CONCLUDING REMARKS

1. The effect of t e m p e r a t u r e on t h e inviscid stability of two- dimensional wake flows is both stabilizing and destabilizing. A s the t e m p e r a t u r e i n c r e a s e s , the c r i t i c a l M a c h number i n c r e a s e s , and the r a n g e of M a c h n u m b e r s o v e r which subsonic d i s t u r b a n c e s c a n e x i s t a l s o i n c r e a s e s . However, as long a s the r e l a t i v e M a c h number i s below the c r i t i c a l M a c h number t h e n e u t r a l inviscid wave number will d e c r e a s e d t h i n c r e a s i n g t e m p e r a t u r e .

2, The numerical calculations indicate t h a t a heated wake will be m o r e s t a b l e than a cool one if the r e l a t i v e M a c h number i s l e s s than the c r i t i c a l M a c h number. F o r a typical hypersonic blunt body wake, using G a u s s i a n p r o f i l e s f o r t e m p e r a t u r e and velocity i n the Dorodnitsyn-Howarth variable, the maximurn d i m e n s i o n l e s s spatial amplification r a t e is constant i n the downstream direction and o c c u r s a t one p r e f e r r e d frequency, T h i s r e s u l t i s s i m i l a r to t h a t f o r the i n c o m p r e s s i b l e f l a t plate wake.

3. The inviscid stability problem f o r a x i - s y m m e t r i c c o m p r e s s i b l e wake flows is d i r e c t l y analogous t o the two-dimensional problem i n a

t r a n s f o r m e d orthogonal velocity space, except f o r a delta function singularity a s s o c i a t e d with the Reynolds s h e a r stres's n e a r the c r i t i c a l point. T h i s s t r e s s is a l w a y s a destabilizing influence. It i s a l s o found that a n e c e s s a r y and sufficient condition f o r the existence of n e u t r a l subsonic d i s t u r b a n c e s is that f o r s o m e w =

-

^c_R

<

_'/M

[ i n a s y s t e m fixed i n the fluid a t r e s t ]

,

the gradient df the density- vorticity product i n a c e r t a i n d i r e c t i o n m u s t vanish.

4. F o r i n c o m p r e s s i b l e axi- s y r n m e t r i c wake flow (using a Gaussian), the only m o d e s that a r e unstable a r e the n 2 1 and n

=

m o d e s . F o r slowlying varying t e m p e r a t u r e p r o f i l e s t h e s a m e m o d e s a r e unstable. However, f o r a "top-hat" t e m p e r a t u r e profile, the n = 0, 1 m o d e s a r e unstable. By physical a r g u m e n t s i t is shown t h a t t h e n = 1 mode should be the m o s t unstable mode f o r wake-type flows.

108 REFERENCES

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1959.

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pp. 731-742, November, 1951.

26. Pai, Shih-I. : On the Stability of a Vortex Sheet i n a n Inviscid

C o m p r e s s i b l e Fluid. J o u r n a l of the Aeronautical Sciences, Vol. 2 1, No. 5

,

pp. 325-328, May, 1954.

27. Pai, S. I. : On t h e Stability of A x i s y m m e t r i c a l Flows. G e n e r a l E l e c t r i c M i s s i l e and Space Division, Space Sciences Laboratory, R62SD75, July, 1962.

28. Pai, S. I. and Li, H. : On t h e Stability of A x i - s y m m e t r i c a l Wakes of a Binary Mixture of C o m p r e s s i b l e Fluids. G e n e r a l E l e c t r i c M i s s i l e a n d Space Division, Space Sciences Laboratory, R62SD79, August, 1962.

29. Savic, P. : On Acoustically Effective Vortex Motion i n G a s e o u s J e t s . Philosophical Magazine and J o u r n a l of Science, Vol. 32, pp. 245-252, 1941.

30. Tatsurni, T. and Gotoh, K.: The Stability of F r e e Boundary L a y e r s Between Two Uniform S t r e a m s . J o u r n a l of Fluid Mechanics,

Vol. 7, P a r t 3, pp. 433-441, March, 1960,

31. Tatsumi, T. and Kakutani, T. : The Stability of a Two-Dimensional L a m i n a r Jet. J o u r n a l of Fluid Mechanics, Vol. 4, P a r t 3,

pp. 261-275, 1958.

32. L e e s , L e s t e r a n d Lin, Chia Chiao: Investigation of the Stability of the L a m i n a r Boundary L a y e r i n a C o m p r e s s i b l e Fluid. NACA Technical Note No. 11 15, September, 1946.

