I~

i

I

~) \ N/ ) ~ 20 0 2 \0\ L-\1S

NASA/T 11- 2002- 211750

Elliptic Relaxation of a Tensor Representation for the Redistribution Terms in a Reynolds Stress Turbulence Model

J. R. Carlson and T. B . Gatski Langley R esearch Center , Ha mpton , Virginia

July 2002

https://ntrs.nasa.gov/search.jsp?R=20020061791 2020-07-23T18:08:29+00:00Z

The ASA STI Program Office .. . in Profile

Since its founding , NASA has been dedicated to the advancement of a ronautics and space science. The NASA Scientific and Technical Information (STI) Program Office plays a key part in helping NASA maintain this important role.

The ASA STI Program Office is operated by Langley Res arch Center, the lead center for

ASA's scientific and technical information. The ASA STr Program Office provides access to the ASA STI Database, t he largest collection of aeronautical and space science STI in the world . The Program Office is also ASA's in titutional mechanism for disseminating the results of its research and development activit ie . These results are published by AS in the NASA STI Report Series, which includes the following report types:

• TECH ICAL PUBLICATIO . Reports of completed research or a major significant phase of research that present the results of ASA program and include extensive data or theoretical analysis . Includes com pilations of significant scientific and technical data and information deemed to b of continuing reference value. ASA counterpart and peer-reviewed formal professional papers, but having less stringent limitations on man uscri pt length and extent of graphic presentations.

• TECHNICAL MEMORA D M. Scientific and techn ical findings that are preliminary or of specialized intere t, e.g ., [uick release report , working papers, and bibliographies t hat contain minimal annotation . Does not contain ext nsive analysi .

• CO TR CTOR REPORT. Scientific and technical finding by NASA-sponsored contractors and grantees.

• CONFERENCE PUBLICATION. Collected papers from cientific and techn ical conference, symposia, seminars, or other meetings sponsored or co- ponsored by NA A.

• SPECIAL P BLICATIO . cientific, technical, or historical info rm ation from NASA programs, projects, and missions, often concerned with su bjects having substantial public interest .

• TECH rCAL TRANSLATIO . English­language translations of foreign scientific and technical material pertinent to

ASA's mission.

Specialized service that complement the STI Program Office's diverse offerings include creating custom thesauri, building customized databases organizing and publishing research results ... even providing vid eos.

For more information about the NASA STr Program Office, see t he following:

• Access the ASA STr Program Home Page at http://www. sti.nasa.gov

• E-mail your question via the Internet to help@sti.n asa.gov

• Fax your question to the ASA STI Help Desk at (301) 621- 0134

• Phone the NASA STI Help Desk at (301) 621- 0390

• Write to: ASA STI Hel p Desk ASA Center for AeroSpace Information

7121 Standard Drive Hanover, MD 21076- 1320

NASA/T 11- 2002- 211750

Elliptic Relaxation of a Tensor Representation for the Redistribution Terms in a Reynolds Stress Turbulence Model

J. R. Carlon and T. B. Gatski Langley R e earch Center. Hampton, lfirginia

National Aeronautics and Space Administration

Langley Research Center Hampton , Virginia 23681-2199

r July 2002

_J

Available from:

SA Center for AeroSpace Information (CASI) 7121 Standard Drive Hanover, MD 21076- 1320 (301) 621- 0390

ational Technical Information ervice ( TIS) 52 5 Port Royal Road Springfield , VA 22151- 2171 (703) 605- 6000

I I

l i I

i (

Elliptic R elaxation of a Tensor Representation for t he Redistribution Terms in a

R eynolds Stress Turbulence Model

J .R. CA RLSON A D T.B.GATSAJ Comp1tiational A10deLing fj Simulation Branch

NASA Langley R esearch Center) Hampton) VA 236 1) USA

Abstract

A formulation to include the effects of wall proximity in a second-moment closure model that utilizes a tensor representation for the red ist ri bution terms in t he Reynolds stres eq uations is presented. The wall-proximity effects are modeled through an elliptic relaxation process of the tensor expansion coef­fi cients that properly acco unts for both correlation lengt h and t ime scales as the wall is approached. Direct numerical simulation data and Reynolds stress solu t ion u ing a fu ll differential approach are compared to the tensor repre­sentation approach for the case of fully developed channel flow.

1. INTRODUCTION

The theoretical development of higher order clo ure models, such as Reynolds stre s models , have primarily been formulated based on high Reynolds number as­sumptions. The influenc of olid boundaries on the e closure mod Is has usually been accounted for through either a wall fun ction approach or a modification to the high Reynolds number form of the pressure-r lated correlations and tensor dissipation rate and predicated on the near-wall asymptotic behavior of the various velocity second moments (So et al. 1991 , Hanjalic 1994).

