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
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 JohnsonKing 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 equation turbulence model based on the turbulent kinetic energy equation for

both attached and detached flows. Simpson made two relevant conclusions. 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 averaged conservation equations of mass, momentum and energy and the turbulent 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 defined. 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 calculations 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 friction 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," AGARDograph 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 Turbulence 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 Program 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 Simpson 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