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Covering the general principles of gas dynamics to provide a working understanding of the essentials of gas flow, the contents of this book form the foundation for a study of the specialized literature and should give the necessary background for reading original papers on the subject. Topics include introductory concepts from thermodynamics, including entropy, reciprocity relations, equilibrium conditions, the law of mass action and condensation; one-dimensional gasdynamics, one-dimensional wave motion, waves in supersonic flow, flow in ducts and wind tunnels, methods of measurement, the equations of frictionless flow, small-perturbation theory, transonic flow, effects of viscosity and conductivity, and much more. The text includes numerous detailed figures and several useful tables, while concluding exercises demonstrate the application of the material in the text and outline additional subjects.
Advanced undergraduate or graduate physics and engineering students with at least a working knowledge of calculus and basic physics will profit immensely from studying this outstanding volume.
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| ISBN-13: | 9780486316857 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/09/2013 |
| Series: | Dover Books on Aeronautical Engineering |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 464 |
| File size: | 28 MB |
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Elements of Gasdynamics
By H. W. Liepmann, A. Roshko
Dover Publications, Inc.
Copyright © 1985 H. W. Liepmann and A. RoshkoAll rights reserved.
ISBN: 978-0-486-41963-3
CHAPTER 1
Concepts from Thermodynamics
1.1 Introduction
The basis of any physical theory is a set of experimental results. From these special primary observations, general principles are abstracted, which can be formulated in words or in mathematical equations. These principles are then applied to correlate and explain a group of physical phenomena and to predict new ones.
The experimental basis of thermodynamics is formalized in the so-called principal laws. The law of conservation of energy, which thermodynamics shares with mechanics, electrodynamics, etc., is one of these principal laws. It introduces the concept of internal energy of a system. The other principal laws of thermodynamics introduce and define the properties of entropy and temperature, the two concepts which are particular and fundamental for thermodynamics.
The principles laid down in these fundamental laws apply to the relations between equilibrium states of matter in bulk. For instance, thermodynamics yields the relation between the specific heats at constant pressure and at constant volume; it relates the temperature dependence of the vapor pressure to the latent heat of evaporization; it gives upper bounds for the efficiency of cyclic processes, etc.
Fluid mechanics of perfect fluids, i.e., fluids without viscosity and heat conductivity, is an extension of equilibrium thermodynamics to moving fluids. The kinetic energy of the fluid has now to be considered in addition to the internal energy which the fluid possesses when at rest. The ratio of this kinetic energy per unit mass to the internal energy per unit mass is a characteristic dimensionless quantity of the flow problem and in the simplest cases is directly proportional to the square of the Mach number. Thermodynamic results are taken over to perfect fluid flow almost directly.
Fluid mechanics of real fluids goes beyond classical thermodynamics. The transport processes of momentum and heat are of primary interest here, and a system through which momentum, heat, matter, etc., are being transported is not in a state of thermodynamic equilibrium, except in some rather trivial cases, such as uniform flow of matter through a fixed system.
But, even though thermodynamics is not fully and directly applicable to all phases of real fluid flow, it is often extremely helpful in relating the initial and final conditions. This complex of problems is best illustrated with a simple example. Assume a closed, heat-insulating container divided into two compartments by a diaphragm. The compartments contain the same gas but at different pressures p1 and p2, and different temperatures T1 and T2. If the diaphragm is removed suddenly, a complicated system of shock and expansion waves occurs, and finally subsides due to viscous damping. Thermodynamics predicts the pressure and temperature in this final state easily. Fluid mechanics of a real fluid should tackle the far more difficult task of computing the pressure, temperature, etc., as a function of time and location within the container. For large times, pressure and temperature will approach the thermodynamically given values. Sometimes we need only these final, equilibrium values and hence can make very good use of thermodynamic reasoning even for problems that involve real fluid flow.
In fluid mechanics of low-speed flow, thermodynamic considerations are not needed: the heat content of the fluid is then so large compared to the kinetic energy of the flow that the temperature remains nearly constant even if the whole kinetic energy is transformed into heat.
In modern high-speed flow problems, the opposite can be true. The kinetic energy can be large compared to the heat content of the moving gas, and the variations in temperature can become very large indeed. Consequently the importance of thermodynamic concepts has become steadily greater. The chapter therefore includes material that is more advanced and not needed for the bulk of the later chapters. Articles that are starred can be omitted at first reading without loss of continuity.
