R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 24

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Since at each point of A Β thereexist two characteristic directions σ and σ~ that are distinct from eachother and from the direction of the tangent to AB, it follows that a straightline of direction σ~ through P i and another line of direction σ through Pmust have an intersection P' at a distance from A Β of the same order ofmagnitude asP1P2 . W e shall see how u' and v', the values of u and υ atP ' , can be computed by using the compatibility relation ( 5 ) .Neglecting terms of higher order one can substitute in (5) for the dif­ferential quotient du/θσ the quotient (u — U\)/(PiP ) when the transitionfrom P i to P' is considered, and the quotient (u — u )/(P P')for thetransition from P to P' and similarly for dv/da. Using the first form of (5)and the third member of (4c) for K, one gets for u' and v' the two linearrelations2il++2r42fff222Δ (η' -u) +A (u'u) +1 2( 5)-l2(Δx(Δ23 23 2+Δ , β ί α η φ " ) (υ' -+Δ1 3tan φ )(υ' -+ν,) =(Β α-A b)(P P'),ν) =(Β,α-Αφ)(Ρ Ρ'),2γ112with the determinant Δ ι Δ ι (tan φ — tan φ ) .

Here all coefficients are tobe evaluated at the same point, say P i . Without loss of generality it maybe assumed that both φ-values are different from 0° and 90° at P i . Thenaccording to ( 2 ' ) , Δ and Δ are not zero, and also the factor (tan φ —tan φ~) in the expression for the determinant has a finite value. Thus, inthe case Δ ^ 0, the system (5') can be solved for u' and v'. T h e sameargument holds in the case Δ ^ 0 if the second form of the compatibilityrelation (5) is used. Finally, if A = Δ = 0, it is seen from Eq. (2') thatone value of tan φ, say tan φ , equals a /a\ = b /bi and the other equalsα /α = 6 /6 .

Then, using once the first and once the second form of ( 5 ) ,one finds the equations23Χ 3-++2 4Χ 23 4? 4i2+4(u'3-4u )Lx223=(B a2-A b)(P P'),2(υ' -xv )K2=(B,a-A,b)(P P ),2fwhereΚ = Δΐ3 (tan φ~ — tan φ ) ,+L =Δ2 4(cot φ+— cot φ~),which can also be solved for u' and v'.If the whole arc A Β is subdivided (Fig. 45b) into η small segments by asequence of points P i , P , · · · , one can find, in the manner just described,212 2II.G E N E R A LTHEOREMSFOREAC HCOUPL E Ρ » , P,+iA NE WPOINT Ρ,·AN DTH EVALUESO F uAN D νA TTHISPOINT.THES EPOINTS MAYB EJOINE DT OFOR MA NE WCURV E A'B',WHER E AA'{BB')HA STHEDIRECTIONO FONEO FTHECHARACTERISTICSPASSINGTHROUGH A (B).STARTINGFRO MTH ECROS SCURV E A'B'AN DTH E u,v-VALUESCOMPUTE DFO RITSPOINTS P',ON ECANCONTINUE I NTH ESAMEWAYAN DOBTAINANE WCROS SCURV EA B",ETC .

TH EPROCEDUR ESTOP SAUTOMATICALLYWHE NA POIN TI SREACHE DWHERE ACHARACTERISTICDIRECTIONCOINCIDESWITHTH EDIRECTIONO FTHERESPEC ­TIVECROS SCURVE . I FAL LFUNCTIONSINVOLVEDAN DTHEI RDERIVATIVESAR ECON­TINUOUS,THISCA NHAPPE NONLYA TA FINITEDISTANCEFRO MAB. U PT OTHISDIS ­TANCE ANETWOR KO FCHARACTERISTICSWIT HON ESE TO FDIAGONALCURVE SSUC HA SΑ'Β', A"B \· · · , TOGETHE RWIT HTH EVALUE SO F uAN D νA TEAC HCROSSIN GPOINT,CA NB EDEVELOPED.r,nIFA S η —* ooAL LDISTANCES P,P +iTEN DT OZERO ,TH EVALUE SO F uAN D νTENDT OSOLUTIONSO FTH EGIVE NSYSTE M( 1 ) I NA CERTAI NNEIGHBORHOODO FAB(O FTHETYP EDESCRIBE DI NTH ETHEOREM),SATISFYINGTHEGIVENBOUNDARYCONDITIONSO N AB.