A broader ba ed attempt to account for the proximity of a. solid boundary is the elliptic relaxation approach introduced over a decad ago (Durbin 1991) and further develop d for second-moment closures (Durbin 1993a, Wizman et al. 1996·

fan ceau and Hanj ali c 2000 , Manceau , Carlson and Gatski 2001 ). In it two-equa­tion form th v 2

- f model ha been applied to a variety of flows (e .g. , Durbin 1993b, 1995; Pettersson Reif et al. 1999). The new approach outlined here introduces a ten­sor representation for the combined ffects of a near-wall velocity-pressure gradient correlat ion and ani otropic dissipation rate that asymptotes to a high Reynold num­ber form away from olid boundaries through an elliptic equation for th polynomial expansion coefficients . The development of a generalized methodology for determin­ing the polynomial expansion coeffici ents of representations for the turbulent stress anisotropies by (Gat ki and Jongen 2000) is extend d to an elliptic relaxation proce­dure for these expansion coeffi cients.

Although the material presented her introduces tensor representations and a ten­sor projection methodology into the ellipt ic relaxation formulation , thi work can al 0

be viewed as an intermediat step between a fully explicit ellipt ic relaxation algebraic Reynolds stress formulation and the full diff rential ellipt ic relaxation Reynolds str s formulation.

1

The predictive capabi li ties of the new model ar assessed through comparisons with direct num rical simulation channel flow data (Moser et al. 1999). The e com­parisons include both mean and turbulent flow quantities .

2 . Theoretical Background and Development

In thi section, a mathematical framework is developed for the Reynold stress transport equations and the corresponding lliptie relaxation equation wh n a tensor representation of the redistribution terms is u ed in the formulation. The m thod­ology introduces a set of llipt ic relaxation equations for the polynomial expansion coefficient of the chosen r presentation. The T - f mod 1 uses th r distributive terms in the ellip tic equation , while th T - n model use the expan ion coeffici nts in the ellipt ic equations. Both models use the Reynolds stres transport equations.

2.1 Transport Equations

The transport of th Reynolds stresses Tij (= -tli Uj ) is governed by the equation

(1)

where Ui i. the mean v locity, <Pij is the pressure redistribution term, C-ij is the tensor dissipation rate, and Dr; and Dij are the turbulent transport and viscous diffusion resp ctively. In the d velopment outlined here, it is best to have <Pij given by

(2)

so that the trace of the pressure redistribution t I'm is zero . In the appli cation of the ellipt ic relaxation method, it is also necessary to account for the ffect of th dissi­pation rate anisotropy as th wall is approached. Thi accounting for the di ipation rate anisotropy is accomplished (e.g., Manceau 2000) by a relaxation of the dissipa­tion rate anisotropy to its wall valu , which i a umed to be equal to the Reynolds stre ani otropy. This assumption allow the Reynolds stress tran port equation in (1) to b written as

(3)

wh r (4)

with the Reynolds str ss anisotropy bij and di. ipation rate anisotropy elij defined as

el . . = E:ij _ 6ij ~J 2 E: 3 . (5)

The original scaling of the relaxation function I ij was solely through the turbulent kinetic energy X ; how ver , Manceau, Carlson and Gatski (2001 ) have recently shown

2

that an added dissipation rate factor, c, to the scaling (c f{ lij) eliminates an unwanted amplification effect inherent in the original scaling.

Equation (3) is closed when the model for the turbulent transport 'Dl is used. In previous elliptic relaxation studies that used the Reynolds stre s transport equations, the viscous diffu ion and turbulent transport terms were modeled as

with <7J\ = 1.0 and CJ.! = 0.15. Th composite time scale

[ ( //)1/2] 7 c = ll1ax 7, GTX ~ ,

J{ 7=-,

c

(6)

(7)

where CT ! { = 6 determines the switch to th Kolmogorov time scale (// / c )1/2 0 that the turbulent t ime scale will not vanish as the solid boundary is approached. Away from the boundary, the composite time scal asymptotes to the inertial scale Il.'-;c.

In the two-dimensional flow considered here, solutions were obtained for th 7 11

and 722 normal Reynolds stresses and the 712 shear stress . A transport equation for the turbulent kin t ic energy was obtained from one-half the trace of Eq. (3) and was solved for in lieu of the third normal stress 7 33)

DI..,." P a (c tlk Of() >7 2} ' - .-= - c +- J.! - 7 c - - +//v l.. , Dt OXI <7[( OXk

(8)

where P = 7ikOUi/OXk. The modeled transport equation for the turbulent dissipation rate c needed for closure is given by

(9)

where <7e = 1.3, Cd = 1.44, Ce2 = 1. 3, with

(1 0)

Note that thi form of the dissipation rate equation (Durbin 1991) has introduced the composit time scale into both th production and destruction of dissipation terms.