1.2 Thermodynamic Systems
A thermodynamic system is a quantity of matter separated from the "surroundings" or the "environment" by an enclosure. The system is studied with the help of measurements carried out and recorded in the surroundings. Thus a thermometer inserted into a system forms part of the surroundings. Work done by moving a piston is measured by, say, the extension of a spring or the movement of a weight in the surroundings. Heat transferred to the system is measured also by changes in the surroundings, e.g., heat may be transferred by an electrical heating coil. The electric power is measured in the surroundings.
The enclosure does not necessarily consist of a solid boundary like the walls of a vessel. It is only necessary that the enclosure forms a closed surface and that its properties are defined everywhere. An enclosure may transmit heat or be a heat insulator. It may be deformable and thus capable of transmitting work to the system. It may also be capable of transmitting mass. Every real wall has any one of these properties to a certain degree. There do not exist perfectly rigid walls, for example, and similarly there is no perfect heat insulator. However, it is convenient to use an idealized enclosure, consisting of parts which have well-defined properties such as complete heat insulation, etc.
For our purposes it is sufficient to deal with fluids only. The systems that we shall consider here are:
(a) A simple, homogeneous system composed of a single gas or liquid.
(b) A homogeneous mixture of gases.
(c) A heterogeneous system composed of the liquid and gaseous phase of a single substance.
1.3 Variables of State
If a system is left alone for a sufficiently long time, that is, if no heat and no mass is transferred to it and no work is done on it during this time, it will reach a state of equilibrium. All microscopically measurable quantities will become independent of time. For example, the pressure p, the volume V, and the temperature θ, can be measured, and in equilibrium do not depend upon time.
Variables that depend only upon the state of the system are called variables of state, p and V are evidently such variables, and these two are already familiar from mechanics. For a complete thermodynamic description of a system, we need new variables of state, foreign to mechanics. Thus it is a result of experience that the pressure of a system is not only a function of its volume. A new variable of state, θ, the temperature, has to be introduced. For a simple system,
p = p(V, θ) (1·1)
Following R. H. Fowler, one states the "zeroth law of thermodynamics":
There exists a variable of state, the temperature θ. Two systems that are in thermal contact, i.e., separated by an enclosure that transmits heat, are in equilibrium only if θ is the same in both.
Consequently, with the help of Eq. 1.1, we can use the pressure and the volume of an arbitrary system as a thermometer.
When we discuss the exchange of work or heat between a system and its surroundings, we find the need for a variable of state E, the internal energy, which measures the energy stored in the system. The first law of thermodynamics introduces E, as will be seen later.
Furthermore we shall find it necessary to introduce a variable of state S, the entropy, which, for example, is needed to decide whether a state is in stable equilibrium. The second law of thermodynamics introduces S and defines its properties.
For a simple system E and S are functions of p, V, θ. But, since p can be expressed by V and θ, using Eq. 1.1, it is sufficient to write:
E = E(V, θ) (1·2)
S = S(V = θ) (1·3)
Relations like Eqs. 1.1, 1.2, and 1.3 are called equations of state. Specifically Eq. 1.1 is called the "thermal equation of state"; Eq. 1.2, the "caloric equation of state." A specific substance is characterized by its equations of state. The forms of these equations cannot be obtained from thermodynamics but are obtained from measurements or else, for a particular molecular model, from statistical mechanics or kinetic theory.
Any variable of state is uniquely defined for any equilibrium state of the system. For example, if a system changes from one state of equilibrium, say A, to another state B, then EB – EA is independent of the process by which the change occurred. The important consequences of this property of the variables of state will become evident later.
One distinguishes between intensive and extensive variables of state. A variable is called extensive if its value depends on the mass of the system. The mass If of a system is thus an extensive quantity, and so are E, V, and S. For example, the internal energy E of a certain mass of a gas is doubled if the mass is doubled; the energy of a system that consists of several parts is equal to the sum of the energies of the parts.
Variables of state that do not depend upon the total mass of the system are called intensive variables, p and θ are typical intensive variables. For every extensive variable like E we can introduce an intensive variable e, the energy per unit mass or specific energy. Similarly we can define a specific volume v, specific entropy s, etc. Specific quantities will be denoted by lower-case letters.
