Acta Mechanica 108, 219-223 (1995)

ACTA MECHANICA 9 Springer-Verlag 1995

Notc

The freezing property (stabilization law) of transonic flows* J. Zierep, Karlsruhe, Federal Republic of Germany (Received November 26, 1993)

Summary.It is well known that for transonic profile flow the local values of the fluid field freeze if M --+ 1. This is true in many cases even in second order. We give a simple heuristic proof for this behaviour. A control by the Prandtl-Meyer-expansion shows the limits of this property.

1 Introduction For transonic profile flow we have the important statement that the local values of velocity, Mach number, pressure . . . . in the neighbourhood of the profile remain constant if the Mach number at infinity (Moo) passes through unity [1]. More precisely this freezing property reads for the local Mach number at the profile M = M(x, M~o): =0

Moreover, in many cases the freezing property is even fullfilled in second order: = 0.

(1.1.2)

Using the transonic similarity parameter (r -= thickness parameter, 7 = cp/c~) -

1

Z -- .C273(7 + 1)2/3,

(1.2)

we can combine (1.1.1, 2) in the more rigorous form

M(x, Mo~) = M(x, 1) + 0(Za).

(1.3)

This important result was published shortly [2], referring to transonic experiments. However, one can find it already in [3] - [5]. The result had not become well known so far in the literature. This may be due to the fact that the proofs use complicate matched transonic asymptotic expansions not easy to reproduce by a reader. A more simple proof of this important result may be desirable. In addition we have to recognize that (1.3) for M~o _< 1 is only valid in front of the shock in the local supersonic region. Behind the shock things differ considerably. * Dedicated to Prof. Dr. Ing. U. Mfiller on the occasion of his 60th birthday.

220

J. Zierep

2 Discussion of the freezing property with the parabolic method We use the so-called parabolic method due to Oswatitsch [6] and combine it with the local linearization given by Spreiter [7]. By this the local Mach number M = M(x, Mo~) on the profile y = h(x)is [8]

M 2 _ - 1 _ gi/3(x, x*) = y 4" 1

3 rc(y 4" 1)

d

x*

d~ _

~

0

d~

dXl

,

(2.1)

where x* = x*(Moo) is the x-coordinate of the sonic point on the profile. We differentiate with respect to Moo at fixed x: 2M

aM

7 + 1 aM~

__ 1 g _ 2 / 3 ( X ' X*)

3

ag(x, x*) dx* Ox* dMoo"

(2.2.1)

Using (2.1) we have

Og(x, x*) dx* _ ax* dMoo

3 d ~r(? + 1~) ~xi

d~ ~/xi~

d~

0

. dxi dMoo"

(2.2.2)

XI=X*

If we take the limit M| -~ 1 we can use the definition of the sonic point on the profile for the parabolic method for sonic speed. For continuously curved convex profiles the first factor is

d

= 0,

0

(2.3.1)

/xi=x *, M~-~I

while at a convex profile edge the second factor is zero (the sonic line originates at the edge !): dx*

dMoo

- O,

(2.3.2)

Therefore (2.2.1) fullfills the freezing property (1.1.1). This result is already given in [8]. By differentiating (2.1) twice with respect to Moo we get 2 (OM~ z 2M a2M ? + 1 \aMos} + ? + 1 aM 2 -

1

2 ( a g aX*~ 2 9 g- 5m(x' x*) \ Ux* aMod

a2g ( dx* ,~2

4" ~g-a/3(x, x*) - ax .2 \dMooJ

4"

1

3

g-2/3(x, x*)

ag d2x *

ax* dM 2"

(2.4)

In the limit Moo ~ 1 the first and the third term on the right hand side of (2.4) vanish due to (2.3.1, 2), only a2g/ax .2 needs additional consideration. Due to (2.2.2) we have a2g ~x .2

3

d

d2

d-~

d~ .

(2.5)

The freezing property (stabilization Iaw) of transonic flows

221

Due to (2.3.1) the first factor in (2.5) vanishes in the limit M~ ~ 1. Therefore (1.1.2) and by this the freezing property is fullfilled for profile flow even in second order. According to (2.4) and (2.5) we recognize immediately that the third derivative ~?3M/c3M~ does not vanish for M~ ~ 1. It is remarkable that this simple approximate method fullfills the freezing property so good.

3 Consequences for the pressure and drag coefficients For the pressure coefficient % on the profile we have with M = M(x, M~)

_ cp

2 yM~

+-~M~ ~-1--

7-1 -1

+~-M~/

t .

(3.1)

Differentiating with respect to M~ and using (1.1.1, 2) leads finally to (~)M~-.

y+14 ( 1 - ~ )

8M~jM,_~

>0'

(3.2.1)

(Y + 1)2 1 + 3?)

2

%*

< 0.

(3.2.2)

The signs <> 0 are valid due to the expansion of the pressure coefficient for sonic flow %*: Cp* --

4 (Mre +~1

1) + 6(7 - 1) -~ q--l ~ (M -- 1)2 + ...

(3.3)

As a rule we have I%*f ~ 1. All together we get 4 (1 + 3L) ( M ~ - 1 / + . . . cp(x, Moo) = ~ + 1 (Moo -- 1) - 2 (7 + 1)2 + %*(x) 1

-

3+57 7 + 1 (M~ - 1) + ~

-

-

}

(Moo - 1)2 + . . . .

