R. von Mises - Mathematical theory of compressible fluid flow, страница 3

W h a t can and must be added to ( I ) and ( I I ) is some assumptionthat specifies the particular type of motion under consideration. This fifthequation will be called the specifying equation. Its general form is(III)F(p, p, q> χ, y, *, t) = o,where it is understood that derivatives of p, p, and q may also enter into F.T h e simplest form of specifying condition results from the assumptionthat the density ρ has a constant value, independent of x, y, z, and t. I t isevident that if ρ is a constant, the number of unknowns reduces to four,and ( I ) and ( I I ) are sufficient.

This is the case of an incompressible fluid.T h e most common form of a specifying equation consists in the assump61.4SPECIFYINGEQUATION7tion that ρ and ρ are variable but connected at all times by a one-to-onerelation of the form(Ilia)F(p,p) = 0.This means that if the pressure is the same at any two points, the densityis also the same at these two points, whether at the same or different mo­ments in time. Examples of such (ρ, p)-relations are(5a)- = constant,Ρ(5b)(5c)— = constant,p"ρ = A - -,ρΒ > 0,7where κ, A, and Β are constants. In the first of the examples (5) pressureand density are proportional; in general, it will be assumed that ρ increasesas ρ increases, and vice versa, so that dp/dp > 0.

If the specifying equationis of the form ( I l i a ) , the fluid is called an elastic fluid, because of the analogyto the case of an elastic solid where the state of stress and the state of straindetermine each other. A large part of the results so far obtained in thetheory of compressible fluids holds for elastic fluids only.T h e special assumption that the specifying equation is of the form ( I l i a )is, however, too narrow to cover, for example, the conditions of the at­mosphere in the large.

I t is well known from thermodynamics that for eachtype of matter a certain relation exists among the three variables, pressure,density, and temperature, the so-called equation of state. Thus the tempera­ture can be computed when ρ and ρ are known. If the atmosphere wereassumed to be elastic, so that a specifying equation of the form ( I l i a ) held,it would then be sufficient to measure ρ in order to know the temperatureas well. This is obviously not the case, so a specifying equation of the form( I l i a ) cannot hold for the atmosphere in general.

[The equation of state isnot a specifying equation, even of the more general type ( I I I ) , since itimplies temperature as a new variable.] In many aerodynamic problems,only comparatively small portions of the atmosphere need be considered,such as the vicinity of the airplane. In such cases, there is no objection inprinciple to the use of a specifying equation of the form ( I l i a ) , if thisbrings the solution of the problem within reach.T h e particular cases corresponding to the examples of (p, p)-relationsgiven in (5) can be interpreted in terms of certain concepts from thermo-8I.INTRODUCTION1F I G . 2.

ρ versus ~ for (1) isothermal and (2) isentropic flow,ρdynamics. For a so-called perfect gas* the equation of state is(6)- =ρVgRT,where Τ is the absolute temperature ( ° F + 459.7) and R is a constant de­pending upon the particular gas. For dry air, if considered as a perfectgas, the value of R can be taken as 53.33 ft/°F. From Eq. (6) it followsthat for a perfect gas the condition (5a) implies a flow at constant tem­perature, or isothermal flow.8The entropy S of a perfect gas is defined by(7)S =gR7 - 1. log — + constant,ypwhere γ is a constant, having the value 1.40 for dry air. Thus the motionof a perfect gas with the condition (5b) as specifying equation and κ = yis isentropic. T h e motion of any fluid having the specifying equation (5b),with κ > 1, will be termed polytropic.If ρ is plotted against 1/p, the specifying equation for an isothermal flowis represented by (1), an equilateral hyperbola (see Fig.

2 ) ; the curve for anisentropic flow is shown as the dotted line ( 2 ) . Whenever the variation of ρand of p is confined to a small range of values, the relevant part of these, (orother)* curves can be approximated by a straight line, giving the third type* A perfect gas is not necessarily inviscid (see [16], p. 83).1.5. A D I A B A T I CFLOW9of specifying condition (5c).

This linearized form of the (p, p)-relation(see Sec. 17.5) often facilitates the solution of a problem. Specifying equa­tions which are not of the form ( I l i a ) will be discussed later (Arts. 2 and 3 ) .In the case of steady flow, which was defined by the added conditiond/dt = 0 for all five dependent variables, it would appear that there aremore differential equations than unknowns. Actually, if t does not occurin the specifying equation, which is true a fortiori when this equation-is ofthe form ( I l i a ) , the system consisting of Eqs. ( I ) , ( I I ) , and ( I I I ) does notinclude t explicitly.

