A recently developed K-tau model for near wall turbulent flows is applied to two severe test cases. The turbulent flows considered include the incompressible flat plate boundary layer with the adverse pressure gradients and incompressible flow past a backward facing step. Calculations are performed for this two-equation model using an anisotropic as well as isotropic eddy-viscosity. The model predictions are shown to compare quite favorably with experimental data

Abid, R.

Speziale, Charles G.

Thangam, S.

English

NASA Technical Reports Server

NASA Contractor Report 187518
ICASE Report No. 91-16
ICASE
APPLICATION OF A NEW K-r MODEL TO NEAR
WALL TURBULENT FLOWS
S. Thangam
R. Abid
C. G. Speziale
Contract No. NAS1-18605
February 1991
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, V'uginia 23665-5225
Operated by the Universities Space Research Association
IWLS/X
Nalional Aeron_utic_ and
Space Adminislralion
Lnngley ReseArch Cenler
Hampton, Virginia 23665-5225
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https://ntrs.nasa.gov/search.jsp?R=19910009084 2020-03-19T19:42:24+00:00Z

Application of a New K-z Model to
Near Wall Turbulent Flows
S. Thangamt§
ICASE, NASA Langley Research Center
Hampton, Virginia 23665
R. Abid
Vigyan Research Associates
Hampton, Virginia 23665
C.G. Speziale§
ICASE, NASA Langley Research Center
Hampton, Virginia 23665
Abstract
A recently developed K-x model for near wall turbulent flows is applied to two severe test cas-
es. The turbulent flows considered include the incompressible flat plate boundary layer with ad-
verse pressure gradients and incompressible flow past a backward facing step. Calculations are
performed for this two-equation model using an anisotropic as well as isotropic eddy-viscosity.
The model predictions are shown to compare quite favorably with experimental data.
t Permanent Address: Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken,
NJ 07030
§ This research was supported by the National Aeronautics and Space Administration under NASA Con-
tract No. NAS1-18605 while the first and the third authors (ST & CGS) were in residence at the Institute
for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center,
Hampton, VA 23665.

