R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 33
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Then Ρ corresponds to P'{u\,vi)in the speedgraph (Fig. 63), and the characteristicsare mapped into the ± 4 5 ° lines through P' as shown. Each point Q on PA(or PB)corresponds to a point Q' on P'A'(or P'B')determined by thevalues of u and ν at Q. A single one of these, say v, is sufficient to identifyQ'. Of course, the values of χ and t are known for each point Q' on eithercharacteristic (since they are the coordinates of Q in the 2,2-plane), and theyare uniquely determined if the given values of ν change monotonically alongPAand PB.Then we havealong PA:(48)η = ν — u = ηι,% = v + u = 2v —η,χ+ ui);ν =along Ρ Β:ξ = ν +u = ξι,η = ν — u = 2ν — ξι,i(fiν =+ *?)·T h e boundary conditions determining the flow pattern must be rewrittenwith ν (or u) as the independent variable, and V(u,v)tion. Thus we may consider either dV/duvt/5 as given along P'A'andas the unknown func= χ — ut or — dV/dv = at =P'B'.(a) Given χ — ut along the characteristics.
T h e data may be assumed inthe formχ — ut = α(ν),ν = J(£ +ut = β(ν),ν = £(& +alongP'A',alongP'B',(49)χ -τ;)where a(v) and β(ν) are given functions. Of course, a(v )and this= β(νι),xvalue may be taken to be zero by locating the origin suitably on the £-axis.When expressions (49) are substituted in (47), and ^ ( ^ ι ) is abbreviatedto g'i, etc., we find that / and g must be such thatf,{= «W-f&{Uν*giv)~,vVgz+ A νi v )=-β{ν)where ν =4,9ν6+ % vd-(ξ+1*),21where ν =vl&+,).2T h e first of Eqs. (50) is an ordinary first-order differential equation determining/', while the second determines g'.
Both are easily integrated. If weIII. O N E - D I M E N S I O N A L174FLOWintroduce the functionsAGO(51)f a(v) dv,=B(v)=f β(ν) dv,JviJviand if c and k are constants of integration, the solutions of (50) are/'({)=-2v A(v)+ g [ -22vg[ + cv ,ν = ±(£ +2),Vl(52)g'( )=2v B(v) + / i - 2vf" + kv\ν =+ η),as can be verified by differentiation.Of the six constants /(, /(', g[, gi, c, and k, which appear in (52), onlytwo can be chosen arbitrarily. W e get two conditions restricting the constants if we evaluate (52) for ξ = ξι and η = ηι, to obtain2v(530/ ί = g'i -g[ = f[2vig ( + cvi ,f2-2vifi+kv .2Furthermore, differentiating Eqs.
(52) and then setting ξ = ξ ι , η =ηιwe obtain(53")fi=-gi+ cvi,gi =-fi+kv ,xwhere we have used a = βι = 0. T h e four conditions (53) still leave somefreedom of choice. For example, all conditions are satisfied if we choosex26fi= g[ = 0,fi= gi =2Cvi,c = k = 4C,where C is an arbitrary constant. Then Eqs. (52) take the form/ ' ( ξ ) = -2v A(v)+ 4Cv(v -g'( )+ 4Cv(v -2(54)v=2v B{v)2ν = J(f +Vi)+ν =ym),n),where the integrals A and Β are known functions of ν; when ν is expressedin terms of ξ and η respectively, we have/'(f) =-\(ί +m)(ίι +i?) +2+C(* +-ίι),, ) ( , -m).(54')ff'Gi)=I2Cfe+In terms of ξ and η the integrals A and Β have upper limits (ξ + ηι)/2and (ξι + η)/2, respectively. Equations (540 solve the problem. B y differentiation, we find / " and g" and thus χ — ut is given explicitly as a function of u and v, using the first equation (47); this expression is independentof C.
T o evaluate vt by means of the second Eq. (47), it is necessary tointegrate the expressions (54) to find / and g, the constants of integrationbeing chosen so that the second equation (47) is correct at P'. A n examplewill be studied in Sec. 13.5.12.6VALUES G I V E N ON TWO175CHARACTERISTICS(b) Given t along the characteristics. Here we assume that the values ofat (equal, in our case, to vt/S) are given functions of ν on two characteristics intersecting at P'(u vi).These data may be written in the formuν = i(£ +t = {p)a(55)t =j 8( ,)t>= ΐ(vξ ι)alongP'A\)alongΡ'Β'.ηι+ηB y choosing the x-axis through Ρ we can suppose that t = 0 at υ =i.e., that α(^ι) = β(ν )λtions for g(vi), g'(vi)v\,= 0.
If again g , g[, etc., are written as abbreviaxetc., the conditions (55) may be written, using (47),yas(56)3/ -along ΡΆ',3vf + v f"= \v a(v)A{v)3gi + 3vg[ -v g"2and an analogous equation, w i t h / , g, and a replaced by g, /,and β, respectively, along P'B'.(57)-h2fj «(*)£ (l -=\If we now definedz,Β (υ) = i £(l -β(ζ)dz,the general solution of (56) is(58)/ ( { ) = 4v A(v)-Agi + 2g[v -2gW++c v\2where ν = J(f + m), and Ci, c are constants of integration. A n analogousequation gives 0(77), introducing two more constants of integration k ,2xk.2A s in the preceding case, the constants are not independent, and sixconditions may be found restricting the ten constants / ι , / 1 , /f, gi, g[,g", Ci, c , ki , and k , by computing the values /, /', f" at ν = Vi from(58) and equating them t o / i , /ί, /f, etc.