2.2 Elliptic Relaxation M ethodology : 1 - f Model

The rescaled elliptic relaxation equation is driven by the high Reynolds number form of the pressure-strain rate correlation II and a contribut ion from the Reynolds stress ani otropy 2ccbij (away from the wall th di ssipation rate is assumed to be isotropic dij = 0) . This combination re ults in an elliptic relaxation equation for i ij given by (d. Manceau and Hanjali ' 2000)

(1 - L2\72) i ij = c ~\.- (IIf.i + 2Ecbij ) == ji~ (11 )

3

where

( 12)

and the relaxation cales are defin d as

(13)

with CL = 0.16 and CLf{ = O. Previou implem ntation of th ellipti c relaxation procedur . (Man eau and Hanjalic 2000) using the Speziale, Sarkar and Gatski (S G) pressur strain rate mod 1 (Spezial et al. 1991 ) used th full nonlinear form. Th linear form of the G model impl m nted here i given by

(14)

with Cf = 1. , ct = 3.4, C2 = 0.37, C3 = 1.25, and C4 = 0.4. Note that inc th linear form of th pre sur -strain rate mod I is used here, th valu for CL differ. from that u d previously (CL = 0.2, see Man au and Hanj alic 2000) for the form of the ellipt ic relaxation quation given in (11) .

Boundary conditions are need d for t he I ij and are determin d in t he vicinity of the wall , by the balance of the redi tributive t rm by the viscou diffusion of th Reynold tres es resulting in Table 1. Only the 22- and 12-compon nt of f have determinate solutions to the near-wall balance of th stress t ranspor t equations. For th remaining compon nts III = h3 = - 122/2 ar used as boundary conditions to en­sure that I ij i traceless (Manceau, arlson and Gatski 2001 ). Symmetry condition were appli d at the cent rline.

In the current work , one of t he goals i to develop a methodology for incorporating at nsor repr ntation for the relaxed r distribution function ! ij . Onc d v loped and validated t his arne proc dur can 1 u d in conjunction with ten or r pr ntations for the Reynold stres ani otropies as well. Such a combination would then yield an llipti c relaxation expli cit algebraic stress mod 1. The detail of the r pr entation

for the R ynolds stress ani otropy will not b addressed in th current work, but deserves furth r work . As will be di scussed in Sec. 2.3 , such a representation would b consistent with a linear pressure- train rate model.

2.3 Representations and Elliptic Relaxation: T - n Model

lthough the elliptic r laxation formulation has aIr ady been appli d within a full diE rential Reynolds str s model, a question aris s about what role tensor r pr en­tations can play within th framework of the ellipt ic r laxation procedur .

4

,~

I

Table 1. Boundary Condit ions for the i ij Tensor

Component Wall Centerline

ill 1

- 2" i 22,w Symm t ry

i 22 -201/2T221 1

Symmetry 62 y4 w· 1

/33 1

- 2"i22,tv Symmetry

i 12 -201/2T1 211

0 62 y4 tv 1

The different ial elliptic r laxation equation for i ij can be obtained from the inte­gral expression (e .g. Manceau and Hanj ali t 2000 )

i ij (x ) = 6 (X) ~{ (X ) {<pij (X) - 2c(x ) [dij(x ) - bij (x )]} = k d3x' [;~~~~~}) l Gn(X, x'),

(15 ) where

( ' ) ( ) 2 ap ( ') () 2 ap ( ' ) 2 r () ,,2 ap ( ' ) Fij X, X = - Ui X \7 ~ x - Uj x \7 ~ x + -:-OijUk x v ~ X UXj UXi 3 UXk

