Subgrid-scale modeling suggested by a two-scale DIA

Journal of Wind Engineering and Industrial Aerodynamics, 46 & 47 (1993) 69--76 Elsevier

69

Subgrid-scale modeling suggested by a tw0-scale DIA A. YOSHIZAWA Professor, I n s t i t u t e of Industrial Science, University of Tokyo, 7-22-1, Roppongi, ~linato-ku, Tokyo 106, Japan

Abstract Subgrid-scale (S6S) modeling in large eddy simulation is studied using the results of a two-scale I)IA about the effects of h e l i c i t y controlling the energy cascade process. Such effects are closely related to v o r t i c i t y effects on the SGS Reynolds stress that are lacking in the SGS-viscosity representatiom These results are applied to the improvement of the SGS models of the Smgorinsky type. Discussion is also made about the relationship with the present two-equation model with the Smagorinsky model and the one-equation model based on the SGS energy.

I.

INTRODUCTION

Turbulence modeling is roughly c l a s s i f i e d into two categories. One is the Reynolds-mean modeling [1] and another is the subgrid-scale (SGS) modeling [23 based on the f i l t e r i n g procedure that is widely used in large eddy simulation (LES). A prominent difference between two types of modeling lies in the difference between the spatial scales to be modeled. In the SGS modeling, the spatial scales to be modeled are only the ones that are smaller than the f i l t e r width. As a result, the components of motion to be d i r e c t l y dealt with are always highly time-dependent and three-dimensional. This situation makes sharp contrast with the counterpart in the Reynolds-mean modeling. In the l a t t e r , the moan quantities can become steady and one- or two-dimensioual, as in channel and boundary layer flows. The smallness of the spatial scales of motions to be modeled in the SGS modeling is a great merit for constructing models with higher universality. At the present stage, however, the SGS models based on the SGS-viscosity approximation, whose representative is the Smgorinsky model, are not in a state so optimistic as have been anticipated. For instance, the Smgorinsky model with the Smugorinsky constant fixed cannot simulate some different types of fundamental flows with sufficient accuracy. As the cause of this d i f f i c u l t y , we can consider two factors. One is the SGS-viscosity representation for the SGS Reynolds stress and another is the way of estimating the SGS energy dissipation rate that cannot be captured using an adopted f i l t e r . In this work, we shall study the ~ S modeling from the viewpoint of rectifying the above-stated shortcoming. For this purpose, attention will be focussed on the results of a two-scale DIA (TSDIA) [3,4] about effects of h e l i c i t y .

0167-6105/93/$06.00 © 1993 - ElsevierSciencePublishersB.V. All fights reserved.

70

2.

FUNDAMENTAL EQUATIONS

Throughout this work, we consider a fluid motion at very low Mach numbers to neglect f l u i d compressibility. Such a motion of a viscous fluid is described by the Navier-Stokes equation

(1)

~ ) u / 0 t + V ' ( u u ) : - Vp + ~)Au, with the solenoidal condition V ' u pressure divided by constant fluid component of V-(uu) is defined by In LES based on the $6S models, f i l t e r function G as

= 0. Here u is the velocity vector, p is the density, and v is the kinematic viscosity. The i ( 0 / a xj)ujui. we smooth out small-scale fluctuations using a

(2)

fF ~ (f} : f G ( y - x)f(y)dy

(subscript F means f i l t e r i n g ) , where f denotes ( ~ p , w ) and the fluctuation f' around f,~ is given by f' = f f~ [w (= V × u) is the v o r t i c i t y ] . We apply Eq. (2) to Eq. (1) to obtain (3)

OuF/Ot + V'(UFUF) : - VPF + V~UF + V'(R~ + L + C). Here the SGS Reynolds stress Rvij, the Leonard term L~j, and the cross term C~j are defined by R,;~j - - ( u i ' u j ' } , Li.i - -

({UF~UFj} --UFiUFj), Cij = -- ({UF~Uj') + {Oi'UFj)),

(4--6)

respectively. The terms Lij and Cij are the q u a n t i t i e s i n t r i n s i c to the f i l t e r i n g procedure and may be called the double f i l t e r i n g effects [5]. Of Eqs. (4)-(6), Lij can be computed in the course of L ~ and needs no modeling. On the other hand, the modeling of R~ij and C~j is necessary; specifically, the former is a central problem in the ~ $ modeling.

