R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 31
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(7.29)]. T h e detailed discussion of the differential equations (17)and of the corresponding flow patterns is left t o the reader.223. Interchange of variables. SpeedgraphA s has been seen in Sec. 1, Eqs. (1) and (4) may serve as the equationsgoverning the one-dimensional nonsteady flow of an inviscid elastic fluid:dudp . dplTx Tp+UX+dtΛ°>=(18)du , a dp , dudxρ dxdtwhere a = dp/dp is a known function of p.* This flow is the centered simple wave of Sec. 13.2 [see E q .
(13.13)].12.3I N T E R C H A N G E OF V A R I A B L E S . S P E E D G R A P H161T h e further argument is greatly simplified if the variable ρ is replaced bya certain function of p, namely,fp(19)ν =JPl14a ,dva= -,upρ- dp,ρd a ddp=- _ρ ovwith an appropriately chosen lower limit pi in the integral;* there is a oneto-one correspondence between ν and p. T h e new variable ν has the dimensions of velocity. For example, in the case of the polytropic relation (3),we choose pi = 0 and find/—~Fi(19 )2ν = VKCu—D/2Pτκ— 1and, in particular, if κ = y =κ — 12a=τκ —,α = — —Iν,21.4,(19")ν = 5α.If an isothermal condition ρ = constant · ρ holds, we take pi = 1 and have(19"')α = constant,ν = a log p.When the substitution (19)dv Eqs.dv (18) readdu is made,u —dxdxdt(20)dudvdua —dxdx~dt++++T o each couple x,t there corresponds a pair of values of u and υ determinedby (20).
This mapping of the 2,^-plane onto the w,?;-plane will be called thespeedgraph transformation of the one-dimensional flow. T h e speedgraph isanalogous in many respects to the hodograph of a two-dimensional steadyflow (see Sec. 8.2). In the hodograph the lines of constant speed q, and consequently of constant M a c h number Μ, are concentric circles. Here, in thepolytropic case, the straight lines through the origin in the u,y-plane arethe lines on which the Mach number is constant, since Μ=\u\/a equals2/(κ — 1) times | u \/v.
In particular, the rays v/u = ± 2 / ( κ — 1) separatethe subsonic region in the middle from supersonic regions on either side(Fig. 59).B y adding and subtracting Eqs. (20) we find that(21)(M±a ) _+ ~ ==F|_(«±e)- + - J.N o w the left-hand member is the rate of change of u in the respectivecharacteristic direction dx/dt =(u ±a ) , while the bracket on the right is* N o t e that ν is defined also for a particlewise (p,p)-relation.162III.ONE-DIMENSIONALFLOWsupersonicF I G .
5 9 . Regions in which Μ ^ 1.the same for v. Thus, along the (u + a)-characteristic in the z,£-plane wehave du = — dv, and along the other characteristic du = dv. This showsthat the two sets of characteristics dx/dt = u ± a in the x,t-plane are mappedinto the T 4 5 ° lines(22)ν + u = constantandν — u = constant,respectively, in the speedgraph plane.In the two planar first-order equations (20), the coefficients depend onlyon the dependent variables u and v\ hence we proceed as described in Sec.10.6 and interchange the dependent and independent variables.
T h e Eqs.(20) may be identified with Eqs. (10.1); if the present t replaces y thenthe coefficients of (10.1) area\ = bz = a,aaz = b\ = u,4= 6 = 1,2a2= 64 = 0,and the transformed equations (10.22) read(23)dta dvdtdt , dxu - — r - _ = 0 ,duduΛdtdx— — — = 0.dvdudvThis system, in contrast to (18) and (20), is linear, so that the characteristics are independent of the unknowns. T h e characteristic directions canbe computed as in Sec. 10.1 and are found to be dv/du = ± 1 in agreementwith (22).T o obtain from (23) a single second-order equation for one unknown, wecan eliminate χ by differentiating the first equation with respect to ν andthe second with respect to u. Remembering that α is a function of ν only,we obtainu(24)Λ--a__!_(da\dtT h e equation for x, obtained b y an analogous elimination of t, is similar,except that the first partial derivative of the unknown, on the right, is re-12.3I N T E R C H A N G E OF V A R I A B L E S .SPEEDGRAPH163placed by a more complicated first-order expression.
