Read an Excerpt
Viscous Hypersonic Flow
Theory of Reacting and Hypersonic Boundary Layers
By William H. Dorrance Dover Publications, Inc.
Copyright © 2017 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-82258-7
CHAPTER 1
An Introduction to Hypersonic Heat-transfer Problems
1-1. General Considerations. This book is written in an era of greatly accelerated application of the fruits of scientific research to engineering problems. Not the least evident of these applications are those related to the development of space-exploration vehicles and long-range missiles. Early in the development of such devices, problems arose which could be solved only by applying the results of the most advanced research in gas dynamics or, in some cases, by furthering the research in this time-honored field of science. Before engineers can reliably design devices to survive flight through or into an atmosphere at hypersonic speeds, they must somehow provide for, avoid, or otherwise accommodate the enormous heat-transfer rates to the vehicle engendered by such flight speeds. This book deals with the details of some of the theories used to cope with this problem.
In particular, this book is concerned with viscous-gas-flow problems involving reacting gas mixtures. Starting from the statement of the boundary-layer equations for a mixture of gases, the theory is developed for both the laminar and turbulent boundary layers. Solutions for heat transfer to and from two- and three-dimensional shapes are developed taking into account the effects of dissociation of the boundary-layer gas, the effects of chemical reactions other than dissociation, the effects of mass transfer, the effects of a melting surface, and interactions between the boundary-layer gases and the surfaces over which they flow. An effort is made to provide fundamental ideas for the research scientist who wishes to push beyond the present state of this science while also providing some results which can be applied by those engaged in the design of hypersonic vehicles. The book is intended to cover hypersonic and reacting boundary-layer theory only. Those interested in problems of hypersonic inviscid flow and experimental techniques will not find those subjects covered in this book. Rather, it is intended that the various problems associated with a hot, reacting, compressible boundary layer will be thoroughly treated in the present book.
1-2. The Hypersonic Heating Problem. The balance of this book beyond the present chapter is devoted to the details of hypersonic or reacting boundary-layer theory. It is the purpose of the present chapter to provide a less detailed, broader description of the "hypersonic heating problem" and its implications.
When an object penetrates or flies within an atmosphere at hypersonic Mach numbers, considerable drag work is done by the object upon its surroundings. Such an object might be a long-range ballistic or boost-glide missile, a reentering satellite, a meteor, or any other object of extraterrestrial origin encountering an atmosphere. The drag work done ultimately shows up as heat. This heat, in turn, is found in the increased temperature of the air in the wake of the object or in heat which was transferred to the object itself. This drag work may be done over a long period of time, as by a boost-glide vehicle, or over a short period of time, as by a ballistic missile nose cone, reentering earth satellite, or meteor. Let us calculate, to a first approximation, the magnitude of the heat transferred to a hypersonic object owing to the viscous heating of the boundary-layer air.
Newton's first law applied to an object decelerating within an atmosphere at hypersonic speed in the absence of a thrust force yields
[MATHEMATICAL EXPRESSION OMITTED] (1-1)
where W = weight of the object, assumed constant to the approximation of this analysis
V = flight velocity
g = gravitational constant
D = total drag force or resistance due to atmosphere and where we have neglected the contribution due to gravity as being much less than drag in the region of interest. Furthermore, define a heat-transfer coefficient [bar.C]H by the relation
[MATHEMATICAL EXPRESSION OMITTED] (1-2)
where Q = heat transferred to the object
ρ = atmospheric density
S = area of object exposed to high-temperature gas
ΔI = enthalpy potential representing temperature difference which results in heat transfer from hot gas to cooler body
That is, define ΔI as
[MATHEMATICAL EXPRESSION OMITTED] (1-3)
where I∞ is the stagnation enthalpy defined as
[MATHEMATICAL EXPRESSION OMITTED] (1-4)
Iw is the enthalpy of the air mixture at the object's surface temperature as given by
[MATHEMATICAL EXPRESSION OMITTED] (1-5)
where Cp is the specific heat at constant pressure of the boundary-layer gas mixture and T∞ is the ambient gas temperature before being heated by the passage of the object.
Equation (1-2) indicates that the heat-transfer rate is directly dependent upon body geometry, density or flight altitude, and flight velocity. In this book we are concerned chiefly with problems related to high heat-transfer rates which, in turn, occur within the relatively dense atmosphere under most circumstances. Thus we deal with continuum-flow problems exclusively.
