The primary objective of the Center for Modelling of Turbulence and Transition (CMOTT) is to further the understanding of turbulence theory for engineering applications. One important foundation is the establishment of a data base encompassing the multitude of existing models as well as newly proposed ideas. The research effort described is a precursor to an extended survey of two equation turbulence models in the presence of a separated shear layer. Recently, several authors have examined the performance of two equation models in the context of the backward facing step flow. Conflicting results, however, demand that further attention is necessary to properly understand the behavior and limitations of this popular technique, especially the low Reynolds number formulations. The objective is to validate an incompressible Navier Stokes code for use as a numerical test-bed. In turn, this code will be used for analyzing the performance of several two equation models

Steffen, C. J., Jr.

English

NASA Technical Reports Server

Center ]or Modellin9 of Turbulence and Transition
Research Briefs - 1991 N93-15799
Two Equation Modelling and the
Pseudo Compressibility Technique
C. J. Steffen, Jr.
1. Motivation and Objective
The primary objective of the Center for Modelling of Turbulence and Transition
(CMOTT) is to further the understanding of turbulence theory for engineering
applications. One important foundation for this research is the establishment of
a data base encompassing the multitude of existing models as well as newly proposed
ideas. The research effort described in the next few pages is a precursor to an extended
survey of two equation turbulence models in the presence of a separated shear layer.
Recently, several authors have examined the performance of two equation models
in the context of the backward facing step flow. Conflicting results, however, demand
that further attention is necessary to properly understand the behavior and limita.
tions of this popular technique, especially the low Reynolds number formulations.
The objective of this research is to validate an incompressible Navier Stokes code
for use as a numerical test-bed. In turn, this code will be used for analyzing, the
performance of several two equation models.
2. Work Accomplished
To date, the validation of the incompressible code DTNS is complete. The details
of this validation study are documented in reference[l]. The code is based upon
the pseudo-compressibility technique and incorporates the approximate factorization
scheme for time integration. Two laminar benchmark flows are used to measure the
performance and implementation of the numerical methods. The classic solution of
the Blasius boundary layer is used for validating the flat plate flow, while experimental
data is incorporated in the validation of backward facing step flow.
An initial result for the standard high Reynolds number k-¢ equations has also been
calculated to demonstrate an initial performance level of the solution technique for
the turbulence equations.
2.1 Numerical Method
The absence of a variable density in the continuity equation governing incompress-
ible flow complicates the numerical integration procedure. One solution technique
that has been well received is that of pseudo-compressibility. This idea was first
put forward by Chorin[2] and enables the equations to be solved using the prim-
itive variables. Recently, Chang and Kwak[3], Rizzi and Eriksson[4], Kwak and
Chakravarthy[5], Michelassi and Shih[6], and Turkel[7] have found this method suit-
able for resolving incompressible flow. This particula.r implementation has been vali-
dated by Gorski[8-10] for several different benchmark flows.
95
https://ntrs.nasa.gov/search.jsp?R=19930006610 2020-03-17T08:38:21+00:00Z
Christopher J. Steffen, Jr.
Examine the system of equations involved in the pseudo compressibility method
and notice that they differ from the incompressible Navier Stokes equations by the
addition of a time dependant pressure term in the continuity equation.
10p + Ou Ov
+P-e:e uv =o
O( 1 Or)O( 10_y)_'4" uv Re_xx + v2+p l_e --0Y\
(1)