33. Landau, L. : Stability of Tangential Discontinuities i n C o m p r e s s i b l e Fluid. Akedemiia Nauk S. S. S. R., Comptes Rendus (Doklady), Vol. 44, No. 4, pp. 139-141, 1944.

34. Miles, John W. : On t h e Disturbed Motion of a Plane Vortex Sheet. J o u r n a l of F l u i d Mechanics, Vol. 4, P a r t 5, pp. 538-552, September, 1958.

35. Hatanka, Hiroshi: On the Stability of a S u r f a c e of Discontinuity in a C o m p r e s s i b l e Fluid. J o u r n a l Soc. Sci. Culture, Japan, Vol, 2, pp. 3-7, 1947: cited by Applied Mechanics Reviews, Vol. 2 , p. 897, July, 1949.

36. Batchelor, G. K. a n d Gill, A. E. : Analysis of the Stability of Axi- s y m m e t r i c J e t s . J o u r n a l of F l u i d Mechanics, Vol. 14, P a r t 4, pp. 529-551, Dec., 1962.

37. Gill, A. E. : On the O c c u r r e n c e of Condensation on Steady A x i s y m m e t s i c J e t s , J o u r n a l of Fluid Mechanics, Vol. 14,

P a r t 4, pp. 557-567, December, 1962.

38. Squire, H. B. : On the Stability f o r Three-Dimensional Disturbances of Viscous Flow Between P a r a l l e l Walls. P r o c e e d i n g s of the Royal Society of London, S e r i e s A, Vol. 142, pp. 621-628, 1933.

39. Dunn, D. W. and Lin, C. C. : The Stability of the L a m i n a r Boundary L a y e r i n a C o m p r e s s i b l e Fluid f o r the C a s e of Three-Dimensional Disturbances. J o u r n a l of the Aeronautical Sciences, Vol. 19, No. 7, p. 491, July, 1952.

40. Case,

K.

M. : Stability of Inviscid P l a n e Couette Flow. The P h y s i c s of Fluids, Vol. 3, pp. 143-148, March-April, 1960.

41. Case, K. M. : Stability of a n Idealized Atmosphere. I. Discussion of Results. The P h y s i c s of Fluids, Vol. 3, No. 2, pp. 149-154,

, March-April, 1960.

42. Case, K. M. : Edge E f f e c t s and the Stability of P l a n e Couette Flow.

The P h y s i c s of Fluids, Vol. 3, No. 3, pp, 432-435, May-June, 1960.

43. Case,

K.

M. : Hydrodynamic Stability a n d the Inviscid Limit.

J o u r n a l of Fluid Mechanics, Vol. 10, P a r t 3, pp. 420-429, May, 1961.

44. Case, K. M. : Hydrodynamic Stability a n d the Initial Value P r o b l e m . P r o c e e d i n g s of Syrnposia i n Applied Mathematics, Hydrodynamic Stability, A m e r i c a n M a t h e m a t i c a l Society, Vol, 13, pp. 25- 33, 1962.

45. Lin, C. C. : Some M a t h e m a t i c a l P r o b l e m s i n the T h e o r y of t h e Stability of P a r a l l e l Flows. J o u r n a l of F l u i d Mechanics, Vol. 10, P a r t 3, pp. 430-438, May, 1961.

46. Whitham, G. B. : G r o u p Velocity and E n e r g y Propagation f o r T h r e e - Dimensional Waves. Communications on P u r e a n d Applied Mathematics, K. 0. F r i e d r i c h s A n n i v e r s a r y I s s u e , Vol. 14, No. 3, pp. 675-691, August, 1961.

47. von Karman, Th., and Rubach, H. : Uber d i e M e c h a n i s m u s d e s F l u s s i g k e i t s und Luftwider standes. Physik. Z ,

,

13:49, (1 912).

48. Roshko, A. : On the Development of Turbulent Wakes f r o m Vortex S t r e e t s . NACA R e p o r t 1191, 1954.

49. Taneda, Sadatoshi: Studies of Wake V o r t i c e s (11), E x p e r i m e n t a l Investigation of t h e Wake Behind C y l i n d e r s and P l a t e s a t Low Reynolds Numbers. R e p o r t s of R e s e a r c h Institute f o r Applied Mechanics, Japan, Vol. 4, No. 14, pp. 29-40, October, 1955.

50. Schlichting, Hermann: Boundary L a y e r Theory. T r a n s l a t e d by J. Kestin. 4th Edition, McGraw-Hill, 1960.

51. Dewey, C. F.

,

Jr. : M e a s u r e m e n t s i n Highly Dissipative Regions of Hypersonic Flows: I. Hot W i r e M e a s u r e m e n t s inLow Reynolds Number Hypersonic Flows. 11. The Near Wake of a Blunt Body a t Hyper sonic Speeds. California Institute of Technology, Ph. D.

