R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 82
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T h e graphs of u, p, and ρ versus χ for any t w o solutions are,respectively, translations of each other in the ^-direction.For ra < 0 the same result holds with the particles passing from a state1 at χ = + o o to a state 2 at χ =— o o . For ra = 0 the gas is at rest withuniform pressure and density. W e now combine these results and interpretthem in terms of the original system of Eqs. (42). Let Ui, p\, pi and u ,2p , p be two sets of values of u,p, ρ depending on the time t, and S an arbitrary22line in the x,t-plane, whose slope is denoted by dx/dt = c(t).
Then the shockconditions(14.2) and (14.9) are not only necessary but also sufficienttions for there to exist shock transitioncondisolutions T ( s , β), with respect to S andthese two states, of the equations governing the one-dimensionalnonsteady flowof a perfect viscous gas in simply adiabatic motion. For any t w o such solutions the graphs of u, p, and ρ versus χ at each time t are translations ofeach other in the z-direction.A similar result holds in the case of steady plane flow.
W e consider acurve S in the #,i/-plane and this time introduce an orthogonal system ofcoordinates in which one set of coordinate lines are the normals of S, seeFig. 165. A coordinate line of the other set then cuts these normals at a constant distance from S. Thus for one coordinate, a, at a point Ρ we may take440V. I N T E G R A T I O N T H E O R Y A N D S H O C K Sdistance from Ρ to S along the normal, and for the other, β, arc length alongS from a fixed point P0to the foot of this normal.
T h e equations governingthe steady plane flow of a perfect viscous gas in simply adiabatic motion arethen written in terms of these new coordinates and thecorrespondingcomponents of velocity u and ν; a viscosity coefficient μο is introduced whenthe two-dimensional analogue of the viscosity assumption (11.6) is used toexpress the stresses in terms of the velocity gradients. In order to defineshock transition solutions of this system of equations the normal coordinate a is magnified in the ratio 1: μ and then every term having a factor μ00is replaced by zero. Three of the equations in the resulting system are foundto be equivalent to Eqs. (11.8); the fourth expresses the fact that the component of velocity, v, in the β-direction (i.e., parallel to S) remains constantyF I G .
165. Orthogonal curvilinear coordinates in the z,?/-plane.along an α-line. From this it is easily seen that the shock conditions,Eqs.(22.3) and (22.16), are not only necessary but also sufficient conditions for theexistence of shock transition solutions, with respect to S and two sets of valuesof u, ν, ρ, ρ (depending on β), of the equations governing the steady plane flow ofa perfect viscous gas in simply adiabaticmotion.6. Asymptotic solutions of the equations of viscous flowIn order to understand the role of these shock transition solutions it isnecessary to investigate more carefully the connection between the viscoussolution of a problem and its solution by means of the principles establishedin Sees. 14.2 and 22.2.
Suppose that a specific problem, in the χ,ί-plane, say,has been solved by means of the equations governing the simply adiabaticflow of a perfect viscous gas, for all small values of μ ^ 0. W e denote this0family of solutions by S(x, t \ μ ) , where as before the single symbol stands0for the set of state variables. When χ and t are replaced in it by μ $ +0f^)24.6441ASYMPTOTIC SOLUTIONSand β, respectively, in accordance with (39), this same family will be denoted by S(s, β; μ ) . T h e dependence of the family on k is not indicatedsince for simplicity we may suppose that μο/k, which is proportional to thePrandtl number P, has the same constant value for all μ . Suppose alsothat the same problem is solved according to the principle given in Sec.14.2; the asymptotic solution S (x, t) satisfies the equations for strictlyadiabatic flow of a perfect inviscid gas at all points of the ζ,ί-plane concerned, except for lines S, across which the discontinuities in u, ρ and ρsatisfy (14.2) and (14.9).000On the basis of the discussion in Sec.
14.1 we expect that for sufficientlysmall μ , the viscous solution S lies close to S except in the neighborhoodof S, where it changes rapidly in the normal direction, the total changebeing approximately equal to the jump in S across S [governed by Eqs.(14.2)]. Within the transition region S(x, t; μ ) has derivatives which areunbounded as μ —> 0. I t is plausible however that derivatives of S(s, β; μ )are bounded. Then the right members of Eqs. (42) are of order μο, and wemay expect S to lie close to a shock transition solution T(s, β) of (42) inthis region. N o w from A r t .