3.

RESULTSOF h n0-SCALE DIA

The cross term Cij is a peculiar quantity in the sense that the counterpart in the ensemble-mean procedure i d e n t i c a l l y vanishes, whereas the $(]S Reynolds stress RFI j haS the nonvanishing counterpart in the ensemble-mean procedure which is usually called the Reynolds stress. This fact s i g n i f i e s that the ensemble-mean results about the l a t t e r , which have been obtained using two-point turbulence theories, are applicable to the modeling of RFij. In what follows, this point will be c l a r i f i e d . 3. 1 Ensemble-mean r e s u l t s For the SGS modeling, we s p e c i f i c a l l y use the r e s u l t s of a two-scale DIA [3,4] (TSDIA) that has been developed for the study of a turbulent shear flow under the ensesble-mean procedure <. > . According to the TSDIA results, the ensemble-mean Reynolds stress R~ij is written as

71

-

(7)

(2/3)we.VH6 ~]

(subscript E denotes ensemble mean). Here KE, SEij, WE, and HE are the ensemble-mean turbulent energy, v e l o c i t y strain, v o r t i c i t y , and turbulent h e l i c i t y , respectively, which are defined by KE :

, S ~

: Ou~/Ox~ + 0 U E i / 0 X j ,

OJE : ~

)~ UE, HE :



(8-11)

(uE is the ensemble-mean velocity and u" = u - u~). We use the ensemble-mean charact e r i s t i c legnth g E and the KE d i s s i p a t i o n r a t e ~ E to express KE, VE (eddy viscos i t y ) , and ~E as KE = Cl ~ r 2/a S z / 8 , C1 = 0.665; ~E : C2 ~ 4 / 3 ~ E l / 3 E = C3 g 8 / 3 s E-~/S, C~ = 5 , 9 0 × i 0 -4.

C2 :

0.0349; (12-14)

From Eq. (12), we have E :

C~K3/2/

~ ~,

C* :

C~ - 3 / ~

: 1.84,

(15)

which r e s u l t s in ~ E = C5KE2/~ E, C5 : C2C44/3 = 0 . 0 7 8 7 ;

~ E = C6ZE4/ e E, C6 : C3C48/3 : 0.003. (16,17)

Here we note the following two points. One is Eq. (16), which is the s o - c a l l e d eddy v i s c o s i t y in the two-equation model. The numerical factor Ca has already been optimized as 0. 09. Another is the t u r b u l e n t - h e l i c i t y (HE)/related terms in RE~j [Fxl. (7)], which are connected with the mean v o r t i c i t y wE. In the familiar eddyv i s c o s i t y representation for REIj, the mean v e l o c i t y - s t r a i n tensor SEij plays a centr a l role. An a r b i t r a r y mean v e l o c i t y gradient a u ~ j / 0 x~ is divided into two parts as 0UEj/0Xl = (1/2)SEij + (1/2) WEij,

(18)

where coEij is the mean v o r t i c i t y tensor defined by o~Ej -= 0 u ~ j / 0 x ~ -

0u~i/0xj

:

e~j~w~.

(19)

In the study of turbulent shear flows with nonvanishing 0uE~/Ox~, i t is well recognized that v o r t c i t y bears an important role in close r e l a t i o n to v o r t i c a l flow structures. I t s mean part, however, is separated from REIj under the eddy-viscosity approxisation. Some of the major shortcomings of the approximation a r i s e from t h i s point (for more d e t a i l s , see Ref. 4). Specifically, Eq. (7) shows that e f f e c t s of h e l i c i t y can contribute to RE~, except the case of two-dimensinal mean flow in which O~E is always normal to VH~. 3. 2 SGS r e s u l t s A method for deriving the SGS r e s u l t s with the aid of the ensemble-mean counterparts was given by the author [6] who introduced a s t a t i s t i c a l f i l t e r . According to

72

the method, the transformation of the ensemble mean to SGS results can be done using the replacement ~.:

•

2h,

(20)

where h is the representative f i l t e r widtk Eq. (20) s i g n i f i e s that the largest wavelength of SGS eddies in a rect-angular region with the side length h, which has a great influence on the grid-scale (GS) motion through RFi~, is 2h. Under Eq. (20), we obtain Kv = CvhZ/:~ v 2/'~, Cv v = Cgh8/a e v -1/~, from Nqs. (12)-(14).