Thesis, June, 1963.

52. Chapman, D. R. ; Kuehn, D. M.; and Larson, Hp Ko : Investigation of Separated F l o w s i n Supersonic and Subsonic S t r e a m s with

E m p h a s i s on the Effect of Transition. NACA Report 1356, 1958.

53. L a r s o n , H. K. : Heat T r a n s f e r i n Separated Flows. J o u r n a l of the A e r o s p a c e Sciences, Vol. 26, No, 11, pp. 731-738,

November, 1959.

54. McCarthy, John F., Jr.: Hypersonic Wakes. GALCIT Hypersonic R e s e a r c h P r o j e c t , Memorandum No. 67, J u l y 2, 1962.

55. Kubota, T. : L a m i n a r Wake with S t r e a m w i s e P r e s s u r e Gradient

-

11. GALCIT I n t e r n a l M e m o r a n d u m No. 9, April, 1962.

56. Pai, Shih-I.. : Viscous Flow Theory. I

-

L a m i n a r Flow.

D. Van Nostsand Company, Inc., 1956.

57. Tollmien, W. : Asymptotische Integration d e r Storungsdifferential- gleichung e b e n e r l a m i n a r e r Stromun e n bei hohen Reynoldsschen Zahlen. Z. angew. Math. Mech. 25727, pp. 33-50, 70-83, 1947.

58. Heisenberg, W. : On the Stability of L a m i n a r Flow. P r o c . Int. Congr. Math., pp. 292-6, 1950.

59. Reshotko, Eli: Stability of the C o m p r e s s i b l e Boundary L a y e r . GALCIT Hypersonic R e s e a r c h P r o j e c t , Memorandurn No. 52, J a n u a r y 15, 1960.

\60. Slattery, R. E. and Clay, W. G, : R e e n t r y P h y s i c s and P r o j e c t P r e s s P r o g r a m s . Semmiannual Technical S u m m a r y Report to the Advanced R e s e a r c h P r o j e c t s Agency (U), pp. 11-11 to 11-17, Lincoln Laboratory, M a s s a c h u s e t t s Institute of Technology, 30 June 1962.

61. Ince, E. L. : O r d i n a r y Differential Equations. Dover Publications, Inc., 1956.

62. Benney, D. J. : A Non- L i n e a r T h e o r y f o r Oscillations i n a P a r a l l e l Flow. J o u r n a l of F l u i d Mechanics, Vol. 10, No. 2, pp. 209-236, M a r c h , 1961.

APPENDIX A

BOUNDARY CONDITIONS F O R THE AXI-SYMMETRIC PROBLEM

F o r the axi- symmetric wake the boundary conditions on the a x i s a r e d e r i v e d f r o m the p u r e l y kinematic condition that all d i s t u r b a n c e a m p l i t u d e s a n d the v o r t i c i t y d i s t u r b a n c e m u s t be finite t h e r e , r e g a r d l e s s of the viscosity o r the c o m p r e s s i b i l i t y of the fluid. The t h r e e components of v o r t i c i t y fluctuation a r e

F o r n = 0, the continuity equation [Eq. (2. 38)] shows that q r

-

^r^{a s r}^--a^{0 i f}^q ^and ^S a r e to be finite on the a x i s and q

(b- '

r

is to be finite on the a x i s Eq. (A. 3)]

.

T h e r e f o r e , q c ( o ) = ^%(ul =o;

**?*c.),**

T r ( o 1

,

s L d a n d

eCOj

a r e a r i b t r a r y .

F o r n

#

0, l e t qr + r €

A

^{a s}^r^--r⁰

.

Then f r o m Eq. ( 2 . 38),

Substituting Eq. (A. 4) into Eq. (A. 3 ) one obtains

Then € = n

-

¹and f o r

F r o m the (#)-momentum equation, n ( 0 )

=

0 when n

#

⁰

,

and f r o m Eq. (A. l ) , qx- r ( n

#

0) o r qx(0) = 0. Therefore, s ( 0 ) = e(0) = 0 when n

#

0. In addition, Eq. (2. 40) shows that ~ ' ( 0 )

=

0 when n

>

APPENDIX B

TWO- DIMENSIONAL WAKE MODEL

The m e a n flow quantities a r e a s s u m e d to satisfy the boundary l a y e r equations. Using Kubota's f o r a z e r o external p r e s s u r e gradient, the following s e t of equations a r e obtained for the c o m p r e s s i b l e wake behind a flat plat'e o r hypersonic vehicle:

