its application to large eddy simulation of -...

Science in China Set. G Physics, Mechanics & Astronomy 2004 Vo1.47 No.4 463-476 463

An improved dynamic subgrid-scale model and

its application to large eddy simulation of

rotating channel flows

LIU Nansheng, LU Xiyun&ZHUANG Lixian

Department of Mechanics and Mechanical Engineering, University of Science and Technology of China,Hefei 230026, China

Correspondence should be addressed to Lu Xiyun (email: Au @ustc.edu.cn)

Received January 12, 2004

Abstract A new dynamic subgrid-scale (SGS) model, which is proved to satisfy theprinciple of asymptotic material frame indifference (AMFI) for rotating turbulence, isproposed based on physical and mathematical analysis. Comparison with directnumerical simulation (DNS) results verifies that the new SGS model is effective for large

eddy simulation (LES) on rotating turbulent flow. The SGS model is then applied to theLES of the spanwise rotating turbulent channel flow to investigate the rotation effect onturbulence characteristics, budget terms in the transport equations of resolved Reynoldsstresses, and flow structures near the wall regions of the rotating channel.

Keywords: large eddy simulation (LES), dynamic subgrid-scale (SGS) model, rotating turbulent channel flow,asymptotic material frame indifference (AMFI).

DOI: 10.1360/03yw0228

Rotating turbulence occurs extensively in nature and engineering circumstances.Meanwhile, understanding physical mechanisms of the rotating turbulence is importantto the fundamental research of turbulence. The turbulent flow in rotating frames under-

goes two kinds of Coriolis force effects. First, a secondary flow is induced in the casethat there is a mean vorticity component perpendicular to the rotating axis. Second, there

are augmenting or suppressing effects on the turbulence if there is a mean vorticitycomponent parallel to the rotating axis. Both the two effects profoundly affect not onlythe mean flow field, but also the turbulence intensities and coherent structures in the

wall region. According to the Taylor-Proudman theorem, turbulence subject to strongrotation will undertake a transition toward the two-dimensional state and eventually to

relaminarization, which is observed in experiments; thus the turbulence dissipation ratebecomes trivial and the turbulent eddy viscosity vanishes, which represents the depend-

ence of the turbulent eddy viscosi妙on the imposed rotations''. Meanwhile, the strongrotation suppresses the nonlinear energy cascade from large to small scales throughphase scrambling 121. Although the angular velocity of rotating frame disappears in the

Copyright by Science in China Press 2004

万方数据

464 Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vo1.47 No.4 463-476

transport equation of turbulent kinetic energy, it has been recognized that the rotationexhibits a significant influence on turbulence production and dissipation. However, some

physical mechanism of rotating turbulent flow is still unclear and needs to be investi-gated further.

In the past decade, large eddy simulation (LES) method becomes an efficient toolfor the prediction of complex turbulent flows. The rationale of this approach is to com-pute the large-scale components directly and model the subgrid scales via SGS model.So, a reasonable SGS model is crucial in the LES method. Extensive efforts have been

taken to develop and improve the SGS model. Thus, the construction of reasonable SGS

model, which can account for rotation effect on turbulent flow, is still a challengingproblem.

Bradshaw[31 elucidated the similarity among the effects of rotation, streamlinecurvature and thermal stratification on turbulent flows and defined an equivalent

gradient "Richardson number" to identify either an augmenting or suppressing effect onthe flow subject to system rotation. Fu and Wang [4] proposed a second-moment closuremodeling of turbulence in a non-inertial frame. Rubinstein and Zhou151 constructed aSGS model by direct interaction approximation, which reproduces the fact that turbulent

eddy viscosity vanishes in the limit case S2 -4 -. Unfortunately, the model coefficients

in those models are determined empirically.

For turbulent flow in rotating frame, Spezialel6-sl indicated that the SGS stresstensor formation is dependent on the frame of reference but the divergence of SGS stress

tensor is independent of the frame of reference, and required that the SGS model shouldbe compatible with the principle of material frame indifference(MH).Shimomuraet al.""" further claimed that the principle of MFI should be imposed not only on thedivergence of SGS stress tensor but also on the SGS stress tensor itself in the limit of

SZ---) -，where SZ is the angular velocity of the reference frame. This constraint requires

that the dependence of the model equation for the SGS stress tensor tends to disappear as

Qapproaches infinity, which is referred to as the asymptotic material frame indifference

(AMFI) of the SGS model equation. The principle of AMFI is theoretically proved andregarded as a constraint to the SGS stress closures for rotating turbulent flow. By exam-

ining some typical SGS models, it is found that most SGS models, e.g. Smagorinskymodel"", dynamic Smagorinsky mode11121 and dynamic mixing mode11131, are inconsis-tent with the principle of AMFI. Thus, the motivation of this study is to develop a rea-

sonable SGS model that is consistent with the principle of AMFI and accounts for theinfluence of the system rotation on the turbulent flow.