v

velocity -pre s ure gradient cor rela tion

+~ { 21/ ( aUi (X)\72aUj (X' ) + aUj (x )\72 aUi (Xl )) 2 aXI aXI aXI aXI

, v J

tensor d issipa t ion ra te

- ~~~) JUi(X)\72Uj(X/ ) ~ Uj( X)\72Ui(X/ ) ]> } ( 16)

R eynolds st ress tensor

and Gn(X, X' ) is approximated by the free-space Gre ns funct ion Gn(x , x /) = (47rr t 1 with r = Ilx' - xii . The 6ij contributions to both the di ipation rate and Reynolds stress anisotropies cancel so that the only remaining contributions are the tensor dissipation rate and Reynold stress tensor. The tensor funct ion i ij and Fij can be r presented by polynomial expan ions of basis tensors just as the associated Reynolds stress anisotropy tensor bij has b en. For such a basis given by Ti~m\X) (m = 1, .. . , ), t he following repre ent a.tions ar assumed:

i ij(x ) = L l( x )Tij )(x ) (17) 1= 1

5

I

L

' ) Fij (X X' ) _ L ( ') (n)( ) F~· ( X X = - "V X X T· · X tJ ' ( ) ]. - ( ) In, tJ . c X 'l X n=l

(1 )

A tensor scalar product (denoted by [ : ]) b tween each basis t n or TSm)(x ) and the I' pres ntations given in Eqs. (17) and (1 ) can b formed, and this lead (using matrix notation for conv ni nce) to

N L I(X) [T (I)(x ):T(m)(x )] = in d3x' [F*(x , x'):T(m)(x )] Gn(x , x') 1= 1

Sinc the funct ional dependency of the indicat d calar product depends solly on x , Eq. (19) can be reWl'itt n a

(20)

The modeling of the scalar funct ion In (X x') follows that establi hed previously for the lliptic relaxation approach, that is

In(X, X' ) = I n(X' x' ) xp ( - {n) . (21)

wh re, in gen ral, the In coefficient can have an associated length cale uniquely defined by the form giv n in Eq. (13) .

With thi mod 1, Eq . (20) can be rewritten as

( )=1 d3 I (' ,)exp(-r/Ln ) n X X n X , x .

n 4nr (22)

This equation lead directly to the differential counterpart

(23)

wher ~ (x) are the xpansion coeffici nts from the t n or repr ntation of a qua i­homog neous form of f . ince the di i1 ation rate is assumed to b i otropic, f i compo d of the qua i-homogeneous form of th pressure-strain rate orr lation and a contribution due to the Reynolds stre s ani otropy. The r ultant expr ion for ~ (x) is given by

N 1 L h(x ) [T (n)(x ):T(m)(x )] = _ [(rrh(x ) + 2ccb(x )) :T(m)(x )] n E(x )I'l (x )

n = l

[rr~ (x ) :T (m)(x ) ]

c(x )I'l-(x )

6

--- ---------

1TI = 1, .. . " (24)

wher the quasi-homogeneous form of the pressure- train rate model II~ i given by

II~ = - £c ( Cf - 2 + ct ;) b + k C2 S + k 3 ( bS + Sb - ~[b:S] I)

-I{C4 (bW - Wb). (25)

ote that a comparison of Eqs. (14) and (25) shows that the return-to-isotropy term proportional to b has been modified. The factor Cc now influences the ntir term and the contribution from the Reynold stress anisotropy 2ccb to th r laxation function f i now included in this (slow) term contribu tion to II~ .

One of the improvement in the current lliptic relaxation formulation is that the ca.led relaxation function f d fined in Eq. (4) is 0 (1) in the log-layer region. This scaling negate the adv rse influence of th llipti c operator in the log-layer that occurred in the original (Durbin 1993a) formulation. In ord r to retain this benign ffect in the tensor repre entation formulation us d h re, it is necessary to ensure that

the expansion coeffici nt n also have this n utral effect. Previous representations for the Reynold str ani otropy t nsor hav used ba is

ten ors of the form S , SW - WS and S2 - [S:S]I /3. In th log-lay r, wher the velocity gradient has a y-l behavior, this choice of basis tensor would r quire that th corresponding expan ion coefficients 1, 2 , and 3 have a y, y2, and y2 b havior, resp ctively, in that region to ensure that f b haves as 0 (1). Unfortunately, given that behavior of the n , th amplification effect would now ffect th n and the sought-after O( 1) behavior for the f is 10 t. For th fully dev loped channel flow of interest, this problem can be asily circumv nt d by using a normaliz d ba is set of the form

T (l) = S* T (2) = S*W " - W *S" , , (26)

wh re S '" = S/{S2}1/2 and W * = W /{S2P/2 . Thi normalization now makes the b havior of both th expan ion coefficients and basis t nsors 0 (1) in th log-layer, which then preclude any adv r effect of the lliptic operator in the relaxation quation (23) .

Boundary conditions for the n expansion coefficients are r quired. Con istent with th boundary conditions for the tensor fun bon i ij, the COlT sponding n bound­ary conditions are Ii ted in Table 2 a functions of Tij (se Appendix A for details).

