R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 43

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Numerical method of integrationIn all the examples we have discussed, solutions could be given con­sisting of regions of uniform flow and simple waves, separated, respec­tively, by straight shock lines or straight characteristics. I t is clear thatcases with general boundary conditions cannot be treated in this way. Forexample, in the Riemann problem considered in Sec. 14.6 the solution wasobtained only up to the moment when the shock wave to the right meetsthe simple wave to the left [point E(x0for t >t,0,t ) in Fig. 79].

T h e flow after this,0is determined by the conditions u = 0 at χ = 0 and χ =I,together with the conditions within the fluid for t = to, namely,for χ < xo :u = f(x),ρ = g{x),ρ =for χ > Xo :u = 0,ρ = pi ,ρ = pi .h(x);In these conditions xo,h are the (known) coordinates of E, and p piuthe(known) values of ρ and ρ after the shock. T h e functions/(x), g(x), and h{x)are constant to the left of AD,but then continue with the nonconstantvalues of u, p, and ρ determined b y the equations for the simple centeredwave. I t can be anticipated that, for t > U, a curved shock line will develop,starting at E, but neither its shape nor the flow pattern on either side of itcan be expressed in terms of elementary functions.

In such cases one has toresort to numerical methods of integration, that is, one substitutes finitedifference equations for the differential equations of the flow and thensolves the resulting algebraic problem b y the methods of practical analysis.W e shall sketch this procedure here for the case of one-dimensional nonsteady flow of a viscous fluid, neglecting gravity and heat conduction. In­cluding heat conduction, however, would not essentially change the setup.If we write σ rather than σ , Eqs. (2) to (6) of A r t . 11, with k = 0, giveχdp 1Su^_ du,d{p -σ)15.4N U M E R I C A L M E T H O D OF225INTEGRATIONT h e first is the equation of continuity, the second is Newton's equation fora viscous fluid, the third expresses the fact that the motion is (strictly)adiabatic, and the last represents the usual (Navier-Stokes) assumption asto the form of the viscous stresses.

T h e boundary conditions will be statedlater.T h e third equation in (10) may be simplified by subtracting from it utimes the second equation, giving7 -1 \dtρ dt)dxΓand then replacing dp/dt by its value from the first equation, giving(ID^ =^dtdx[(γ- 1)σ-γρ].T h e first equation in (10) is automatically satisfied if we introduce theparticle function ψ [see (12.11)]. This function is constant along particlelines: ά-ψ/dt = 0. Then the second equation (10) becomes*±du( 1 2 )eip-^+dx dt=0dxT h e system (10) is thus replaced by (11) and (12), together with σ=μodu/dx.T o carry out numerical integration, we let u„(t), p (t),vand x {t)vdenotethe values of u, p, and χ as functions of t on particle lines corresponding toequally spaced values of ψ: yp (t) — ψ -ι(ί)νv= ra = constant.

Then all de­rivatives with respect to χ are replaced by difference quotients, e.g., dp/dx be­comes (pv— p -\)/{xv—vso that our three equations yield for eachXy-ι),value of ν the equationsdt(13)%=(ItK+1~U'[(7 -IV, "7PJ,Xv+\ — %vUy+lσν=μο— UV—.These equations can be interpreted as follows: in a tube of finite lengthand unit cross-sectional area, the continuously distributed fluid mass isconsidered as broken up into a finite number, say n, of mass points, each ofmass ra.

T h e abscissa of the I>th mass point at time t is x (t),vis(14)„,(*) =—.and its velocity226III. ONE-DIMENSIONALFLOWtFIG. 85. Discrete particle model for one-dimensional nonsteady viscous flow.T h e pressure and the viscous stress are considered constant in the intervalbetween the *>th and (v + l ) s t particle and denoted b y p and σ „ . Then thevfirst equation (13) is N e w t o n ' s Second L a w for the vth mass point; thesecond equation (13) gives the rate of change of p as determined b y thevcondition of adiabatic flow, while the third expresses σ in terms of theνvariables x u .ViI n the x,2-plane the motion is represented b y η distinctvparticle lines (Fig. 85), each defined by one of the functions x (t),ν = 1,2,v• · · , η .

T h e figure also includes t w o curves marked 0 and η +x-axis; these lines Xo(t) and x +i(i)are supposedngiven1 on theas boundaryconditions.If the value of σ„ from the last equation (13) is substituted into theother two equations, and u is expressed in terms of x b y (14), we have, forvveach ν (v = 1,2, · · · , n ) , a system of two simultaneous ordinary differentialequations in x and p , the first equation being of second order in x , thevvvsecond of first order in p .