= 22/3Ci

C9 :

:

1.06; Vv-

C8h4/3

e v t/:~, C8 - 2 4 / 3 C 2

=

2a/3C3 : 3.75X10 3,

0.0879; (21-23)

Using the replacement (20), Eq. (7) is cast into the SGS form as

RF~j = - (2/3)KvG~j + vvSv~.i - (2/3)~v.VH~6,j].

V,~[wv~(OHv/Oxj) + o v i ( O H v / O x j ) (24)

In Eqs. (21)-(23), KF etc. are the f i l t e r i n g - r e l a t e d counterparts of KE that are obtained by replacing u~ and u" with uv and u'. Specifically, SF~j and ~ , are the 6S velocity s t r a i n and v o r t i c i t y , respectively. Here we should note that s ,~ and HF are yet to be determined. For the l a t e r SGS modeling, we give the transport equations for KF and Hr. We neglect the d o u b l e - f i l t e r i n g effects to obtain DEF/Dt = R F i j ( 0 H F j / 0 X i )

(25)

-- ~ F ÷ ~ ' T F ,

I)HF/Dt = RFij(0 w , . j / 0 x i ) - w~+(0/axj)R~+~

eHF + V'(KF~oF + THe),

(26)

with D/Dt - a / a t + uF-V. Here ~Hs [= 2 9 ( ( a u / / 0 x l ) ( a w j ' / a x ~ ) } ] is the destruction rate of HF, whereas TF and TH, are the KF and Hv transport rates, respectively, both of which are of third order in u" and o)" (for more details, see Ref. 4). In Sec. 4, we shall propose some SGS models of d i f f e r e n t levels on the basis of these results.

SGS MODELS The S(IS models that will be constructed using the TSDIA r e s u l t s in Sec. 3 can be c l a s s i f i e d according to the number of the transport equations for the SGS quantities like KF. 4.1 Two-equation SGS models The relationship between KF and s v is given by Eq. (21); namely, e v is written as e F = C, OEF3/2/h, C,o - Cv- s / 2 = 0 . 0 1 6 .

From Fxts. (22) and (27), ~ v is given by

(27)

73

~

(28)

: C~b-K~~/2, C~ : C8C~o~/s : 0.0854, ~ = C~2ha/K~~/2, C12 = C9C~o-~/a = 3.86X10 -a.

(29)

When the KF transport equation (25) is combined with Eq. (27), TF is the sole quantity to be modeled in the former, except the dependence of RF~j on Hr. The KF transport r a t e TF is most simply modeled as TF = (~)F/aK)VKF,

(30)

o . ~ I,

using the concept of gradient diffusion. In Eq. (26) for HF, e H~ is the counterpart of ~ F in the K~ equation. In the modeling based on the ~ HF equation, the resulting syste~ is of three-equation type and is too conrplieated in LF~ in which flow f i e l d s are always highly time-dependent and three-dime~sional. In order to reduce the computational burden, we model ~ HF as (31) using the $6S time scale K~/~ F [4,7].

