A k-epsilon model is proposed for turbulent bounded flows. In this model, the turbulent velocity scale and turbulent time scale are used to define the eddy viscosity. The time scale is shown to be bounded from below by the Kolmogorov time scale. The dissipation equation is reformulated using the time scale, removing the need to introduce the pseudo-dissipation. A damping function is chosen such that the shear stress satisfies the near wall asymptotic behavior. The model constants used are the same as the model constants in the commonly used high turbulent Reynolds number k-epsilon model. Fully developed turbulent channel flows and turbulent boundary layer flows over a flat plate at various Reynolds numbers are used to validate the model. The model predictions were found to be in good agreement with the direct numerical simulation data

Shih, T. H.

Yang, Z.

English

NASA Technical Reports Server

NASA TechniCal Memorandum 105238
ICOMP'91- 16; CMOTT-91'06
74-7"7
A k-e Modeling of Near
Wall Turbulence
-\ _:, m3x
._ (NASA-TM-IO523d) A k-EPDILON MOOTLING OF N91-32@60
NEAR WALL TURbULENCe (NASA) 14 p CSCL 20O
.... J= : t'
-_ Uncles
_3134 00457_7..
Z. Yang and T.H. Shih
Institute for Computational Mechanics in Propulsion
and Center for Modeling of Turbulence and Transition
LeWis Research Center
Cleveland, Ohio
Prepared for the
Fourth International Symposium on Computational Fluid Dynamics
Davis, California, September 9- i2, 1991
fiJ/ A
https://ntrs.nasa.gov/search.jsp?R=19910023146 2020-03-17T14:54:06+00:00Z

