A one-equation turbulence model based on the turbulent kinetic energy equation is presented. The model is motivated by the success of the Johnson-King model and incorporates a number of features uncovered by Simpson's experiments on separated flows. Based on the results obtained, the model duplicates the success of algebraic models in attached flow regions and outperforms the two-equation models in detached flow regions

Hassan, H. A.

English

NASA Technical Reports Server

V
t
NASA COOPERATIVE AGREEMENT NCCI-22
Computational Fluid Mechanics
Progress Report No. 18
For the Period December 15, 1987 to June 14, 1988
Prepared by
H. A. Hassan
Project Coordinator
Department of Mechanical and Aerospace Engineering
North Carolina State University
Raleigh, North Carolina 27695-7910
-CB-183022) CG«POIATJOK?.L flOID 388-24.915
KICfliKICS .trcgress Report , 15J Dec. 1987 - 14
Jun . 1S88 {Uorth Carolina Stele Univ.)
16 p CSCL 20D Unclas
G3/34 01U8116
https://ntrs.nasa.gov/search.jsp?R=19880015531 2020-03-20T07:07:46+00:00Z
The major accomplishments during this period are summarized in the
enclosed abstract which was submitted for the Aerospace Sciences meeting.
Other approaches to turbulence modeling are being pursued.
A One Equation Turbulence Model
for Transonic Airfoil Flows
R. A. Mitcheltree*
North Carolina State University , Raleigh, North Carolina
M. D. Salas*
NASA Langley Research Center, Hampton, Virginia
H. A. Hassan*
North Carolina State University , Raleigh, North Carolina
Abstract
A one equation turbulence model based on the turbulent kinetic energy
equation is presented. The model is motivated by the success of the Johnson-
King model and incorporates a number of features uncovered by Simpson's
experiments on separated flows. Based on the results obtained , the model
duplicates the sucesses of algebraic models in attached flow regions and out
preforms the two equation models in detached flow regions.
Introduction
This work is the first phase of an on going research effort designed to develop
turbulence-models for separated turbulent flows over transonic airfoils. It
'Research Assistant, Mechanical and Aeiospace Engineering, Student Member AIAA.
* Head, Theoretical Aerodynamics Branch, Transonic Aerodynamics Division, Associate
Fellow AIAA.
* Professor, Mechanical and Aerospace Engineering, Associate Fellow AIAA.
•.-'-86,;.-.' ' ' . -
is generally accepted that algebraic turbulence models are not suited to
describe non-equilibrium turbulent flows. This was clearly brought out by
the extensive work of Simpson and his associates (see reference 1 for a
summary ). The next logical step is to use a higher order closure model.
Unfortunately, the work of Patel, Rodi and Scheurer [2] showed that most of
the fc —c models and related two equation models are unable to predict some
simple flows. Moreover, the work of Coakley [3] showed that predictions
of two equation models are not much better than algebraic models in the
presence of separation.
The recent model of Johnson and King [4], which is based on a simplified
version of the tubulent kinetic energy equation, preformed well in regions
where non-equilibrium turbulent effects are important. However, it did
not always preform well in regions where equilibrium turbulent flow exists.
This negative result does not diminish the significance of their contribution.
Rather, it suggests that more physics exists in the turbulent kinetic energy
equation that is yet to be explored and exploited.
Because of the above, an effort was undertaken to develop a one equa-
tion turbulence model based on the turbulent kinetic energy equation for
both attached and detached flows. Simpson made two relevant conclu-
sions. First, scaling within and outside separated flow regions is not the
same. Second, the eddy viscosity assumption is incorrect in the backflow
region of a separating boundary layer. Thus, the model developed retains
the accepted form of the turbulent kinetic energy equation but adjusts the
•tubulent shear stress and the scaling depending on the local flow conditions.
Most of the previous work using one equation models employed the
same length scale for turbulent stresses and turbulent dissipation. This led
to numerical difficulties in the near wall region with the result that most
researchers lost interest in such a model. The difficulties can be avoided
by using two different length scales in the near wall region, one for the
