An unsteady finite difference procedure was used to calculate the steady inviscid flow over an 11% thickness ratio NASA supercritical profile of LWP 505 at M sub infinity = 0.80 and alpha = 0. An attempt is made to include the viscous effects using a modified form of Head's entrainment method to calculate the turbulent boundary layer. The attempt to predict the viscous effects using the compressible form of Head's integral method with a modified auxiliary equation for the form factors was unsatisfactory. Though a reasonably separation bubble was established on the lower surface, a grossly exaggerated displacement effect resulted downstream of the shock on the upper surface. There clearly is substantial further effort required to evolve a satisfactory boundary layer procedure, which must then be coupled in a still unproven manner with the inviscid procedure

Magnus, R.

Yoshihara, H.

English

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3 NASA CR-111888
CALCULATION OF THE PLANAR SUPERCRITICAL FLOW
OVER A NASA SUPERCRITICAL PROFILE (U)
By H. 'Yoshihara and R. Magnus
CLASSIFICATION CRANGE
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(NAFA—CF{-111888) CALCULA'iION OF THE PLANAR	 479-15904
SUPERCRITICAL FLOW OVER A NASA SUPERCRITICAL
f ROFILE ((;eneral DyUa©icS/CCnvair) 	 11 FA02/MF A01	 CSCL 01A	 Unclas0/02 35056
Prepared under Contract No. NA°? -9308 by
CONVAIR AEROSPACE DIVISION OF GENERAL DYNAMICS
San Diego, California
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NATIONAL AERONAUTICS AND SPACE ADMINiSTRATIU.q
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UNCLASSIFIED
CALCIJLATION OF THE PLANAR SUPERCRITICAL FLOW
OVL'R A NASA SUPERCRITICAJ. PROFILE
By H. Yoshihara and R. Magnus
Convair. Division of General Dynamics
San Diego, Calif.
SUMMARY
An unsteady finite difference procedure is used to calculate the steady
inviscid flow over a 1.1% thickness ratio NASA supercritical profile of
LWP 505 at M = 0.80 and a - 0. An attempt is made to include the viscous
effects using a modified form of Head's entrainment method to calculate the
turbulent boundary layer. The results of the inviscid calculations compared
to the experimental results of Whitcomb indicate a significant influence of
viscosity, though the comparison is partially obscured by t.h-_ unknown wind
tunnel wall interference effects. The effects of viscosity were manifested
in three ways. Firstly, the experiments clearly indicated the presence of
shock-induced separation at the base of the shock. The resultant flow dis-
placement effect produced a significant upstream movement of the shock.
Secondly, the displacement effect of the boundary layer, primarily on the aft
upper surface, reduced the effective aft camber which in turn resulted in
changes of the "plateau" pressures on both the upper and lower surfaces charac-
teristic of a decreased flap effect. Thirdly, the severe adverse gradients
on the aft portion of the lower surface resulted in a significant thickening
or a local separation of the boundary layer that led to decreased over-
pressures.
The attempt to predict the viscous effects using the compressible form
of Head's integral method with a modified auxiliary equation P the form
factors was unsatisfactory. Though a reasonable separation buc..jle was es-
tablished on the lower surface, a grossly exaggerated displacement effect
resulted downstream of the shock on the upper surface. Here in these explora-
tory calculations the experimental pressure distribution was simply used to
check the boundary layer equations. There clearly is substantial further
effort required to evolve a satisfactory boundary layer procedure, which must
then be coupled in a still unproven manner with the inviscid procedure.
UNCLASSIFIED
^.w
UNC! ASSIFIED
INTRODUCTION
The calculation of the planar supercritical flow over airfoils is uni-
versally recognized as a horrendous task. Sources of the difficulties are
manifold. At the high Reynolds numbers of pertinence the resulting boundary
layer is turbulent, and is sufficiently thin that the classical boundary
layer concept would be applicable were it not for the presence of the termi-
nating shock wave. The presence of the shock wave can so drastically alter
the "effective" shape of the profile through the displacement effects of the
boundary layer that the inviscid solution no longer serves directly as a
viable basis to obtain the pressure distribution. In short the inviscid arid
boundary layer flows are strongly coupled. Despite the strong coupling with
proper ingenuity it would not be a hopeless task to compute the coupled flow
if methods were available to calculate both the inviscid f'le ,,r over a given
shape, and the turbulent boundary layer flow, possibly separated, in the
presence of a prescribed pressure distribution.
In the present study the objective will be to carry out the inviscid
calculations for a prescribed airfoil, namely the 11% thickness ratio NASA
supercritical profile of LWP 505, at a
	