I NSUC HA NEIGHBORHOODEXISTENC E(AN DUNIQUENESS )O FTHESOLUTIONISGUARANTEED. IF , I NADDITION,TH EEXISTENC EO F AREGULARBOUNDEDSOLUTIONI SASSURE DBEYON DTHISNEIGHBORHOOD oi AB,OU RPROCEDUR E(PRO ­VIDEDI TDOE SNOTBREA KDOWN)WIL LFURNIS HA NAPPROXIMATIONT OTHISSOLU ­TION.TH EACCURAC YO FTH EPROCEDUR EWIL LINCREAS E IF ,A S ηGET SLARGER ,AL LTHEDISTANCE S Ρ»Ρ»+ιAR EMADESMALLER. O NTH EOTHE RHAND,EVE NI FTH EMECHANICSO FOU RMETHODDOE SNOTMEE TAN YOBSTACLE ,W ECANNOTB ESUR ETHATTHEAPPROXIMATIONCONVERGEST OASOLUTIONI NTHEREGIO NCOVERE DUNLES SWEKNOWFRO MELSEWHER ETHATSUC HASOLUTIO NEXISTS.t4844THECONSTRUCTIONCA NB EEXTENDE DT OTHEOTHE RSID EO F ABB YINTERCHANG­INGTH EROLE SO F σ AN D σ~.TH ESPECIA LCIRCUMSTANCESPREVAILINGI NTH ECASEO FLINEAREQUATIONSWILLB EDISCUSSE DI NTH EFOLLOWINGSECTIONS .Theorem B.SUPPOS ETW OARC S A ΒAN DA CI NTH E £,2/-PLANEAN DTH EVALUESO F uAN D νALONGTHE MAR EGIVE NI NSUC HA WA YTHATA TEAC HPOINTO FAB (AC)TH EDIRECTIONO FTH ECURV ECOINCIDESWITHTHE σ ( σ ~ )CHARACTERISTICDIRECTIONDETERMINEDB Y x, y, u,AN D νAN DTHAT,LIKEWIS EA TEAC HPOINT,u, v,AN DTHEI RDERIVATIVESSATISF YTH EAPPROPRIATECOMPATIBILITYRELATION.++Then a solution, assuming the given values along A Β and AC, exists in aneighborhood of the point A in a region bounded by segments ΑΒι, ACi ofthe characteristics AB, AC and the two other characteristics through Bi and C\respectively.A SFA ROU TA STHESOLUTIONEXISTS—THATIS ,A TMOS TI NA CHARAC­TERISTICQUADRANGLE ACDB—ITI SUNIQUELYDETERMINEDB YTH EGIVENVALUESALONG ABAN D AC.(SE E FIG .

46) .NOT ETHATI F ACURV EI SGIVENI NTH EFOR My = y(x)AN DTH EVALUESO F uAN D νAR EGIVENB Y u = u(x)AN D ν = v(x),ONLYON EO FTHETHRE EFUNCTIONS MAYB EASSUMEDARBITRARILY. I NFACT ,I FTH ECURVEIST OB E ACHARACTERISTIC,TH ESLOP EO FTHECURVEMUSTB E AGIVENFUNC ­TIONO F x, y, u,AN D νACCORDINGT O(2') ,AN DA SECON DRELATIO NBETWEE N10.3TWO IMPORTANT123THEOREMSthe three functions is the compatibility relation.

I n the linear case, y(x) byitself is determined by a differential equation, namely Eq. ( 2 ' ) , and thuseither w or y can be chosen, the other variable being determined b y thecompatibility relation.45W e again show how an approximate solution can be built up from thegiven " c o m p a t i b l e " values of u and ν along AB and AC (compatible witheach other and with the curves). L e t P i P and P i P (Fig.

