multiple solutions of the transonic potential flow equation

8/17/2019 Multiple Solutions of the Transonic Potential Flow Equation

1/7

MULTIPLE SOLUTIONS OF THE TRANSONIC POTENTIAL

FLOW EQUATION

John Steinhoff* and Antony Jameson**

Grumman Aerospace Corporation

Bethpage, New York 11714

Abstract

The two-dimensional transonic potential flow

equation, when solved in discrete form for steady

flow over an airfoil, has been found to yield more

than one solution in cer tain band s of angle of

attack and Mach number. The most str ik ing ex-

ample of t hi s

s

th e appearance of nonsymmetric

solutions with large positive or negative lift, for

symmetric airfoi ls at zero angle of attack. The

behavior of the se anomalous solutions is exam-

ined as grid size s varied by large factors and

found to be not qualitatively different from that

of %ormalrf solutions (outside th e nonuniqueness

band). Thus it appears that the effect is not due

to discretization error, and that the basic tran-

sonic potential flow partial differential equation

admits nonunique solutions for certain values of

angle of attack and Mach number.

Introduction

The full potential equation is widely used for

computing transonic flow over aircraf t. It ex-

presses conservation of mass, neglecting effects

due to viscosity, vorticity and entropy produc-

tion. For flow without massive separated regions,

where the shocks are not too strong, the equation

is a good approximation to the Navier-Stokes equa-

tions outside of a thin bo undary layer and wake

region. In combination with boundary layer

methods, it has been used to give very accurate

solutions for flow over airfoils in two dimensions.1

Recently, the authors discovered that in

cert ain bands of angle of attack and Mach num-

ber, the full potential equation, when solved in

discrete form for steady flow over airfoils, yields

multiple solutions. These solutions have ver y dif-

fere nt values of lift and drag . Preliminary resu lts

appear in Ref 2 ) .

The purpose of this paper is to present re-

sult s of numerical experiments designed to tes t

whether the non-uniqueness appears a s a result

of the discretization procedure, or whether the

continuum problem admits a corresponding non-

uniquen ess. Even if thes e solutions are caused

by discretization error, it may be important to

know that a small perturbation can result in a

very different solution.

@

The paper consists of three basic par ts. In

the first, details of the difference scheme are

described. In the second, results ar e presented

for a symmetric Joukowski airfoil at zero angle

of attack

a)

and fixed free-stream Mach number

(M,) Here, in addition to th e expected sym-

metric solution, two (mirror-image) unsymmetric

solutions appear with large abs olute value of lift

*Staff Scientist, Research Department

**Professor, Dept. of Mechanical and Aerospace

Engineering, Princeton University

Copyright American Instituteof Aeronautics and

Astronautics Inc. 1981 All rights resewed.

coefficient (CL). In this section we attempt to

validate these solutions by checking that bound-

ary conditions and the Kutta condition are satis-

fied, and by st udying the variation of the solu-

tions

as

the computational grid and artificial

viscosity are varied. In the third p ar t, the be-

havior of the llanomalous so lutions is stud ied as

M, and a are varied.

Difference Scheme

In a curvilinear coordinate system, the full

potential equation can be written3

where

h

=

det

H)

The isentropic relation for th e density

where q is the speed and

y

is the ratio of specific

heat s. The basic transformation from physical

(x, y) to computational

X , Y

frames is ex-

pressed by the matrix

The physical velocities ar e derived from a

potential,

and the contravariant velocities required for the

transformed flux balance equation are


2/7


3/7

tion on grids with

96

x

2 4 , 1 2 8

x

32 , 192

x

4 8 ,

256

x

6 4

and

384 192

cells. For comparison, CL

is also plotted for the same airfoil at M m

. 7 5 ,

2 0 ,

where only one solution was found.

It can

be seen that CL converges at comparable rates for

the two solutions.

The reason tha t CL decreased

as the grid was refined for the

a 0

solution was

that the lower shock, which was weaker than the

upper one, became stronger and moved backward,

increasing the magnitude of the negative contribu-

tion of the lower part of the airfoil to CL.

The up-

per shock did not change as much as the grid was

refined. For the

a

= O case, there was only a

single shock located at the upper surface, which

moved backward as the grid was refined.

The artificial viscosity in these calculations

had the second order form defined in the previous

section.

It was verified that the unsymmetric

solutions could also be obtained with the first order

accurate form of artificial viscosity obtained by

setting

=

0 .

In order to check whether the solution is a

peculiarity depending on the form of the gr id , cal-

culations were also performed on

a

''C mesh gen-

erat ed by mapping to parabolic coordinates. The

solution is displayed in Fig. 6 for a grid with

128

x

32

cells. With this type of mesh, the cell

width near the trailing edge was 400 times the width

on the 256 x 6 4 0 mesh. It can be seen that the

unsymmetric solution persists.

Hysteresis

The unsymmetric solutions have been found

only in a narrow Mach number band.

In the case

of the

1 1 . 8

hick Joukowski airfoil at zero angle

of attack, only the symmetric solution could be

found below

. 8 2

and above

- 8 5 .

The lift coeffi-

cient for the unsymmetric solution at

0

is plot-

ted in Fig. 7 for Mach numbers between these

Limits. In this band th e non-uniquenes s was as-

sociated with a hysteresis.

In Fig.

8 ,

CL as a

function of i is plotted for a series of Mach num-

bers . The curves for Mach number less than

. 8 2

show th e expected behavior. Above

. 8 2 ,

on the

othe r hand , we find the hysteresis. On the

. 8 3 2

Mach number curve, the two intercepts of the

CL axis at

a =

0 correspond to the two unsym-

metric solutions discussed in the previous section.