Continuity

Momentum

E n e r g y

with the boundary conditions

where

'T = P r a n d t l number

=

constant d* = c h a r a c t e r i s t i c body dimension

**TY/*: fey:**

= constant

:

Chapman-Rubesin r e l a t i o n

hi ^, ^G'

⁼^constant

The above equations a r e l i n e a r i z e d by using Oseen type v a r i a b l e s

W : I - U

^{< (}

I

Retaining the lowest o r d e r t e r m s , the following equations a r e obtained

with the boundary conditions

W(O,Y)

⁼

By using Laplace t r a n s f o r m s , the solutions of Eqs. (B. 4) subject to the boundary conditions, Eqs. (B. 5) a r e obtained, a s folldws:

Drag The m o m e n t u m t h i c k n e s s ( o r d r a g coefficient,

='

3 ~ u e s x ~ ~

a e i s given by

r; ^u;

and the net h e a t t r a n s f e r r e d to the body by

-m

= constant

where H* = stagnation enthalpy.

L e t

Then f r o m Eqs. (B. 7) and (B. 8 )

If the initial conditions a r e assumed to be point s o u r c e s (delta functions), i. e.

,

3 ) =

R s(9)

then f r o m Eqs. (B. 10) and (B. 11)

The solutions then become

(B. 11)

(B. 12)

Let the c h a r a c t e r i s t i c length scale, L*, of the mean flow field be defined a s

F o r a flat plate incompressible wake,

F o r convenience, the following notation i s adopted:

so that

(B. 14)

(B. 15)

(B. 1 6 )

(B. 17)

V* i s the velocity defect of the wake and L* i s the Y position a t which

The Reynolds number of the wake i s

APPENDIX C

METHOD O F SOLUTION O F TATSUMI AND KAKUTANI~' FOR SMALL a R The O r r - S o m m e r f e l d equation c a n be e x p r e s s e d i n the following f o r m ~ q .

[

(3.211 :

subject to the boundary conditions, Eqs. (3. 3) a n d (3. 7),

- J L m

^y ^{- A}^-f

9 - "

^C

^'f4a,

- T < a + d L - c 4 R c ) L W Anti- s y m m e t r i c Disturbances

S y m m e t r i c D i s t u r b a n c e s

Tatsurni a n d ~ a k u t a n i ~ expand the solution i n p o w e r s of a R a s follows:

w h e r e

Substituting Eq. (C. 5) into Eq. (C. l ) , and matching powers of i a R, the following equations r e l a t i n g the (b(n)ls

,

a r e obtained

The solutions of Eq. (C. 6) a r e

The solutions of Eq. (C. 7) can be found by the method of variation of p a r a m e t e r s and a r e

f o r n >/ 1

,

j = 1 , 2 , 3, 4 . The general solution of Eq.

(C.

1) is

where the C . ' s a r e a r b i t r a r y constants.

Since the solution Eq.

(C.

8) m u s t satisfy the outer

(C. 10)

boundary condition Eq.

[

(C. 2 )

I , C,

C4 ⁼ ^O

^andfor a non-trivial solution,

$bl

^and

$b3

m u s t s a t i s f y the following eigenvalue equations:

Anti- s y m m e t r i c disturbances [Eq. (C. 3)]

Symmetric disturbances [Eq. (C. 44

(C. 11)

(C. 12)

F o r convenience, l e t A($:,:") ), B($'*' )

c ( @ ~

^'n' ) and D($ ^'n) )

be the t e r m s i n the b r b t s of the solution $I,'*'

[

^{Eq. (C.}^9)]

,

respectively, where the lower l i m i t is taken t o be infinity, s o that

and f u r t h e r introduce the notation

(C. 13)

*

This definition d i f f e r s f r o m that of Eq. (6. 3), Reference 31, by the factor ( l / i a ~ c ) .

Then

(C. 15)

Substituting Eq. (C. 5) into Eqs. (C. 11) and (C. 12) and using

Eq. (C. 15), the eigenvalue relations can be reduced to the following f o r m

Symmetric Disturbances

Anti- s y m m e t r i c Disturbances

~0 (C. 17)

- 1

f **( i * a I n** I"'

n = I

n 21

-

^In)

-

^{c i}^*QJ"

J (+,I I +y)LA~~n~"' ⁽⁰³¹

la = I

Eqs. (C. 16) and (C. 17) a r e then expanded and only t e r m s of the third and lower o r d e r i n

i

a R a r e retained. The quantities i n Eq. (C. 14)

2 w e r e evaluated using Eq. (C. 9) with w =

+

^{e - Y}

Since the complex wave speed i s of o r d e r unity, o r l e s s , then

p -

a will be of the o r d e r of a R. In o r d e r to be consistent with the approximations used, the coefficients in the eigenvaluk equations were

expanded i n powers of

P -

a = ^{a Q}( d complex) and t e r m s of the o r d e r G 3 and higher w e r e neglected. The complex eigenvalue equations then

r O ^(C.¹⁶⁾

become

Anti- Symmetric Disturbances

Svmmetric Disturbances

& ( z + G ) ~ ( L A R )

I

+ + A ( ~ + G )

(c.

