Testing and Development of LES Models for Use in ... - CiteSeerX

Testing and Development of LES Models for Use in Multidimensional Modeling Eric Pomraning and Christopher Rutland Engine Research Center University of Wisconsin  Madison Introduction In this paper, several traditional eddy viscosity LES turbulence models are examined for use in internal combustion engine modeling. These models were implemented into the KIVA combustion simulation code and their results are compared to experimental data. In addition, a new one equation noneddy viscosity dynamic model has been developed. Preliminary results of this model will also be shown.

Background LES is based on the decomposition of a flow variable into resolved (filtered) and sub-grid scale terms. The resolved (or filtered) terms are solved for directly using transport equations, while the sub-grid terms and their effects on the resolved scales are modeled. The decomposition into a resolved and sub-grid is accomplished by filtering the flow variable. A filtered variable is defined by

φ (x ) =

z

V

G( x , y )φ ( y ) dy ,

(1)

where G(x,y) is the filter function which must satisfy

z

V

G ( x , y )d y = 1.

(2)

Applying the filtering definition, for a uniform grid, to the incompressible Navier-Stokes equations gives

¶ ui ¶ ui u j ¶ P ¶ τ ij ¶ 2 ui + =+ν , ¶t ¶x j ¶ xi ¶ x j ¶ x j ¶x j

(3)

where the sub-grid scale stress tensor is

τ ij = ui u j - ui u j .

(4)

It is necessary to model the sub-grid scale stress tensor since it can not be easily determined directly from the resolved field. This approach differs from the more traditional Reynolds Averaged Navier-Stokes (RANS) models in that the ensemble average in the RANS model is replaced by a spatial average in a LES model. The ensemble averaging in RANS models tends to smear out flow structure. Since the ensemble averaging has been removed in a LES model, unsteady flow behavior can be more accurately simulated. This makes LES methods more suited for complex highly time dependent flows such as in an internal combustion engine.

Eddy Viscosity Models Typically, the LES equations are closed by modeling the stress tensor with an eddy viscosity. Three viscosity based LES models were implemented into KIVA and compared to experimental data. A backward facing step at a Reynolds number of 42,000 was chosen as the test case for the LES models (Eaton, Johnston and Westphal, 1986). The computational grid was three dimensional and consisted of approximately 175,000 cells. It should be noted that for this Reynolds number and the grid resolution the simulations would be better described as Very Large Eddy Simulation (VLES) instead of Large Eddy Simulation. Traditional LES simulations are highly resolved.

The first model implemented was the Smagorinsky model. The scale stress tensor is modeled as

1 τ ij = -2Cs2 D2 S Sij + δ ij τ kk , 3 where the filtered rate of strain tensor is defined as

Sij =

F GH

I JK

1 ¶ ui ¶ u j + , 2 ¶ x j ¶xi

(5)

(6)

and the magnitude of the rate of strain tensor is

S = ( 2Sij Sij )1/2 .

(7)

The parameter, Cs , is a user specified constant that may vary significantly depending on the flow and grid resolution. A significant drawback of the Smagorinsky model is that it requires a priori knowledge of the flow to set Cs correctly. As can be seen from Figure 1c, the results for the Smagorinsky model were poor (Cs =0.09). It should be noted that it may have been possible to obtain better results by adjusting the model coefficient. The second eddy viscosity model implemented into KIVA was the dynamic Smagorinsky model (Germano, 1991). Instead of using a universal model coefficient, the model coefficient is dynamically determined as a function of space and time from the resolved field. This approach is based on an assumed scaling between resolved and sub-grid scales and a mathematical identity that arises. These models offer the advantage of not requiring a priori knowledge of the flow to set the flow coefficient. By defining a new filter level (the ‘test’ filter), a stress tensor at the ‘test’ level can be defined by

Tij = u i u j - ui u j .

(8)

The ‘grid’ level stress tensor and the ‘test’ level stress tensor are related by the Germano identity

Lij = Tij - τ ij ,

(9)

where the Leonard stress term, Lij , is defined by

Lij = u i u j - u i u j .

