Table Of ContentON COMPRESSIBLE LAMINAR BOUNDARY
LAYER WITH SUCTION
By
Vi-Chang L iu
A d i s s e r t a t i o n s u b m itte d in p a r t i a l f u l f i l l m e n t
o f th e r e q u ir e m e n ts f o r th e d e g re e of
D o c to r o f P h ilo so p h y in th e
U n i v e r s i t y o f M ichigan
1950
Committee in ch arg e:
P r o f e s s o r A rnold M. K u^the, Co-Chairman
A s s o c ia te P r o f e s s o r J u l i u s D. S o h e tz e r, Co-Chairman
A s s o c ia te P r o f e s s o r Mark V. M orkovin
A s s o c ia te P r o f e s s o r R o b ert C. F„ B a r te ls
j
1
ACKNOWLEDGMENTS
The au th o r would lik e to ta k e th is oppor
tu n ity to express h is s in c e re g r a titu d e to P ro fesso rs
A„ M. Kuethe, J . D. dch etzer, R. 0. F. B a rte ls and
M. V. Morkovin fo r t h e i r v alu ab le c ritic is m s and
suggestions about h is workj p a r t i c u l a r l y to P ro fesso r
A. M. Kuethe fo r h is kind e f f o r t to make t h is work
p o s s ib le .
Acknowledgment is e lso made to his w ife, Hsi-Yen,
who typed the p relim in a ry m anuscript in the turm oil of
h e r m edical c a re e r.
ABSTRACT
An exhaustive study has been made of the exact
mpthods of solvin g the com pressible lam inar boundary
la y e r eauatlons of the flow along a f l a t p la te with
su o tlo n . The ex isten ce of " sim ila r p r o f ile s " in the
flow is considered as the key to the exact so lu tio n of
th e boundary la y e r equ atio n s.
Two g en eral types of boundary la y e r flow problems
have been discu ssed: (1) Flow along a f la t p la te with
constant tem perature and with su c tio n , (2) Flow along a
f l a t p la te with an a r b i t r a r y a n a ly tic d is tr ib u tio n of
wall tem perature and with su c tio n .
"S im ilar p r o f ile s " is defined as a p ro p erty of
a c e rta in type of boundary la y e r flow In which the v e lo c ity
or tem perature d is tr ib u tio n (or both) can be expressed
as fu n ctio n (or fu n ctio n s) of a sin g le v a ria b le .
The ex isten ce of such s im ila r p r o f ile s in the
boundary la y e r flow makes i t p o ssib le to reduce i t s
a sso c ia te d boundary la y e r equations to an ordinary d i f f e r
e n tia l eouation.
In the second ty re of boundary lay e r flow as above
mentioned, i t is not expected th at sim ila r tem perature
p r o f ile s w ill e x is t.
i l l
A c o m p a tib ility co ndition f o r the existenoe of
sim ila r v e lo c ity p r o f ile s and sim ila r tem perature p ro file s
in the f i r s t type of boundary la y e r flow (as above mentioned)
and s im ila r v e lo c ity p r o f ile s alone In the second type of
boundary la y e r flow has been developed. For the boundary
la y e r flow over the p la te w ith constant tem perature and
with su ctio n , P ra n d tl number and s p e c ific h e ats a re assumed
to be c o n stan t. Through sim ila r p r o f ile s transform ation,
the boundary la y e r equations are reduced to a system of two
ordinary n o n -lin e a r d if f e r e n c ia l eouations which are solved
by a sp e c ia l method of successive approxim ation. For the
boundary la y e r flow of the second type, one a d d itio n a l •
assumption is ma.de, namely, p. = CT. In th is case the momen
tum equation alone is reduced to an ordinary d i f f e r e n t i a l
equation of B lasius type while the energy eoufltion becomes
a p a r t i a l lin e a r d i f f e r e n t i a l eouation. The l a t t e r is
solved by method of se p ara tio n of v a ria b le s .
The th re e co n fig u ra tio n s of suction d is trib u tio n
which are compatible with the ex istence of "sim ila r pro
f ile s " as above mentioned are as follow s:
(1) Uniform ta n g e n tia l v e lo c ity ,a n d zero normal
suet Ion.
(2) Uniform ta n g e n tia l v e lo c ity ,a n d normal v elo city
p rescribed p ro p o rtio n al to the wall tem perature (for th is
c a s e ,s im ila r p ro file s e x ist a t I n f in ite d istan c e downstream
from ending edge of the p la te .)
iv
(3) Uniform ta n g e n tia l v e lo c ity ; normal
v e lo c ity p rescrib ed p ro p o rtio n a l to the w all tem perature
and in v ersely p ro p o rtio n a l to th e souare root of d istan ce
from the leading edge of the p la te .
0
An extensive se t of c a lc u la tio n s has been made
f o r d if f e r e n t types of boundary co n d itio n s in which the
in te n s ity of suction and the w all tem oerature a re v aried .
R esults i n d ic a t'3 th a t normal suction is more
e ffe c tiv e in reducing th e th ic k n e ss of boundary la y e r than
#
ta n g e n tia l su c tio n . Thin boundary la y e r IS d e s ira b le in
lam inar s t a b i l i t y c o n sld era tio n and superaerodynamlc wind
tu n n el d esig n .