T h e second degreeterms of (24) again show that the characteristics of the present problemare dv/du = =b 1.Alternatively, by the use of a dp = ρ dv, Eqs. (23) may be rewritten inthe form4-(x-=ut)du(25)0,ρ dVA ( - ) +±2( ) = o.xutdvρ duT o integrate the first equation, we introduce a function U of u and ν bysettingfoa\(26)p/,adUldUut = - -—,t = - ρ dvρ duthe second equation then shows that U must satisfyχ -;^£-^=a\1(1-^)^1dvdudv ) dv '(26')22Similarly, the second equation (25) is satisfied if we set(27)χ -ut = —,* =-—,α dvand then the first equation supplies the condition,Kd V _ a Vdv}2du2=_1/a \15_da\dVdv)dv 'Also, either χ — ut or pt may be eliminated from the system (25), and itis seen that pt and χ — ut satisfy the same second-order equations as Uand V, respectively.I t is also of interest to study the relations between the functions U andV on the one hand, and the previously introduced potential and particlefunctions Φ and φ on the other.
From the definition (11) it follows thatdt_du(28)δψ __dvdxP~pdupudtdu. — (χ duut) + pt,dtdx''d- — pu= — [p{x — ut)]dvdvdv— - (x — ut).aI t is easily seen that the two expressions on the right are the derivativeswith respect to u and v, respectively, of a dU/dv — U. Thus, except for anadditive constant, which has no significance,164III. ONE-DIMENSIONAL(29)φ =FLOWψ - υ .αI n exactly the same way we find that(30)Φ =ν-—+ - IP-—)-V.dua\2/ dvEquations analogous to the first parts of (28) can be written for οΦ/du andοΦ/dv.
When these equations are compared with (28) we can verify, using(25), thatd^-Αΐο ^+Β^ί(31)— = Λ — + Β —,dvdudv22where(310Λ, = B2= -,A2-== BxlΡ-ap\Also, using α dp = ρ dv and dP = a dv, we haveMdvi _ ^i?dw= ο— dv— =du1'- — (Pa p\2a \Λ2/Ρ--Vl4- — ^2Λdv)dv JThus, when Φ is eliminated from (31), the result isKdv}1dua\2^ dv)dv'Elimination of ψ leads to a similar equation for Φ, but with a more complicated first-order expression on the right.All these equations, (24), (26'), (27'), and (32), for t, U, V, and ψ, respectively, are of similar form, differing only in the factors ± (1 =b da/dv)on the right.
In the polytropic case these factors are all constants, since(19') gives da/dv = (κ — l ) / 2 . T h e right-hand members consist then of thederivative of the unknown with respect to v, multiplied, respectively, bythe factorsK+113 — #c 1κ —1νκ —3 -1νκ —κ11νwhich in the particular case of κ = 1.4 are(330-,4-- ,4?.κ +11κ —1 ν12.4G E N E R A L I N T E G R A L I N T H E A D I A B A T I C CASE165I t will be shown in the next section that the general integral of the equation can be given in simple form in the cases (33') and in certain other casesalso.4.
General integral in the adiabatic caseI n the preceding section we have seen that, if the (p,p)-relationhas16the form ( 3 ) , and the polytropic exponent κ the value 1.4, the second-orderdifferential equations for t, U, V, φ, χ — ut, and pt have the common form(34)^1— —1dvwith η =η =2 for z=n-2ndu2U or pt, η =dZndvν2171—2 for V or χ — ut, η = 3 for ψ, and— 3 for t. I t can be shown that, if η in (34) is any integer, the generalintegral of (34) can be given in a simple form. This result applies, moregenerally, whenever κ has any value which leads to an even integer forthe quotients (3 — κ)/(κ — 1) and (κ +l)/(κ— l).Moreover, the formu1 8las can be extended in a certain way to cover the case of any real n.First, if η = 0, Eq.