In this study, a nonlinear SGS model satisfying the principle of AMFI is proposed,

whose coefficients are dynamically determined based on the resolved flow field. Theperformance of the nonlinear SGS model is examined by the LES of the spanwise rotat-ing turbulent channel flow. Then, the rotation effect on turbulence characteristics is

investigated.

Cop州ght by Science in China Press 2004

万方数据

An improved dynamic subgrid-scale model 465

1 Mathematical formulation

The governing equations for theflow (as shown in fig. 1 for the sketch)

LES of the spanwise rotating turbulent channelare the three-dimensional filtered Navier-Stokes

equations. To non-dimensionalize thegovei刀lngequations, the friction veloc

used as the velocity scale, and the half-height of the channel h as the lengthnon-dimensional governing equations are given as

Sc

u7 is.The

au:一一一二=U.

ax(1)

au, au;u; 而 _ 1 a1u:一 +一 =一一一d,+— — 一N -C..'

at ax; ax ’‘Re, axjaxj(2)

where the overbar denotes the resolved vari-

able,万is the sum of the pressure and cen-

trifugal force. The non-dimensional parame- +1-

ters in this problem are the rotation numberand Reynolds number, which are defined as

从=292h/u, and ReT=uzhlv, respec-

tively.瓦(i=1, 2, 3) is the resolved velocity -1-

and is represented as, for writing convenience, Fig. 1

U(y)

口

Sketch spanwise rotating turbulent

u, v and w in

wall-normal (x2,

(XI,the streamwise(

or y) and spanwise

or x), channel.

(z, or x3) directions in rotating Cartescan coordinate

system. In eq. (2),乓=uiui一utul isthe subgrid-scale turbulent stress and needs to be

modeled by SGS model.

2 Nonlinear dynamic SGS model for rotating turbulence

2.1 Nonlinear SGS model functional form in inertial frame

To specify the model's functional form, we neglect the turbulence historical effects

and assume that z,} depends instantaneously on local au; /axe and some scalars. Un-

der this hypothesis,几can be written in the following functional form:

(3)

where S represents a scalar group and will appear in the coefficients of the model. Here,

we further decompose au; / axe into strain tensor弓and rotation tensor风，

www.scichina.com

万方数据


(4)

Then, based on the tensor operation rule, an asymptoticexpansion on eq.(3) aboutSi j

and风 is taken. The symmetrictensor几 thus can be written as

气一二Tkk j氏二一2v,3ij+v6 (Sik @kj+S jk @ki )

·:〔、、一合Skl S0 '5ij)·二〔Wik COkj一合}1k }kl }ij (5)

、l

es

，/

Compared to the second-order SGS model proposed by KosovicE141, this nonlinearSGS model is of perfect functional form, which contains all the second-order nonlinear

terms of瓦and风·In an inertial frame, the effect of the system rotation on Zij is

represented implicitly by瓦and风·

2.2 Nonlinear SGS model functional form in rotating frame

Further, it is needed to deduce the functional form of the nonlinear SGS model in

non-inertial frame. By taking orthogonal rotational frame transform on eq. (5), thenonlinear SGS model functional form in rotating frame is then written as

(弓):=-2代021s*._，__，2j2，_，_，，Si j+Vb万V ik叭

一，一，、，Q2+S ik然i )+咋不丁

1乙一'Pik Coki一喜Sii Ojlk O)kl戈 J

/了les;\

- 嵘

+

、!