~oreover the H~ transport r a t e T.F is modeled

as

THF : ( v J a H ) V H ~ ,

aH ~ 1.6,

(32)

in a similar way to Eq. (30). S u m r i z i n g the above r e s u l t s , we have reached a SGS model of two-equation type. I t consists of the uF equation (3), the RF expression (24), the KF transport equation (25), the HF transport equation (26), and the a u x i l i a r y r e l a t i o n s (27)-(32), except the Leonard and cross t e m s Lij and Cia. 4.2 0ne-equation SGS models Let us introduce the further approximation to the foregoing two-equation model to derive some simplified versions. One of such models is obtained by t o t a l l y neglecting e f f e c t s of SGS h e l i c i t y (HF). The r e s u l t i n g model is the so-called one-equation model based on the KF transport equation [8]. Such a SGS model has already been applied to channel and mixing layer flows [9,10]. At the present stage, however, i t is not c l e a r whether the one-equation model can r e a l l y overcome the d i f f i c u l t y in LES using the hmgorinsky zero-equation model with the S~agorinsky constant fixed ( t h i s d i f f i c u l t y w i l l l a t e r be referred to in more d e t a i l ) . In order to derive another one-equation model, we assume the balance between the KF production and d i s s i p a t i o n rates as (33) in Eq. (25), where RFIj is given by Eq. (24)." e f f e c t s of HF on RFIj to obtain 8 F = Ci4h2S s,

with

C14 : C8 3 / 2

:

0.0261,

In the e s t i m t e of e ~, we neglect the

(34)

74

s - [s~,~/2] '~,

(35)

M]ere use has been made of Eq. (22). Kv

= C , 5 h 2 S ~, C , 5

: C 7 C 1 4 u/a

- 0.

From Eqs. (21)-(23) and (31), we. have

0933; v

F : Ct6hZS,

C,~

= CsC~

'/3 - 0 . 0 2 6 1 ;

(36,37)

~F = C,;h2/S, C,; - C,CL~ '/~ - 0.0126;

(38)

e,,F - C~HF8, C,~ : C,:,C,~/C,~ : 0.280.

(39)

The one-equation model using the HF transport equation consists of the uF equation (3), the RF expression (24), the HF transport eqnation (26), and the a u x i l i a r y r e l a tions (31), (32), and (36)-(39). ~ 3 Zero-equation SGS models In order to further simplify the one-equation models, we assume the balance between the HF production and destruction rates as

RFIJ(O(A) F j / 0 X i ) -

(40)

(t) Fi(0/OXj)[~ji ~" ~HF.

We combine Eq. (39) with F~t. (40) and neglect the HF e f f e c t s on RFIj on the left-hand side of the l a t t e r equation. As a r e s u l t , we obtain H, = C , ~ S - ' [ v F S F ~ j ( 0 ~ F j / 0 x ~ ) C,9 = 1/C,a = 3.57.

~Fi(D/0xj)

~SFjl

+ (2/3)wF'VK~],

(41)

Finally, the zero-equation model with the h e l i c i t y e f f e c t s included consists of the u~ equation (3), the RF expression (24), and the a u x i l i a r y r e l a t i o n s (36)-(38) and (41). The simplest zero-equation SGS model with both the KF and H~ e f f e c t s dropped is the so-called Smagorinsky model. I t is given by the uF equation (3) and the SGS Reynolds stress RFIj : - (2/3)KF6ij ÷ ~ F S F i j ;

~F

=

(Csh)~S, Cs

=

el6

1/2

=

0.16,

(42)

with Eq. (36) for KF.

4. 4 Leonard and cross terms We have already mentioned t h a t the Leonard term L~j [Eq. (5)] can be computated in the course of LES, whereas C,j needs to be modeled. Here i t should be noted that the retention of one of Lij and Cio leads to the v i o l a t i o n of the Galilean invariance i n t r i n s i c to the original Wavier-Stokes equation (1). Considering this situation, a model s a t i s f y i n g the invariance in the combination with L~j is given by [11] C~j = - ((UFi}(UFj) - UFiUFj).

(43)

Almost a l l of the current LES's are based on the Smagorinsky model or its combination with the Leonard and cross terms Lij and Cij. The relationship of the type of a f i l t e r and the retention of L~j and Cij with the accuracy of computed results was discussed in detail by Piomelli et al. [12] in LF~ of a channel flow.

75

5.