Introduction
Two-equation turbulence models have become increasingly popular for the calculation of prac-
tical aerodynamic flows which can have important applications in the design of advanced aircraft.
The reason for this popularity is clear: two-equation models incorporate substantially more turbu-
lence physics -- and require less ad hoc empiricisms -- than the older algebraic eddy-viscosity
models without most of the added difficulties that arise from the computational implementation of
second-order closure models. Among the existing two-equation models, the K-e model is probably
the most popular; its successes and limitations have been fairly well documented [1,2]. One major
difficulty with the K-e model that has yet to be fully resolved involves its application to near wall
turbulent flows where inaccuracies and numerical stiffness problems can arise (see Patel, Rodi and
Scheuerer [3] for an interesting discussion of these problems). Recently, Speziale, Abid and
Anderson [4] developed a new K-x model for near wall turbulent flows wherein modeled transport
equations are solved for the turbulent kinetic energy K and the turbulent time scale ¢. While the
use of the turbulent time scale had been proposed earlier within the context of two equation models
(cf., Reynolds [5] and Zeierman and Wolfshtein [6]), none of these previous studies rendered a
model that could be integrated directly to a solid boundary with the no-slip condition applied. Spe-
ziale et al. [4] demonstrated that their new K-x model yielded excellent results -- and was compu-
tationally robust -- when integrated directly to the wall in an incompressible flat plate boundary
layer with zero pressure gradient. The K-'t model alleviates the problem of the lack of natural
boundary conditions for the dissipation rate in the K-e model since the turbulent time scale "t van-
ishes identically at a solid boundary.
The purpose of the present paper is to provide a more comprehensive testing and evaluation of
the K-x model. Two test cases are chosen to evaluate the model: (i) the incompressible flat plate
boundary layer with adverse pressure gradients, and (ii) incompressible flow past a backward fac-
ing step. Unlike in the previous study [4], the K-'t model will be solved with an anisotropic as well
as isotropic eddy viscosity. It has been demonstrated during the past decade that, in some cases,
the predictions of two-equation models can be enhanced considerably by the use of an anisotropic
eddy viscosity [7,8]. The results obtained from the K-'_ model will be documented in detail and
recommendations will be made for future research.
The K-x Model
! I I
In the standard form of the K-¢ model, the Reynolds stress tensor "cij =- u i uj (where u i is the
fluctuating velocity) is of the form
where the eddy viscosity is given by
vr = cW'.K (2)
-1-
-- 1 , ,
and u i is the mean velocity, K =- _ u i u i is the turbulent kinetic energy, e =- v au i/axj aui /axj
is the turbulent dissipation rate, x =-K/e is the turbulent time scale, and Cla is a dimensionless con-
stant that assumes a value of 0.09. In (2),fl a is a wall damping function that is needed for asymp-
totic consistency [4]; it is taken to be of the form
3.45 h . y+f_ = 1 + = ltanl_ (-7-_)
_Re t )
(3)
g'c
where Re t - is the turbulence Reynolds number and y+= yux/v is the usual wall coordinate
V
given that u_ is shear velocity and v is the kinematic viscosity. In an attached boundary layer, the
shear velocity is usually taken to be 4_o/9 (where x o is the wall shear stress); for more complicat-
ed turbulent flows with separation, the shear velocity is assumed to be proportional to K 1/2. The
wall damping functionfl a goes to unity sufficiently far from the wall. In the anisotropic version of
the K-x model, nonlinear corrections are added to the eddy viscosity model as developed by Spe-
ziale [8]. This leads to the following Reynolds stress model
zO 2K8(/_ 2Ct_fgK,_i j 2 2 2= 4CDCV;Kz 1 *- 1 _
-- (SikSkj--_ SklSkl_ij + Sij - -_ Skk_ 6 ) (4)
where
l (aU i aUj_
__ aSij _ aSij au i _ auj_
S ij - -_ + U k a x'--Tk _kk S k j - _kk S k i
(5)
(6)
are, respectively, the mean rate of strain tensor and its frame-indifferent Oldroyd derivative while
CD is a dimensionless constant that assumes a value of 1.68. Of course, the standard eddy viscosity
model (1) is recovered in the limit as CD -_ O.
The Reynolds stress models (1) and (4) are solved in conj unction with modeled transport equa-
tions for K and z which take the form [4]:
OK OK aui g a [( V VT')aK 1at +u'-- - + +taxi -'Cijaxj _ "_i -_-K)_iiJ
(7)
-2-
0-_+Ui- =OXi
"f _'U i _ V( VT _ _'C l
(Cel-1)--1;"--+ (Ce2f2-1) + [_v+K vaxj _// %2 JaxiJ
(8)
where
"e.  qE1e...y .lf2 = I 1 -_exp _, 36 )J (9)
is a wall damping function that is needed for asymptotic consistency. The constants Cel and Ce2
assume the values of 1.44 and 1.83, respectively, whereas for their preliminary calculations, Spe-
ziale et al. [4] chose
°xl = (Yx2 = GK = 1.36 (10)
with the understanding that these constants could be fine tuned to a non-equivalent set of values
when a future optimization over a variety of turbulent flows is conducted. When oxl, ox2 and o K