¹⁹⁾

where

The asymptotic behavior of Eqs. (C. 18) and (C. 19) was determined by a t r i a l and e r r o r method. The c o r r e c t limiting p r o c e s s e s and

reduced equations a r e a s follows f o r c

I

= 0 :

- -

3 ³ ^{(C. 20)}

F

² ⁴⁸

magin nary part:

-

^Gr

+ [

_-L

^-

_{L A ]}

R - ~ G R ~

Symmetric d i s t u r b a n c e s Ci; ^-,^{- 0}

,

- ^-

^CT

^,

^CK^-L^o

R e o l p a r t CR + ²3 - - d 5 ~ [ - 3 -

8 1 p 2 = Q

3 2

I r n a g i n a r y p a r t

GI c [ f i - **$ * t P ] R** -

d Z r x 8~ (C. 21)

The equations a r e solved simultaneously and the r e s u l t s a r e given in Table 3. 2.

*

Since the coefficient of (iaR) i n Eq. (C. 3 19) i s r e a l , the coefficient of R 3 i n Eq. (C. 21) Real part] i s z e r o to the o r d e r of the approximation used.

Eq. (C. 18) w a s solved graphically and a minimum c r i t i c a l Reynolds number was found

a able

^3.1

^] ^.

*

Since the profile w

=

e-Y w a s used in these calculations, the L sign of c a s computed f r o m Eqs. (C. 20) and (C. 21) m u s t be changed

to conform to the notation i n the r e s t of the text. T h i s was done i n these tables.

APPENDIX D

SOLUTION O F THE INVISCID EQUATIONS FOR AMPLIFIED DISTURBANCES

F o r amplified subsonic disturbances, the solution of Eq. (4. 14) and Eq. (4. 17) i s r e g u l a r everywhere on the r e a l axis. Since G i s

singular a t the a x i s [ G - ( l / a )

1,

i t is convenient to m a k e the following transformation

Eq. (4. 7) then becomes

Eq. (D. 2) i s a complex equation. I t s r e a l and imaginary p a r t s a r e

The boundary conditions a s

)7

^---,rn a r e

I

C

[/ ^-

^M^'⁽ ^CnZ-

: ^] ⁾ ^: ^c

^{(D. 4)}

Using a power s e r i e s expansion about the axis, and satisfying the condition n(0) = G

,

where

(D. 5)*

*

P r i m e s (') indicate differentiation with r e s p e c t to

4

2 w*" C Z

c, ⁼

(I 4- cRIZ ,- CTZ

T h e r e a r e only two i n t e g r a l c u r v e s that will simultaneously s a t i s f y the boundary conditions a t the a x i s and a t infinity f o r a given s e t of eigen values; a

,

c R and cI

.

^Thesea r e sketched below.

Sketch D. 1

If the given s e t i s not consistent, the boundary conditions will not be satisfied and the i n t e g r a l c u r v e s oscillate v e r y rapidly n e a r the axis.

F o r t h i s reason, the integrations w e r e s t a r t e d f r o m the a x i s and infinity and the values of H and H w e r e compared a t a point within

R I

the domain. The matching point was taken to be the point a t which

The calculation procedure used to obtain the inviscid amplified solution for the given profiles w ( q ) and T(

9

), the relative Mach number, M, and the wave speed,

'

i s a s follows:

Integration f r o m Infinity to the C r i t i c a l Point and f r o m the Axis to the Critical Point

(1) A s s u m e a value of a and c and evaluate the boundary con- I

dition a t infinity f r o m Eq. (D. 4) and the boundary condition for a s m a l l positive value of f r o m Eq. (D. 5).

( 2 ) Continue the calculation of H and H by the simultaneous

R I

integration of Eq. (D. 3 ) to the c r i t i c a l point,

( 3 ) Compare the values of H and HI a t

q c

obtained f r o m the R

inner and outer integrations.