0
= 0 and M = 0.80, and attempt to
extend the results to include the viscous effects.
THE INVISCID CALCULATIONS
To compute the inviscid flow the u:.stead.y finite difference procedure
as described '-n Ref. 1 will be used. Isere the coexistence of ;subsonic and
supersonic flows with their highly differing mathematical characters in the
steady state makes the unsteady approach more tractable than the steady from
a calculational point of view, since in the latter representation the character
of the equations remains hyperbolic in both the subsonic and supersonic flow
regimes.
In Figure 1 is shown the resulting Mach wave pattern for the supercritical
profile at M = 0.80 and a = 00 , whereas in Figure 2 the resulting chordwise
distribution Hof the pressure is shown. Also shown in Fig. 2 are some experi-
mental results as obtained by Whitcomb (Ref. 2) for several geometric value:.
of a. The pressure distributions for the several values of a are shown
here since the precise value of the angle of attack is not known because of
the disturbing influence of the wind tunnel walls. If tests had been carried
out additionally for geometric a of a	 o	 'y	 	 pproximately 2 , the plateau values
F	 `t
M!
UNCLASSIFIED
	 2
UNCLASSIFIED
of the pressure on both the upper and lower surfaces would agree closely with
the calculated result. The immediate conclusion might then be that there is
a minus 20 correction in (Y due to wall effects for this case. That this
may not be a completely correct conclusion may be suggested by the lowness
of the trailing edge pressure, and the fact the shock pressure rise falls
significantly short of that for the normal shock. The latter clearly indicates
the presence of a separation ')ubble immediately downstream of the shock which
then leads to a significant boundary layer displacement effect in the vicinity
of the upper surface trailing edge as manifested by the depressed trailing
edge pressure. The added displacement effect of the boundary layer here gives
rise to a lessening of the effective trailing edge angle; that is, a lessening
of the aft camber. This in turn leads to increased plateau pressures on the
upper side and decreased plateau pressures on the lower side as would result
from a decreased flap effect. In short, the modifications of the pressure
distribution due to viscous effects on the airfoil surface are of the f^eme
general character as that due to wall interference effects.
Aside from the viscous effects just described a more serious consequence
of the separation bubble is that the shock wave at the surface is now oblique
instead of normal as in the unseparated case, and the resultant shock pressure
rise will then be significantly less than the normal shock pressure rise. As
a result the shock wave will move upstream to a position sufficiently far up-
stream to allow a sufficient "run" of su b sonic recompression to attain, the
required trailing edge pressure. suite apart from the above effect there is
a countering influence on the shock movement by the depression of the trailing
edge pressure. Thus, for a depressed trailing edge pressure there would be
less subsonic recompression required, and as a result the shock wave would not
have to migrate as far upstream as for a higher trailing edge pressure.
Finally in Fig. 2 it is seen that there is a significant depression of
the overpressure: on the lower aft surface caused by a separation or a thicken-
ing of the boundary layer resulting from the severe adverse suberitical pressure
gradient caused by the aft camber.
The above results clearly indicate that inviscid results must be supple-
mented by the addition of the viscous effects.
VISCOUS CALCULATIONS
There presently does not exist a :suitable procedure to calculate a
turbulent boundary layer in the presence of a prescribed external pressure
URLASSiFIED
	 3
UNCLASSIFIED
distribution containing adverse pressure gradients sufficient to lead to
separation. An attempt will be made however to extend an existing method for
this purpose. Of' the many procedures one of the more promising methods is
due to Head (Ref. 3) and Green (Ref. 4), and this procedure will be adopted
for our further considerati , in. Another widely used procedure is due to Nash
and Macdonald (Ref. 5). The latter method is an extremely simple procedure
and is based upon the use of the classical streamwise momentum integral
equation which results in the following ordinary differential equation for
the momentum thickness e:
dtnu
dx
-- (H+2-Me2 ) e ^ e +2Cf	 (1)
Here If is the usual form factor If	 M and u are the Mach number
and velocity in the external inviscid flow, and C is the wall skin friction
coefficient. In the Nash-Macdonald method Eq. (1) issupplemented by two
additional equations in order to obtain a fully determined system, assuming
for the time being that u e (x) and Me (x) are prescribed. First of the
required equations expresses C as a function of the Reynolds number based
upon e, and one or more of the other dependent variables appearing in (1).
A suitable form is that given in Ref. 5, or the well known Ludwieg-Tillmann
formula in one of its many variant forms. The second auxiliary relation
suggested by Nash and Macdonald is based upon the assumption that the flow
is locally an "equilibrium boundary layer;" that is one that fulfills the
condition
G(TT) = 6.1 (Tr + 1.81) 1/2 - 1.7	 (2)
where
	