(3.4)

This agrees with [2] and after correction of misprints with [3]. Due to (3.2.1, 2) cp first increases at Moo = 1, reaches a maximum at slightly supersonic Moo in order to decrease again. Just the same is true after integration for the drag. This is confirmed completely by experiments [1]. For a wedge 0 _< x _< 1 with apex angle 0 we end up in reduced variables with (1.2)

(7 + 1)1/3 05/~

(Moo-l) c~ = C,v = 4 (Y + ])2/3 02/3 +cw*

= 2Z

1---(M~-1)+

1+37 2

(Y + 1)5/3 0 2 / 3

(M~o - -

1) 2 + ...

(Moo-l) 2+

02/3( 1 + 27) (7 + 1)1/3 + "" { 02/3 1 Z2~4-/3(2+37) + C,,* 1 -- Z (7 + 1)1/~ + ~ (7 + 1)2/3 +

"

- - •2

} ....

(3.5)

222

J. Zierep

4 Control of the freezing property by Prandtl-Meyer-Expansion It is desirable to control the discussed freezing property by using an exact solution of the gasdynamic equations. To do this we study an expansion of a transonic slightly supersonic flow along a curved convex wall (Fig. 1). Along the characteristics r / = const, ~ = const we have the following relation connecting the flow direction angle O and the Mach number M (c~ = Mach angle) [81: ~)-- 1/1 2

i

0=+__ / ~ - _ - ~ 7 / ~ l { a r c t g J T - l ( M 2 - 1 ) -

d,

= const, dxx = tg (9 - a)

g {arctg ~ -

(4.1)

1 - arctg ~ } . dy

= const, dxx = tg (O + c0, In transonic approximation 2 1 O = + 3 7 + 1 { ( M ~ - I ) 3 / 2 - ( M 2 - 1 ) 3/2} + ....

(4.2)

We proceed along t / = const from 1 --+ 2. We keep 0 = 0 2 fix and study the dependence of M = M z as function of M~. After this we come to the limit M~ ~ 1. In this case t / = const steepens but this right hand running characteristic originates in any case from the domain of constant state M --- M~ >_ 1, O = O~ = 0:

c3 (4.2) 90 - 2 1 OM---~ "" -3 7 + 1

3 M ~ ( M 2 ~ _ 1),/2 _

r 3M2(M2 2 _ 1) 1/'2O--M~-~J ~

(4.3.1) (~M2 _ aMo~

/~-2 _- 1 M~o ~/il

- 1 i2

with (4.2)

I/M2

2 -

1 =

-

1 ) 3/2 -

3

+

1) 02} 1/3 .

%/ /

/

/

/

~

M2.~2"~ Fig. 1. Prandtl-Meyer-expansion along a convex curved profile

(4.3.2)

The freezing property (stabilization law) of transonic flows

223

In the limit Moo ~ 1 (0M2" ~

=

'2 l/Moo 1

\OMooJM.~l

(M 2 -- 1)

-- ~_ (7 +

11<}i, 3

IM,~I

=o.

The freezing property is fullfilled in first order (1.1.1). The second derivative leads to 02 0

2 (4.2) Moo

02M2

9

~I2V M22-1

0M2.o

1~-~ - 1 + ~

j

M22(M22 -- 1) T.2.2

7( ~ O M 2

MOO~/M~-ll]/Ma2--10M

|

M2 2 OM2"l Ml/-~227_ I O M ~ j

M 2 2 ( M 2 2 - 1) In the limit Moo + 1 with (4.3.1) (02M2)

~

(1

1

~

)

-+

oo.

(4.4.)

That means that thefreezilN property is in this case no~ futlfilled in second order. This may be due to the fact that upstream is a domain of constant state: M -= Moo > 1, 0 - ,9o~ = 0 with no asymptotic fading away to infinity. The latter behaviour is important especially for all matched transonic asymptotic expansions and for the discussed parabolic method. With other words: the Prandtl-Meyer-expansion is a global uniformly valid solution that fullfills the freezing property in first but not in second order. The matched asymptotic solutions for profile flow on the other side fultfills the freezing even in second order apparently under additional conditions what is not always respected in literature.

References

[1] Liepmann, H. W., Roshko, A.: Elements of gasdynamics. New York London: John Wiley 1957. [2] Cheng, K. : A novel transonic flow feature-local deflection characteristic at Mach Number One. Proc. Int. Congr. Fluid Mech. II, Beijing 1993, pp. 282-286. [3] Guderley, K. G.: Theory of transonic flow. Oxford London New York Paris: Pergamon Press 1962. [4J Diesperov, V. N., Lifshitz, Yu. B., Ryzhov, O. S.: Stabilization law and drag in transonic range of velocities. Symp. Transsonicum II, pp. 156--164. Berlin Heidelberg New York: Springer 1976. [5] Cole, J. D., Cook, L. R : Transonic aerodynamics. Amsterdam New York Oxford Tokyo: North-Holland 1986. [6] Oswatitsch, K., Keune, F.: Proc. of the Brooklyn Polytechnics Conf. on High-Speed Aeronautics 1955, pp. 113. [7] Spreiter, J. R., Alksne, A. : Proc. of the Third US National Congr. Appl. Mech., New York 1958, pp. 827. See also: NACA Report 1359 (1958). [8] Zierep, J.: Theoretische Gasdynamik. Karlsruhe: G. Braun 1976.

Author's address: Prof. Dr.-Ing_ Dr. techn. E. h. J. Zierep, lnstitut f/ir Str6mungslehre und Str6mungsmaschinen, Universit~t (TH) Karlsruhe, D-76128 Karlsruhe, Federal Republic of Germany