(10)

The Leonard stress term can be readily determined from the resolved velocity field in an LES simulation. Thus, the above equation is extremely useful because it relates the unknown stress tensors at the two scales, τ ij and Tij , to a known tensor, Lij . The next step in formulating a dynamic LES sub-grid model requires postulating models for the two stress tensors. Thus, various approaches can be taken. Traditionally, it is assumed that the Smagorinsky model is valid at both filter levels. This gives rise to:

1 τ ij = -2C D2 S Sij + δ ij τ kk 3

(11)

2 1 Tij = - 2C D S Sij + δ ij Tkk . 3

(12)

and

Substituting these modeled stress tensor terms into the Germano identity gives the dynamic model

1 Lij - δ ij Lkk = α ij C - β ij C , 3

(13)

where 2

α ij = -2 D S Sij

(14)

and

β ij = -2D2 S Sij .

(15)

Removing the coefficient from the integral and applying a least squares fit for the coefficient (Lilly, 1992) results in an algebraic expression for the coefficient

C =

F L M I, GH M M JK ij

ij

kl

(16)

kl

where

Mij = α ij - β ij .

(17)

The results for this model were also poor. However, others have reported good results for both the Smagorinsky and the dynamic Smagorinsky model. In our opinion, the reason for the good results is that the simulations were highly resolved. As stated earlier, the current simulations were not well resolved. Since it appears that these models require high resolution, given the current computational power, these models may not be suitable for use in internal combustion engine simulations. The last eddy viscosity model tested was a one equation model. By adding a transport equation for the sub-grid scale kinetic energy, it is reasoned that the accuracy of the LES sub-grid scale stress models will be improved. In addition, the use of a transport equation for the sub-grid scale kinetic energy may allow for coarser grids than can be used for a comparable problem with a zero equation model. The reason for this is that some sub-grid information is available for the formulation of sub-grid scale models, which should serve to improve the modeling of the sub-grid effects on the resolved scales. Based in part on the work of Yoshizawa (1985), a one-equation sub-grid scale model has been proposed by Menon et al. (1996). The modeled sub-grid kinetic energy transport equation is written as

FG H

IJ K

¶k ¶k ¶u ¶ ν tk ¶k + ui = - τ ij i - ε + , ¶t ¶xi ¶x j ¶xi σ k ¶ xi

(18)

where, in this case, the sub-grid kinetic energy is defined as

k=

1 2

cu u - u u h. i i

i

i

(19)

Note, using the Cauchy-Schwarz inequality and the definition of the sub-grid kinetic energy, it can be shown that the sub-grid scale kinetic energy is always positive (Ghosal, 1995). The sub-grid scale stress is modeled as

τ ij = -2ν tk Sij +

2 kδ ij , 3

(20)

where the turbulent kinetic viscosity, ν tk , is modeled as

ν tk = Ck k 1/ 2D . A model for dissipation of the sub-grid scale kinetic energy,

ε = Cε

k

(21)

ε , is given by

3/2

D

.

(22)

The constants in the model are set to Ck = 0.05 , Cε = 1.0 , and σ k = 1.0 (Yoshizawa, 1985). As can be seen from Figure 1, the results using the one-equation model were in good agreement with experimental results. In addition, the calculated flow using the one-equation model is highly unsteady when compared to the k-ε model. A significant drawback of this model is that the user must still specify model coefficients. In general, a weakness of all LES eddy viscosity models is that the viscosity is applied over all wave numbersadditional dissipation is added where it is needed (at the higher wave numbers) and also where it is not necessary (at the lower wave numbers). In addition, a viscosity model is purely dissipative. It is

known that energy flows locally from the sub-grid to the resolved as well as from the resolved to the subgrid with a slight preference for the later. This phenomenon is known as backscatter (Piomelli, 1991). The justification for an eddy viscosity is that it is consistent with the idea of an energy cascade. However, this argument is only valid in the context of an ensemble averaged or time averaged model. In the case of LES, where there is no ensemble or time averaging, this argument is not valid.