ttrrbulent stresses and one for turbulent dissipation. Specification of the
length scales requires close attention to scaling near the wall.
The ,;Navier-Stokes code employed is based on that used by Swanson
and Turkel [5]. The turbulent kinetic energy equation is coupled to the
conservation equations and a four-stage Runge-Kutta time stepping scheme
is employed. The Reynolds averaged Navier-Stokes equations that use the
Johnson-King model or higher order closure models cannot be started from
a uniform flow condition. The usual procedure is to use an ijiitial solution
obtained from a flow solver that incorporates an algebraic turbulence model.
This innefficient starting proceedure is not necessary in the present work.
Approach
The equations governing turbulent flow past airfoils are the Reynolds aver-
aged conservation equations of mass, momentum and energy and the tur-
bulent kinetic energy equation. When Favre's mass averaging and indicial
notation are used the conservation equations are ;
(1)
„• = 0. (2)
(pE) „ + [i,. (pE + P) + ijj + ^ffc* - Ui (f0- - XXf)] „• = 0 (3)
where
 2
fij = fj. \Ui,j +Ujri - o^Jum,m (4)L o J
(5)
where p is the density, Ui is the mean velocity in the direction of Xj, E and
ft. are the total energy and enthalpy per unit mass, P is the mean pressure.
ft , A and T are respectively, the molecular viscosity and conductivity and
temperature. In attached flow regions, the turbulent stress and heat flux
are determined from the eddy viscosity (Bousinessq) approximation.
<8>
Prt is the turbulent Prandtl number. fj,t is the eddy viscosity and k is the
mean turbulent kinetic energy per unit mass.
For detached flow the above, relations are replaced by correlations obtained
from Simpson's experimental data.
The modeled turbulent kinetic energy equation is;
(p~k) „ + Ipujk - v (pic} „• -cfc— pu'fu'l (p~k) ,< ,.,- = -pu'lu'fi^ ~pe (10)
where c is the turbulent kinetic energy dissipation rate.
c/c is a constant (0.1). it is the dissipation length scale and v is the molecular
kinematic viscosity. In a one equation turbulence model the eddy kinematic
viscosity is given by
vt = C^ktp (12)
where CM is a constant and 1M is the turbulent length scale.
Simpson emphasized the importance of normal turbulent stresses in
separating flows regions. Because of this, the governing equations retain all
normal stresses.
The Length Scales
To complete the formulation of the model, the length scales must be de-
fined. The expressions are summarized here and a complete derivation
will be given in the paper. All variables are non-dimensionalized with the
chord being the characteristic length scale and (Poo/Poo)' the characteristic
velocity. Letting
Lt = Cty [l - exp (r*-A
LM = Ciy fl -
F(y) = y | u ,
then
tf = min(Lf,ymaa.)
tp — min (Lp, ymax)
where y^ox is the value of y at which F(y) is maximum, u> is the vorticity
and Re is the Reynolds number. The constants in the definition are
_s
«•/'"' «I — n/\s u
C^ = 0.09
K = 0.41
Ae = 2C,
Ap = 76.
AF = 47.47
Results and Discussion
The results presented here are for a Hate plate at zero angle of attack and
for an NACA 0012 airfoil at a variety of Mach and Reynolds numbers and
angles of attack. Fig. 1 shows flat plate results for a free stream Mach
number, MOO, of 0.5, Re = 106 and an angle of attack, a , of zero. As is
seen in the figure, good agreement is indicated for the skin friction, Fig. la,
velocity distribution in the near wall region, Fig. Ib, and the one seventh
power law, Fig. Ic.
The next set of calculations are for an NACA 0012 airfoil. All calcula-
tions employed a 160 X 79 C-grid. The normal spacing of the first point
off the wall was 2X1Q~B. The first case considered is that for M^ = 0.5,
Re = 2.89JT106 and a = 0.. Fig. 2(a) compares the pressure distribution
with the measurements of Ref. 6. Fig. 2(b) compares the skin friction
coefficient with a boundary layer calculation given in Ref. 7. As is seen
in the figure, good agreement is indicated. The difference in the skin fric-
tion coefficients at the leading and trailing edges are a result of much finer
resolution in the boundary layer calculation. Another comparison with the
experiment of Ref. 6 is given for M^ = 0.756 , a = 0. and R, = 4.01A'106.
Again excellent agreement with experiment is obtained.
All of the above cases are for a = 0. . Fig. 4 compares predictions of
this theory with the experiment of Ref. 8 for a = 1.49° and R* = 9.X106 ..
As is seen in the figure, both upper and lower surface presure distributions