夕/

--*sl --*s&l 乓

1

-3

一 --*s*j

Jsi*

Zr十

.又

+v}02V, (6)

This form

notes the

is proved to satisfy the principle of AMFI. In eq. (6), the superscript‘*’de-variablein rotating coordinate system.(几):=凡一凡fu /3; v,,, vb' and

嵘 are non-dimensional model coefficients andare一------一 _‘_」 __ 二 A2..，1，*‘rUPrcScuLcu "S va=。 Va I S 1,

Vb=02vb /12 and v,=2j2v", respectivelyI s * 1=(可可)“’ △ is the grid filter

size 式is the additional tensor due to the orthogonal rotational frame transform

.尹

、、leeee||，/

+氏*Sij 'Qa《一0i SZj (7)

which indicates that弓is the function of the angular velocity。of the rotating frame.Then, the nonlinear SGS model in the inertial frame can be rewritten as

Cop州ght by Science in China Press 2004

万方数据

An improved dynamic subgrid-scale model

(rij )E=-2var1713ij ，Q2__ _一、+Vb百(Sik叭+Sjk叭‘)

，Q2+Vb二下丁

1L

‘__ I___、，二，厂__ I___、!Coik Coki一百d ii O)lk Okl}+Vc△一}Sik Sk，一二6ii Skl Skl\ j 了 \ j Z

467

(8)

Detailed derivation can be found in footnote 1) and is thus omitted here. In this

study, LES is carried out in the rotating frame. So, the nonlinear SGS model functionalform (6) is used to model the SGS stress.

2.3 Determination of the coefficients in the nonlinear SGS model

According to the dynamic procedure proposed by Germano et al.' 121 to determinethe model coefficients, the test-filtered SGS stre

modeled by a functional form similar to eq. (6):

ss inrotating coordinates叮can be

(叮):一2 V,入，1 -}!* I"2j2 1 S可，2K2，二，二，

+咋- 哆ik叭i 1乙

二*二。、，02+S ik叭i )+咋不二

1乙} Sik ski一喜Sii CON司\ 一 J 一 /

之生*止生* 1。一一口。 3

之生*止生*

Sik Ski Skl Skl)·(Vb一‘)(2'j’一 (9)

/了一

‘、

2二

△ 代 +

where the superscript 'A' denotes the test-filtered variable, and△ is the test filtersize.

Substituting eqs. (6) and (9) into the Leonard stress, L=ui uj二

i.e. 一u;u

叮一弓，we then obtain

Kii=V +V 'Bijb+Icii,V, (10)

where

Kij=

八

_ *_ * 气二ui uj一ul uj Aj=22j2.s * I sib -2A2.* *s Is ii，

/.十

t、

+

Bij=一 (Sik式 + S jk Coki卜一粤成，病 j

矛-12

+鬓{ * *Coik Coki一粤4iOIkWkl I十 1乙\ j j

1) Liu Nansheng, Direct numerical and large eddy simulations on rotating turbulence, Dissertation of Univer-sity of Science and Technology of China, 2003.

www.scichina.com

万方数据

468 Science in China Set. G Physics, Mechanics&Astronomy 2004 V61.47 No.4 463-476

、、!

./一蚤、ASkl SO- * - * - *之生*

Sik Ski I。

一一d“ 3

SO Skl

/了1卫1

、、

2 二△

+

、、十

./

According to Lilly's proposal''', the model coefficients, Va，v;, and v,'，can be

determined in a least-square method'. It is proved that, as I .(2 I�) -, Va一O(1),

Vb一1*O(1/0Z), v,,一。(1). So, the functional form of the nonlinear SGS model ineq. (6) is independent ofQin the limit case U21-4 -; it means that the model satisfies

the principle of AMFI.

3 Numerical method

To perform LES calculation, the factional-step method, developed by Rai andMoin[161 and Kim and Moin[171, was employed to solve the filtered Navier-Stokes equa-tions. Spatial derivatives are discretized by a second order central difference. The con-vective and viscous terms are treated by Adams-Bashforth and Crank-Nicholson

schemes, respectively, and a third order Runge-Kutta scheme is used to advance in timethrough three sub-steps.

By use of the nonlinear SGS model described above, the LES is carried out for the

spanwise rotating turbulent channel flow. Periodic boundary conditions are imposed inthe streamwise and spanwise directions, no-slip and no-penetration conditions are em-ployed on the walls of the channel. The channel flow is driven by a constant pressuregradient in the streamwise direction.

4 Results and discussion

Here, fully developed turbulent flow is through a channel that is assumed to rotatein the clockwise direction with the rotating axis along the spanwise direction at an angu-

lar velocity口(as shown in fig. 1). The Reynolds number is Re,=194 and the rotation

number N=0.1-0.5, which is defined as N=2.(A1U�� U�, is the bulk mean velocity.