The equival nce of the elliptic relaxation of the xpansion coefficients n given by Eq. (23) with the ellipti c relaxation of th function i ij given by Eq. (11) can be readily shown with the curr nt normalized ba. is . Th solution to Eq. (24) i easily obtained as

(27)

7

I ~

Table 2. Boundary Conditions for /3n

/3n Wall Centerline

/31 - 20.)21/2 Td(1) [llc:T (1)] h

E. 2 y4 E. f{ w (1)

/32 -151/2

T22I(1) [llc:T (2)] h

2 4 E. f{ E.wY(l)

/33 -301/2

T22 1(1) [llc:T (3)] h

2 4 E. f{ E.wY(l)

If the tensor r presentation Eq. (17) is applied to f i} then the /3n solution from Eq. (27) would yield for the components of .m

(fh f h fh fh) ( /3f h + /3; /3h + /3; _ /3; ) 12 , 11' . 22 , 33 = .)2' - 2 6' 2 6' 3

(2 )

A comparison of the right-hand side of Eq. (28) with t he right-hand side of Eq. (11) shows that the two are equivalent. (The reader hould recall from the discussion following the definition of ll~ in Eq. (25) that the form of the slow term was slightly modified from the definition given in Eq. (14). vVith this change taken into account , the exact equival nce Eqs. (11) and (2 ) holds.)

3. R esults and Discussion

All flow calculations were carried out on fully developed turbulent channel flows. The equations that were solved were scaled in wall units with friction Reynolds number Re-r based on channel half-height and friction velocity at the wall. A one­dimensional finite-difference algorithm de cribed in Appendix B was us d for all com­putation .

As shown in Sec. 2.3 , the representation methodology that has been developed rields an ellipt ic l' laxation formulation qui valent to the ellipt ic relaxation of the ten­

sor fun tion f ij . Whil such ten or projection methods have been used in conjunction with nonlinear algebraic equations, the application here also validates its u e with differential operators.

Figures 1 - 3 show the predi ctive accuracy and equivalence of both the Tij -

f ij and Tij - /3n approaches . The flow fi ld is the fully develop d channel flow at R e-r = 590 (Mos l' et al. 1999). The figure include both a linear and log cale in th wall normal direction. As can be seen from Fig. 1 for th mean velocity both the distribution acros, the channel and the near-wall a ymptotic b havior agre with

- .--- - -- ---------

the direct numerical simulation (DNS) data. Excellent agreement with the D S data

25

20

15

::::>

10

5

------

'tij-t ij

t{(~n DNS

10'

Y

10'

(a) ](/ (b)

](1 J(Y 1 0-' '-7-'-'--'.J...U..<"-;;--,-,-.w..u.uL.,-~....w.......w..,:-,-'--W..u.uJ

10-' 10° 10' j(j J(Y y

Figure 1. Mean velocity di tribution acro s channel at R eT = 590: (a) log-linear scale· (b) log-log scale.

across the channel is also shown for the shear tress profile (Fig. 2); however , the asymptotic approach to the wall is greater t han the th oretical estimate of O(y3). The di screpancy becomes apparent for values of y < 1. This result is in contrast to the predictions for the turbulent kinetic energy shown in Fig. 3. In this case

1 1if

't ij-!;j 10·' 0.8 ------ 'ti,.-P"

DNS 10.2

0.6 .. ~!:: 10-3 ~-

0.4 ( a) 10-1 (b)

0.2 10-5

10.6

0 100 200 300 400 500 600 10" l if 10' 1d l d Y Y

Figure 2. Turbulent shear stress distribution across chann 1 at R eT = 590: (a) linear scale; (b) log-log scale.

the near-wall asymptotic behavior is consistent with the D S results but the overall values ar slightly lower across the channel than the D S data. Overall , the predictive results for the mean velocity, Reynolds shear stress and turbulent kinetic en rgy are quite exceptional and show that the method can be cali brated to provide excellent predictions of thi, fiow field . In actuality, since the models are formally equivalent, no changes are required in any of the calibration constants.

Since a full diff r ntial R ynolds str model i used for the turbulent velocity field , it i possibl as well as insightful to furt h r examine the component stress predictions. F igures 4 and 5 show the T11 and T 22 component stresses . Since the

9

5

2

1

--- Vlij ti(~n DNS

( a)

100 200 300 400 500 600 Y

(b)

1 0.3 '-;--'--'-'--'-U.li'-;;-''-'--'..L.U..U-'-;-L....L..>...Lll.L~'---'-'~ 1 0·' j(1 10' J(i 1 d

y

Figur 3. Turbul nt kineti n rgy di tribution across channel at R eT linear cal ; (b) log-log cale.

590: (a)

near-wall a ymptotic b havior O(y2) is dominated by the Tn (and T33) omponents, it is not urprising to e from Fig. 4b that th near-wall asymptotics closely match th D IS re ult . The O(y4) behavior that charact ri zes th D IS results for the T2 2

component (s Fig. 5b) are very closely replicat d by the predictions. Fig. 4a shows that acros th channel pr dieted results were lower than th D results for the Tn

