R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 37

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From these data we knowQQ194III. ONE-DIMENSIONALFLOWthe functions a(v) and β(ν) of E q . (12.49), namely,(17)χ — ut = a(v) along CF,χ — ut = β(ν) alongCFi.T h e formulas developed in A r t . 12 then supply an explicit solution of theproblem,* giving, for example, the coordinates of the point G, etc.If the two simple waves to the left and to the right are symmetrical cen­tered waves, the computation can be carried out easily. B y shifting thex-axis if necessary, we may place the centers on the x~axis and use thesepoints as A and B.

I t was shown in Sec. 2, E q . ( 1 1 " ) , that along a crosscharacteristic for a centered wave the value of χ — ut is proportional to the— (3 — κ)/2(κ — 1) power of v, e.g., for κ =1.4, it is proportional to v~ .2In deriving this equation it was assumed that the center of the wave wasat χ = 0. Here, if we choose the /-axis through C, whose ordinate is t , theQcenters are at χ = ztaoto = =bi> Zo/5. Then the value of χ in E q . ( I I " ) is to0be corrected by this amount, so that for the left-hand wave(18)A(v)=f= -4^{va(v)dv-v )\0I t is not necessary to examine β(ν), etc., because of the symmetry of thewhole problem, which means that the functions / and g of Sec. 12.6 areidentical.

For convenience we choose the constant C in E q . (12.54) tobe ζ Λ / 2 0 . Then since ξι = η = vo, E q . (12.54) givesλM=\votoviv -vo) (v-F)= jg'iv)= I v tov(v -Vo) (v-= ^*{(I-2vo )2(19)0-vo ).2B y differentiation and integration we derive from (19)(20)/ "=^ ( 3 ξ2/ - g t f - O *- ^ ) ,and analogous expressions for g" and g. If these are introduced in (12.47)we find, after some algebraic rearrangement,x (21Γ)1=ut = ^(u* -383v* -vo )2πΛ•»l£ 1/ ν " ° +1 +42mV+%2) +* Since /(£) in (12.47) is determined by a (v) alone, see (12.54'), it represents the in­fluence of the wave on the left, and, similarly, ς ( η ) that of the wave on the right.14.1N O N E X I S T E N C E OF S O L U T I O N S195These two equations supply the values of χ and t corresponding to eachpoint (u,v) of the rectangle A'E'G'E'i. In particular, corresponding to G'we find the x,t-values for the " e n d " of the penetration.T h e results expressed in (20) and (21) can be checked by computing thevalues of χ — ut and t along the characteristics CF and CFi as functions ofv, setting u = dz(v — v ), and comparing them with the values along crosscharacteristics given by (11') and ( 1 1 " ) , corrected for the fact that thecenters are not at the origin.

Moreover, χ — ut satisfies the differentialequation (12.34) with η = — 2 , and t satisfies the same equation withη = -3.0Article 14Theory of Shock Phenomena1. Nonexistence of solutions. Effect of viscosityAssume that at t = 0 the interval A Β of the z-axis is occupied by a fluidmass in a state of rest and at uniform pressure: u = 0, ν = v = constant.If we draw through A and Β (see Fig. 74a) the two straight lines of slope± α ο (where a is the sound velocity corresponding to v and the slopes aremeasured from the £-axis) and the vertical particle lines representing thestate u = 0, ν = v for all points (x,t) within the triangle ABC, this solutioncertainly satisfies the general differential equations (11.2), (11.3), and (11.4),regardless of whether viscosity and heat conduction are admitted, and alsosatisfies the initial conditions at t = 0.

In the theory, developed in A r t .12, of an elastic, inviscid and nonconducting fluid, the solution is uniquelydetermined within the characteristic quadrangle, of which ABC is the upperhalf, by the conditions along AB. This means that the differential equa­tions of an inviscid and nonconducting fluid have no solution in ABC con­sistent with the given initial conditions other than the solution u = 0,ν = v everywhere in ABC. [Such a fluid is necessarily elastic by virtue ofthe entropy being initially the same for all particles (see Sec. 2.3).]00000On the other hand, it is undoubtedly possible to subject an actual fluidmass between A and Β to some additional conditions. For example, the fluidmay be contained in a cylindrical tube with a solid piston initially at rest atthe point A.