The computational domain is 4 7rh X 2 h X 2 7ch in the streamwise, wall-normal and

spanwise directions, respectively, and is resolved by the corresponding grid size 97 X 81

X 65 with uniform grid in the streamwise and spanwise directions and stretched grid in

the wall-normal direction. To achieve a high resolution, five grid points are located in

the wall regions within y+<10.AccordingtothepresentandpreviousoNsdata['8''9},10. According to the present and previous DNS databoth the computational domain size and grid number are sufficient.

Following the terminology"o,211, the lower wall and the upper wall of the rotatingchannel are termed suction wall and pressure wall. In addition, the DNS performed byKristoffersen et al.1191 and the present DNS and LES are represented as DNS-1, DNS-2and NSM, respectively.

1) See footnote 1) on page 467.


万方数据


4.1 Validation of nonlinear SGS

model

24

20 A'曦减.， Figs. 2 and 3 show the profiles of

mean velocity and turbulence intensi-ties in the rotating channel. Here, the

mean velocity(ui) is the ensembleaverage of the resolved velocity砚，

and the corresponding velocity fluctua-

tion is calculated by u=Ui一(ui)·Comparisons are taken among the re-sults of DNS-1, DNS-2 and NSM for N

=0.1, 0.2 and 0.5. As shown in fig. 2,

although the mean velocity of NSM is

slightly smaller than that of DNS, these

=-tz=犷渡盆手16

悦

E3 N= 0, DNS-1 .NO N= 0.1, DNS-1.N》N= 0.2, DNS-1*N叼N=0.5, DNS-1闷N

=0,DNS-2 气乙

︿谓

DNS-2

DNS-2

DNS-2

N= 0, NSM

一一一 N= 0.1, NSM一 “一 .N= 0.2, NSM- ·’- N = 0.5, NSM

们

y

01-1 -05 0.5 1.0

Fig. 2. Streamwise mean velocity profiles.

profiles of }u;) are compatible with each other.

Fig. 3(a) and (b) show the profiles of the streamwise turbulence intensities. Near

the pressure wall,(u rms)predicted by NSMis in good agreement with those of

DNS-1 and DNS-2. Near the suction wall,(u声刀之s)of NSM is slightly larger than theDNS results. Globally, the NSM results agree well with the DNS data. In fig. 3(c) and (d)

the distributions of(1'':>an d< ,rms C}'rms)calculated by the LES and DNS agree wellover the channel.

Moreover, we have compared other turbulent quantities with the DNS results and

can confirm that the present LES coupled with the nonlinear SGS model is able to pre-dict turbulence characteristics of the rotating turbulent flow. Further, turbulence statistics

and turbulence production and dissipation rates near the walls of the rotating channel are

analyzed in the following.

4.2 Skewness and flatness factors

Figs. 4 and 5 show the profiles of the skewness and flatness of the streamwise andwall-normal velocity fluctuations, respectively. The skewness factor S and flatness factorF can be viewed as the measurement of extent to which the probability density functionof turbulence fluctuation deviates from normal distribution. For Gaussian distribution, S

=0 and F=3. Thus, the fact that S(u;')<0 indicates that the velocity fluctuation with

u;'< 0 is dominant in probability; otherwise, the velocity fluctuation with u;'> 0 be-comes dominant. And the flatness factor represents the intermittent character of wall

turbulence. Compared to the non-rotating case, both S(u ') and S(v ') near the suc-

tion wall are strengthened in fig. 4. The fact that S(u’)>0 and S (v ')<0 indicates

that the main contribution to u ' comes from the sweep events, which are related to the

www.scichina.com

万方数据

470 Science in China Ser. G Physics, Mechanics&Astronomy 2004 Vo1.47 No.4 463-476

(a) (b)

.‘

r w- }彭叭

了子 Q 4

一二尹

乌卜︸

-t卜︸

奋卜︸

|卜卜

气乙

帐生n

4

，j

，‘主

降

找 _.二二 .止色

.花二了兮二二爪二 ·.一

搏协偏‘

L

﹁，.l浏..I

-0.5 0ylh

:炙二.·尹

_ _一上一一一一

0.5 1.0

OL0.80 0.85 0.90 0.95

y1h

.

‘‘......口....