For such a system the assumption d/dt — 0 at t = 0leads to a solution totally independent of t. Thus, the assumption of steadymotion is a boundary condition only, according to the definition at thebeginning of this section. T h e same is true in the case of a plane motion,which is defined by the conditions q = 0 and d/dz = 0 for all other vari­ables, provided ζ does not occur explicitly in the specifying equation.z5. Adiabatic flowIn many cases the specification of the type of flow is given in thermo­dynamic terms.

I t is then necessary, in order to set up the specifyingequation ( I I I ) , to express these thermodynamic variables in terms of themechanical variables. T w o examples have already been mentionedabove. If it is known (or assumed) that the temperature is equal at allpoints and for all values of t, then the equation of state, which is a relationbet\veen T, p, and p, supplies a relation of the form ( I l i a ) , F(p, p) = 0;or, if the condition reads that the entropy has the same value everywhereand for all times, the definition (7) supplies another relation of the type(Ilia).The most common assumption in the study of compressible fluids is thatno heat output or input occurs for any particle.

If this refers to heat trans­fer by radiation and chemical processes only, the flow is called simplyadiabatic. If heat conduction between neighboring particles is also excludedwe speak of strictly adiabatic motion. In order to translate either assumptioninto a specifying equation, the First L a w of Thermodynamics must beused, which gives the relation between heat input and the mechanicalvariables.

If Q' denotes the total heat input from all sources, per unit oftime and mass, the First L a w for an inviscid fluid can be written(8)where c is the specific heat of the fluid at constant' volume, and quantityof heat is measured in mechanical units. T h e first term on the right repre­sents the part of the heat input expended for the increase in temperature;the second term corresponds to the work done by expansion.

Equation (8)v10I.INTRODUCTIONis equivalent to the more familiar equationdQ = c dT+ ρ dv,vwhich is derived from (8) by multiplying by dt and writing υ (specific vol­ume) in place of 1/p.If it is known that a flow is strictly adiabatic, i.e., that the total heatinput from any source (conduction, radiation, etc.) is zero, then Q' has tobe set equal to zero in (8) while Τ may be expressed in terms of ρ and ρ bymeans of the equation of state.

Finally, an expression for c in terms of thevariables T, p, and ρ is needed. For a perfect gas, where the equation ofstate is ( 6 ) , it is generally assumed that c is a constant given byvv(9)cvwhere, for dry air, y =(6) and (9)gR=l'1.4. W e note for further reference that from Eqs.c.r =(9')- J - P1ρ7 -Thus, from ( 6 ) , ( 8 ) , and (9)>0J_gRd_y-lgRdt\p)=v_11 \P dtρ dp\ _ρ dpp dt)p dt2V (\dp1Ρ dpp dt2/1 dp _~ 7 =ΛΛ __ Ί7 — 1 ρ \p dt2dp\ρ dt / 'or(10)ι<y-?'(iog!L)1 ρ dt \pV7 -holds for a perfect inviscid gas. W i t h the assumption that Q' = 0, Eq.

(10)re'duces to the specifying equationholding for strictly adiabatic flow of a perfect inviscid gas.B y means of (6) and ( 7 ) , Eqs. (10) and (11) may be expressed in terms ofthe entropy S, giving(12)Q' = Τ ^,^= 0 when Q' = 0.2.1SOMETRANSFORMATIONS11Thus, S (or p/p ) keeps the same value for each particle at all times whenQ' = 0. Nevertheless, this constant value of S may be different for distinctparticles, so that strictly adiabatic flow need not also be isentropic in thesense of Sec. 4.

Only under the additional assumption that all particleshave a common value for S at some particular time t, does the condition(11) lead to a (p, p)-relation of the form ( 5 b ) . Actual cases of fluid motionwill be met later (Sec. 15.6) in which the specifying equation is exactly E q .(11), while p/p has different values for different particles. T h e specifyingequation (11) is not of type ( I l i a ) ; a perfect inviscid gas in adiabatic flowdoes not necessarily behave like an elastic fluid.y9yArticle 2Energy Equation. Bernoulli Equation1. Some transformationsNewton's equation (1.1), the equation of continuity ( l .

I I ) , and an appropriate form of specifying equation ( l . I I I ) constitute a complete basisfor the theory of inviscid compressible fluid flow. In the present article ascalar equation, known as the energy equation, will be derived from Eqs.(1.1) and ( l . I I ) , independently of Eq. ( l . I I I ) . This equation holds in allcases, regardless of any hypothesis that the fluid is elastic or the flow adiabatic, etc., since these assumptions enter only with Eq. ( l . I I I ) .