SOME PROBLEMS OF COMPRESSIBLE BOUNDARY LAYERS

Abstract

She present thesis is an attempt to study some of the unsteady flow problems of compressible boundary layers past a wedge* In general it deals with the effeets of non* linear law of viscosity JJ~ * constant fey where u ie the coefficient of viscosity and T is the temperature, <a and 4 are the arbitrary constants* Prom this we can obtain the simple linear law (« • o, « « 1), power law («C • o) and the Sutherland*s formula (co • 0.5) based on the kinetic theory of gases as special eases* In Sutherland*s formula < depends on the nature of the fluids and for air which is to be taken 114aK* In power law o*5 < m < 1, end the different values of to correspond to different temperatures, the temperature 52°F corresponds to am e*8* These two well known laws of viscosity have been used by so many authors Cope, Hartree, Euoken end Von Karmsn etc* assuming the different values to o and «<* Here the two parameters are connected in one formula and the various results are discussed for their different values* The above mentioned non-linear formula verify the results obtained by the various authors* The study of the effects of non-linear law of viscosity amounts to the study the effeets for different values of ra and «< • It contains five chapters dealing with the effects of non-linear law of viscosity in compressible boundary layers* It deals with ths problems of general (11) character under which some particular problems special eases* The stream function, the velocity vector, the wall temperature, the skin friction, heat transfer from the wall and the boundary layer thickness are the important quantities associated with boundary layer which are obtained a#* ^^••^p QWMW^Va ^^^Mm^^m^i^W mW^fMw "•» w*A^S Mr1^ ^*SW «S*^Pe*^^S SmnS^MnltiJPWW^P SSMJ •*•*• w^*^w ^jp^m*^^vMt ^n<^a 'iwiF v*vw sjmS> w wp** «e»es w *^anfc ^p iaov ^<»Aa**s^ w^P*fc*^ ™^e* the thesis, we obtain systems of general differential equations in a single variable* For the complete study of a particular problem, we have solved these equations by numerical integration for various values of the parameters and numerieal results are tabulated in the tables given at the end of each chapter* In most of the chapters the techniques used are the extensions of the techniques followed by Lighthill end .Sanaa who have solved the unsteady two-dimensional boundary layer equations. Each chapter begins with its introduction and contains different sections dealing with particular typee of problems coming under the chapter* The first chapter is on the effects of non-linear law of viscosity in compressible boundary layer on a hot flat plate when a progressive sound wave is propagated in the main stream* This chapter is divided into three sections* m section (la), the introduction of the chapter and in section (lb) -the basle equations and the general analysis are given. In section (lo), solutions of unsteady boundary layer equations are studied when the body (flat plate) is at rest end the main stream is fluctuating on account of the sound wave* The solutions are developed for large as well (lii) as for small time*. The expressions for skin friotion coefficient, heat transfer and the boundary layer thickness are obtained and their variations*are illustrated graphically. She ranges of validity of the two solutions are determined from the graphs* The second chapter deals with the effeets of non-linear law of viscosity in combined forced and free convection on a wedge shaped body moving in a non-uniform free stream and contains four sections* In section (11a) the Introduction of the chapter and in section (lib) the basic equations and ^p^**^p •^•^svsi^p^*^wwp* vMvv^MMMky wp4SJi" tifj^*> ^pw«^p *i*t *FWPWi*^e*j ^M^pas* ipiww*!^ *.^^w**^#iwy ^f^s^s^^^f**.^w**^^ *Hes*^^ given* la section (lie) the solutions of steady equations are given* In section (lid) the solutions of unsteady boundary layer equations for a moving body (wedge) are given* The solutions are developed for large times as well as for orall times assuming the relative temperature difference (wall and the main etreem) to be small constant whose sign determines the nature of the flow l*e* whether the flow is aiding or opposing* She parameter controlling the forced and free convections is the ratio of Oarshoff*s number and the square of the Reynold*s number* Finally the expression for steady and unsteady part of the skin friotion, heat transfer and the boundary layer thickness are obtained and their variations are shown graphically* She ranges of validity of the large w *SmHV^B* •* WSaSrVa "<•* W^mSS*» *WIS*W*» Wa»^S W^"^J^P'0)*<*w S*"BIW w*r ^r••* *•» »•<•» ™*»<bP» *ani9 ^HfWW ^F^WWW*w^^B*^^^^w from the graphs* She third chapter deals with the effects of non-linear law of viscosity in a compressible boundary layer on a moving body with uniform temperature and it contains five sections* (iv) In section (Ilia) the introduction of the chapter and In section (Illb) the basic equations and ths general analysis are given* In section (Ills) the solutions of unsteady boundary layer equations for a moving body (wedge) are obtained* later on the whole problem is analysed In detail for a flat plate* In section (Hid) the problem is analysed for a converging canal with linear law of viscosity* m seotion (Hie) the solutions of unsteady boundary layer equations for a moving converging canal are obtained* She solution is developed for large as well as for small times in both ths above problems and ths expressions for skin friotion and boundary layer thickness are obtained* The variations of various quantities are illustrated graphically and the ranges of validity of the two solutions are determined MfJAHj hue) ^^liTrains'm She fourth chapter deals with the effects of non-linear law of viscosity in compressible boundary layer on a wedge at rest with unsteady suction and Injection* It contains three sections. In seotion (IVa) the introduction of the chapter and in section (IVb) the basic equations and the general analysis associated with the steady and unstsady equations are given* In section (lYo) the solutions of unstsady boundary layer equations ere given when the body (wedge) is at rest together with unsteady suction or Injection perturbing about a zero mean* The problem of flat plate is analysed in detail* The two different solutions are developed one for large times and the other for small times* The skin friotion end the boundary layer thickness are obtained and their (v) variations are shown graphically* i3ie ranges of validity of small times and large tines solutions are determined from the graphs* She fifth chapter deals with the compressible boundary layer with linear law of viscosity on a moving body vtfien the temperature gradient of the wall is prescribed and contains five sections. In seotion (Va) the introduction of the chapter end in section (lb) the basic equations and the general analysis for steady and unsteady equations are given* In seotion (To) the solutions of unsteady boundary layer equations for moving body (wedgs) are given* later on the problem is analysed in detail for flat plate* In section (Vd) the analysis is given for a converging canal* In section (Ve) the solutions of unsteady boundary layer equations for a moving body (converging canal) together with a steady temperature gradient at the wall are given* In the case of converging canal the non-dimensional ratio of the Hussein number and square root of the Reynold's number is assumed to be small, to make our analysis easier* In both the oases solutions for large times as well as for sr all times are developed* She skin friction, heat transfer, wall temperature and the boundary layer thickness are obtained end their variations arc illustrated graphically* She ranges of validity of the two solutions ere determined graphically* In whole of the thesis, the meet of ths numerieal work has been carried out at im 1620 Computer which is available at the Structural Engineering Research Centre, Roorkee, end part of the numerieal work has been carried out (vi) at IBM 7044 Computer at the Indian Institute of Technology, Kanpur. She programs for the computer were prepared by the author himself Whatsoever the author has investigated In this thesis is interpreted graphically with the aid of numerical results given in the various tables at the end of each chapter