‘.1

I

C

-

习一中_尹;

.far叮一’一、·

.d._Ik}.-K赶 D >4-3}1 ii, }_p. s D -'-OL70'

一腼卜卜

(c)

.支.A 、娜

i> 亡卜言冷

哎J

n

rJ

‘

:

，‘

，‘

1

_E 1.0

少‘吸略。.5, a0.5 VO

二:o‘e}-.:,

0

ylh

0.5 -0.5 0

y/h

0.5 1.00l--l

n

1

L

Fig. 3. Distributions of the root-mean-square values of velocity fluctuations. (a) Streamwise component; (b) localdrawing of the streamwise component near the pressure wall; (c) wall-normal component; (d) spanwise component.All legends are the same as fig. 2

— N=0

=0.1

=0.2

=0.5

(a) (b)N=0

N=0.1

N=0.2

N=0.5

N

N

N

小洛写

歇川盯扮

2 一

、知代蓄‘;二孟;，.，.

-2L

一1.0 -0.5 0.5 1.0

一 L

-1.0 -0.5 0.5 1_0八U

y

八们

y

Fig. 4. Profiles of the skewness of velocity fluctuations. (a) Streamwise; (b) wall-normal.

high-speed elongated streaks. Similar behavior near the pressure wall is also found sinceS(V)>0 and S (v‘)>0.Fig. 5 shows that F(u') and F(v') increase near the

suction wall for the rotating case; it is evident that turbulence becomes highly intermit-tent. However, the turbulence intermittent behavior becomes weak near the pressure wall.

It is needed to note that S(v') and F(v') near the suction wall do not vary mono-

tonically with the rotation number. This fact was also revealed for turbulent rotating pipe


万方数据


flow based on DNS results and can be explained as the inclination of near-wall vortical

structures induced by the rotation effect1221.

90 90— N

60

=0

二0.1

=0.2

=05

(a) (b)

60

N

N

N

N=0

N = 0.1

N= 0.2

N = 0.5

饭

1

、飞入︾

蓦30 一L?301﹂刀

o月

0

二

L

p卜””.以

0匕

一1_0 -05 0.5 -0.5 0.5 1.0

Y

(b) wall-normal.

nU

y

Fig. 5. Profiles of the flatness of velocity fluctuations. (a) Streamwise;

4.3 Budgets of resolved Reynolds stresses

The resolved Reynolds stress budgets provide the detailed information of rotationeffect on the dynamical characteristics of turbulence, e.g. production and dissipation rateof turbulent kinetic energy, redistribution of turbulent kinetic energy due to Coriolisforce, and viscous and turbulent diffusion of resolved scale motions near the walls. By

taking ensemble average on eq. (2), the transport equation of resolved Reynolds stressescan be obtained and written as

，、a(ul)，一，一， )一 +(“ “。

‘axk “了

一“

一ui

了.了、

221||||1又

「‘一‘， _、aV au;

(Kez Va+1)-ax丁axk

一从。，{。，:lu ‘\“几\ uil+-'ilk(uk喇‘。+H;i , (11)

，

k

where vQ is the eddy viscosity, P' is the modified pressure fluctuation P’二p' +

味Sli / 3.Eq. (11) is normalized by the velocity scale proposed by Mansour et al.""These terms on the right-hand side of eq. (11) are the production rate (P,,, referred to asPR), turbulent diffusion(几，TD), velocity-pressure correlation (VPG), viscous diffusion

(D,,, VD), dissipation rate (e,,, DS), Coriolis force term (N,,, CO), and H;} being the con-vection term of mean flow and the additional terms due to the nonlinear SGS model.

Here, the VPG term can be further decomposed into the pressure-velocity diffusion (II,,,PV) and the pressure-strain correlation (X,, PS),

www.scichina.com

万方数据

472 Science in China Ser. G Physics, Mechanics&Astronomy 2004 Vo1.47 No.4 463-476

3)

a)，

了‘、

6

、、、、矛产.了产

、、十

少2

a(P'u}) ax ax;

、、、2口.了尹

Fis. 6 shows the mofiles of the budLyet terms in the (u: W11 equation. In fi". 、尸 ‘ ~ 、 “ ，二 ‘口

all the terms in the(u i u 2) equation are suppressed strongly in the region of y+<10(i.e. near the suction wall), and only the PV and PS terms are dominant there. The CO

term with positive value in most part of the rotating channel plays as a source term to

(u 2 u 2) and redistributes turbulent kinetic energy from the horizontal components tothe wall-normal component. In fig. 6(b), it is evident that the redistribution of turbulenceenergy is more activated in the core region of the channel. Fig. 6(a) and (b) show that the

TD term plays an important role to the (u 2 u 2) budget in the core region of thechannel, and the TD term is enhanced in the pressure wall region while suppressed in thesuction wall region due to the system rotation effect.

