ON 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.