Here, z and y are the independent variables and Re refers to the Reynolds number.
The constant /3 is known as the pseudo compressibility parameter. This system
is hyperbolic in nature while the incompressible flow equations are elliptic. The
pseudo sound speed, c = v/_ +/3, is governed by the value of the parameter /3,
whereas the physical sound speed is infinite. Chang and Kwak [3] have shown that
for/3 > 0 the finite speed pseudo waves vanish as time progresses and yield the proper
incompressible solution at the steady state limit. It is through this parameter/3 that
the convective and acoustic waves are decoupled, and thus convergence is governed. In
choosing an optimum value for this parameter, the goal is to avoid giving the viscous
effects time to react to the passing of the nonphysical transient pressure waves. Thus
a lower bound on the acoustic speeds translates into a loiver bound on 3. However,
an upper bound on /3 is strictly scheme dependent.
The approximate factorization is rather straight forward. Equations (2) can be
_(q) + _x(fl+gl)+ _y(f2+g2)= 0
, fl = u2+p , t'2 =
11 v
q ._
rewritten as
V
UV
v2+p
_J
or in generalized coordinates (_(x, y), T/(x, y)) as seen here in equation (3).
q
q--- J
-- _xfl +_f2fl-- , =
J
Ggl +_g2 __i
gl -- , g2 --J
j _ ,7)
O(.r, v)
_fl + _vf2
J
r/zgl + 7}vg2
J
(2)
(3)
96
Two Equation Models and Turbulent Flows
The third order accurate Chakravarthy and Osher TVD scl_eme is used to discretize
the convective terms while central differencing is used for the viscous fluxes. Thus
the approximate factorization of this method is expressed in terms of flux Jacobians
A and B as
{V O-+ 0 + _ O-} _, Of(_'+_5")Of_+_)
_-_-+-_-A +-b-_-A +-_-A,, Aq = +
O_ Or/ (4)
V - 0 + _ O- __-/+ -_B+ + _--_B + _-_Bv}Aq = VAq *
where V is the cell volume, and At = _ is the timestep. The matrices Av and By
are the viscous flux Jacobians. The interested reader is encouraged to see reference[9]
for more details.
The k-e equations are solved in a similar manner as that seen in equation 2.
Consider the standard high Reynolds number form of the transport equations of the
turbulent kinetic energy and dissipation of turbulent genetic energy as in equation
nO _x (ft+gt) + N(f_+g2) = stc_gt(qt) + 0 - t t'
with
[vk]' V_
' st='_e C,I_P C2_Re
(5)
vt
vt=u+_
crk
__L (6)
zJ, = v+
O"t
and the production of turbulent genetic energy defined as shown below.
Again, we can translatethese equations in a manner similarto that of equation (3)
and create the following system for generalized coordinates:
o=
_(q ) + _-_f_+gl) + _--_ (8)
This can be solved via approximate factorization as shown here in equation 9.
{_ O-t+ O+At_ O- t c9(_+'_2) S--(+_-A +-if(- +-ff(A"-°"Ht} A-_'-- 0(_+_'1)+0( c_9,l +
v O-B+ O+ -
_'+ c_9,} +_I/B-+ "_-_/B"-oqHt}A_-=_ A'qt*
(9)
97
Christopher J. Steffen, Jr.
Again, the scheme is written in terms of the turbulence flux Jacobians and the
approximate source Jacobian, H t. The coefficients a_ and aT/are provided to enhance
the diagonal dominance of the implicit method. While the convective flux terms are
linear, the linearization of the source terms is not straight forward. Furthermore, the
low Reynolds number formulation of equation 5 includes other nonlinear correction
terms in the source vector S t . The interested reader is encouraged to see the recent
paper by Michelassi and Shih[6].
2.2 Discussion
Overall the performance for laminar flow documented in reference[I] is very en-
couraging. The fiat plate results shown agree with the theoretical Blasius solution
for several different grid configurations. The laminar back facing step flow was also
well resolved both in terms of the primary recirculation zone reattachment length,
and corresponding velocity profiles as seen in figures 1 and 2. However, performance
of the two dimensional flow code drops off dramatically when the Reynolds number
corresponding to the onset'of 3D flow is exceeded. Armaly et al [11] observed this
phenomenon occur at a Reynolds number of 400, based upon the hydraulic diameter
of the inlet channel. Again, the numerical results bear this finding out.
Figure 1 Reattachment length predictions----compari_n with experiment [11]. xl, x2, and x3 are the
