accurate estimation of subgrid turbulent kinetic energy, more specifically, the ratio q/Q.

FIG. 7. A priori tests of the filtered flame speed ¯ S for various models for u 8 /S 0L 50.5, p51 with the cutoff filter; DNS result: —; Model 1: – • – ; Model 2A: ---; Model 2B: ••• . Phys. Fluids, Vol. 9, No. 12, December 1997

FIG. 8. A priori tests of the filtered flame speed ¯ S for various models for u 8 /S 0L 52.0, p53 with the cutoff filter; DNS result: —; Model 1: – • – ; Model 2A: ---; Model 2B: ••• .

V. RESULTS OF LES RUNS

We now present results of actual LES using the suggested dynamic models; this may be called a posteriori testing. The initial flow field of the DNS is truncated to 323 ¯ 5x iniresolution and used as the initial field for ¯ u , and G 0 tially. Only the stronger turbulence case (u 8 /S L 52) with the cutoff filter is presented here. Figure 11 shows the turbulence and scalar energy spectra of the DNS and various LES averaged over time after the scalar field is fully developed. Two subgrid kinetic energy models are tested: Model 1 ~a poor model! and Model 2B ~the best model!. The LES energy spectrum is in fair agreement with the DNS, although the subgrid-scale model may be slightly over-dissipative. The scalar energies of the two

FIG. 9. A priori tests of the filtered flame speed ¯ S for various models for u 8 /S 0L 50.5, p52 with the Gaussian filter; DNS result: —; Model 1: – • – ; Model 3: – •• – . Im, Lund, Ferziger

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FIG. 10. A priori tests of the filtered flame speed ¯ S for various models for u 8 /S 0L 52.0, p52 with the Gaussian filter; DNS result: —; Model 1: – • – ; Model 3: –••–.

LES cases agree fairly well with the DNS, despite the disagreement of the results for the flame speed. Finally, the volume-averaged turbulent flame speed, rep¯ u & is plotted as a function of time in Fig. resented by ¯ S ^ u ¹G 12. Results from the DNS are plotted as the solid line. As was expected in the previous section, Model 2B gives the best result in this case since the scalar spectrum falls more rapidly with wave number for stronger turbulence. To illustrate the dependence on the exponent p, results for Model 1 with p51 are also shown. This simple model may overpredict the turbulent front speed by as much as a factor of two; significant improvement is obtained by merely switching to the appropriate value (p53). It appears that the exponent p depends on the turbulence intensity and spectrum and on the type filter used. Further study is needed to determine the sensitivity of p on these variables, especially in higher Reynolds number flows.

FIG. 12. Volume averaged turbulent flame speed as a function of time, for u 8 /S 0L 52; DNS: —; LES with Model 1, p51: – •• – ; LES with Model 1, p53: – • – ; LES with Model 2B with p53: ••• . The large-scale eddy turnover time is approximately 1.25 in the time unit shown.

VI. CONCLUDING REMARKS

Dynamic models for LES of the G-equation of turbulent premixed combustion have been proposed and tested. Several such models were tested in forced homogeneous isotropic turbulence. The results indicate that, unlike the case for Smagorinsky’s model applied to the momentum equation, the estimate of the subgrid kinetic energy is crucial to accurate prediction of turbulent flame speed. For the cases studied here, the extended Smagorinsky model overpredicts the flame speed, while the exponential curve fit gives excellent agreement. Determination of the most appropriate model type awaits further examination in turbulent flows at sufficiently high Reynolds numbers. Tests based on DNS results also show that ¯ S /S 0L is not 0 necessarily a linear function of q/S L ; it rather depends on the filter type and turbulence intensity relative to the flame speed. From the differences between the spectra for u 8 /S 0L 50.5 and 2, it appears that the inability to fit ¯ S /S 0L as 0 a function of q/S L with a fixed value of p is mainly due to the non-similarity between the turbulence and scalar energy spectra shown in Fig. 2. A modification is proposed to improve the model; one can free the exponent p and use the dynamic procedure to determine it. This requires two levels of test-filters and complicates the procedure. Finally, we remark that, although G is treated as a passive scalar in the present study, the concept can be extended to include heat release. The challenge is a numerical issue of how to properly capture discontinuities in hydrodynamic quantities across highly corrugated flames while resolving the small-scale turbulence. Methods designed to resolve this issue have been proposed;22,23 they contain an implicit high order hyperviscosity, but further work is needed. ACKNOWLEDGMENTS

FIG. 11. Comparison of averaged energy spectra obtained from DNS and various LES models for u 8 /S 0L 52 after the g-field was fully developed; DNS: —; LES with Model 1, p53: – • – ; LES with Model 2B with p53: ••• . 3832


The authors would like to thank Professor A. Bourlioux of CERCA and l’Universite´ de Montreál for the stimulating Im, Lund, Ferziger


discussions during the CTR Summer Program in 1996, and Dr. G. R. Ruetsch and Dr. N. S. A. Smith for helpful comments.

APPENDIX: ALTERNATIVE DERIVATIONS OF THE SECOND GERMANO IDENTITY

Instead of using the mathematical identity, Eq. ~15!, two alternative relations are derived here. ~1! Since the average speed of the fronts at any filtered ¯ u & ,2 requiring the conservalevel can be computed by ^¯ S u ¹G tion of the average speed at the grid- and test-filtered level, we obtain ¯ u & 5 ^ SC u ¹G C u&. S u ¹G ^¯

~A1!

Substituting Eq. ~17!, Eq. ~A1! becomes ¯ u & 5 ^ @ 11C ~ Q/S 0 ! p # u ¹G C u&, ^ @ 11C S ~ q/S 0L ! p # u ¹G S L

~A2!

such that ¯ u & # 5S 0 p @ ^ u ¹G ¯ u & 2 ^ u ¹G C u & 2 ^ q p u ¹G C u& #, C S @ ^ Q p u ¹G L ~A3! provided that C S is a function of time only. ~2! We use the same physical requirement of conservation of the average speed, yet in a slightly different manner as used by Bourlioux et al.:19 ¯^ u ¹G ¯ u&, S 0L ^ u ¹G u & 5S or ¯ S S 0L

5

^ u ¹G u & ¯ u& ^ u ¹G

~A4!

K S DL

5 11C S

q

S 0L

p

~A5!

.

Similarly, at the test-filtered level we have SC

5 0

SL

K S DL

Q A L ^ u ¹G u & 5 11C S 0 5 AC SL ^ u ¹GC u &

p

.

~A6!

Eliminating ^ u ¹G u & from Eqs. ~A5! and ~A6!, finally we obtain ¯ u & # 5S 0 p @ ^ u ¹G ¯ u & 2 ^ u ¹G C u & 2 ^ q p &^ u ¹G C u& #, C S @ ^ Q p &^ u ¹G L ~A7! where LHS terms are different from that of Eq. ~A3!.


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