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Mathematical Aspects of Subsonic and Transonic Gas Dynamics
By Lipman Bers Dover Publications, Inc.
Copyright © 2016 Lipman Bers
All rights reserved.
ISBN: 978-0-486-81633-3
CHAPTER 1
THE DIFFERENTIAL EQUATIONS OF A POTENTIAL GAS FLOW
2. DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS
We shall recall briefly the derivation of the potential equation of a compressible fluid flow. A detailed description will be found in many texts (e.g., Courant and Friedrichs).
Basic equations
The flow of a perfect fluid is described, in the so-called Eulerian representation, by giving the density ρ and velocity components, u1, u2, u3 as functions of the Cartesian coordinates x1, x2, x3 and time t. A complete description requires also the knowledge of two other thermodynamic variables, say, pressure p and temperature, or pressure and entropy. We deal, however, with the case in which the flow is adiabatic and isentropic. In this case, pressure is a definite function of density; for an ideal gas
(2.1) p = const. ργ
where γ > 1 is a constant (ratio of specific heats). The standard value of γ for air is 1.4. It will be convenient not to restrict oneself to this pressure density relation but to consider also a general barotropic fluid, i.e., a fluid in which
(2.2) p = p(ρ),
p(ρ) being some sufficiently smooth increasing function.
The velocity components and the density are connected by the continuity equation which expresses the law of conservation of mass:
(2.3) [MATHEMATICAL EXPRESSION OMITTED]
On the other hand, the velocity components must obey the Euler equations of motion which are the analogue of Newton's second law. If one neglects body forces, in particular gravity, these equations read
(2.4) [MATHEMATICAL EXPRESSION OMITTED]
Note that in view of relation (2.2), the pressure can be eliminated from the Euler equations. The equations of motion and the continuity equation obey the relativity principle of classical mechanics, i.e., they hold in every inertial coordinate system if they hold in one.
Speed of sound
The derivative dp/dρ is positive, and we will denote it by c2. It is seen at once that c has the dimensions of a velocity; as a matter of fact, it is to be interpreted as the speed of propagation of small disturbances in a flow, and is, therefore, called the local speed of sound.
To verify this statement, it is sufficient to consider propagation of small disturbances in a fluid at rest since, given any continuous flow, we may introduce an inertial coordinate system in which the velocity vector vanishes at a given point (x0, y0, z0) at a given time t, and the fluid may be assumed to be approximately at rest in a small neighborhood of (x0, y0, z0), and in a sufficiently small time interval around t0. Thus we consider the case in which the flow variables may be written in the form
(2.5) ui = εu'i, ρ = ρ0 + ερ', p = p0 + [epsion]p'
where ρ0, p0 are constants and the parameter ε is so small that its square may be neglected. In view of the relation between pressure and density, we have
(2.6) p' = c02ρ' + O (epsilon]2)
where c0 is the value of the speed of sound corresponding to the value ρ = ρ0 Introducing (2.5) and (2.6) into the equations of motion and the continuity equation, and neglecting terms containing ε to a power higher than the first, we obtain the linear system of the partial differential equation
[MATHEMATICAL EXPRESSION OMITTED]
Eliminating all but one variable by differentiation, we obtain for the density perturbation the classical wave equation
[MATHEMATICAL EXPRESSION OMITTED]
which, as is well known, describes the propagation of disturbances with the speed c0. The same equation is satisfied by the velocity components.
Thus, the propagation speed of small disturbances in a fluid obeying equations (2.3)and (2.4) depends on the local value of the density. The character of the flow depends essentially on the dimensionless number (Mach number) M = q/c where q2 = u12 + u22 + u32. The flow is called subsonic (supersonic) if M< 1 (M > 1).
Steady potential flow
We shall deal exclusively with steady flows, i.e., with solutions of the equations of motion which do not depend upon time. Furthermore, we shall assume that the flow is irrotational, i.e., that the vorticity vector
[MATHEMATICAL EXPRESSION OMITTED]
vanishes identically. There are good physical reasons for considering such flows since it can be shown from the Euler equations that the line integral of the vorticity over a closed material curve, i.e., over a curve the points of which move with the velocity components u1, u2, u3, is constant (Kelvin's theorem). Thus a flow which started out as irrotational will remain so. Kelvin's theorem holds also if there are body forces present, provided that these forces are conservative.