component . For the T22 componen , however, the predicted p ak valu was higher than the D S results, but th predicted values wer lower over the remainder of the channel a n in Fig. 5a.

3

2

1

--- Vlij t~tPn DNS

(a)

00 100 200 300 400 500 600

Y

(b)

10.3 '-;--'--'-'-..L.U..U'-;;-''-'--'..L.U..U-'-;-,--,-,..u..u.~"'--'-~ 1~ 1d ld 1d ld

Y

Figure 4. Reynold normal stres omponent T1l di tribution a ross chann 1 at R eT = 90: (a) linear scale; (1 ) log-log scal .

An int r sting assessment of how well th llipti c relaxation formulation models the redistribution t rms across th channel an b obtained from Eq. (4) . Th quantity c:K!ij obtained from th expli cit repr ntation given in Eq. (1 ) and th ellipti cally r lax d n from Eq. (23) ar plo t d in Fig. 6 along with th quantity <Pij -2c: (dij -bij ) obtain d from the DNS data. As Fig. 6a how , the d\.J12 compon nt produce th corr sponding DNS l' ult very well in the near-wall r gion and in the

10

-~ ~ ~------

1.4 ]0'

1.2 Vfij 10°

------ V~n 10" . 1

DNS 10.1

0.8 p'

0 ~: 10.3 0

0.6 0

10-1 (b) 0.4 ~ ]0.5

0.2 10.6

00 100 200 300 400 500 600 10.7 0

10" J(I 10' 1d 1d Y Y

Figur 5. Reynolds normal stress component T22 distribution across channel at ReT = 590: (a) linear scale; (b) log-log scale.

outer layer r gion toward the centerline. Between these two regions, the peak value of the computations greatly exceeds that of the DNS. The normal c; f{ ill and c; ]{ h2 components show an even poorer prediction of the DNS results. In the cases, only the outer layer region is correctly predicted; whereas, over the rest of the channel the qualitative and quantitative predictions are generally poor. 'While the results of this a pTio1'i validation of the ellip tically relaxed function c; f{ f ij are disappointing, it is clear that the actual predictions of the fully modeled s t of equations are generally v ry good. Thus other modeled terms in the formulation are able to account for any discrepancies in the prediction of the redistribution term.

As Fig. 6 shows, all components of the elliptically relaxed r distribution term correctly reproduce the D S data in th outer layer of th channel flow but differ extensively from th Dl S data when reproducing the inner layer. Since the ellipt ic operator term (- L2\,12) is responsible for the deviation of th {3n from their quasi­homogeneou {3~ forms, it is worthwhile to quantify the size of th region acros the channel that is affected by this term. Figure 7 shows the distribution of -L2\,12{3n acro s the channel for the three expansion coefficients (n = 1, 2,3) at three different values of ReT ' In the inner layer, the wall uni t scaling basically collap es the results for all values of ReT, with the exception of the {31 component where the r suIt in the near-wall region show some dependence on R T' this sensitivity to ReT is not found in the other components as Figs. 7b and 7c show. The effect of the ellip tic operator falls to zero at y (wall unit) values around 102

. The overshoot in the outer layer shown in all the figures is attributed to the asymptotic behavior of the nergy dissipation rate c; in this r gion . Both \,12 nand c; decrease (L incr as s); howev r , the dis ipation rate c; decr a es faster (L increases faster) than the orresponding decrease in \,1 2{3n . The variation with ReT in thi , r gion is not surprising sinc the wall uni t scaling is not the proper scaling for this region.

11

0.3

0.25 ''ti{PII ( a) DNS

0.2

0.15

0.1

0.05

go·' 10° 10' ](y Y

0.1

0.05 't«(~11 (b) DNS

0 0

-0.05

-0.1

-0.15

-0.2 , lO" 10° 10' 1d 1rY

Y

0.1

"ti/PII (c)

DNS

0.05

o 0

100 1d

Figure 6. Compari on of predict d redistribution tenn components with D at R eT = 590: (a) 12-com.ponent , (b) ll-component, (c) 22-component. omponents Tij - n r suIts are Eh .. -f ij, and D S re ults are <pij -2c (dij -bij ) .

12

- -- ----_ .. -~----~---

data For all

1.5 Ret=395

(a) ------ Ret=590

1 - -- _ ._._-- Ret =2000

~ ., '1/\

d '1/ \ ~r>

0.5 ~.'

~/

f'~ // :: .:: .......

f · . \ I.

0 / \

-0.10

., 10° 10' 1d ](1 104

Y

Ret =395 (b) 1

------ Re,=590 ----------- Re,=2000

o · ~r> 0.5 f'~

0

-0.10

., 10° 10' 1d ](1 1 0~

Y

1

Re, =395 (c) ------ Re

t=590

0.5 - ---- ----- Rer =2000

.., c::l. ~r>

0 \ i f'~ \ . . , \ . . , \ .

-0.5

y

Figur 7. Effect of ellipt ic operator across channel at di:ili r nt R T . (a) ,aI-coefficient , (b) TCO fficient , (c) 3-coefficient.

13

4. Summary