Approximate formulas f o r rap id p re d ic tio n of skin
f r ic tio n c o e ffic ie n t and heat t r a n s f e r c o e ffic ie n t have
been d e riv ed . The r e s u lts c a lc u la te d by using th ese approxi
mate form ulas show e x c e lle n t agreem ent with the "exact
values" from laborious ste p -b y -s te p in te g ra tio n .
TABLE OF CONTENTS
No. Page
ACKNOWLEDGMENT S' ..................................................... i i
ABSTRACT .......... H i
LIST OF TABLES ............................................................... viL
LIST OF ILLUSTRATIONS ............. viii
1 INTRODUCTION .................................................... 1
2 NOMENCLATURE ........................................... 3
3 THE COMPRESSIBLE LAMINAR BOUNDARY LAYER
EQUATIONS IN TWO-DIMENSIONAL FLOW .......................................... 7
if. A BRIEF REVIEW OF THE PREVIOUS WORKS ON
COMPRESSIBLE LAMINAR BOUNDARY LAYER FLOW ........................... 16
5 STATEMENT OF THE PROBLEM . .................................... 30
6 SIMPLIFICATION OF BOUNDARY LAYER
EQUATIONS WITH SUCTION .......................... 32
7 "SIMILAR PROFILES" TRANSFORMATION ............................................. 36
g SOLUTION OF THE BOUNDARY VALUE PROBLEMS ............................. E5
9 THE APPROXIMATE FORMULAS FOR SKIN FRICTION
COEFFICIENT AND HEAT TRANSFER COEFFICIENT ........................ 65
10 RESULTS AND DISCUSSION ......... 69
11 CONCLUSIONS ...................................................................................... 9^
APPENDIX .......................................... 96
BIBLIOGRAPHY ............. 123
LIST OF TABLES
JABL£ PACE
I C alcu latio n of ■«," w ith Various
Values of c. and c, . 96
I I C alcu latio n of Yc , Y« ', X , Y*. w ith
99
C6 = 2 . c, = o .
I l l Values o f 'fo) w ith c,= o . 100
IV V elocity and Temperature D istrib u tio n in the
Boundary Layer along a Heat In su la te d P la te
w ith Various Values of and c, . 101
V V elocity and Temperature D istrib u tio n in the
Boundary Layer along a Cold P la te (tv* 14 )
w ith Various Values of c* and c, , 102
VI V elocity and Temperature D istrib u tio n in the
Boundary Layer along a P la te w ith Non-uniform
Surface Temperature D istrib u tio n . 107
LIST OF ILLUSTRATIONS
1 V elocity D istrib u tio n s in the Boundary Layer
Flows along a Heat In su la te d P late w ith Normal
and T angential Suction.
2 Temperature D istrib u tio n s in the Boundary Layer
Flows along a Heat In su lated P late w ith Normal
and T angential Suction.
3 V elocity D istrib u tio n s in the Boundary Layer
Flows along a Heat In su la te d P late w ith D ifferent
Degrees of In te n s ity of Normal Suction.
M- Recovery Factors in the Boundary Layer Flows
along a Heat In su lated P late w ith D ifferent
Degrees of In te n s ity of Normal Suction.
5 Temperature D istrib u tio n s in the Boundary Layer
Flows along a Heat In su la te d P late w ith D ifferen t
Degrees of In te n s ity of Normal Suction*
6 V elocity D istrib u tio n s in the Boundary Layer
Flows along a Cold P late w ith Normal and Tan
g e n tia l Suction.
7 Temperature D istrib u tio n s in th e Boundary Layer
Flows along a Cold P la te with Normal and Tan
g e n tia l Suction.
8 V elocity D istrib u tio n in the Boundary Layer Flow
along a P la te w ith V ariable Surface Temperature
D is tr ib u tio n .
Temperature D istrib u tio n in the Boundary Layer
9
Flow along a P late w ith V ariable Surface Temper
a tu re D istrib u tio n .
10 Comparison of R esults of Skin F ric tio n Coeffi
c ie n t and Heat T ransfer C oefficient C alculation
by Approximate Formula w ith those by Exact
Method.
viii
1. INTRODUCTION
The Idea of boundary la y e r c o n tro l by auction
or In je c tio n has been su c c e ss fu lly a p p lied in various
f ie ld s : (1) In in c re a sin g s t a b i l i t y of lam inar boundary
la y e r, (2) in re ta rd in g boundary la y e r flow sep ara tio n ,
and (3) in sw eat-cooling of combustion chambers in Jet
motors and ro c k e ts.
Experim ental r e s u lts in d ic a te th a t the boundary
la y e r s tr u c tu r e d e f i n i te l y in flu en ces shock wsve form ation
(Refs. 32, 33)» In f a c t, so much so t h a t , fo r th e sake of
sim p lify in g the experim ental in v e s tig a tio n of the shock
wave, It would be a d v isa b le to remove the boundary lay er
i f i t is p o s s ib le .
Experim ental evidence In d ic a te s th a t surface
suction is a p o te n tia lly powerful means of Increasing the
s t a b i l i t y of the lam inar boundary la y e r.
A knowledge as a c c u ra te as p o ssib le , of th e
v e lo c ity and tem perature d is tr ib u tio n in the lam inar
boundary la y e r with suction forms the s ta r tin g p o in t fo r
the s t a b i l i t y In v e s tig a tio n .
In view of the e x tra o rd in a rily com plicated equa
tio n s of the boundary la y e r flow, exact so lu tio n to the
boundary la y e r equations with any sp e c ia l suction d i s t r i
bution w ill command I n t e r e s t.