It is easy to see that in a potential flow the quantity
[MATHEMATICAL EXPRESSION OMITTED]
is a constant (Bernoulli's law). It follows from Bernoulli's theorem that in a potential flow the density is a given function of the speed. The units may be chosen so that ρ = 1, c = 1, for q = 0. Then
(2.7) ρ = ρ(q)
is a decreasing function defined implicitly by the relation
(2.7a) [MATHEMATICAL EXPRESSION OMITTED]
and
(2.8) c2 = -ρq/ρ'(q), M2 = q/ρ dρ/dq
In particular, for a flow governed by the adiabatic pressure density relation (2.1), we have
(2.9) [MATHEMATICAL EXPRESSION OMITTED]
(2.10) [MATHEMATICAL EXPRESSION OMITTED]
Thus the flow is subsonic if the speed q is less than the critical speed
qer = (2/γ + 1)1/3,
and since ρ must be positive, only speeds below the maximum speed
qmax = (2/γ - 1)1/2
are possible.
Since the vorticity vanishes, the velocity components are partial derivatives of a not necessarily single-valued function (x1, x2, x3), called the velocity potential:
ui = [partial derivative]φ/[partial derivative]xi, i = 1, 2, 3
The continuity equation can now be written in the form
[MATHEMATICAL EXPRESSION OMITTED]
or, by (2.7) and (2.8)
(2.11) [MATHEMATICAL EXPRESSION OMITTED]
This is the basic potential equation of gas dynamics. It should be remembered that the expression c2 occurring in this equation is a given function of the speed
[MATHEMATICAL EXPRESSION OMITTED]
The equation is therefore nonlinear, more precisely quasilinear, since the second derivatives enter in it in a linear way.
Subsonic and supersonic flows
The type of a quasilinear equation depends upon the solution considered. For equation (2.11), the discriminant equals 1 – M2. This is seen at once at a point at which the velocity vector has the direction of the x1-axis, so that u1 = q, u2 = u3 = 0, and it is sufficient to consider this case since equation (2.11) is rotationally invariant. Thus, the equation is elliptic if the flow is subsonic, and hyperbolic if the flow is supersonic. At points at which the local speed equals the local speed of sound (M = 1), the equation is of parabolic type.
We shall always assume that the density speed relation (2.7) has the general character of the adiabatic density speed relation (2.9). More precisely, we assume that the function ρ(q) is defined, positive, sufficiently smooth, and nonincreasing in an interval
0 ≤ q < qmax ≤ + ∞
and that there exists a value qcr, 0 < qcr ≤ qmax such that M2< 1 for q< qcr and M2 > 1 for q >qcr, M2 being defined by (2.8). Every such function ρ(q) corresponds to a pressure density relation p = p (ρ) with p' (ρ) = c2; see (2.8).
A very simple but extremely basic distinction between subsonic and supersonic flows follows from the relation
(2.12) 1 - M2 = 1/ρ d(qρ)/dρ
which shows that the quantity qρ(q) (mass velocity) is an increasing function of the speed for subsonic speeds, and is a decreasing function in the supersonic range. Hence, in a small tube, in which the velocities in a cross section may be assumed, as a first approximation, to be constant, the speed will increase or decrease in a narrowing section of the tube according to whether the flow is subsonic or supersonic.
Plane flows
We shall be concerned almost exclusively with two-dimensional flows, i.e., with flows for which the velocity component u3 vanishes. For a two-dimensional flow the velocity potential may be considered as a function of the two variables x1x2. From now on we write
(2.13) [MATHEMATICAL EXPRESSION OMITTED]
and / = /(x, y). The potential equation takes the form
(2.14) [MATHEMATICAL EXPRESSION OMITTED]
On the other hand, the continuity equation in a two-dimensional flow reads
(2.15) (ρu)x + (ρv)y = 0
so that there exists a function (stream function) ψ(x, y) such that
(2.16) ρu = ψy, ρv = -ψx.
The velocity potential and the stream function are connected by the first-order equations
(2.17) ρφx = ψy, ρφy = -ψx
in which ρ = ρ(q) is a given function of q = | /x - i/y |. Elimination of ψ from this system leads back to the potential equation (2.14). If we try to eliminate the potential, however, we encounter the difficulty of ρ being a two-valued function of the mass flow q = (ψx2 + ψy)1/2. Thus we can write down a second-order quasilinear differential equation of elliptic type satisfied by the stream function of a subsonic flow, and another such equation, this time of hyperbolic type, which is satisfied by the stream function in a supersonic flow. There exists, however, no single second-order equation satisfied by the stream function.
Weak discontinuities and shocks
A few words ought to be said concerning possible discontinuities of a solution. In the subsonic domain the potential of a compressible flow satisfying equation (2.14) will be as smooth as the equation permits. This is, of course, a consequence of a general property of solutions of elliptic equations; see Art. 7. In particular, if the density speed relation is analytic, the potential will be a real analytic function. If the density speed relation is such that the function c = c(q) occurring in (2.14) has a certain number of continuous derivatives with respect to the argument, every solution will be continuously differentiable a definite number of times. We lose no generality, therefore, assuming from now on that the potential of a subsonic flow is at least twice continuously differentiable.