1.4 The First Principal Law
Consider a fluid contained in a heat-insulating enclosure, which also contains a paddle wheel that can be set into motion by a falling weight. The pressure of the system is kept constant. The temperature θ and the volume V are measured initially (state A). The weight is allowed to drop a known distance, and θ and V are measured again after the motions in the system have died down and a new state of equilibrium B has been established.
In this way a certain amount of work W, equal to the decrease in potential energy of the weight, has been done on the system. Conservation of energy requires that this work is stored within the system. Hence there exists a function E(V, θ) such that
EB – EA = W (1·4)
It is also possible to use work to produce an electric current and to supply this work to the system in the form of heat given off by a heating coil. Both of these experiments were performed by Joule in his classical studies on the mechanical equivalent of heat. A given amount of work done on the system yields the same difference in internal energy regardless of the rate at which the work is done and regardless of how it is transmitted.
One can furthermore relax the condition of complete heat insulation and allow also the passage of a certain amount of heat Q through the enclosure. Q can be defined calorically by the change in temperature of a given mass of water, or one can use Joule's experiments to define Q entirely in mechanical terms. It is important, however, to define Q and W in terms of changes measured in the surroundings.
We can thus formulate the first law:
There exists a variable of state E, the internal energy. If a system is transformed from a state of equilibrium A to another one, B, by a process in which a certain amount of work W is done by the surroundings and a certain quantity of heat Q leaves the surroundings, the difference in the internal energy of the system is equal to the sum of Q and W,
EB – EA = Q + W (1·5)
It is often convenient to discuss a simple idealized enclosure, the cylinder-piston arrangement of Fig. 1.1. The cylinder walls are assumed rigid. We can assume them to be heat-insulating or. capable of heat transmission, depending on the process that we wish to study. Work can be done by the surroundings only by the displacement of the piston. W is defined as in mechanics in terms of a force vector F and a displacement dr,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1·6)
The force acting on the piston is parallel to the displacement; thus, introducing the pressure p and the piston surface area A, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1·7)
with the convention that dV is positive if the volume of the system increases. It is not difficult to show that Eq. 1.6 leads to Eq. 1.7 even in the case of pressures acting on a deformable enclosure of any shape. (Shear forces can be introduced also; this is done later in discussing the equations of motion of a real fluid in Chapter 13.) For a small change of state, we can write Eq. 1.5 in differential form,
dE = dQ + dW (1·8)
or, using Eq. 1.7,
dE = dQ - pdV (1·8a)
Equation 1.8a can also be written for unit mass:
De = dq – p dv (1·8b)
Now E is a variable of state, whereas Q and W depend on the process followed in changing the state. This is sometimes indicated by writing δW and δQ instead of dW and dQ. We shall not follow this custom here.
1.5 Irreversible and Reversible Processes
A change of state of a system is possible only by a process for which
ΔE = Q + W
The first law does not restrict the possible processes any further.
Now in the paddle wheel experiment of Joule it is evidently impossible to reverse the direction of the process. One cannot induce the wheel to extract the energy ΔE from the system and to lift the weight. The process is irreversible. It is very easy to find other similar situations, and indeed all natural or "spontaneous" processes are irreversible. If one scrutinizes these irreversible processes, it becomes evident that the deviation of the system from equilibrium during the process is of primary importance. A motion like the stirring of a fluid, sudden heating, etc., induces currents in the system. The term current refers to the flux of a quantity like heat, mass, momentum, etc. A current of heat flows, if there exists a finite temperature difference; a current of mass flows, if there exist differences in concentration of one component; a current of momentum flows, if there exist differences in velocity.
A system is in a state of equilibrium if it is free of currents. A process leading from one state to another is reversible if the system remains during the whole process in equilibrium; i.e., if the work W and the heat Q are added in such a way that no currents are produced. Such an ideal reversible process can actually be closely approximated in an experiment. For example, instead of using the paddle wheel, W could be transferred to an insulated system by a slow displacement of a piston, so that the pressure and temperature remain uniform within the system during the whole process. (Exercise 1.9 gives a simple and instructive example of an irreversible process.)