46) be t w o in­finitesimal elements in the characteristic directions. If u and ν are knownat Ρ2 and P , then the same argument as for theorem A shows that thevalues of u and ν at the point P (intersection of a σ through P and σ~through P ) can be computed. T h e point P combined with a subsequentpoint on AB, represents the same situation and allows us t o compute u andν at a subsequent point on the characteristic A'B', etc.

Thus, at most in awhole quadrangle determined b y AB and AC, the network of characteris­tics can be constructed and for each cross point the values of u and ν canbe computed.233+4324Often, the actual boundary problems occurring in fluid dynamics aremore complicated than the cases envisaged in the t w o theorems above. Atypical example is indicated in Fig. 47. Here the values of both variables,u and v, are prescribed along a noncharacteristic arc AB, exactly as intheorem A, but, in addition, either u or ν or a linear combination du +c v is prescribed along lines AAi and ΒΒχ.

In many cases the process ex­plained above leads t o a solution. First, the values interior t o ABC m a ybe found as above, where AC and BC are the characteristics through A andΒ respectively. N e x t , let Pi be a point of AC near A, and suppose that theother characteristic segment through P i meets AAiatP . Then the valuesu and v at P satisfy the compatibility relation along P\P ; and thisequation, together with the value of c u + c v prescribed at P , is sufficientto determine u and v . N e x t , one computes the values of u and ν at P ,from the t w o relations along the characteristic segments P P and P*P\,then at P , etc., until the entire wedge adjacent t o AA\ is filled up.24622222x22242245F I G . 46.

Illustration of the characteristicboundary-value problem.124II. GENERALTHEOREMSAF I G . 47. Illustration of a more general boundary-value problem.Obviously this procedure fails if Λ Α ι or any part of it, or ΒΒι , lies insidethe triangle ABC, or if the characteristic segments through Pi , throughP , etc., fail to meet ^4^4i. I t will be shown later that for quite simpleand physically admissible boundary conditions of this kind, no solutionexists (see Sees. 14.1 and 22.1).44.

The linear caseT h e two theorems A and Β of the preceding section have been dis­cussed for the general case of a planar system ( 1 ) . T h e step-by-stepprocedure outlined above can be considered as a method of approximateintegration. A rigorous proof of the two theorems requires convergenceconsiderations, which can be supplied in the case of A for a sufficiently smallneighborhood of the arc A Β and in the case of Β for a sufficiently small neigh­borhood of the point A. In the particular case that the equations (1) arelinear differential equations, and not merely planar, an existence proof canbe given for the region consisting of the whole characteristic quadrangle.For convenience we shall develop the argument for equations which arehomogeneous also, i.e., linear differential equations in which the right-handmembers are of the form a{x,y)u + β(χ,ν)ν, but it will easily be seen thatthis restriction is not necessary.If the left-hand members of Eqs.

(1) are linear, i.e., if the at and hi donot depend on u and v, and if the system is hyperbolic, Eq. (2') determinesthe two values of tan φ as explicit functions of χ and y. These two ordinarydifferential equations, dy/dx = tan φ and dy/dx = tan φ~, can be inte­grated, giving the equations of the two sets of characteristics in the form+K J!J)X=constantandg(x,!j)= constant,10.4125L I N E A R CASEwhich form a nonsingular curvilinear mesh, at least in a bounded region ofregularity.L e t ξ and η be new variables defined by(6)ξ = K*,y),ν =g(x,y).In terms of the new variables the system (1) is still linear, e.g., the coeffi­cient of du/d£ in the first equation is a\ d£/dx + a βξ/dy, etc.

T h e charac­2teristics of the transformed system in the ^,77-plane are the lines ξ =stant and η =con­constant. T h e linear relation between the derivatives in acharacteristic direction will, in this case, involve derivatives with respectto £ only or derivatives with respect to η only. These relations will thereforebe of the form/_(7)0, du., dv,,fdu.,tdvb — +1 — + a — = a,N3σξ2σξ04 —σηση=,/b,where the coefficients are functions of £ and η only. Since the equations (7)are merely suitable linear combinations of the equations ( 1 ) , they will alsobe homogeneous in the same sense as equations ( 1 ) . T h e equations (7) maybe further simplified by introducing new unknowns U and V, takingU = a[u + αίν,(8)V = bu +2b[v.Then Eqs.

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