are identically equal, this K-x model reduces to an equivalent K-e model which can be simply ob-
tained by replacing 't with K/e in (7) and replacing the x-transport equation (8) with the equivalent
equation
_)e 3e e O_i
+ U.- - Cel--X..--
taX i K 'J aXj-- - cJ: +azii +< )azii] (11)
where o e is equal to the common value of Oxl, ox2, and o K given in (10).
Discussion of Results
The first problem that we will consider is the incompressible flat plate boundary layer with ad-
verse pressure gradients -- the Samuel and Joubert [9] test case. This represents a rather severe
test since the adverse pressure gradient is strong enough so that the flow is not that far removed
from separation. In Figure 1(a), the mean velocity profiles predicted by the standard K-x model
using the eddy-viscosity representation (1) are compared with the experimental data [9] at two sta-
tions: x = 1.76 m and x = 2.87 m (this corresponds to free-stream Reynolds numbers Re x in the
range of 2x106 to 4x106). It is clear from these figures that the model compares quite favorably
with the experimental results. In Figure l(b), the skin friction predicted by the K-x model is com-
pared with the experimental data. Again, the agreement between the model predictions and the ex-
perimental data is extremely good. Unlike most other two-equation models, the K-'_ model does
not overpredict the skin friction for this adverse pressure gradient test case (see, Rodi and Scheuer-
er [10] for an interesting discussion of this issue). It should be noted that the use of an anisotropic
eddy-viscosity does not yield a significant change in the results for this case.
-3-
The second problem to be considered is incompressible turbulent flow past a backward-facing
step -- the same test case considered by Driver and Seegmiller [ 11 ] in their recent experiments.
For this flow configuration the aspect ratio H1/H is 8 and the Reynolds number based on the inlet
free stream velocity and step height is approximately 38,500 (see Figure 2). The calculations were
conducted on a 200x100 nonuniform mesh using a finite volume method (see Thangam and Hur
[12]). Here, we use the anisotropic eddy-viscosity model (4) since it has been recently demonstrat-
ed that such models give rise to improved predictions in recirculating flows [8,12]. In Figure 3(a),
the computed streamlines obtained from the anisotropic K-x model are shown. They indicate re-
attachment at approximately 6 step heights downstream of separation -- a result that is in agree-
ment with the experimental data [11]. In Figure 3(b), the mean velocity profiles obtained from the
anisotropic K-x model are compared with the experimental results; again the agreement is extreme-
ly good. In Figure 4(a), the profiles of the turbulence intensity obtained from the anisotropic K-x
model are compared with the experimental data. While there are some discrepancies between these
results in the separation and recovery zone, on balance, the agreement is fairly good. In Figure
4(b), the profiles of the Reynolds shear stress obtained from the anisotropic K-'r model are com-
pared with the experimental data [11]. Again, while there are some discrepancies in the results
within the separation and recovery zone, the model predictions are still satisfactory. There is no
question that this near wall model does a substantially better job than the standard K-¢ model which
underpredicts the reattachment point by an amount of the order of 25-30%.
Conclusion
The near wall K-x model of Speziale et al. [4] has been tested in two applications involving
incompressible turbulent flows and the results obtained are quite encouraging. The standard K-x
model -- with an isotropic eddy viscosity -- yielded excellent results for the challenging incom-
pressible adverse pressure gradient case of Samuel-Joubert [9]. For the separated flow considered
-- namely, turbulent flow past a backward facing step -- excellent results were obtained from the
K-x model based on an anisotropic eddy-viscosity. In fact, we believe that these may be the most
accurate calculations for this backstep problem that have yet been obtained from a two-equation
model without the ad hoc adjustment of constants.
Future calculations are planned for supersonic boundary layers with streamline curvature and
separation (shock-induced or otherwise). For these flows, the K-x model will most probably have
to be applied with the anisotropic eddy viscosity model and it may become necessary to incorporate
dilatational effects into the model. After these calculations are completed, and refinements are
made accordingly, we will have a much better idea of the range of applicability of the K-x model.
Nonetheless, at this point the prospects appear to be quite promising.
-4-
[11
[21
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[lO1
[11]
[12]
References
Launder, B.E., "Phenomenological Modeling: Present and Future," Proc. Whither Turbulence
Workshop (ed. J.L. Lumley), Lecture Notes in Physics, Vol. 357, pp. 439-485. Springer-Ver-
lag, Berlin, 1990.
Speziale, C.G., "Analytical Methods for the Development of Reynolds Stress Closures in Tur-
bulence,"Ann. Rev. of Fluid Mech., Vol. 23, pp. 107-157, 1991.