A k - e MODELING OF NEAR WALL TURBULENCE
Z. Yang and T.H. Shih
Institute for Computational Mechanics in Propulsion and
Center for Modeling of Turbulence and Transition
NASA Lewis Research Center
Cleveland, Ohio 44135
ABSTRACT
A k - e model is proposed for turbulent wall bounded flows. In this model, the turbulent
velocity scale and turbulent time scale are used to define the eddy viscosity. The time scale
is shown to be bounded from below by the Kolmogorov time scale. The disslpatlon equation
is reformulated using this time scale, removing the need to introduce the pseudo-dissipation.
A damping function is chosen such that the shear stress satisfies the near wall asymptotic
behavior. The model constants used are the same as the model constants in the commonly
used high turbulent Reynolds number k-e model. Thus, when it is far away from the wall,
the proposed model reduces to the standard k-e model. Fully developed turbulent channel
flows and turbulent boundary layer flows over a flat plate at various Reynolds numbers are
used to validate the model. The model predictions are found to be in good agreement with
the DNS data.
INTRODUCTION
Because of a wide range of scales involved in a turbulent flow, DNS (direct numer-
ical simulation) is limited to flows of moderate Reynolds number and simple geometry.
Turbulence modeling is the only viable approach for the calculation of turbulent flows of
engineering interest. In turbulence modeling, k-e model is the most used model in engineer-
ing calculations and model constants have been computationally optimized, giving what is
called the Standard Model (e.g., Spalding and Launder [1], Rodi [2]). The Standard Model
is devised for high turbulent Reynolds number flows and is traditionally used in conjunction
with a wall function when it is applied to wall bounded turbulent flows. Since universal
wall functions do not exist in complicated flow situations, it is necessary to develop a form
of the k - e model such that the equations can be integrated directly down to the wall.
Jones and Launder [3] made the first proposal for the low Reynolds number k - • model
and the equations were successfully integrated to the wall. A number of models have been
proposed since then. A critical evaluation of the pre-1985 models was made in the paper by
Patel et al. [4]. They pointed out that it is important for the damping function in the eddy
viscosity to satisfy the near wall asymptotlcs. More recent models could be found in Shih
[5]. Of the models proposed, a pseudo-dissipation is introduced near the wall. However,
the definition of the pseudo-dissipatlon is quite arbitrary, except for the constrains on the
wall and far away from the wall. In addition, the model constants used are different from
the Standard Model, making the near wall models less capable of handling flows containing
both high Reynolds number turbulence and near wall turbulence.
In the present paper, we try to provide remedies for these two shortcomings stated
above. In section 2, we list the general form of the low Reynolds number k - e models. We
then propose in section 3 a new model. The proposed model is validated in section 4 for
fully developed turbulent channel flows and turbulent boundary layer flows over a flat plate
at different Reynolds numbers. Section 5 presents conclusions and possible future works.
EXISTING NEAR WALL k - e MODELS
In turbulence modeling, the instantaneous quantities of an incompressible flow are de-
composed into the mean and the fluctuations, i.e. fii = Ui ÷ ui, _ = P ÷ p. The mean
field Ui satisfies the following continuity equation and the Reynolds averaged Navier-Stokes
equation:
Vl,i = 0 (1)
Ui + UjVl,j = -1Pi + vUi,jj- < uiuj >j (2)
P
where the Reynolds stress term, - < uiuj >, has to be modeled.
In an eddy viscosity model, the Reynolds stress is related to the mean field by
- < u uj >= .r(V ,j + vj, ) - 2k6 j, (3)
where VT is the eddy viscosity and k is the turbulent kinetic energy. From a dimensional
reasoning, the eddy viscosity is given by
VT _,_ utlt_
where ut and It are the turbulent velocity scale and turbulent length scale respectively.
In the framework of the k-e model, It _ k3/2/e and ut _ k1/2. In the case of near
wall turbulence, the above is modified by introducing a damping function fz and a pseudo-
dissipations _. The general forms for the eddy viscosity and the transport equations for k
and e are (Patel et al. [4]),
= (4)
2
k + ujk,_ : [(_ + :)k j],_- < _uj > u_,_- _+ D (5)
e e_+ uses = [(_ + )_,s],s+ clll_rV_,ju,,s - c_I2¥ + E (6)
where
_=E-D
Various models use different sets of model constants c_,, ¢rk, a,, cl, e2 and different
damping functions f,, fl, f2 as well as different forms of D and E. Far away from the wall,
the damping functions .f_,, fl, f2 approach to 1 and the near wall terms D, E are negligibly
small so that the model equations become the high Reynolds number forms of k - • model.
THE PROPOSED MODEL
The turbulent length scale is characterized by the size of the energy containing eddies.
Near the wall, these eddies would have a size of O(y). Following Patel et al [4], the turbulent
velocity field has the following expansions near the wall:
u' = uly + u2y 2 +...
V t = v2y 2 q-...
w r - wly+w2y _+... (_)
where ut, v2, wl are non-zero in general. Thus, as the wall is approached, both the turbu-
lent length scale and the turbulent velocity scale approach zero. However, the turbulent
time scale, which is given by the ratio of the scale of the energy containing eddies to the
turbulent velocity scale, approaches to a non-zero value. We expect this time scale to be the
Kolmogorov time scale because viscous dissipation is dominating near the wall. We thus
have, for the turbulent time scale,
Tt = k. + T_ (8)