Suppose we start moving the piston to the right according to anarbitrary law χ = f(t) subject only to the restriction that df/dt = 0 at t = 0.This law may be represented by the curve AD in Fig. 74a, and this curve isnecessarily the particle line for the fluid particle initially at A. Nothing canprevent us from moving the piston so that the point D falls within the tri-196III.

ONE-DIMENSIONALFLOWangle ABC?Thus, a contradiction is apparent. T h e velocity at D must bethat determined by the slope of the tangent to the curve AD at D and notthe value u = 0 determined by the solution given above.N o t e that this contradiction would not occur if the fluid were consideredto be incompressible, so that the whole fluid mass moved with the piston,behaving as a rigid body. In fact, the triangle ABC, in which the solutiongiven above is determined, would not exist, since for an incompressible fluidthe sound velocity is infinite, causing C to fall on the z-axis.9T h e curve AD can cross over into the area ABC only if the velocity of thepiston exceeds, at some time, the value a determining the slope of AC.Thus one might suppose that the inconsistency is restricted to the occur­rence of supersonic velocities.

One can, however, superimpose on the flowof Fig. 74a a uniform motion with velocity u toward the left, i.e., u nega­tive. This is shown in Fig. 74b. Here the characteristic triangle is no longerisosceles, the sides AC and BC having slopes u + a and u — a , respec­tively. This time the initial velocity of the piston must be u rather than 0.T h e curve AD can intersect AC if at some time the velocity of the pistonexceeds u + α , and if we take u > — a then both | u | and | u + α |are subsonic.W e see then that the differential equations for the theory of an inviscid andnonconducting fluid admit no solution corresponding to certain boundary con­ditions which can be enforced by simple physical arrangements.00000000000000In considering the physical example mentioned above, we cannot giveup the condition of at least simply adiabatic flow, nor can we restrict thetC(a)tC(b)AΒFIG.

74. Enforceable boundary conditions for which no inviscid solution exists.014.1N O N E X I S T E N C E OF197SOLUTIONSfreedom of choosing the boundary conditions; therefore the only way out ofthe impasse is to take viscosity and/or heat conduction into account.One could object, for example, that for dry air under normal conditionsthe viscosity coefficient is very small, and that the influence of viscosity maybe negligible. This objection, however, is invalidated by the results of A r t .11.

There it was seen that a particular type of flow, much different from thatoccurring in an inviscid fluid, is possible in the viscous case, the differencebecoming more pronounced, the smaller the value of the viscosity coeffici­ent. W e refer to the flow pattern with rapid change from a velocity valueUi to a smaller value u , accompanied by similar changes in ρ and p. A s wepass to the limit μ = 0 in (11.26), we see that the extent L of the transi­tion region also tends to zero. In other words, the theory of a viscous com­pressible fluid yields, in the limit of vanishing viscosity, flow patterns withabrupt changes in the state variables, no indication of which is given in thetheory based throughout on the assumption μ = Ο. Since the forces causedby viscosity are proportional to the product of μ by velocity derivatives,here du/dx, it is understandable that even a very small value of μ, when com­bined with an exceedingly large value of du/dx, may produce a considerableeffect upon the flow pattern.* A l l this suggests the following conception ofthe flow patterns possible in a fluid of low viscosity.2040One may expect that in a fluid of low viscosity the ideal theory, assumingμ = 0, will supply a reasonable approximation to the actual flow in general,but that the regions in which such approximations are valid are separatedfrom each other by steady or moving thin layers within which occur rapidchanges in the state variables, as suggested by the theory for a viscous fluid(partly) developed in Art.

11. Numerous observations of air in motionconfirm this conception in all respects. In particular, it is observed that flowpatterns of this combined type occur in cases, such as the example men­tioned at the beginning of this section, for which there exists no consistentsolution based only on the ideal fluid theory.41This phenomenon of an almost abrupt change in the values of the statevariables is known as shock.A shock theory for one-dimensional flow in a form accessible to mathe­matical treatment is therefore based on the following principles.

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