11 we know that any fixed proportion of thetotal change in T(s, β) takes place in a distance of order μ [see the thickness estimate (11.54), where according to (11.26) L is proportional to μ ].Hence for successively larger proportions to be realized the interval mustbe of lower order than μ , and this will be the case for S also.0000000000W e now formulate this conjecture more precisely: Let ά(μ ) be a distancewhich tends to zero more slowly than μο , i.e., μο/ά(μ ) —» 0. Then by taking μsufficiently small we can ensure, with any preassigned accuracy, that(a) At points displaced more than ά(μ ) from S, the viscous solutionS(x, t; μ ) approximates S (x, t), and(b) At points displaced less than ά(μ ) from S, the viscous solutionS(s, β; μ ) approximates T(s, β), a shock transition solution withrespect to S and the two sets of values of u, ρ, ρ attained on S by S .This of course has not been demonstrated in general. In the preceding section we have however proved the existence of shock transition solutionspostulated by ( b ) .
In other words, it has been shown that" (b) is not selfcontradictory; in addition we have made it plausible. T h e exact solution ofSec. 11.4 provides us with the only concrete example known so far for whichthis conjecture can be verified. T h e problem consists in finding a flow forwhich u, ρ, ρ take on prescribed constant values Ui, pi, pi and u , p , pat χ = — oo and χ — + ° ° , respectively, for all t, these values satisfying theshock conditions (44) with m > 0, and also T ^ Τι .
In addition we requirethat u = %(ui + u ) at χ = 0, say, in order to fix S(x, t; μ ) . The solutionS Or, t) consists of constant state values 1 to the left, and constant statevalues 2 to the right of S: χ = 0. T h e shock transition solution T(s, β) is000000000502220022442V. I N T E G R A T I O N T H E O R Y A N D S H O C K SS(s, β; μ ) itself. This example does not depend on time, but one whichdoes can be derived from it by superimposing a constant velocity c on thewhole flow.T h e situation in the case of steady plane flow is essentially the same. Itsexposition is complicated however by the occurrence of a second type ofrapid transition region known as the boundary layer.0These results apply equally well when μ and k are functions of T, provided suitable bounds are put on their variations.0Article 25Transonic Flow1.
O n some additional boundary-value problemsI n Arts. 17 and 20 we collected various examples of flows obtained bymeans of the hodograph method. Some of them, source flow, vortex flow,spiral flow, etc., were primarily examples of solutions of the basic equationsand there was no attempt to satisfy boundary conditions. (Of course it waspossible to find an a posteriori interpretation, e.g.
for the flow between t w ostreamlines considering the streamlines as walls.) A n outstanding exampleof an exact solution of a given boundary-value problem is Chaplygin's jetproblem. Various investigators have followed along similar lines.In A r t . 21 we explained the general methods of Bergman and Lighthill,which enable us to generate stream functions reducing, as q —> oo, to thestream function of a given incompressible boundary-value problem.
W eshall discuss here some further results concerning flow around a profile andadd some remarks on channel flow.m(a) Remarks on flows past a profile. E v e n in the problem of flow withoutcirculation past a circular cylinder the elegant solutions by means of anintegral formula such as found in (21.23) or (21.37) are lost, if westudy the flow beyond Μ — 1.In Sec. 21.3 we started with series expansions of the hodograph potentialw ( f ) ; in the case of a circle three such series were needed to cover the f-plane.W e also found that the compressible flow could not be correctly obtainedby "translating" each of these series into its compressible counterpart, b ymeans of the method invented by Chaplygin for the jet problem, or b y anequivalent method using a different factor / (n, n ) .
T h e series obtained inthis manner were not analytic continuations of each other; a correct procedure was to start with an appropriate branch of the compressible flow and025.1SOME A D D I T I O N A LPROBLEMS443F I G . 166. Streamlines and lines of constant speed for flow around a circle-likeprofile derived from incompressible flow around a circle by Τ . M . Cherry.
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