where
Tk = ck(_) 1/2 (9)
is the Kolmogorov time scale and ck is a constant of order one.
Using this turbulent time scale and k 1/2 as turbulent velocity scale, the turbulent
Reynolds number is
R_ = ltu----At= kTt (10)
l/ l,/
3
andthe eddyviscosityis
VT= c_,f_,kTt (11)
The transport equation for k remains the same as eq (5) (with D = 0, since we are
solving the real dissipation.) The transport equation for • is
+ Ujej = [(u + UT)ej],j + (CI_'TUi,jUi,j - C2f2e) 1 + E (12)
l't
where
f2 = 1.0 - 0.22exp(- 3_ ) (13)
E = vVTVi,jkVi,jk (14)
Since Tt is always a positive nonzero quantity, the above formulation removes the singu-
laxity of the high Reynolds number k - e model when the wall is approached, f2 and E are
of the form used by Jones and Launder [3] for the low Reynolds number model, in wh:ch f2
is used to capture the final decay of the homogeneous turbulence and E is used to model
turbulence in the buffer layer.
Near the wall, the shear stress - < uv > should behave as O(y3). Thus we would
require a damping function which has a near wall behavior of O(y). The damping function
is assumed to be of the following form
f,u ---- 1.0 -- exp(aly + -_- a2(y+) 2 -_- a3(y+) 3 --_ aa(y+) 4) (15)
with the constants are determined by calibrating with the DNS data for 2D channel flow at
Rer = 180. The calibration gives
al = 4x10 -3,a2 = 5x10 -5,az = -2x10 -6,a4 = 8x10 -s
The model constants used are the same as those in the Standard Model. i.e.
cz = 0.09, C1 = 1.44, C2 = 1.92, ak = 1.0, a_ = 1.3
Thus, far away from the wall, f2, f_, become 1, the Kolmogorov time scale is much smaller
than k/e, and E is much smaller than the other terms on the right hand of • equation, our
model would reduce to the standard k - e model, ensuring the good performance of our
model in the high turbulent Reynolds number limit.
MODEL VALIDATION
We use 2D fully developed channel flows and turbulent boundary layer flows to validate
the proposed model. These flows axe attractive for model testing, because they have self-
similar solutions so that the initial conditions do not have to be accurately specified. These
flowsareverysimple,and solutionscanbe foundveryefficiently;yet, the effectsof the wall
on turbulent shear flow are still present. In addition, DNS data providing detailed flow
information is available for these flows. We will compare the model predictions with the
DNS data.
The boundary condition for • on the wall is determined by applying k equation to the
wall, which gives
•_ = vk_. (16)
In this study, the following boundary condition for •, which is mathematically equivalent
to the above but computationally much more robust, is used.
dkl/2 2
• = (17)
ay
An implicit finite difference scheme is used to solve the momentum equation and the
transport equations for k and e. The coefficients for the convective terms are lagged one
step in the marching direction, and the source terms in the k and • equations are Iinearized
in such a way that numerical stability is ensured.
A variable grid spacing is used to resolved the sharp gradient near the wall. The grid
distribution is controlled by 5yi/Syi-1 = a. Both a and the total number of the grid, N, are
varied to ensure the grid independence of the numerical results. The marching step size,
5z, is also varied to ensure accuracy. It is found that N = 150, a = 1.03, and 5z = 0.05 are
sufficient to give a solution with a less than 0.5% error.
Computations are carried out for 2D fully developed turbulent channel flows at Re_. =
180 and Rer = 395. Fig. 1 shows the mean velocity profile for Rer = 180, together with the
DNS data of Kim et al [6]. In this figure, and in the following figures, both the dependent
and the independent variables are normalized by ur and v. The agreement between model
prediction and DNS data is excellent. This may not be unexpected since the constants in
the damping function are calibrated for this case. We then show the results for 2D channel
flow at a different Reynolds number, Re_. = 395. Fig. 2 to Fig. 5 shows the profiles of the
mean velocity, shear stress, turbulent kinetic energy and the dissipation, respectively. The
predictions of Jones and Launder [3] and Chien [7] are also shown. These predictions are
compared with the DNS statistics for this case (Mansour [8]). Overall, the proposed model
gives a better prediction.
Predictions of the mean velocity, shear stress, turbulent kinetic energy and dissipation
rate for turbulent boundary layer over a flat plate at Reo = 1410 are shown in Fig. 6 to
Fig. 9. The model predictions are compared with the DNS data of Spalart [9]. Similar
computations are made for Reo = 670, where the DNS data is also available (Spalart
[9]). The results for Reo = 670 are not shown here, due to space limitation. The model
predictions for the skin friction are shown in Fig. 10, together with the DNS data of Spalart
[9] at Reo = 670 and Reo = 1410.
In the above computations, ck = 1.0 is used. The constant ek was varied in the range of
0.5 - 3.0 and the solutions were found to be quite insensitive to ek in this range. As more
flow situations are tested, the value of ck could be optimized by fine tuning.
CONCLUSION
We have presented a k - e model for wall bounded flows. We have shown that near the
wall, the time scale is characterized by the Kolmogorov time scale rather than zero. By using
this time scale and a turbulent velocity scale, the singularity in the standard dissipation
equation is removed as the wall is approached and the equation can be integrated down
to the wall. The use of this time scale also renders pseudo-dissipation unnecessary, which
contains an arbitrary element in its definition.
The proposed model uses the same set of model constants as that used in the Standard