The changes of state discussed here lead from one static condition of the system to another. It is often much more convenient to consider processes that proceed at a steady rate. This is true for many measurements in thermodynamics and is essential for fluid mechanics. Thus, instead of dealing with a paddle wheel in a closed "calorimeter," as in Joule's experiment, we may consider a heat-insulated duct in which a fluid flows at a steady rate through a turbine wheel or fan. The system consists now of a certain mass of fluid which passes through the fan. Instead of dealing with a system before and after the motion of the paddle wheel, we now deal with the fluid upstream and downstream of the fan. Our definitions of thermodynamic equilibrium can be extended to this case easily. For direct comparison with thermodynamic processes like Joule's experiment, we have to require the fluid to flow very slowly so that its kinetic energy is negligible. In the next chapter we shall drop this restriction and extend the same considerations to high-speed fluid flow.
(Continues...)
Excerpted from Elements of Gasdynamics by H. W. Liepmann, A. Roshko. Copyright © 1985 H. W. Liepmann and A. Roshko. 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.
Table of Contents
Chapter I. Concepts from thermodynamics1.1 Introduction
1.2 Thermodynamic Systems
1.3 Variables of state
1.4 The first principal law
1.5 Irreversible and reversible processes
1.6 Perfect Gases
1.7 The first Law applied to reversible processes. Specific Heats
1.8 The first Law applied to irreversible processes
1.9 The concept of Entropy. The Second Law
1.10 The Canonical equation of state. Free energy and free enthalpy
1.11 Reciprocity relations
1.12 Entropy and transport processes
1.13 Equilibrium conditions
1.14 Mixtures of perfect gases
1.15 The law of mass action
1.16 Dissociation
1.17 Condensation
1.18 Real Gases in Gasdynamics
Chapter 2. One-dimensional gasdynamics
2.1 Introduction
2.2 The continuity equation
2.3 The energy equation
2.4 Reservoir conditions
2.5 Euler's equation
2.6 The momentum equation
2.7 Isentropic conditions
2.8 Speed of sound; mach number
2.9 The Area-velocity relation
2.10 Results from the energy equation
2.11 Bernoulli equation; dynamic pressure
2.12 Flow at constant Area
2.13 The normal shock relations for a perfect Gas
Chapter 3. One-dimensional Wave motion
3.1 Introduction
3.2 The propagating shock wave
3.3 One-dimensional isentropic equations
3.4 The Acoustic equations
3.5 Propagation of Acoustic Waves
3.6 The speed of sound
3.7 Pressure and Particle Velocity in a sound wave
3.8 "Linearized" shock tube
3.9 Isentropic Waves of Finite Amplitude
3.10 Propagation of Finite Waves
3.11 Centered Expansion Wave
3.12 The Shock Tube
Chapter 4. Waves in supersonic flow
4.1 Introduction
4.2 Oblique shock waves
4.3 Relation between beta and theta
4.4 Supersonic flow over a wedge
4.5 Mach lines
4.6 Piston analogy
4.7 Weak oblique shocks
4.8 Supersonic compression by turning
4.9 Supersonic expansion by turning
4.10 The Prandtl-Meyer function
4.11 Simple and nonsimple regions
4.12 Reflection and intersection of oblique shocks
4.13 Intersection of Shocks of the same family
4.14 Detached shocks
4.15 Mach reflection
4.16 Shock-expansion theory
4.17 Thin airfoil theory
4.18 Flat lifting wings
4.19 Drag reduction
4.20 The Hodograph Plane
4.21 Cone in supersonic flow
Chapter 5. Flow in ducts and wind tunnels
5.1 Introduction
5.2 Flow in Channel of Varying Area
5.3 Area Relations
5.4 Nozzle Flow
5.5 Normal Shock recovery
5.6 Effects of second throat
5.7 Actual performance of wind tunnel diffusers
5.8 Wind tunnel pressure ratio
5.9 Supersonic wind tunnels
5.10 Wind tunnel Characteristics
5.11 Compressor Matching
5.12 Other wind tunnels and testing methods
Chapter 6. Methods of measurement
6.1 Introduction
6.2 Static pressure
6.3 Total pressure
6.4 Mach number from pressure measurements
6.5 Wedge and cone measurements
6.6 Velocity
6.7 Temperature and Heat transfer measurements
6.8 Density measurements
6.9 Index of refraction
6.10 Schlieren system