0.04

(a) Suction side

0.020.4匡一.0.2忆.，一’‘’、’、

Pressure side

k-- ---一 _. ...口 . ...口二口.. . ...口 .

.....-..⋯ ⋯ ‘ 全r ，户J毛二JJ护，产. .. . . .. .. 二共Fi-Is +} -!!

二二耘:月冷二

PSDSTDVDCO

1 ·····⋯⋯PS 1 一 -一少

一。.02卜 — — ID I 一 ··一 yll } 一一一一CO I -·一·一一 PV

-0.2

-0.0410, ‘9，

少

一一二笠」一。 10,

、、

.、一、⋯_.-厂 N= 0.5

一一一一」一一一一一一一一一一一

100 10'

犷

Fig. 6. Profiles of the budget terms in the(W2' W,) equation at N=0.5. (a) Near the suction wall; (b) near thepressure wall. y'=(1一lyl)Re,

Fig. 7 exhibits the profiles of the budget terms in the( u 1 u 1今equation. In fig.7(a), the peak of the PR term is damped near the suction wall and the position of thepeak shifts away from the suction wall of the channel. The most intensive turbulence

production is located at y+二45, where the contributions of all these terms in the

(ui u1今equation are comparable. Compared to the non-rotating channel flow, it is lo-

cated at，+二12 1171 . Fig. 7(b) shows that the PR term is strengthened near the pressure

wall. The peak of the PR term is located at，+二5.2，where the balance of the (IF,' w j)budget is achieved by the interaction of the PR, DS, VD and TD terms. It is evident (fig.

7(a), (b)) that the contributions of the VD and DS terms are dominant to the (w,' u 1今budget. Even though the CO term contributes little to the (u,' u 1今budget, it plays animportant role in redistributing the turbulent kinetic energy near the wall regions. In


万方数据


most region of the rotatingenergy from the stream w tse

channel, the CO term with negative value drains turbulencefluctuation to the wall-normal component.

0.04

(a) Suction side (b) Pressure side

寡犷五内助

0.02

0声二石峨盆.比自曰.‘苗 . ....曰.. . -

— PR

·····⋯ ⋯ PS— ·— DS— — TD

左二

-0.02、·勺

.，.~

‘ 、

、

、

、

} Y ， } J “.，

-1卜 /‘口户

l0]

y+

-0.04下乞a

VD

CO

10

Y.

N一0.5、-----七---

102-2

N= 0.5一曰I一一一一一

100 102

Fig. 7. Profiles

pressure wall.

of the budget terms in the(厅1‘百1) equation at N=0.5. (a) Near the suction wall; (b) near the

4.4 Flow structures

Coherent structures always get much attention in the study of wall turbulence, sincethe turbulence wall structures are responsible for the burst events, turbulence production

and dissipation in turbulent boundary layer. Based on the patterns of near-wall velocity

and vorticity fluctuations, Kim and Moin1171 investigated the coherent structures near thewall, e.g. elongated high- and low-speed streaks, sweep and ejection events. The flowstructures predicted by the numerical simulation [171 are quite similar to those observed inexperiment performed by Runstadlerr241. For the spanwise rotating turbulent channelflow, since turbulence intensity is enhanced near the pressure wall and suppressed near

the suction wall, the flow structures near the walls will change due to the rotation effectcorrespondingly.

Fig. 8 shows the contours of the wall-normal velocity fluctuation u二in the (y+,z+) plane near the pressure and suction walls at N=0.5. Compared to the non-rotatingflow structure (not shown here), the velocity fluctuation u二shown in fig. 8(a) becomes

more intensive, which indicates that the wall-normal turbulence intensity is enhanced

near the pressure wall. Since the contours of u二with positive and negative values are

related to the in-rush and out-rush motions, respectively, it is indicated in fig. 8(a) thatturbulence sweep and ejection events become more active near the pressure wall. In the

region between the positive and negative values of u三，the Helmholtz instability of the

shear layer appears, which is responsible for the generation of the streamwise vorticalstructures in the pressure wall region. According to fig. 8(b), the turbulence energy pro-

duction, the sweep and ejection events are suppressed near the suction wall. The flowpatterns with absent streaky structures are well consistent with the suppression of theproduction of the streamwise vortices.

www.scichina.com

万方数据


认

nU

八钊

八曰

nU

︵11

4

，、

，‘

1

十气

Fig. 8. Contours of the wall-normal velocity fluctuation in the y"-z' plane at N = 0.50.2. (a) Near the pressure wall; (b) near the suction wall.