Star ting from the upper righ t hand part of this

curve, which corresponds to the upper shock

being close to the trailing edge, we can generate

the rest of the curve by incrementally de-

creasing an d, at each st ep , computing a new

solution from the previous one. Once we de-

crease a below about

.14O

th e solution flips to

the negative CL one. We can then generate this

part of the cur ve in the same way. The sym-

metric solution (CL =

0 ,

=

0)

proved to be un-

stable to small perturbations, and went over to

either t he positive or negative CL

a

=

0)

solution.

Below the Mach number where hysteresis ap-

pears, it can be seen that the change in CL for a

small change in a at = increases as we increase

M and becomes large a s we approach a critical

value, Merit .

Thinking of a as a function of CL, we see

that the slope of

a

vs. CL at CL

=

0 goes smoothly

to zero at Merit, and th ere i s the possibility th at

it can become negative beyond Merit, where the

hysteresis is found.

The hysteresis occurs in a rather narrow

band of angle of attac k. If this band were to be-

come smaller as the mesh was refined, there is a

possibility that in stead of a non-unique solution

there would be a rapid switch from large negative

to large positive Cr as the angle of attack passed

through zero.

Refinement of the mesh did not in-

dicate any significant nar rowing of th e band, how-

ever. It s width was also insensi tive to th e far field

stre tchi ng and placement of the o uter boundary.

Also, the value of Merit was not sensitive to mesh

refinement or far field stretching.

A final result is presented in Fig.

9

Values

of CL vs . for an RAE

2822

airfoil at

. 7 2 5

Mach

number were computed using a C mesh with the

fir st ord er form of artificial viscosity. A similar

hysteresi s band can be seen. Similar resu lts have

also been found for an

NY U 8 2 - 0 6 - 0 9

airfoil.

Here,

a conservative finite differen ce rath er th an finite

volume scheme was used (details of this scheme

are presented in Ref.

5 ) .

Conclusions

Multiple solutions of the disc rete full poten-

tial equation have been found for steady two di-

mensional flow over airfoils . The most st riki ng

example is the appearance of unsymmetric solu -

tions with large positive or negative values of

lift for a symmetric airfoil at zero angle of attack.

This occurs only in a narrow band of Mach num-

ber, between

. 8 2

and

. 8 5

in the case of an

1 1 . 8

thick Joukowski airfoil.

Extensive numerical experiments, including

the use of alternative g rids and extreme grid re-

finemen t, have confirmed the persi stence of these

solutions. Hence, it appears likely that they

actually correspond to a non-uniqueness of the

continuum problem, and are not a consequence

of discretization er ro r. Since the rearward shock

wave in the unsymmetric solution is not quite

at the trailing edge, it seems that the non-uni-

queness is not caused by interferen ce between

a shock wave at the trailing edge and the Kutta

condition.

The multiple solutions may have a physical

counterpart. The Mach number band in which

they appear is just the band in which an airfoil

of this thickness typically experiences buffeting,

with the upper and lower shocks alternately

reaching a forward and rearward position similar

to our positive and negative lift solutions. While

buffeting may be triggered by boundary layer

separation, this raises the question of whether

an instability of th e outer inviscid part of the

flow may also be a contributing facto r to thi s

phenomenon.

The non-unique solutions were found by

a sophisticated iterative method, and it is not

known whether they correspond to stable equi-

librium points of t he tr ue time dependent equa-

tion. Also, it is not known whether they are

only associated with the potential flow approxi-

mation, or whether a similar non-uniqueness can

also occur in solutions to the Euler or Navier

Stokes equations. An investigation of these


4/7

questions would shed more light on the possible

physical significance of th is phenomenon.

References

1

Melnik R W Chow, R and Mead,

H .R

Theory of Viscous Transonic Flow over

Airfoils at High Reynolds Number, AIAA

Paper 7-680, 1977.

2.

Steinhoff, J.S. and Jameson, A., Non-

Uniqueness

in

Transonic Potential Flows ,IT

Proc. of GAMM Speciali st Workshop fo r

Numerical Methods

in

Fluid Mechanics,

Stockholm, September 1979.

3.

Jameson, A. and Cau ghey , D.A., A Finite-

Volume Method for Transonic Potential Flow

Calculations ,Ir Pro c. of AIAA 3rd Computa-

tional Fluid Dynamics Con feren ce, pp. 35-54,

Albuquerque,

N

.M., June 27-29,

1977.

4.

Jameson

,

A.

,

A Multi-Grid Scheme for

Transonic Potential Calculations on Arbitrary

Grids

,

Proc. of AIAA 4th Computational

Fluid Dynamics Conference, pp

122-

146,

WiUiamsburg, Va. , Ju ly 23-25, 1979.

5.

Bauer, F., Garabedian, P., Korn,

D .

and

Jameson , A . , Supercritical Wing Sections

I

Sp rin ger Verla g, New Yo rk, 1975.


5/7

MQCH =

0 832 ALPHA 0 0

MACH

0 832 RLPHR 0 0

Fig 1 Symmetric Solution

Fig 2 Unsymmetric Solution


6/7

Fig omputational Grid

0 100 200 300

F I N E G R l O I T E R T I O N S

Fig 4 ResidualDecay

Fig 5 Li f t onvergence


7/7

MACH

0 840

AI PHA 0 0

Fig

6

Unsymmetric Solution

Parabolic Coordinates

Fig Hysteresis

Fig

8

Unsymmetric Solution

Fig 9 Hysteresis R E

2822

r

0

Airfoil

multiple solutions of the transonic potential flow equation

Documents