Model, and when it is far away from the wall, the proposed model reduces to the Standard
Model. Thus, the proposed model would be applicable in both the near wall turbulence
and the high Reynolds number turbulence.
The damping function is chosen in such a way that the near wall asymptotics for the
shear stress is satisfied, a feature which is very important in predicting the near wall prop-
erties (Patel et al [4]). The form of the damping function is assumed to be a function of
y+ with the constants calibrated with the DNS data for a particular case. This y+ depen-
dence of the damping function must be viewed as a drawback of the model, for y+ is not
clearly defined, or even does not exist for complex flows (comer flows, flows with separa-
tion, for example.) In addition_ y+ makes the modeling equation not invariant in coordinate
transformations. Currently, effort is being made to remove these deficiencies.
ACKNOWLEDGEMENT
The DNS data used was kindly made available to us by Dr. N. Mansour of NASA
Ames Research Center. Help from Dr. A.T. Hsu of Center for Modeling of Turbulence and
Transition, NASA Lewis Research Center on numerical computation is also acknowledged.
This work is supported by Institute of Computational Mechanics in Propulsion, NASA
6
Lewis Research Center.
REFERENCES
.
.
.
4.
5.
6.
7.
8.
9.
B.E. Launder and D.B. Spalding,Computer Methods in Applied Mechanics and
Engineering,3, 269 (1974).
W. Rodi, Turbulence Models and Their Application in Hydraulics, (Book Pub. of
Int. Asso. for Hyd. Res., Delft, the Netherlands, 1980).
W.P. Jones and B.E. Launder, Int. J. Heat Mass Transfer 16, 1119 (1973).
V.C. Patel, W. Rodi, and G. Scheuerer, AIAA J. 23, 1308 (1985).
T.H. Shlh, NASA TM 103211, (1990).
J. Kim, P. Moin, and R. Moser, J. Fluid Mech. 177, 1119 (1987).
K.Y. Chien, AIAA Journal 20, 33 (1982).
N.N. Mansour, Private communication, (1991).
P.R. Spalart, J. Fluid Mech. 187, 61 (1988).
20,0 ! I
10.0
5.0
0.0 | !
| 0 0'0 101'° Y+ !02.0
Figure 1. Mean velocity profile for channel flow, Rer = 180.
] 0 5.0
25.0 ! I
20.0
U
15.0
10.0
5.0
0.0
10°'°
fl I
f
J
_." - - - - Jones & Launder
.v4(* - Chien
- Present
I I
101'° 102.0
i/÷
Figure 2. Mean velocity profile for channel _ow_ ReT = 395.
10 3.0
1,0 ! I !
¢ DNS
.... Jones & Launder
.... Chlen
-- Present
5,0 I I I
4.0 ¢ DNS
.... Jones & Launder
k .... Chlen
-- Present
3.0
2.0 '_
_" <" _"'-'...- ._ .
1.0 !
0.0
0.0
.-- I I I ________ I ,
100.0 200.0 300.0 400,0
!/+
I"igure 4. Turbulent kinetic energy for channel flow, Re, = 395.
0.4
0.3
0.2
O.I
0.0
0,0
o DNS
Jones & Launder
.... Chten
-- PresenL
./%
,/
'I
't
, I I ,I , i ,I =
25.0 50.0 75.0 100.0
it+
Figure 5. Dieslpation rate for channel flow_ Rer = 395.
U
25.0
20.0
15.0
10.0
5.0
0,0
I 0 °'°
I I I
ff_
•, L . 0 DNS
.... Jones & Launder
Chien
Present
I I !
t 0_'° 10='° 103.o
_+
Figule 6. Mean veloclty pzo/_le for tlat plate boundary layer, Re0 = 1410.
10 4.0
10
0.8
-< uv >
I
200.0
0 DNS
.... Jones & Launder
• , .... Chien
0 - -- Present
0 • •
400.0 600.0
y+
Figure 7. Turbulent shear streu for flat plate boundary l-yer, Re¢ = 1410.
50
4.0
k
\
\
\
\
\
\
X
\
0 DNS
.... Jones & Lounder
.... Chien
--- Present
3.0
2.0
1,0
0.0
0.0
,_0 O" .
"'. _%_._ 0
200.0 400.0 600.0
I/+
Figure 8. Turbulent kinetic energy for fiat plate boundary layer, Re. = 1410.
1!
0.4 l I I
0.3
0.2
0.I
0.0
0.0
,/,
/
0 DNS
Jones & Lounder
.... Chien
Present
0
I i I . , i ,I i
25.0 50.0 75.0 100.0
y+
Figure g. Dissipation rate for fist plate boundary layer_ Ree = 1410.
6.0 I I 1
4.0
3.0
4O0
o DNS
.... Jones & Lounder
k,_ .... Chien
_. _
L I l __ L
800 1200 1600
Ree
Figure 10. Skin friction coemcient for flat plate boundary layer.
2000
12
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE
September 1991
4. TITLE AND SUBTITLE
A k-e Modeling of Near Wall Turbulence
6. AUTHOR(S)
Z. Yang and T.H. Shih
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135 - 3191
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National Aeronautics and Space Administration
Washington, D.C. 20546-0001
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AGENCY REPORT NUMBER
NASA TM- 105238
ICOMP- 91 - 16; CMOTT- 91 - 06
11. SUPPLEMENTARY NOTES
Prepared for the Fourth International Symposium on Computational Fluid Dynamics, Davis, California, September 9-12, 1991. Z. Yang and
T.H. Shih, Institute for Computational Mechanics in Propulsion and Center for Modeling of Turbulence and Transition, Lewis Research Center,
Cleveland, Ohio (work funded under Space Act Agreement C-99066-G). Space Act Monitor, Louis A. Povinelli, (216) 433-581 g.
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13. ABSTRACT (Maximum 200 words)
A k-e model is proposed for turbulent wall bounded flows. In this model, the turbulent velocity scale and turbulent time
scale are used to define the eddy viscosity. The time scale is shown to be bounded from below by the Kolmogorov time
scale. The dissipation equation is reformulated using this time scale, removing the need to introduce the pseudo-
dissipation. A damping function is chosen such that the shear stress satisfies the near wall asymptotic behavior. The
model constants used are the same as the model constants in the commonly used high turbulent Reynolds number k-e
model. Thus, when it is far away from the wall, the proposed model reduces to the standard k-e model. Fully developed
turbulent channel flows and turbulent boundary layer flows over a flat plate at various Reynolds numbers are used to
validate the model. The model predictions are found to be in good agreement with the DNS data.
14. SUBJECT TERMS
Turbulence model; K-epsilon
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A k-epsilon modeling of near wall turbulence

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