Patel, V.C., Rodi, W., and Scheuerer, G., "Turbulence Models for Near-Wall and Low Rey-
nolds Number Flows: A Review," AIAA J., Vol. 23, pp. 1308-1319, 1985.
Speziale, C.G., Abid, R., and Anderson, E.C., "A Critical Evaluation of Two-Equation Mod-
els for Near Wall Turbulence," A/AA Paper 90-1481, 1990.
Reynolds, W.C., "Fundamentals of Turbulence for Turbulence Modeling and Simulation,"
Lecture Notes for Von Karman Institute, AGARD Lecture Series, No. 86, pp. 1-66. NATO,
New York, 1987.
Zeierman, S. and Wolfshtein, M., "Turbulent Time Scale for Turbulent Flow Calculations,"
AIAA J., Vol. 24, pp. 1606-1610, 1986.
Yoshizawa, A., "Statistical Analysis of the Deviation of the Reynolds Stress from its Eddy
Viscosity Representation," Phys. Fluids, Vol. 27, pp. 1377-1389, 1984.
Speziale, C.G., "On Nonlinear K-l and K--eModels of Turbulence," J. Fluid Mech., Vol. 178,
pp. 459-475, 1987.
Samuel, A.E. and Joubert, P.N., "A Boundary Layer Developing in an Increasing Adverse
Pressure Gradient," J. Fluid Mech., Vol. 66, pp. 481-493, 1974.
Rodi, W. and Scheuerer, G., "Scrutinizing the K-e Turbulence Model Under Adverse Pressure
Gradient Conditions," ASME J. Fluids Engrg., Vol. 108, pp. 174-179, 1986.
Driver, D.M. and Seegmiller, H.L., "Features of a Reattaching Turbulent Shear Layer in Di-
vergent Channel Flow," AIAA J., Vol. 23, pp. 163-171, 1985.
Thangam, S. and Hur, N., "A Highly Resolved Numerical Study of Separated Flow Past a
Backward Facing Step," Int. J. Engrg Sci. (in press).
-5-
¢<3
>,,
(o)
1.0
0.8
0.6
0.4
0.2
0.0 i
0.0 0.2
(b)
4.0-
3.5-
3.0
2.5
"- 2.0
x
C)
1.5
1.0
0.,5
0.0
0.0
X = 2.87 m
0
_L.._ I I I I I --_ I
o_ 06 08 _,o o.o o_ o_ o.o 08
_/u_ _/u_
I I I I I I t I
1.0 2.0 3.0 4.0
X(m.)
I
1.0
Figure 1. Comparison of the results of the K-¢ Model and experiments for the flat
plate boundary layer with an adverse pressure gradient (-- K-_ Model,
o experiments of Samuel & Joubert [9]) :
(a) mean velocity, and (b) skin friction coefficient
.6-
H1
_\\\_ T _ _ _ H2
L L _1F "!
'[
x
Figure 2. Schematic of the diagram for turbulent flow past a backward facing step
-7-
3 •
Y/H
0 5
X/H
I
lO
Y
H
X/H : 1.00
5
(
(
2 - C
0
B
- ib
ql
ii
0 I till!
0 1
2.00
0
• C
O
-y
.R
-iI_ t 1 I 1 I
0 1
4.00
)
!
I
(
m
M
-_(I I
0 1
8.00
)
)
• I
p /I I
0 1
12.00
I
I
/q
I I !
0 1
20.00
- I
i
- I
I
- !
x i
0 1
Figure 3. Results obtained from the anisotropic K-_ Model for incompressible
turbulent flow past a backward facing step with H1/H = 8, Re = 38,500. ( _ K-¢
Model, o experiments of Driver & Seegmiller [11]):
(a) mean velocity streamlines, and (b) mean velocity profiles.
-8-
YH
X/H : 1.00
5
o
-c
2 -_)
3
,')
0 / ' '
0.0 0.I 0.2
2.00
o
-o
-C
i)
-t)
I I
0.1 0.2
4.00
t:l
0.1 0.2
8.00
o
-c
I b
i
0.1 0.2
12.00
i
o
o
"C
()
0.1 0.2
20.00
IC
l°
1
_/ I I I
0.1 0.2
Y
H
X/H : 1,00
5
2
2.00
Q
Q
I I I J I
i: __oU211/2
4.00 8.00 1 2.00
0 I I I I I I I I I I
0.0 0,01 0.02 0.01 0,02 0.01 0.02
r
)
0.01 0.02
I
I
r
J
4_1 I I I I
0.01 0.02
J fl
iI I I I I
0.01 0.02
20.00
Figure 4. Results obtained from the anisotropic K-_ Model for incompressible
turbulent flow past a backward facing step with H1/H = 8, Re .= 38,500. ( _ K_
Model, o experiments of Driver & Seegmiller [11]):
(a) turbulence intensity profiles, and (b) Reynolds stress profiles.
-9-
Report Documentation Page
t_lOq31 _Or_ut(5 ,_'X_,
1. Report No.
NASA CR-187518
ICASE Report No. 91-16
4. Title and Subtitle
2. Government Accession No.
APPLICATION OF A NEW K-T MODEL TO NEAR WALL
TURBULENT FLOWS
7. Authoris)
S. Thangam
R. Abid
C. G. Speziale
9. Performing Organization Name and Address
Institute for Computer Applications in Science
and Engineering
Mail Stop 132C, NASA Langley Research Center
Hampton, VA 23665-522_
12, Sponsoring Agency Name and Address
National Aeronautics and Space, Administration
Langley Research Center
Hampton, VA 23665-5225
3. Recipient's Catalog No.
5. Repo_ Date
February 1991
6. Performing Organization Code
8. Performing Organization Repo_ No.
91-16
10. Work Unit No.
505-90-52-01
11. Contract or Grant No.
NASI-18605
13. Type of Report and Period Covered
Contractor Report
14. Sponsoring Agency Code
15. Supplementa_ Notes
Langley Technical Monitor:
Michael F. Card
To be submitted to AIAA Journal
Final Reoort
16. Abstract
A recently developed K-T model for near wall turbulent flows is applied to two
severe test cases. The turbulent flows considered include the incompressible flat
plate boundary layer with the adverse pressure gradients and incompressible flow
past a backward facing step. Calculations are performed for this two-equation model
using an anisotropic as well as isotropic eddy-viscosity. The model predictions
are shown to compare quite favorably with experimental data.
17. Key Words(SuggestedbyAuthor{s))
near wall turbulence; two-equation models;
adverse pressure gradients; flow separa-
tion
18. Distribution Statement
34 - Fluid Mechanics and Heat Transfer
Unclassified - Unlimited
19. SecuriW Cla_if. (of this repot)
Unclassified
20. SecuriW Cla_if. (of this pa_)
Unclassified
21. No. of pages
11
22. Price
A0 3
NASA FORM 1626 OCT 86
NASA-Langley, 1991




Application of a new K-tau model to near wall turbulent flows

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