H	
d d,n 
u 
and
	
J` f,/ 2	 dx
112(H - ^.)
G = ( C^2 )	
`\ H
with
H ￿ 	 H + 1	 - 1
1 + 0.178 M 2
e
UNCLASSIFIED
	 4
UNCLASSIFIED
.,br a given u (x) and M (x) the differential equation (1) for 8 can
be integrated usinge (2) and the eskin friction law, and one can obtain quite
simply the chordwise distribution of e(x), C f(x), as well as 8 (x). For
an unseparated boundary layer use of the above procedure has been found to
yield a satisfactory distribution for A(x), but is inadequate to yield an
acceptable distribution for 8 * (x) (Ref. 5). (The general experience among
most integral methods has been that all procedures appear to yield a reasonable
distribution for e, but the determination of 6 * is a much more delicate
undertaking.) For the separated case it can be expected that the Nash-
Macdonald procedure would simply be grossly inadequate to predict 8*(x).
(Note that the authors of Ref. 5 do not claim the applicability of their
procedure to the .3eparated case).
A more sophisticated approach is the use of the compressible form of
Head's procedure (Ref. 3) given by Green (Ref. 4). The basic equations in
this procedure in addit-on to Eq. (1) include the integral form of the conti-
nuity equation, namely,
dL	
dtnu
dX - F
e + ( Mew - 1) A	
dx 
e
Here	 b
0 = 8 - b ,	 f p u dy , and
o e e
(3)	 ^
v Fe = - u +	 (Entrainment function)dx
e
where p is the density, b the boundary layer thickness, v the transverse
velocity, and the subscript e denotes the local inviscid value at y = b.
In the Head-Green method the basic differential equations (1) and (3)
must be supplemented by three additional equations. These are first a skin
friction law, for which we use the Felsch modification of the Ludwieg-Tillmann
law. (Ref. l). The second is an expression; for the entrainment function F
and the third is a relation of the form H = H(H l ) where H = A/6. It is e
in the last relation that an attempt was made to derive an expression valid
for the separated case using available experimental data. For this purpose
the experiments of Green (Ref. 4) and Seddon (Ref. 6) were used to derive the
UNCLASSIFIED
	 5
UNCLASSIFIED
relation H = H(Hl ) despite the fact that in the latter experiments it can
be clearly shown that there is serious departure from the planar condition
due to the influence of the sidewall boundary layer displacement effects
(see Ref. 1).
To test the resulting set of equations calculations were carried out where
an existing experimental pressure distribution from LWP 505 was used. The
results showed a reasonable a * upstream of the abrupt adverse gradients
on both the upper and lower surfaces as would be expected. On the lower
surface the abrupt recompression led to a local separation bubble that appeared
to be reasonable, but downstream of the shock on the upper surface the resulting
distribution of 8 grew excessively towards the trailing edge.
This unreasonable growth of a* as the trailing edge was approached
could be directly traced to the inadequacy of the auxiliary relation H = H(H1)
used which yielded near-singular values of H for the values of }fl that
were being predicted.
Despite this shortcoming the resulting a * , with the nreasonable values
arbitrarily modified, was used to obtain a modified airfoil shape, and the
	