primary reattachment, secondary detachment, and secondary reattachment lengths, respectively.
20
18
16
14
12
Oq
_10
X
8
6
4
2
0
0 2O0 400
Re D
600 800
Experiment:
xl
x2
v x3
2D Calculation:
O x I
O x 2
A xta
98
Two Equation Models and Turbulent Flows
Figure 2 RED=389 velocity profiles---comparison with experiment [11].
E
x/s=0.0
r
2.55
%
4- °%
, or"
0
0 i . 0
--4 -
I 1 1 1 1 I _ 1 1 I [
0 80 0 80
u (crn/sec)
3.06
2, -" 2
= JPl I
i ID I '
o- _ i 01
-J " I
-2 .... 21
-4 .e i -4]
0 80
4.18
%
4- %,
--
- w
-
_ w
I_ I I I I
0 80
13.57
4r _
2 _ ,4v
0-
- ID
-2
-4
I I_lll I I I
0 80
The current thrust of effort is in establishing a comparison between the standard
high Reynolds number form k-¢ model and a version which incorporates the low
Reynolds number corrections, such as that of Jones and Launder[12]. This model
is particularly convenient in that the damping functions are independent of the
parameter, y+. Thus, multiple walls bounding the computational domain do not need
special attention. An initial calculation using the high Reynolds number approach
agrees with the published results of Speziale and Thangam[13]. However, the Jones
and Launder model is proving to be quite challenging.
3. Future Plans
Eventually, several two equation models will be examined for their behavior in the
back facing step flow. The following is a partial list of the two equation models of
primary interest:
1. Jones and Launder
2. Chien
3. Yang and Shih
4. Shih and Lumley
The current research effort will result in an increased understanding of the need
for proper modelling of the near wall region. However, the increase in accuracy
maybe offset by an unwarranted increase in effort. Further numerical experiments
are necessary before any conclusions can be drawn.
4. References
] Steffen, Jr., C. J., "An Investigation of DTNS2D for Use as an Incompressible
Turbulence Modelling Test-bed," NASA Technical Memorandum 105593 (1992).
99
Christopher J. Steffen, Jr.
2 Chorin, A. J., "A Numerical Method for Solving Incompressible Viscous Flow
Problems," J. Comput. Phys., 2, 12-26 (1967).
3 Chang, J. L. C. and Kwak, D., "On the Method of Pseudo Compressibility for
Numerically Solving Incomprpssible Flows," AIAA paper 84-0252 (1984).
4 Rizzi, A. and Eriksson, L. E., "Computation of Inviscid Incompressible Flow with
Rotation," J. Fluid Mech., 153, 275-312 (1985).
5 Kwak, D., Chang, J. L. C., Shanks, S. P., and Chakravarthy, S. R., "A Three
Dimensional Incompressible Navier-Stokes Flow Solver Using Primitive Variables,"
AIAA J., 24, 390-396 (1986).
6 Michelassi, V. and T.-H. Shih, , "Elliptic Flow Computation by Low Reynolds
Number Two Equation Turbulence Models," NASA Technical Memorandum
105376 (1991).
7 Turkel, E., "Preconditioned Methods for Solving the Incompressible and Low Speed
Compressible Equations," J. Comput. Phys., 72, 277-298 (1987).
8 Gorski, J. J., "Incompressible Cascade Calculations Using an Upwind Differenced
TVD Scheme," Advances and Applications in Computational Fluid Dynamics, 66
of ASME-FED, edited by O. Baysal, 61-69 (1988).
9 Gorski, J. J., "TVD Solutions of the Incompressible Navier-Stokes Equations With
an Implicit Multigrid Scheme," Proceedings of the AIAA/ASME/SIAM/APS 1st
National Fluid Dynamics Congress, 1,394--401 (1988).
10 Gorski, J. J., "Solutions of the Incompressible Navier-Stokes Equations Using
an Upwind Differenced TVD Scheme," Proceedings of the Ilth International
Conference on Numerical Methods in Fluid Dynamics, 278-282 (1988).
11 Armaly, B. F., Durst, F., Pereira, J. C. F., and SchSnung, B., "Experimental and
Theoretical Investigation of Backward-Facing Step Flow," J. Fluid Mech., 127,
473-496 (1983).
12 Jones, W. P. and Launder, B. E., "The Prediction of Laminarization with a Two-
Equation Model of Turbulence," Int. J. Heat Mass Transl., 15, 301-314 (1972).
1:3 Thangam, S. and Speziale, C. G., "Turbulent Separated Flow Past a Backward
Facing Step: a Critical Evaluation of Two-Equation Turbulence Models," NASA
Contractor Report 187532 ICASE Report No. 91-23 (1991).
100


Two equation modelling and the pseudo compressibility technique

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