The situation is different in a supersonic flow which is governed by a hyperbolic equation. It is known that a solution of a quasilinear hyperbolic equation may possess discontinuities of the second derivatives. These discontinuities, however, must occur on certain lines, called the characteristics, which are determined by the equation and the solution.
In the case of a compressible flow, these weak discontinuities are discontinuities in the derivatives of the velocity components. The characteristics, called in this case Mach lines, are the solutions of the ordinary differential equation
(2.18) (c2 - u2) dy2 + 2uv dx dy + (c2 - v2) dx2 = 0.
Thus, the characteristics depend upon the flow considered.
It is seen at once, by considering a point at which v = 0, u = q, that, through every point in the supersonic region, there pass two Mach lines and that the Mach lines intersect the streamlines at the angle
(2.19) ±α = ± arc tan 1/√ M2 - 1.
At a point on a sonic line the Mach angle α is π/2; at such a point the two characteristics are tangent to each other.
It is necessary to consider also another kind of discontinuity. The basic equations (2.4) are derived by neglecting the viscosity of the fluid. In some problems this neglect is not justified and this is reflected in the nonexistence of a continuous solution. In the flow of a real fluid there might exist narrow regions in which a very rapid change of velocity and density, accompanied by dissipation of energy, take place due to viscosity effects. In the theory of an ideal gas such a region is represented by a surface, in the case of a two-dimensional flow by a line, across which the velocity vector, the density, and the pressure experience jump. These jumps are not arbitrary, they are governed by certain relations (shock conditions) which are derived from mechanical and thermodynamical considerations. In a steady flow shocks occur only at supersonic speeds, though the flow may become subsonic upon crossing the shock. In general, the flow will be neither isentropic nor potential after crossing the shock. For so-called weak shocks, however, the change in entropy and the resulting deviation of the flow from irrotationality may be neglected. In fact, the shock relations show that the change in entropy is of the third order in the strength of the shock, this strength being measured, for instance, by the jump in the density.
For a weak shock there are only three shock conditions. The first requires that the tangential components of the velocity be continuous across the shock line. The second, resulting from the law of conservation of mass, requires that the product of the normal component of the velocity by the density (Fano number) be continuous across the shock. Note that these conditions may be satisfied without continuity of the velocity components because the speed q is a two-valued function of the mass flow ρq. The third condition, a consequence of the second law of thermodynamics, demands that the density decrease across the shock. In particular: a supersonic flow may become subsonic upon crossing the shock line, but not vice versa. Whereas the shock conditions are derived for a fluid possessing thermodynamic properties, the conditions for a weak shock, mentioned above, are purely kinematical and may be applied to any barotropic flow.
We shall be concerned mostly with shockless flow and, in particular, with conditions under which a shockless flow becomes impossible.
Boundary conditions
We mention now some typical boundary value problems for equation (2.14). The most important, from the point of view of aerodynamics, refers to the flow around an obstacle (which is thought of as a cross section of an airplane wing). The solution is to be determined in the domain exterior to the obstacle P, and is to describe a flow which is uniform at infinity. In other words, we prescribe the value of the velocity at infinity
(2.20) [MATHEMATICAL EXPRESSION OMITTED]
On the profile P we have the boundary condition
(2.21) [partial derivative]φ/[partial derivative]n = 0
expressing the fact that the obstacle is a stream line. It may also be written in the form
(2.21a) ψ = const.
These two conditions are, however, not sufficient to determine the flow. In fact, we have no reason to assume the velocity potential to be single-valued, and we might, therefore, want to prescribe the value of the circulation
(2.22) [MATHEMATICAL EXPRESSION OMITTED]
where [??] is any simple closed curve around the profile.
From the physical point of view a different way of determining the flow is preferable. We shall assume, unless otherwise stated, that the profile is a smooth curve, except for one protruding curve or cusp, called trailing edge, and require that the velocity be continuous at the trailing edge (Kutta-Joukowski condition). If the profile has no corners or cusps the Kutta-Joukowski condition consists in requiring that the velocity should vanish at a given point of the obstacle. The fact that the Kutta-Joukowski condition determines the circulation uniquely is, of course, by no means obvious. For a purely subsonic flow it can be proved, as we shall see later.
(Continues...)
Excerpted from Mathematical Aspects of Subsonic and Transonic Gas Dynamics by Lipman Bers. Copyright © 2016 Lipman Bers. Excerpted by permission of Dover Publications, Inc..
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