are well predicted by the present one equation model.
Simpson noted that in the backflow region of a separating boundary
layer, the eddy viscosity assumption is incorrect. Because of this, Simpson's
measurements are used to develop correlations for the turbulent stresses in
the backflow region. Unfortunately, there is scatter in the data . Bacause
of this, the results presented in Fig. 5 ( M^ = 0.799 , R* = 9.XW* ,
a = 2.26° ) should be considered preliminary.
Details of the separated models together with comparisons for the RAE
2822 and the Cast 10 airfoils will be presented in the paper. Based on the
results obtained to date, it is concluded that one equation models, properly
formulated, are capable of out preforming two equation models at reduced
computational cost.
References
[1] Simpson, R. L. ," Two-Dimensional Turbulent Separated Flow," AGAR-
Dograph NO. 287 Vol.1, 1985.
[2] Patel, V. C., Rodi, W., and Scheuerer, G., "A Review and Evaluation
of Turbulence Models for Near Wall and Low^Reynolds-Number Flows,"
AIAA Journal , Vol. 23, 1985 , p.1308.
[3] Coakley, T. J., "Numerical Simulation of Viscous Transonic Airfoil
Flows," AIAA Paper 87-0416, Reno, Nev., 1987.
[4] Johnson, D. A., and King, L. S, u A Mathematically Simple Tur-
bulence Closure Model for Attached and Separated Turbulent Boundary
Layers," AIAA Journal , Vol. 23, No. 11, Nov. 1985, pp. 1684-1692.
[5] Swanson, R. C., and Turkel, E., " A Multistage Time-Stepping
Scheme for:the Navier-Stokes Equations," ICASE Report No. 84-62, Feb.,
1985.
[6] Thibert,J, J., Granjacques, M., and Ohman, L. H., NACA 0012
Airfoil, AGARB Advisory Report No. 138, Experimental Data Base for
Computer Program Assessment, May 1979.
[7] Miner, E. W., Anderson, E. C., and Lewis, C. H., " A Computer Pro-
gram for Two-Dimensional and Axisymmetric Nonreacting Perfect Gas and
Equilibrium Chemically Reacting Laminar, Transitional, and/or Turbulent
Boundary Layer Flows," NASA CR-132601, May, 1971.
[8] Harris, C. D., "Two-Dimensional Aerodynamic Characteristics of
the NACA 0012 Airfoil in the Langley 8-Foot Transonic Pressure Tunnel,"
NASA TM 81927, 1981.
Acknowledgments
This work is supported in part by NASA Cooperative Agreement NCCI-22.
The authors would like to express their appreciation to Dr. Roger Simp-
son for providing us with his tabulated measurement and to Dr. Charles
Swanson for allowing us to use his 1984 Navier-Stokes Code. One author
(H.A.H.) would like to acknowledge helpful discussions with Dr. Roger
Simpson and Dr. Tom Coakley.
x 10
20
10 -
5
3
— Correlation, M^ « 0.
Cf = 0.0592/ee^-2
O One Eq. Model
O O o o o o
105
Rea
Figure la. Turbulent flat plate flow ( M = 0.5, Re = 10
comparison of skin friction distributions.
106
1.0
0.8 —
0.6
0.4 —
0.2
0.0
0.0
r\ r\
0.2
— 1/7-th Power Law
O One Eq. Model
0.4
u/u«
Figure Ib. Turbulent flat plate flow ( M = 0.5, Re = 10 )
:' comparison of velocity profiles.
25—i
20-
15 —
10-
5 —
- Law of The Wall,
O One Eq. Model
w 0.
101 102 102
Figure Ic. Turbulent flat plate flow ( M = 0.5, Re = 106)
comparison of near wall velocity profiles.
-1 —
O Experiment, AGARD AR-138
— One Eq. Model
0 —
1 —
0.0 0.2 0.4
X/C
0.6
I
0.8
Figure 2a. Turbulent flow over a NACA 0012 airfoil ( M = 0.5.
Re = 2.89xl06 ): pressure distributions.
1.0
103
Boundary Layer Code
One Eq. Model
0.0 0 . 2 0 .4
I ' I
0.< O.I
x/c
1.0
Figure 2b. Turbulent flow over an NACA 0012 airfoil ( M = 0.$.
Re = 2.89xl06 ): skin friction distributions.
-1.5—1
-1.0
-0 .5 —
cp o . o —
0.5 —
1.0 —
O Experiment, AGARD AR-138
— One Eq. Model
1.5
0.0 0.2 0.4 0.6 0.8 1.0
'i{?•
X/C
Figure 3. Turbulent flow over an NACA 0012 airfoil ( M = 0.756J,
a= 0 *, Re = 4.01x10 ): pressure distributions.
-1
O Experiment
— One Eq. Model
0.0 0 .2 0.4
x/c
Figure 4. Turbulent flow over an NACA 0012 airfoil ( M = 0.7,
a =1 .49* , Re = 9.xlO ): pressure distributions.
1.0
-1 .5
-1.0
- 0.5
0 .0
0 . 5
1 .0
1.5
One Equation Model
— Coakley: q-u;
O Experiment
1
 I ' I ' I
0 . 0 0 . 2 0 . 4 0.6 0.« 1.0
x/c
t
Figure 5a. Turbulent flow over an NACA 0012 airfoil { M = 0.799,
qc=.2.26* , Re = 9.x]0 ): pressure distributions
— 1.5
- 1.0
- 0 . 5
0 .0
0.5
1.0
1.5
— One Equation Model
-— Coakley: Johnson-King Model
G Experiment
0.0 0 . 2 0 .4 0.6 O.I 1.0
x/c
Figure 5b. Turbulent flow over an NACA 00]2 airfoil ( M = 0.799,
ot = 2.26* , Re = 9.x]0 ): pressure distributions


Computational Fluid Mechanics

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server