Since these conditions are homogeneous (for the first and second derivatives of A and Β vanish at ν = v ), apossible choice is to have all ten constants vanish. Then the solution takesthe form2227x/(ξ)= 4ιΛ4(.)(59)g( )v= 4v B(v)Af4= -v0v2= -v*J(v - z) a(z) dz,ν = J(£ +(y -V = * ( ί ι + η).771),Vlfζ)β(ζ)άζ,From the formulas (59) and their derivatives, χ and t are expressed interms of u and υ by means of (47).These solutions hold for (u,v) in the rectangle A'P'B'Cin the speedgraph plane, corresponding to a curvilinear quadrangle in the x,^-planebounded by characteristics.176III.ONE-DIMENSIONAL7.
Analytic solution: given u and ν at t =FLOW0This is the initial-value problem discussed in Sec. 5; the solution maybe given analytically in simple form in the case of a polytropic gas with κ =1.4. There are given two functions(60)u = u(x),αύν = ν(χ),xύ b,representing the velocity and density distributions at t = 0, along aninterval of the x-axis from χ = a to χ = b. W e assume that Eqs. (60) givea one-to-one correspondence between the segment A Β of the x-axis and anarc A'B' (Fig. 64) in the speedgraph plane which at no inner point has atangent in the ± 4 5 ° directions.
In a contrary case such as the one explained at the end of Sec. 5 (see Fig. 62), the solution is obtained in severalsections, each part satisfying either boundary conditions of the type hereconsidered or those of the characteristic initial-value problem, for whichthe analytic solution has been given in the preceding section.Under the above hypotheses, to each point P' interior to the speedgraphtriangle A'B'Cthere correspond exactly two points on A'B': P[, havingthe same value of ξ = ν + u as P', and P , having the same η = ν — u.T h e points P i and P correspond to two distinct points on AB, whose^-coordinates we designate as χι(ξ) and χ (η), respectively.
Analytically,is the inverse of the function ξ = ν (χ) + u(x), and χ (η) the inverseof η = ν (χ) — u(x). Each point P' in A'B'Ccan thus be characterized bythe two quantities or coordinates'' χι(ξ) and χ (η). T h e ^points on theboundary A'B', and these points only (which correspond to t = 0 ) , arecharacterized by #i(£) = χ (η). This suggests using xi(£) and χ {η) to determine the form of the functions/(ξ) and g(v).2222α222When t = 0 Eqs. (47) have the formf ' - Q ' (61)3(/ +g) -ΜΓv(f"-g")+ 9') + v\f=v\+ g") =0,where υ is a given function of x, from (60).
T h e values of u as a given function of χ also enter into (61) by way of the arguments ξ and η of / and g.νuF I G . 64. Location of the points P[ , P' in the analytic solution of an initial-valueproblem.212.7GIVENINITIALVALUES177I t is easily verified that the left-hand member of each of these equationsis not changed if we add α£(B ++0£ Co t o / and — α207?(Bo — i? <3 to g,+020where G o , (Bo, and 6 are any constants. This fact suggests that, as in the0well-known method of " variation of parameters'',/(ξ) and 0(77) should beset up in the form,(62)29/«)x= <*[*(*)] +g( )+= -α[χ(η)]V?e[x(£)],+ η®[χ(η)]-ηβ[χ(η)].T h e new α, (Β, C are functions of a single variable, t o be chosen so thatthe conditions (61) are satisfied. These conditions apply only for t = 0,and therefore for χ (ξ) = χ(η), so that we may work with G , (B, (B simplyas functions of χ and writeβ'_d& _dd d£dxάξ dxdd__j_ ^dd άη_άξad ^,_άη άχ^άη'etc.
Then for points on the boundary arc A'B' we have/(63)= a + f(B + £ e,2= -g/' =(Β + 2£C +/" =2e +α + T?(B - 7? β,2Fi,</' =(Β -z,</' =-127,6 +F ,22e + z ,2where the F and Z are the functions of χ defined bytt(υ' + u') Y =a' +(υ' +Υ[1ξ β',{©' +w') F 2 = -(ι/ -2a'+ η<&' -η β',2(64)u')Z1=2{β',(Β' ++(ι/ -u')Z=2F2+(Β' -2776'.When the expressions in (63) are inserted into the conditions (61), thesereduce to the conditionsYi ~ Y2 (65)3(F+Xv(ZxF ) -Z ) = v*x,-2v(Z, + Z ) = 0.22Equations (64) and (65) are six linear relations among the seven variablesα', (Β', C , Fi , F , Zi , Z .
T o eliminate α', (Β', β', we multiply the22equations in the second line of (64) by — ν and add all four equations together, obtainingv(Y[++ Z ) -ΥΪ) = v' HZ,(F2+ F )]x2(66)+The values Zxu' HZ,-=b Z may be taken from (65), yielding2Y'l+Y'2-2L (VYl+γ)2= -v'xu ,Z ) 2(Fx - F ) ] .2178III. ONE-DIMENSIONALFLOWThis is a linear first-order differential equation for the function Fi +F ,2and its general integral is(67)Fi +F=2ν j-xu dx =ν f-χ du =-v V(x),2where the lower limit of integration is arbitrary. On the boundary A'B ',which corresponds to t = 0, the expression dV/du — χ — ut reduces to x,and dV'/dv — —at to zero; thus the integral of χ du, which we have calledV{x), is the value of V(u,v) at the point of A'B'corresponding to thepoint χ on A B.1Since only six relations govern our seven variables, one choice is open.W e decide to set(68)Fx = F= - t2V(x),Y[ = Y ' = 2'V(x)vv--lfxu'.Then Eqs.
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