Now, because by our definition of the hypersonic speed range we are concerned with values of flight Mach number exceeding 6, then, for air
[MATHEMATICAL EXPRESSION OMITTED]
where
[MATHEMATICAL EXPRESSION OMITTED]
and M is flight Mach number, where
[MATHEMATICAL EXPRESSION OMITTED]
Since Tw is of the order of T∞, Eq. (1-3) becomes
[MATHEMATICAL EXPRESSION OMITTED] (1-6)
in view of the orders of magnitude of the various terms involved in ?I. Calculations of the type to be described in later chapters show that the heat-transfer coefficient is proportional to the skinfriction coefficient [MATHEMATICAL EXPRESSION OMITTED] where [MATHEMATICAL EXPRESSION OMITTED] is defined by the equation
[MATHEMATICAL EXPRESSION OMITTED] (1-7)
where Df is the drag force due to viscous shear stress at the exposed surface of the object and
[MATHEMATICAL EXPRESSION OMITTED] (1-8)
For a smooth flat plate the constant of proportionality is 1/2. For convenience we assume 1/2 here. Thus, Eqs. (1-1), (1-2), and (1-6) can be combined to give
[MATHEMATICAL EXPRESSION OMITTED] (1-9)
where D is the total drag, that is,
[MATHEMATICAL EXPRESSION OMITTED]
Dp is the drag force due to normal stresses over the exposed surface, and we assume to a first approximation that the ratio of [MATHEMATICAL EXPRESSION OMITTED] to [bar.C]D is independent of velocity. Integrating Eq. (1-9) we obtain
[MATHEMATICAL EXPRESSION OMITTED] (1-10)
where the subscript f denotes the final value of heat transferred to the body after it has spent its kinetic energy and the subscript i denotes the initial value. We assume that Vf = 0 and Qi = 0 to a first approximation.
Thus we see that the heat transferred to an object decelerating at hypersonic speeds within an atmosphere is directly proportional to the product of its initial kinetic energy and the ratio of the friction drag coefficient to the total drag coefficient. We note that only half of this product represents heat transferred to the object. The other half is transferred to the surrounding cooler gas. Equation (1-10) immediately suggests that, if Qf is to be minimized, the ratio of [MATHEMATICAL EXPRESSION OMITTED] should be minimized; that is, the fraction of the total resistance to motion represented by friction drag should be minimized. Equation (1-10) was used to calculate total heat transferred to decelerating bodies as a function of initial velocity. The results of this calculation are shown in Fig. 1-1. Some typical values of the ratio [MATHEMATICAL EXPRESSION OMITTED] are given in Table 1-1.
Figure 1-1 shows that the total heat transferred to a decelerating body can exceed the heat capacity of most known refractory materials if the body is streamlined and the initial velocity is high enough. The heat capacities of several refractory materials, including the heats of fusion and evaporation, are given in Table 1-2. Without regard to the unfavorable manufacturing characteristics of some of these materials, it is apparent that few of them provide sufficient heat capacity for use as a structural material of a streamlined decelerating object without precautions being taken somehow to avoid or dissipate the heat. Furthermore, the theoretical maximum heat capacity of any material is never realized in applications of the type being discussed here owing to the limitations of materials with finite thermal conductivity in distributing the heat uniformly through the material.
Of course, steps can be taken to alleviate this problem. This book will present the details of some of these steps as they relate to the reacting boundary layer surrounding the object.
1-3. Heat-transfer Effects upon Hypersonic-vehicle Design. It is too early in this book to describe the details of the interaction of the hot gas boundary layer with the surface material of the object over which it flows. This will be done in later chapters. However, we have seen enough to anticipate some of the steps which can be taken by a designer of a hypersonic vehicle to deal best with the hypersonic heating problem. Equation (1-10) and Fig. 1-1 immediately suggest that the object might be made rather blunt and unstreamlined if large amounts of heating of the object are to be avoided. Thus a sphere might successfully serve as the shape of such an object. However, the use of a blunt shape has its disadvantages depending upon the use to which the object is put. For example, a sphere has high-pressure drag accompanied by strong bow shock waves which result in a large volume of gas being heated as the gas passes through the shock wave. This heating can be accompanied by ionization and luminosity of the gas in the wake of the object. Communication with the object may thereby be compromised. Such an object may well be difficult to conceal if that is also an objective.
Streamlining the object, on the other hand, may result in excessively high heat-transfer rates, since deceleration will thereby be delayed until denser regions of the atmosphere are penetrated. The designer finds himself judiciously seeking a compromise between the heat-protection system and the shape of the object. No hard and fast rules are available to a designer, and an optimum compromise is continually being sought as the designer exercises his skills.
Figure 1-2 is a shadow photograph of a model of a typical missile nose-cone shape in flight down a ballistic range. The nose is blunted in order to reduce heat-transfer rates in that region. The aft-cone frustrum provides aerodynamic stability. Note the turbulent nature of the wake. As the turbulent eddies are reduced in size with the passage of time through the effects of viscosity, the temperature of the gas in the wake rises, so that after the object has passed, the average temperature of the gas in and surrounding the wake has increased. Most of the viscous heating of the gas occurs in the relatively thin boundary layer next to the body. Only by detailed analysis of what happens in this boundary layer can the heat transferred to the object be predicted with accuracy.