_ methodology has be n develop d that introduce a polynomial r pI' sentat.ion for the ten or redi tribution function i ii ' An elliptic relaxation equation, analogou to th i ii relaxation equation is formulated for th polynomial expan ion coefficient /3n ­The new prediction m thod is demonstrated on a fully dev loped cha1m 1 flow probl m and gives similar r ults to the previous lliptic relaxation method for iij. formal equivalence j stabli h d between the elliptic r laxation of the ten or function i ij and its tensor repre ntation. lthough the prediction of the m an velocity and turbulent stresse are gen rally accurate over th chann 1, an a prio1'i a essm nt shows that the current formulation doe. not model th r distribution welL uch results are nlightening but ar not uncom_monj th r suIts reflect th fact that in model d

do ur schem , a combination of mod led terms combine to yield predictions of quantities such as the mean velocity and Reynolds str sses.

\iVhile th th or-etical approach d v loped her does not r suIt in a reduction in computational cost , it does introduce a new m thodology that is requisite for dev loping elliptic I' laxation xplicit alg braic stre s models. The next step in the d velopment of such mod Is will be to introduce repre enta ions for the Reynolds tress anisotropi s and analyze the efl cts of modeling the turbul nt transport and

viscous diffu ion term consi tent with the approximation made in the formulation of algebraic tress mod Is_

Acknowledgment

The authors ar ind bted to Dr. Remi Manceau for veral us ful discu IOns during the cour e of this work_

14

Appendix A n Boundary Conditions

The expressions for the /3n boundary conditions are derived from th basis t nsors TSn

) used in the representation of j ij

122 = 2TJ;) + /33 = 3 Tii ) 112 = /31T1(~)

(AI)

Table 1 gives the corresponding boundary condit ions for these I ij components . The boundary condition for /31 is directly proportional to the !I2 boundary condition and is given by

/3 - !I2 ,w - J2I _ -20V2112

TI2 l ,w - (1) - ~ 12 ,w - ? 4

T12 c;Y(l) (A2)

The coeffici nt /33 appears in all three expansions of th diagonal terms of I ij . If I ij is traceless, a unique expression for /33 at th wall will be obtained. From he representation for /33, /33,w can be immediately wri tt n as

(A3)

The representations for ill and 122 can be used to obtain an equivalent expressions for the /33 boundary condition

vVith /33,w known, the representation for either ill or i22 ca.n be used to obtain /32.w'

From the i22 representa.tion , th wall boundary condition on 2 is giv n by

f'33 -15112T22 - f + .,w -- 22 ,tu -2 - - ---=-2 -y74 -<:'10 (1)

(A5)

15

Appendix B Numerical Solution M ethodology

one-dim nsional finit -differ n e code was us d for all computations .. 11 equa­tions were normalized by the bulk vi 0 ity and friction velocity (i. e., wall uni ts) . The differen ing templat was node-centered with clustering close to the wall u ing an exponential tretching function. In t rms of the scaling used for the hannel flow calculations, th 500 node grid had the first point at a height of 0.1 wall uni ts . Th channel Reynold numl r R T determined th channel grid h ight . The f{ and [ quations were impli citly coupled a were the Tij - I ij equations, and for the second

model, the Tij - f3n equations. The variables U, J\", e and Tij w re solved in a time­dependent mod , while th I ij or I equations w r not (i . e., ~t = 0. ) 11 variables w re updat d at each tim tep .

In this app ndix, th terms with the superscript (n + 1) denote variabl that were impli citly solved for and the term with th sup r crip t (n) weI' variabl s used xpli citly at each iteration. The R ynolds stress equations coupled implicitly with ither the I ij or n equations were solv d first with the mom ntum, turbulent kineti c

energy and di sipation rate quation solved cond o An updat d T1 2 wa used in the mom.entum quation bu t th eddy viscosity in th turbulent transport terms of all th equations was not updated until aft l' the completion of each time tep. Typically olutions were re-start d from previou turbulent flow calculations.

Th sym1 01 Y1 denot th h ight of the first node from the wall and ew d not s t he boundary ondition valu for e . The di cr t iz d form of th governing equations ar a follows . For the T - f mod 1;

(B1 )

j.~n+1) _ L (n) 2 d2

f( n +1) = 1 (rr'I + 2c b .. ) (n ) tJ dy 2 ' tJ ern) r{ (n) tJ '-'c tJ .

(B2)

Th boundary ondition w l' implicitly writt n for the I ij as

(B3)

(B4)

_ (n+1)1

I ' 12

I (n +l ) 20 Yl 12 = - (n )2 4 .

w cw Yl

16

- -_.- - ------_I

For th r - (31 model·

Ti~n+l) = Ti~n) + tlt [pi~n) + E(n ) ]{(n ) (t l( n+l)tS))

1=1

imilarly, the boundary conditions were impli citly written for the 1 a