6.11 The knife edge
6.12 Some practical considerations
6.13 The shadow method
6.14 Interference method
6.15 Mach-Zehnder Interferometer
6.16 Interferometer Techniques
6.17 X-Ray absorption and other methods
6.18 Direct measurement of skin friction
6.19 Hot-wire probe
6.20 Shock tube instrumentation
Chapter 7. The equations of frictionless flow
7.1 Introduction
7.2 Notation
7.3 The equation of continuity
7.4 The momentum equation
7.5 The energy equation
7.6 The eulerian derivative
7.7 Splitting the energy equation
7.8 The total enthalpy
7.9 Natural coordinates. Crocco's theorem
7.10 Relation of vorticity to circulation and rotation
7.11 The velocity potential
7.12 Irrotational flow
7.13 Remarks on the equations of motion
Chapter 8. Small-perturbation theory
8.1 Introduction
8.2 Derivation of the Perturbation equations
8.3 Pressure coefficient
8.4 Boundary conditions
8.5 Two-dimensional flow past a wave-shaped wall
8.6 Wavy wall in supersonic flow
8.7 Supersonic thin airfoil theory
8.8 Planar flows
Chapter 9. Bodies of revolution. Slender body theory
9.1 Introduction
9.2 Cylindrical coordinates
9.3 Boundary conditions
9.4 Pressure coefficient
9.5 Axially symmetric flow
9.6 Subsonic flow
9.7 Supersonic flow
9.8 Velocities in the Supersonic field
9.9 Solution for a Cone
9.10 Other meridian shapes
9.11 Solution for Slender Cone
9.12 Slender Body Drag
9.13 Yawed body of revolution in supersonic flow
9.14 Cross-flow boundary conditions
9.15 Cross-flow solutions
9.16 Cross flow for slender bodies of revolution
9.17 Lift of slender bodies of revolution
9.18 Slender body theory
9.19 Rayleigh's formula
Chapter 10. The similarity rules of high-speed flow
10.1 Introduction
10.2 Two-dimensional linearized flow. Prandtl-Glauert and Göthert rules
10.3 Two-dimensional transonic flow. von Kármán's rules
10.4 Linearized axially symmetric flow
10.5 Planar flow
10.6 Summary and application of the similarity laws
10.7 High mach numbers. Hypersonic similarity
Chapter 11. Transonic flow
11.1 Introduction
11.2 Definition of the transonic range
11.3 Transonic flow past wedge sections
11.4 Transonic flow past a cone
11.5 Transonic flow past smooth two-dimensional shapes. The question of shock-free flow
11.6 The hodograph transformation of the equations
Chapter 12. The method of characteristics
12.1 Introduction
12.2 Hyperbolic equations
12.3 The compatibility relation
12.4 The computation method
12.5 Interior and boundary points
12.6 Axially symmetric flow
12.7 Nonisentropic flow
12.8 Theorems about Plane flow
12.9 Computation with weak, finite waves
12.10 Interaction of waves
12.11 Design of supersonic nozzles
12.12 Comparison of characteristics and waves
Chapter 13. Effects of viscosity and conductivity
13.1 Introduction
13.2 Couette flow
13.3 Recovery temperature
13.4 Velocity distribution in couette flow
13.5 Rayleigh's problem. The diffusion of vorticity
13.6 The boundary-layer concept
13.7 Prandtl's equations for a flat plate
13.8 Characteristic results from the boundary-layer equation
13.9 The displacement effect of the boundary layer. Momentum and energy integrals
13.10 Change of variables
13.11 Boundary layers of profiles other than a flat plate
13.12 Flow through a shock wave
13.13 The Navier-Stokes equations
13.14 The turbulent boundary layer
13.15 Boundary-layer effects on the external flow field
13.16 Shock-wave boundary-layer interaction
13.17 Turbulence
13.18 Couette flow of a dissociating gas
Chapter 14. Concepts from gaskinetics
14.1 Introduction
14.2 Probability conc
14.9 Shear viscosity and heat conduction
14.10 Couette flow of a highly rarefied gas
14.11 The concepts of slip and accommodation
14.12 Relaxation effects of the internal degrees of freedom
14.13 The limit of continuum theory
Exercises; Selected references; Tables
1. Critical Data and characteristic temperatures for several gases
2. Flow parameters versus M for Subsonic flow
3. Flow parameters versus M for supersonic flow
4. Parameters for shock flow
5. Mach number and Mach angle versus Prandtl-Meyer function
Charts
1, 2 Oblique shock chart
Appendix, Index