Fig. 9 depicts the contours of the streamwise velocity fluctuation W l' in the same

plane as shown in fig. 8. The contours of u厂with positive and negative values are ar-

ranged alternately along the spanwise direction in fig. 9(a). Based on figs. 8(a) and 9(a),

it is found that the positive ul' is mostly related to the positive u互，which corre-

sponds to the flow rushing towards the pressure wall, i.e. the turbulence sweep events inthe pressure wall region. Due to the solid wall confinement, the sweep events induce two

flows with opposite sign spanwise velocity in the pressure wall region. This dynamicprocess is responsible for the turbulence energy redistribution from the streamwise com-

ponent to the spanwise, termed splattering or impingement effect of high-speed fluid1251.The process is the mechanism to generate the spanwise velocity fluctuation. As shown infig. 9(a), the enhancement of the splattering effect due to the system rotation is thereason of the increase in the spanwise turbulence intensity in fig. 3(d). The contours infigs. 8(b) and 9(b) show that the system rotation suppresses the splattering effect, hencedecreases the generation of the spanwise velocity fluctuation near the suction wall. Thus,the spanwise turbulence intensity is suppressed in the suction wall region of the rotatingchannel.

5 Concluding remarks

In this study, a nonlinear SGS model, which is consistent with the principle ofAMFI, is proposed, and the model coefficients are determined dynamically based on the


万方数据

An improved dynamicsubgrid-scale model 475

认二二飞

.

.

.

.

.…

.

.…

…

气..U”训引引.J鱿彩卜

二:.=口二讨，.I.

甲

.、

，

卜

1

‘卜f:

1.....1

‘..一J

…

引如卜﹄...加目.曰.I曰

.f…

皿比.-. 护

从介拙

‘…

.

.-.--:.品片曰

，..UU:.1

.’

..-.··..--.…卜鱿汁时盯

1，1一J月口俐“臼叮.朋姗汤口..…

..:l加臼阶公毯口曰口脸‘琶日

1.筑

邵

.

:S

断川屠汤川泽续价思洲湘谈

.--…

弓比性“卜陀比日卜“留，侧口叫目训巴巨巴

溉通脾︺即朴

注血﹃朴扛二1:朴二二….

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.…

…

t叫

二.…馆了热二二:ll︸︸︸︸下，

卜日

n钊

﹄、~

汤得

八曰

八曰

内j

Z

+气

‘…

10 二

4a)0

.⋯ ⋯ 侧.气..

300

"1.11、呼

500

z"

吃.竺三石

600

护⋯

““1.介介材

100 200 800 900 1000

法哪腿胭妇怠︾.﹀

一

--.-，-，

七二

、._一/⋯ ，.

.

.

、，..⋯、.声八“

n们

4

伟j

+气

.

.

.

.

.

.

.

.…

…

--·---·---二.、

20

、--..-.…，’..

)0 户..，，、

0

犷(b)、‘..⋯

100 200 300 400 500 Z

生600

Fig叹).5

.9. Contours of the streamwise velocity fluctuation in the y'-z plane at N=0.5 with the increment of contour

.(a) Near the pressure wall(b) near the suction wall.

resolved scale motions. The results calculated by using this new dynamic SGS model is

proved to be in good agreement with DNS data. Then, turbulence characteristics of thespanwise rotating turbulent channel flow are investigated. The skewness and flatnessfactors indicate that the turbulence becomes more intermittent near the suction wall.

According to the budgets of resolved Reynolds stresses, turbulence production, dissipa-tion and diffusion are enhanced in the pressure wall region. The budget terms in the re-

solved Reynolds stresses are suppressed evidently in the suction wall region. The flowstructures near the pressure wall show that the turbulence sweep and ejection events be-come more active and the generation of the streamwise vortices is agitated due to the

enhancement of splattering effect induced by the rotation effect.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos.10302028 and 10125210), the China NKBRSF Project (Grant No. 2001CB409600), the Hundred Talents Pro-

gramme of the Chinese Academy of Sciences, and Specialized Research Fund for the Doctoral Program of Higher

Education (Grant No. 20020358013).