For U, J.,.- and C;

with

(n+l) I '1.12

(n+l)1 - -20J2 y, 1 - ~ (n)2 4

w cw Y1

~(n+1) I 2(n+l)lw '22 = -15 (r Yl

n - 4 cw Yl

(n+1) I (3 (n+l) I = -30 Tn Yl

3 (n)2 4 . w cw Y1

dp 1

d;.r Re.,.

l'.l(n ) - C T(n)~(n ) t - 11 22 ' c .

Th boundary conditions were implicitly wrjtt n for c as

17

(B6)

(B7)

(B )

(B9)

(B10)

(Bll)

(B1L!)

(B15)

References

Durbin, P. A. 1991. ear-wall turbulence closure without 'damping fun ctions' . Th -oret . Comput. FLuid Dyn. 3 , 1- 13.

Durbin, P. A. 1993a. A Reynolds stress model for near-wall turbulence. J. Fluid M echo 249 465-49 .

Durbin , P. A. 1993b. Application of a near-wall turbulence model to boundary layers and heat transfer. Int. J. H eat and FLuid Flow 14 , 316- 323.

Durbin, P. A. 1995. Separated flow computation with the J{ - E - v2 model. AIAA 1. 33 659- 664.

Gatski , T . B. and Jongen , T. 2000. Jonlinear eddy viscosity and algebraic stress models for solving complex t urbulent flows. Progress in A e1'ospace S cien c s, 36 , 655- 682.

Hanjali c, E . 1994. Advanced turbulence closure models: a view of CUlT nt status and future prospects. Int. J. H eat and FLu id Flow 15, 178- 203 .

Manceau , R 2000. Reproducing the blocking effect of th wall in one-point turbu­lence models. PTOceedings of the European Congress Comput. M eth . AppL. S ciences and E ngineering, Barcelona, Spain.

Manceau, R , Carlson , J. Rand Gatski T. B. 2001. A Rescaled Elliptic Relaxation Approach: Jeutralizing the Effect on the Log-Layer. Submitted to Phys . FLuids.

Manceau , R and Hanj ali c, E. 2000. A new form of the elliptic relaxation equation to account for wall effects in RA S mod ling. Phys . Fluids, 12, 2345- 2351.

Moser , R . D. , Eim, J. and Mansour , N. N. 1999. DNS of Turbulent Channel F low up to R er =590. Phys . FLuids, 11 , 943- 945.

Pettersson Reif, B. A. , Durbin, P. A. and Ooi , A. 1999. Modeling rotational effects in eddy-vi scosity closures . Int. J. H eat and FLuid Flow 20 , 563- 573.

So, R M. C., Lai , Y. G., Zhang, H. S. and Hwang, B. C. 1991. Second-order near­wall t urbulence closures : a review. A IAA J. 29 , 1 19- 1 35 .

Speziale, C. G. , Sarkar, S. and Gatski , T. B. 1991. Modeling the pressure-strain correlation of turbulence: an invariant dynamical ys tems approach. J. Fluid M ech . 227, 245- 272.

Wizman, V. , Laurence, D. , Kanniche, M. , Durbin, P. and Demuren, A. 1996. Mod­eling near-wall effect s in second-moment closures by elliptic relaxation. Int. J. H eat and Fluid Flow 17, 255- 266.

1

REPORT DOCUMENTATION PAGE Form Approved OMB No. 07~188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information. including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office 01 Management and Budget, Paperwork Reduction Project (0704--{)188), Washington, DC 20503.

1. AGENCY USE ONLY (Leave blank) 12. REPORT DATE /3. REPORT TYPE AND DATES COVERED

July 2002 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Elliptic Relaxation of a Tensor Representation for the Redistribution 706-31-11-06 Terms in a Reynolds Stress Turbulence Model

6. AUTHOR(S)

J. R. Carlson and T. B. Gatski

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION

NASA Langley Research Center REPORT NUMBER

Hampton, VA 23681-2199 L-18197

9. SPONSORINGIMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORINGIMONITORING

National Aeronautics and Space Administration AGENCY REPORT NUMBER

Washington, DC 20546-0001 NASAfTM-2002-211750

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified-Unlimited Subject Category 34 Distribution: Nonstandard Availability: NASA CASI (301 ) 621--0390

13. ABSTRACT (Maximum 200 words)

A formulation to include the effects of wall proximity in a second-moment closure model that utilizes a tensor representation for the redistribution terms in the Reynolds stress equations is presented. The wall-proximity effects are modeled through an elliptic relaxation process of the tensor expansion coefficients that properly accounts for both correlation length and time scales as the wall is approached. Direct numerical simulation data and Reynolds stress solutions using a full differential approach are compared for the case of fully developed channel flow.

14. SUBJECT TERMS

Turbulence Modeling, Elliptic Relaxation, Tensor Representations

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION OF REPORT OFTHIS PAGE OF ABSTRACT

Unclassified Unclassified Unclassified

NSN 7540-01-280-5500

15. NUMBER OF PAGES

23

16. PRICE CODE

20. LIMITATION OF ABSTRACT

Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39·18 298·102

I

•