(4) Repeat steps (1) through ( 3 ) until the values of H and H

R I

a r e simultaneously matched a t

q c

APPENDIX E

EXPANSION ABOUT CRITICAL POINT

-

AXI- SYMMETRIC CASE

The solution of the inviscid equation [ E ~ . (5. 9 ) ] in the neighborhood of the "singular point" i n the complex r-plane (w

=

c ) is obtained by a Taylor S e r i e s expansion (method of Frobenius). Eq. (5. 9) can be r e w r i t t e n in the following f o r m

where

Let

#

⁼^r

-

^r and a s s u m e a s e r i e s solution of the f o r m

Since (w-c) and T a r e analytic functions of r everywhere i n the finite region of the complex r plane the coefficients of Eq. (E. 1) c a n be expanded i n a Taylor S e r i e s about the point r = r c (w = c):

Y -

- . - [Y, +PC'- x ^*''I 1

w - c ²^w,'

+I

Eqs. (E. 3 ) and (E. 4) a r e substituted into Eq. (E. 1 ) and the coefficient of each power of f i s s e t equal to zero. The two linearly independent solutions, and

9 ,

valid in the neighborhood of the c r i t i c a l point

along the r e a l a x i s a r e a s follows:

where

The coefficient b i s not determined in this method. The proper path 1

f o r analytical continuation of

,

i n passing f r o m

7 >

^Oto

# <

,

l i e s below the point r = r c for wct

>

0 [ ~ ~ ~ e n d i x G

I .

The other disturbance amplitudes can be found i n the neighborhood of the c r i t i c a l point by using Eqs. (5. ll), (E. 5) and (E. 6) :

Note that f o r

7

^' ^O ^1-

7 -

^{\ -}

⁷

7

^{L o}

^{I n} r \ h l ) ! l -

il-r

i n Eq. (E. 8).

1 3 3

APPENDIX F

EXTREMUM O F DENSITY -VORTICITY PRODUCT

F o r the c a s e of n e u t r a l disturbances,

( 2 ~ ' )

m u s t have a t r u e e x t r e m u m a t r = r and not a point of inflection. T h i s c a n be shown

i n exactly the s a m e way a s i n the i n c o m p r e s s i b l e c a s e 36 i n the following way. Add the complex conjugate equations, i n s t e a d of subtracting t h e m ( i n d e r i v a t i o n of Eq. (5. 32) ) to obtain

F o r m o s t p r o b l e m s of i n t e r e s t ,

gR

⁰

,

s o that

and

A n e c e s s a r y and sufficient condition f o r the e x i s t e n c e of n e u t r a l d i s - t u r b a n c e s i s that

and c R = w = c a t t h i s point.

L e t

47 dr/g,

so that

F o r m o s t profiles, (

7

r ~ ' ) ' and hence d 7% changes sign only once i n the infinite interval and f r o m Eq. (F. 7)) (w

-

^{cR) and} ^4' ^w/d7'

m u s t have opposite signs. Therefore, for neutral disturbances, Idw/471 m u s t have a maximum with r e s p e c t to r, i. e., C

I I

(

^{d 3}W/d'3)F=%: and consequently,

( f

w'),=,~ it 0

This r e s u l t cannot be shown f o r amplified disturbances except i n the 3 6

limiting c a s e of incompressible flow

.

135 APPENDIX G

VISCOUS CORRECTIONS IN THE CRITICAL LAYER

In considering inviscid n e u t r a l disturbances, a c r i t i c a l point

o c c u r s in the flow field, a c r o s s which some of the disturbance amplitudes a r e singular Section V. 3

I .

^In^ar e a l fluid these singularities m u s t be smoothed out by the action of viscosity and conductivity i n the

neighborhood of this c r i t i c a l point. These viscous c o r r e c t i o n s a r e important for the amplitude distributions but they do not affect the eigenvalue problem f o r a R

> >

1. However, if aR i s not v e r y much g r e a t e r than unity, the viscous c o r r e c t i o n s around the c r i t i c a l layer m a y

extend to the a x i s and the splitting of the solutions into inviscid and viscous types is not valid. In addition, the t e m p e r a t u r e and density fluctuations a r e singular a t t h i s point, and the t h e r m a l conductivity of the fluid m u s t be included i n the vicinity of t h i s point to smooth out t h e s e discontinuities. It i s to be expected that the viscous solutions for the axi- s y m m e t r i c c a s e a r e s i m i l a r to those for the two-dimensional c a s e except f o r the new element associated with the singularity i n q since

3 the c u r v a t u r e effects i n a thin annulus i n the neighborhood of the c r i t i c a l point a r e unimportant. The incompressible c a s e will be the only one considered here. The compressible problem i s the s a m e a s the incompressible one i n the Tollmien variable59 and will not be discussed.