1
inviscid flow was recalculated in order to obtain a qualitative effect of
the boundary layer displacement. The results indicated the change of the
pressure distribution described earlier for the decreased aft camber.
These highly tentative attempts to incorporate tha viscous effects clearly
point out areas of future effort required to improve the procedure. With a
reasonable amount of additional effort it is not excessively optimistic to
expect that a viable boundary layer procedure can be eventually evolved to
handle the specialized case of shock-induced separations on supercritical
airfoil flows.
Convair Division of General Dynamics
San Diego, California, leg September 1970
UNCLASSIFIED
	
6
1 4
1 2
I#
1 0
0.8
^ 0.
OA-
0 2
-0.2
r
SONIC LINE	 I	 11	 !	 l	 1	 /, l ! I
fill
/	 I	 l	 1/	 '	 /	 // // / /' 11 l !	 ^
	
I	 f	 / 1	 1/	 /'	 r /	 / // ,	 / / l l l
	
I	 !	 '
/
--1--- -- - ^	 l__ ,_ _ 1	 1	 1	 I	 I	 I	 1	 I
0.0	 0.2	 0.4
X/C
	
0.6
	 0.8
	 1.0
Figure 1.. Mach Wave Pattern for the Basic NASA Profile
at Mm =0.80 and a =0.0°
0.4
0.°
0,6
0,
M 1,0	 Off1 I
0,8
0.9
P	
l,n
1,1
1.2
M m 0,80
SYMBOLS
NUMERICAL —0- a = 0.0
FXPERIMENTAL-Ow a GEOM.	 1,5°(NASA)
..^^. a GEOM. = 1.0,.
- 'a* ' aGEOM =0, 5°
1.3
1,4
1, `J
PSTAG
	
1,6 •	 ^	 ^	 ^	 ^	 I	 ^	 I _	 I	 I
	
0,0	 0,1	 0.2	 0,3	 0,4	 0,5	 0.6	 0,7	 0.8	 0,9	 1,0
X/C
Figure 2. Calculated and Experimental Pressure Distributions for the
Basic NASA Profile, M. = 0.80
8
UNCLASSIFIED
FEFERENC
1. Magnus, R.; .vid Yoshihars, H.: Flow Over Airfoils n the Transonic
Regime - Frediction of Buffet O:e^t. AFFD:. Report TR-70 -16, March 1970.
2. Whitcomb, R.: Basic Results of a Wind Tunnel Investigation of an In-cegral
Supercritical Airfoil Section. NASA Langley Report LWP-505, November 1967.
3. Head, M.: Entrainment in the turbulent Boundary Layer. ARC R&M No. 3152,
September 1958.
4. Green, J. E.: The Prediction of Turbulent Boundary Layer Development
n Compressible Flow. J. of F1. Mechs., Vol. 31, Part 4, 1968.
5. Nash, J.; and Macdonald, A.: The Calculation of Momentum Thickness in a
Turbulent Boundary Layer at Mach numbers up to Unity. ARC CP No. 963,
1967.
6. Seddon, J.: Flow Produced by Interaction of a Turbulent Boundary Layer
with a Normal Shock Wave. RAE TM Aero 667, 1960.
UNCLASSIFIEfD--
9
a
-.rte
Response to TO 77-376	 `	 '	 October 28, 1976
As of the moment, statistics on the classified document, is ti,is technology are as
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Responsible
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Calculation of the planar supercritical flow over a NASA supercritical profile

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