We must not overlook the importance of heat-transfer rate to hypersonic-vehicle design, for it is not only the amount but the rate at which heat is transferred to the body which affects the design. For example, a chosen material may have the capacity to absorb the total amount of heat transferred but lack the heat conductivity required to distribute the heat to the interior fast enough to avoid overheating the material near the surface. Various schemes have been devised to deal with this problem, including the use of sublimating or reacting materials which are deliberately allowed to ablateat the surface in an orderly and predictable manner. Such materials may soak up heat in reacting or changing phase and may also provide a "blocking effect" in that they provide a convective current of gas at the surface moving counter to the flow of heat being conducted toward the surface. Again a detailed analysis of the behavior of the boundary-layer gas flow under such circumstances is necessary before accurate heat-transfer rates to the solid undersurface material are known. One purpose of this book is to describe appropriate methods of analysis of this problem.
1-4. The Scope of This Book. Whenever a surface material is heated or cooled by a viscous gas stream, the methods of this book will apply. The heated surface may be that of a rocket nozzle throat, a wind-tunnel wall, or a surface undergoing decomposition owing to phase changes or surface combustion. Many surfacematerial–gas-layer interactions will be analyzed in detail for both laminar and turbulent boundary layers. The effects of surface combustion, gas-layer combustion, surface mass transfer, dissociation, and variation of gas properties with temperature and composition upon the characteristics of both laminar and turbulent boundary layers will all be treated in later chapters of this book. Because the thermodynamic and transport properties of gas mixtures are required to apply the results of the boundary-layer analyses, separate chapters are devoted to each of these subjects. This book is intended to be a reasonably complete reference for the equations and gas property values required to calculate heat-transfer rates to or from a compressible viscous boundary layer under various surface conditions, including those of a chemically reacting surface material.
The subject matter of this book touches on several scientific disciplines, including those of gas dynamics, thermodynamics, thermochemistry, molecular physics, quantum theory, and statistical mechanics. Because of this, an annoying problem of duplication of symbols and nomenclature customarily used in these different scientific disciplines presented itself. Wherever possible, we retained the convention used in the discipline related to the subject under discussion. Some duplication occurs, but whenever it occurs, it is believed that the text makes it obvious which meaning is attached to the ambiguous symbol. Most of the symbols are explained in the appendix.
A special effort was made to cite appropriate references for contributions not yet commonplace in the literature. The open literature was relied on almost exclusively. Occasionally reference is made to an unclassified reference privately published, which, however, should be available in most technical libraries. In some cases these references can be obtained by writing to the agency which published the document. It is regrettable that not all useful references have appeared in the open literature, and it is hoped that by citing them in this manner we may somehow stimulate their publication in the open literature.
CHAPTER 2
The Boundary-layer Equations
2-1. Introduction. The concept of a thin, viscous layer next to a body over which a fluid is flowing is due to Prandtl. According to Prandtl, the fluid velocity relative to the surface of the body increases from zero at the surface to its maximum value away from the surface in a thin region called the boundary layer. This concept is well established and has become a fundamental postulate of fluid dynamics. The concept has been directly confirmed by careful measurements of velocity distribution through the boundary-layer region. Indirect confirmation is provided by the excellent agreement with measurements of solutions to the boundary-layer equations — a set of differential equations derived from the more general equations of motion using the boundary-layer concept to drop certain terms from the more general equations.
In this chapter we shall postulate the existence of a boundary layer and develop the differential equations describing the flow of a reacting gas mixture within the boundary layer. First the origin of the parent equations of gas dynamics will be examined in order to bring out the contributions due to chemical reactions, such contributions being absent in the conventional statement of the boundary-layer equations. A method of seeking similar solutions to the laminary-boundary-layer equations will be presented and used to reduce the partial differential equations to ordinary differential equations. Some particular integrals of the boundary-layer equations will be presented. The appearance of certain dimensionless transport parameters will be discussed, and the significant parameters defined. The equations developed in this chapter will serve as the starting point for the theories presented in Chaps. 3 through 8.
2-2. Gas-dynamics Equations. The fundamental equations of gas dynamics include equations for the conservation of species, conservation of mass, conservation of momentum, and conservation of energy along with the equation of state for the gas mixture. In their most general form they consist of a set of nonlinear partial differential equations with four independent variables: three in space and one in time. The equations follow from the application of the fundamental laws of classical mechanics and thermodynamics to the flow of fluids and gases. With very few exceptions these equations have not been integrated in closed form for boundary conditions appropriate to physically sensible problems. The exceptions are usually rather simple physical situations possessing a natural symmetry which permits the dropping of many terms in the full equations before a solution is sought.
(Continues...)
Excerpted from Viscous Hypersonic Flow by William H. Dorrance. Copyright © 2017 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