The solutions c o r r e c t e d for viscosity a r e given by

[ c o r r e c t e d solutions

₁

= [ + ~ v i s c o u s r e p ~ a ~ e m e n t t e r ~ ] inviscid solution

I -

singular t e r m s

=

[regular inviscid solution]

I

+

[viscous replacement t e r m j ^a

136

where the viscous replacement function i s obtained by solving the full viscous disturbance equations i n the vicinity of the c r i t i c a l point, i. e., retaining only the leading viscous t e r m s i n t h i s region. This function m u s t be such that i t approaches the singular t e r m s in the inviscid

solution "far away" f r o m the c r i t i c a l layer. The viscous replacement 2 2

t e r m s a r e found using the convergent s e r i e s method Introduce the p a r a m e t e r

E = l/(a~) 1/3 ,

a s i n the two dimensional case, and the new independent variable

The m e a n flow quantities a r e expanded in a Taylor s e r i e s about the c r i t i c a l point

Eqs. ( 2 . 3 2 )

-

( 2 . , 3 5 ) then take the following f o r m s

w h e r e TO,

= ,/a2

+ "%/rCx

In o r d e r f o r Eqs. (Go 4)

-

(G. 7) to be consistent, the d i s t u r b a n c e a m p l i t u d e s m u s t be of the foLLowing f o r m

Substituting Eq. (G. 8 ) into Eq. (G. 4 )

-

(G. 7) and eliminating the p r e s s u r e , t h e following z e r o e t h o r d e r equations a r e obtained:

The solutions of Eq. (G. 10) a r e

(G. c)a)

(G.9b)

(G. 10)

(G. 11)

w h e r e

'1'3 y3

3

= ( w c l )

=

( A P N C ' ) (r-rc)

(G. 12)

( 1 )

and

//,,3

^{a n d}

^)-Iy3

⁽²¹

^[ ^f

^{f i}^Z)~''J a r e Hankel

functions of o r d e r (1/3) and the f i r s t and second kind respectively. The a s y m p t o t i c expansions of the Hankel functions of o r d e r (1/3) a r e valid

2 2 i n the following r e g i o n ( L i n ).

The solutions obtained by m e a n s of a n asymptotic s e r i e s (of the full viscous equations) c a n be f o r m a l l y r e l a t e d t o the asymptotic

expansions of the four solutions obtained by the method of convergent 2 2

s e r i e s ( L i n ) The solutions of the inviscid equations a r e two of the asymptotic solutions of the f u l l viscous equations. T h e r e f o r e i n o r d e r that the inviscid solutions r e p r e s e n t valid a s y m p t o t i c solutions of the full viscous equations, the c o r r e c t path of integration around the

singular point should follow the s a m e c r i t e r i o n as Eq. (G. 13), and should l i e below the singular point f o r w _C

' ^>

0 and above f o r wc'

<

.

If c

>

0, t h e singular point of the inviscid equation l i e s above I

the r e a l a x i s , and the effect of v i s c o s i t y c a n be neglected inside the fluid f o r sufficiently l a r g e Reynolds numbers. If c = 0, the two l i n e s

i n t e r s e c t a t a single point on the r e a l axis, and the inviscid solutions can never hold along the e n t i r e r e a l axis. Viscosity cannot be

neglected a t the singular point no m a t t e r how l a r g e the Reynolds

number m a y be. F o r c

<

0 the two l i n e s i n t e r s e c t the r e a l a x i s a t two

I

points and viscosity is important a l l along the r e a l a x i s between the two inter sections.

( 6 )

yv

c a n be determined d i r e c t l y f r o m Eq. (G. 9) and Eq. (G. 11)

(G. 14)

; Yn,

4:;

- -

d, cw: )"3

t

The solutions Eqs. ( G. 11) and ( G.14) a r e identical t o those f o r the two dimensional case.

Rewrite Eq. ( G.9b) in t e r m s of the independent variable

3

(G. 15)

By the method of variation of constants, the solution of Eq. (G. 15) i s

(G. 16)

h l ( z ) h2(z) i s the Wronskian of the functions h ( a ) and

I

h Z ( z )

,

and -(2/3) n

<

a r g (i z )

<

(2/3) rr o r -(7n/6)

<

a r g z

<

(n/6).

O - ^lo) ⁽⁰⁾

F o r q. _X

-

qr* = 1 , q , is a L o m m e l function, L ( Z )

,

.