DISCUSSIONS

The most serious d i f f i c u l t y in LES using the ~magorinsky model is t h a t some d i f f e r e n t kinds of fundamental turbulent flows cannot be simulated with the same Sn~gorinsky constant Cs [Eq. (42)]. Namely, Cs has been optimized as channel flow: 0.1; mixing layer flow: 0.15; isotropic decaying flow: 0.023.

(44)

Such a situation cannot be resolved by incorporating the Leonard and cross terms Lij and Cij. This fact implies that the above-stated difficulty is closely associated with the following two points, as has already been addressed: ( i ) the method of estimating the SGS energy d i s s i p a t i o n r a t e s F; ( i i ) the SGS-viscosity approximation for the SGS Reynolds s t r e s s RFij. As a method for improving the Smagorinsky model from the standpoint ( i ) , the author [13] proposed to include the e f f e c t s of convection and transport in the e s t i mate of e F. AS a r e s u l t , the usual ~mgorinsky medel was changed to a model with a variable Smagorinsky c o e f f i c i e n t . This type of model was r e a l l y applied to LES's of isotropic decaying flow, channel flow, backward-facing flow, flow past a cubic body, etc. with some success [14,15]. Another attempt associated with the standpoint ( i ) is the dynamical SGS model proposed by Germano e t al. [16]. In the model, two d i f f e r ent f i l t e r s are introduced and the similary between the SGS-viscosity representations under these two f i l t e r s is a s s ~ e d . As a r e s u l t , the Smagorinsky c o e f f i c i e n t can be computed automatically in the course of LES. A shortcoming common to the above two attempts improving the Smagorinsky model is that these models are not so robust in the point of numerical s t a b i l i t y as the former. The present work incorporating the e f f e c t s of h e l i c i t y into SGS models s t a r t e d from the standpoint ( i i ) . The most prominent property distinguishing channel flow from decaying flow is the streak structures near the plates of a channel. Such structures also e x i s t in a mixing layer flow [10] though they are less c l e a r compared with the counterpart in a channel flow. Moin [17] performed the d i r e c t numerical simulation of homogeneous turbulence subject to constant mean v e l o c i t y shear to show that the occurrence of streak structures are e n t i r e l y depondent on the strength of mean shear, but not on the existence of a s o l i d wall. These streak structures consisting of streamwise vortices imply that the SGS h e l i c i t y defined by ( u ' . co'} is nonvanishing. Such a nonvanishing SGS h e l i c i t y can give r i s e to the deviation of the SGS Reynolds s t r e s s from the SGS-viscosity representation, as is seen from Eq, (24). The r e l a t i o n s h i p of the h e l i c i t y with the suppression mechanism of the energy cascade was discussed in d e t a i l in Ref. 4. From these discussions, we can consider that h e l i c i t y is a promising concept in overcoming the d i f f i c u l t y in the Smagorinsky model with the fixed model constant. The concept of h e l i c i t y is a useful tool in describing h e l i c a l flow structures in turbulence, as was d e t a i l e d in Ref. 4. Such flow structures are also observed near the apex of a cubic body or a building and cannot always be computed using the current SGS models with s u f f i c i e n t accuracy. The present SGS models with h e l i c i t y e f f e c t s included are expected to contribute to the improvement of LES in these three-dimensional structures.

75 6.

CONCLUDINGREMARKS

In this work, we investigated the SGS modeling from the standpoint of improving the SGS-viscosity approximation for the SGS Reynolds stress. Specifically, we focussed attention on the effects of helicity controlling the energy cascade process to incorporate them into the SG8 modeling. As a result, we proposed a two-equation/type SGS •odel based on the transport equations for the $GS energy and helicity. This equation ms simplified to the one- or zero-equation models by introducing further assumptions. As a result, the relationship of the S~gorinsky model with the two- or one-equation SGS models was clarified. The cause of the d i f f i c u l t y in LES using the S~gorinsky model with the fixed Smagorinsky constant was also investigated in the context of streak structures that are generated by strong mean shear.

Acknowledgments The author is grateful to S. Nisizima and N. Yokoi for useful discussions about effects of helicity. Part of this work was done while the author was in residence at ICOMP (Institute for Computational Mechanics in Propulsion), NASA/Lewis Research Center.

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