A One Equation Non-Eddy Viscosity Dynamic Model To address some of the difficulties in some of the other models a new one equation non-eddy viscosity dynamic model has been developed. Instead of modeling the stress tensor with an eddy viscosity, an attempt is made to estimate the stress tensor directly. A transport equation for the sub-grid turbulent kinetic is added to the dynamic model to enforce a budget on the energy flow between the resolved and the sub-grid scales. To that end, the sub-grid stress tensors models are required to be a function of the sub-grid turbulent kinetic energy. The one equation model for the stress tensors are given by

τ ij = cij k

and

Tij = cij K

(23)

where the ‘test’ level turbulent kinetic energy is defined by

K=

1 ( ui ui - ui ui ) . 2

(24)

The ‘test’ level and ‘grid’ level turbulent kinetic energies are related by the trace of the Leonard term so that another transport equation for K is not required:

K =k+

1 Lii . 2

(25)

Substituting these models for the two stress tensors into the Germano identity gives the following :

Lij = Kcij - k cij .

(26)

It should be noted, the coefficient, cij , is properly left inside the integral indicated by the curved overbar. The result is a set of Fredholm integral equation of the second kind that can be solved using an iterative method. It is important to realize that only under certain circumstances can Fredholm integral equations be solved. A poor choice for a stress tensor model will result in equations that can not be solved. However, for the stress tensor model chosen here, the equations are always solvable. To test the newly formulated model, the sub-grid scale stress tensor model is compared to the filtered DNS field sub-grid scale stress tensor. The DNS case considered is isotropic turbulence in a periodic box at a Reynolds number of approximately 30 based on the Taylor microscale. As can be seen from Figures 2 and 3, there is excellent agreement between the filtered DNS data and the modeled stress tensor. To show the new models capability, an intake simulation is shown in Figure 4. As can be seen there is significantly more structure for the new model than for the k-ε model. It is believed that the increase in structure more accurately simulates an actual intake process.

References Eaton, J., Johnston, J., and Westphal, R., (1986). “Experimental study of flow reattachment in a singlesided sudden expansion”, prepared for Ames Research Center, National Aeronautics and Space Administration, Scientific and Technical Information Office, NASA contractor report 3765 Germano M., Piomelli U., Moin P., and Cabot W. H., “A Dynamic Subgrid-Scale Eddy Viscosity Model,” Physics of Fluids A (1991), Vol. 3, No. 7, pp. 1760-1765. Ghosal S., Lund T. S., Moin P., and Akselvoll K., “A Dynamic Localization Model for Large-Eddy Simulation of Turbulent Flows,” Journal of Fluid Mechanics. (1995a), Vol. 286 pp. 229-255. Lilly D. K., “A Proposed Modification of the Germano Subgrid-Scale Closure Method,” Physics of Fluids (1992), Vol. 4, No. 3, pp. 633-635. Menon S., Yeung P. K., and Kim W. W., Effect of Subgrid Models on the Computed Interscale Energy Transfer in Isotropic Turbulence,” Computer and Fluids (1996), Vol. 25, No. 2, pp. 165-180.

Piomelli U., Cabot W. H., Moin P., and Lee L., “Subgrid-Scale Backscatter in Turbulent and Transitional Flows,” Physics of Fluids A (1991), Vol. 3, No. 7, pp. 1766-1771. Yoshizawa A., and Horiuti K., “A Statistically-Derived Subgrid-Scale Kinetic Energy Model for the LargeEddy Simulation of Turbulent Flows,” Journal of the Physical Society of Japan (1985), Vol. 54, No. 8, pp. 2834-2839.

LES a) One equation eddy viscosity model. Top velocity, bottom sub-grid kinetic energy.

k -ε b) RNG k-ε model. Top velocity, bottom turbulent kinetic energy. 600

800

500

600

400

400

velocity (cm/sec)

velocity (cm/sec)

300 200 100

experiment

0 -100 -200

1-eqn LES k- ε Smagorinsky

-300 -400

200 0 -200 -400

Instantaneous Average

-600 -800

-500

-1000 0

25

50

75

100

125

X (cm)

c) Reattachment length and wall velocity. Figure 1. LES eddy viscosity results.

150

0

25

50

75

100

X (cm)

125

150

175

(a) Determined from the filtered DNS field.

(a) Determined from the filtered DNS field.

(b) Based on the proposed new model.

(b) Based on the proposed new model.

Figure 2. Contour plots of

τ 11.

(a) New one equation LES model.

Figure 3. Contour plots of

τ 12 .

(b) RNG k-ε model.

Figure 4. Intake simulation velocity vectors. Vectors are not scaled by size so that structure can be seen.