The r e a l p a r t of the L o m m e l function is even and the i m a g i n a r y p a r t i s a n odd function of z. The g r a p h s of

L,

(L't ) and

L T

( i t ) a r e shown below

Sketch G. 1

F o r l a r g e values of Z

The viscous c o r r e c t i o n s which apply f o r z = 0(1), r

-

r -- 0 (dR)

-

^'13

will r e m o v e the singularity a t the singular point, and the d i s t u r b a n c e amplitude, q i n the vicinity of r = r will look like

3, ' C

Sketch G. 2

The discontinuity is s m e a r e d out by the a c t i o n of viscosity,

If the phase velocity is taken t o be equal to the velocity of the m e a n flow on the a x i s , then the solution is singular a t that point, and d o e s not s a t i s f y the boundary conditions. Again, a viscous r e p l a c e m e n t

3 7 t e r m m u s t be found. The r e a d e r is r e f e r r e d t o

in^'

^{and Gill} ^{f o r}^a

d i s c u s s i o n of t h i s problem.

TABLE I

NEUTRAL, INVISCID STABILITY CHARACTERISTICS

TABLE I1

AMPLIFIED, INVISCID STABILITY CHARACTERISTICS

C a s e

C a s e 1 2 3 4 5

A U

0 . 6 9 2 0. 285 0 . 1 6 0 0. 083 0 . 0 4 9

A T - 0 0. 50 0. 38 0. 30 0. 20

0 0. 42 0. 17 0. 07 0. 03

1 4 4

T A B L E I1 (CONTINUED)

d C,

- _-

0 0 . 0 4 0 . 1 1 0. 15 0.16 0. 15 0 . 1 1 0.05

0 0.08 0. 13 0.16

4 Cl

- _-

=

0 0.05 0.11 0. 15 0.17 0. 15 0. 11 0. 05

0 0.08 0. 13 0.16 d c,

A- d 4

- 0 . 3 8 -0. 35 - 0 . 3 0 -0. 25 - 0 . 2 0 -0. 2 1 -0. 15 -0. 1 2

- 0 . 3 7 -0.32 -0. 28 - 0 - 2 3 -0.20

- a C~

1 . 4 4 1. 3 4 1. 1 6 0. 9 6 0. 77 0. 58 0. 39 0. 20

1. 59 1. 3 6 1. 17 0. 98

0. 46 0. 20 C a s e

-0, 19 -0.18 -0. 13 0.60

0 . 4 0 0.20 a

1 . 5 1 1 . 4 0 1 . 2 0 1.00 0.80 0 . 6 0 0 . 4 0 0.20

1.63 1 . 4 0 1. 20 1 . 0 0 0 . 8 0

I

^C~

- 0 . 5 6 -0. 53 - 0 . 4 8 - 0 . 4 3 -0.38 - 0 . 3 3 -0. 25 - 0 . 1 6

-0.58 - 0 . 5 4 -0.48 -0.43 - 0 . 3 9

0 0. 03 0 . 0 9 0. 1 4 0.20 1 0. 2 4

0. 27 0. 25

0 0.06 0. 11 0.16 0.20 - 0 . 3 3

- 0 . 2 5 -0. 15

0. 25 0.28 0. 25

WAVE NUMBER

UNSTABLE DISTURBANCE, c l

> ⁰

I N V I S C I D LIMIT

- - - - _ . - - - _ . - - -

UTRAL DISTURBANCE, c(

= 0

RCRIT

Dalam dokumen Harris Gold In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California Thesis STABILITY OF LAMINAR (Halaman 119-159)

-

<

,

=

,

K.

K.

,

,

-

-

-

(b- '

r

.

?*c.),

,

eCOj

#

A

.

-

=

#

,

#

#

=

>

=

TY/*: fey:

:

hi , G'

W : I - U

I

W(O,Y)

='

r; u;

,

R s(9)

[

- J L m

9 - "

C

,

,

(C.

(C.

[

I , C,

C4 = O

$bl

$b3

c ( @ ~

[

,

*

- 1

f ( i * a I n I"'

-

-

J (+,I I +y)LA~~n~"' (031

i

+

p -

P -

I

(c.

I

- -

F

-

+ [

-

R - ~ G R ~

,

- -

,

8

**?*c.),**

**TY/*: fey:**

hi ^, ^G'

r; ^u;

^C

C4 ⁼ ^O

f **( i * a I n** I"'

J (+,I I +y)LA~~n~"' ⁽⁰³¹

^-

- ^-

^,

GI c [ f i - **$ * t P ] R** -

^] ^.

[/ ^-

: ^] ⁾ ^: ^c

c, ⁼

- . - [Y, +PC'- x ^*''I 1

⁷

^{I n} r \ h l ) ! l -

₁

^)-Iy3

^[ ^f

' ^>