Speziale, C. G, Younis, B. A., Rubinstein, R. et al., On consistency conditions for rotating turbulent flows,

Phys Fluids, 1998, 22: 2108-2110.

Greenspan, H. P, The Theory of Rotating Fluids, Cambridge: Cambridge University Press, 1968Bradshaw,P，The analogy between streamline curvature and buoyancy in turbulent shear flow, J. Fluid Mech.,

1969,36:177-191.

Fu, S，Wang, C., Second-moment closure modeling of turbulence in a non-inertial frame, Fluid Dynamics

么

王

或

www.scichina.com

万方数据

476 Science in China Set. G Physics, Mechanics&Astronomy 2004 Vo1.47 No.4 463-476

Research, 1997, 20: 43-65.

5. Rubinstein, R，Zhou, Y., The dissipation rate transport equation and subgrid-scale models in rotating turbu-

lence, NASA/CR-97-206250, ICASE Report No.97-63, 1997.

6. Speziale, C. G., Some interesting properties of two-dimensional turbulence, Phys. Fluids, 1981, 24: 1425- 1427.

7. Speziale, C. G., Closure models for rotating two-dimensional turbulence, Geophys. Astrophys. Fluid Dynam- ics, 1983, 23: 69-84.

8. Speziale, C. G., Subgrid scale stress models for the large-eddy simulation of rotating turbulent flows, Geo-

phys. Astrophys. Fluid Dynamics, 1985, 33: 199-222. 9. Shimomura,丫，A family of dynamic subgrid-scale models consistent with asymptotic material frame indif-

ference, J. Phys. Soc. Japan, 1999, 68: 2483-2486.

10. Kobayashi, H., Shimomura, Y., The performance of dynamic subgrid-scale models in the large eddy simula-

tion of rotating homogeneous turbulence, Phys. Fluids, 2001, 13: 2350-2360.

11. Smagorinsky, J., General Circulation experiments with the primitive equations. I. The basic experiment, Mon. Weather Rev., 1963, 91: 99-112.

12. Germano, M., Piomelli, U., Moin, P. et al., A dynamic subgrid-scale eddy viscosity model, Phys. Fluids, 1991, A3: 1760- 1765.

13. Zang, Y, Street, R. L., Koseff, J. R., A dynamic mixed subgrid-scale model and its application to turbulent re- circulation flows, Phys. Fluids, 1993, A5: 3186-3196.

14. Kosovic, B., Subgrid-scale modelling for the large-eddy simulation of high-Reynolds-number boundary lay- ers, J. Fluid Mech., 1997, 336: 151-182

15. Lilly, D. K., A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids, 1992, A4: 633-635.

16. Rai, M. M., Moin, P., Direct simulations of turbulent flow using finite-difference schemes, J. Comput. Phys.,

1991, 96: 15-53.

17. Kim, J., Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J.

Comput. Phys., 1985, 59: 308-323.

18. Kim, J., Moin, P., Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 1987, 177: 133-166

19. Kristoffersen, R., Andersson, H. L, Direct simulations of low-Reynolds-number turbulent flow in a rotating

channel, J. Fluid Mech., 1993, 256: 163-197.

20. Johnston, J. P., Haleen, R. M., Lezius, D. K., Effects of spanwise rotation on the structures of

two-dimensional fully developed turbulent channel flow, J. Fluid Mech., 1972, 26: 533-557.21. Johnston, J. P., The suppression of shear layer turbulence in rotating systems, ASME J. Fluids Engng., 1973,

195:229-236.

22. Orlandi, P, Fatica, M., Direct simulations of turbulent flow in a pipe rotating about its axis, J. Fluid Mech., 1997,343:43-72.

23. Mansour, N. N., Kim, J., Moin, P., Reynolds-stress and dissipation-rate budgets in a turbulent channel flow, J. Fluid Mech，1988, 194: 15-44.

24. Runstadler, P W., Kline, S. J., Reynolds, W. C., An investigation of the flow structure of the turbulent bound-

ary layer, Dept. Mech. Engng., Stanford Univ, Rept MD-8, 1963.25. Moin, P, Kim, J., Numerical investigation of turbulent channel flow, J. Fluid Mech., 1982,118: 341-377


万方数据

its application to large eddy simulation of -...

Documents