T h e independent variable is t, and the equationsvinclude a constant parameter m. T h e right-hand member of the second equa­tion for ν = η includes x +i and u +i,which are given b y the boundarynncondition specifying x +i(t).T h e first equation for ν =n1 involves σ ,0which is determined from the boundary condition specifying x (t):0μο(^ι — uo)/(xiσ=0— XQ). I t also contains po, and t o make the system of equa­tions complete, we add a (2n + l ) s t equation determining p :0(13')ά^ =^ Ζ ^atThus we have (2n +Χι, X2, · ",xX\[{7-1)σο-7ρο].XQ—1) differential equations for the (2n +1) unknowns; Po, Ph " ',.Ρη as functions of t.nT h e integration of the system (13) requires, in addition t o the boundaryconditions Xo(i) and x +i(t),nu (0),vand p (0).va knowledge of the 3n initial valuesx (0),vN o w the initial conditions of the original problem aregiven as three functions u (x),0p (x),0and po(x).

This last function serves t ogive the value of m and the initial abscissas x (0).yF o r example, one m a y15.5O N A P P L I C A T I O N OF N U M E R I C A Lplot the integral curve Jp (x)dxxx +i(0)n0=0= I, say, and take for x (0)vwith ordinate v\p(l)/(n +χ = X i _ i ( 0 ) to χ =ψ(χ)from χ =x (0)i =0 to χ=0=the abscissa of the point on the curve1). T h e mass in each of the η +Xi(Q),227METHOD1, 2, · · · , η +1 intervals from1, is then ψ(1)/(η+1)and this is taken for the mass m of each of the particles. Finally, we have= Uo[x*(0)], and p„(0) = po[x,(Q)] for ν =u (0)v1, 2, · · · , η.T h e computation procedure is now as follows. For t = 0, the right-handmembers of all equations (13) and (13') are known, and from them followthe initialvalues of dpo/dt, du /dt,vand dp /dt,vwhile dx /dtv=u,(Q).Taking a small time interval At, and assuming that the rates of change re­main constant during this interval, we can compute the values of p , x ,0p , and u at the time t =vyAt for all ν =1,2,v· · · , η.

From the valuesnow found, we can go on to compute the values for t = 2 At, etc. T h e tech­nique of numerical integration provides certain rules on how to improve theaccuracy of the procedure by using successive approximations. Other rulesto be observed concern the order of magnitude of At relative to the size ofthe segments (x„+ihere.— x ), etc. These details, however, cannot be discussedv595. Some remarks on the application of the preceding methodWhen the step-by-step integration method just described is applied tocases where the boundary conditions do not lead to the occurrence of shockphenomena, no serious difficulty arises.

If an empirical value is used forμο, as indicated in Sec. 11.5, one will find, in general, that the influence ofviscosity is practically negligible. Therefore one may omit the σ-terms in(13) and work with the simpler systemdu/vm^=Vv-\ ~IO\(v = 1, 2, · · · , η),V»,(15)dp,~d7 ~r=u,+\ — u,τ "Ί Ρ.(" = 0, 1, · · · , n ) .In particular, when the flow is initially isentropic, this furnishes a numericalapproximation to the solution for which an analytic form was also given inSec. 12.4.

T h e approximation method must be used, however, wheneverthe boundary conditions do not lead to explicit expressions for the functionsf and g appearing in the general integral (12.42). Straightforward compu­tation based on Eqs. (15) will then solve the problem—if a solution exists.W e saw, however, in Sec. 14.1, that a solution of the differential equationsof an ideal fluid does not exist for certain boundary conditions.

In suchcases, pursuing the computation procedure based on (15) would lead toinconsistencies, or to nonconvergence for increasing η and diminishing At.60For these cases one could try to apply the principle developed in Sec. 14.1,228III. ONE-DIMENSIONALFLOWthat is, assume that the whole flow pattern consists of regions in which idealfluid theory applies, separated by shock lines across which occur discon­tinuities governed by the shock conditions. For instance, in the Riemannproblem mentioned at the beginning of the preceding section, one has tosuppose that for / > to the flow pattern includes a curved shock line startingat Ε in an unknown direction (denoted by its slope c, measured from theί-axis).

I t can then be seen that the Eqs. (15), combined with the shock con­ditions, allow us to compute c and the values u and ρ for the time to + At,etc. This procedure, however, is cumbersome and, although some attemptsto find solutions in this way were made, it seems to have been abandoned.*I t must be noted that the Eqs.

(15), or the corresponding differentialequations61dudpTt -ex>( 1 6 )pdpΈ=du=~ΊνΥχ'are not based on the assumption that p/p is an over-all constant (whichwould not be admissible in the case of curved shock lines). In fact, (15) wasderived from (10), with σ = 0, and the third equation (10) then states onlythat p/p remains constant along each particle line (see also the followingsection).yyOn the other hand, using the unabridged system (13) which includesviscous terms, or even equations including heat-conduction terms, hasproved successful. As was mentioned in Sec.

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