(34) is the one-dimensional wave equation (4.6), sothat from (4.7) the general solution is(35)z (u,v)= f(v0+u) +g(v -u),where / and g are arbitrary, sufficiently differentiate functions of a singlevariable. T h e successive derivatives of these functions will be denoted by/ ' , / " , · · · , 0 ' , <?", · · · ·For the sake of abbreviation we introduce the notation(36)ζο = f+ζο = /g,+g ,··· ,ζ= /0where the arguments parallel those in (35). Then each ζfunction of ν +ω,is the sum of aωis(v+1)(v)00gu and a function of ν — u. I t may be noted that zothe derivative of zentiating ζ0+with respect to v, while zis the result of differ(v+2)0twice with respect to either ν or u.W i t h an arbitrary z of the form (35), the general solution of (34) for η a0positive integer can be written in the form(37)z (u,v)n= ZQ +aivz'o +av42219+· · · +avznnM0 fwhere the numerical coefficients a„, depending on n, are found by substituting (37) into ( 3 4 ) :,1V2 -1(n -v\ (2n ( 3 7 , )l)(n l)(2n -2) · · · ( η -1)ν +(* = 2 , 3 ,If η is a negative integer, it is more convenient to set m =writeν +2) · · · (2n -1),n).— η > 0 and166III.
ONE-DIMENSIONAL FLOW(38)= z = t>z-mn1-2m[z„ + fiim + β^ζ'ό +··· +fin-^-W"""],whereft=-l;( 3 8'^ = ( - D '2"( * - 2 ) ( « - 3 ) . . . ( * - r )v! (2m - 3 ) ( 2 m - 4 ) · · · (2m - ν -1)( , = 2, 3, · · · , m -}1)T h e first examples areZl = Zo — ^ζό,z(39)/Z(40)=Z-l-,3= z2= Zo — VZo +0"fvz2202-2 = ^ (Z — MSo),— wo +2-3 =0i» 2o,2, 3 ///— TtV Zo ,\ (Z — Vzi +0il/Zo).In order to prove formulas (37') and (38'), we start b y working out theresult of substituting a single term of (37) into the differential equation:/a2d2n d\,2νox= α,Μ -1 - 2n)tTV)+ 2(p -n)v~W ],v+1)Thus, when the whole of (37) is substituted into (34) and the coefficientof v ~ zv2iv)0ai =equated t o zero, we find— 1;a v(v — 1 — 2n) + a - 2(vv— 1 — n) = 0v X(v = 2, 3, · · · , n),and this recursion formula gives (37').
Formulas (38') can be proved in thesame way. I n either case no terms of higher degree than those given in (37)or (38) are necessary, for the coefficients of such terms would vanish since ηis an integer. This suggests that when η is not an integer, the solution canbe expanded in an infinite series beginning with the terms (37) or (38),the coefficients being determined b y recursion.E v e n in this case the series20reduces t o a finite sum if / and g are polynomials.I t seems that the function best suited for the study of most problems is V.W e repeat E q . (27') for adiabatic flow with κ = y = 1.4, and its solutionfrom ( 4 0 ) :dV_ dV _dvdu2(41)v(42)22_4dFν dv'2_ /(f) + gOt) _ /'(j) + g'fo).v*ξ = V+v2U,η =9V — U.I t still remains t o determine the functions / and g for particular problems.12.4GENERAL INTEGRALI N T H E A D I A B A T I C CASE167If ξ and η are used as independent variables in place of u and v, AVE havedv ~ σξdu ~ σξΘη'so that the differential equation for V becomes(43)2 (dVξ + η\θξ^*Οξβηdvθη121dV\ _δη/Equation (43) is of the form (10.11) with2α = b = ——-,ξ + Vc = 0,and the solution of (10.11) was found in terms of a particular solution of theadjoint equation (10.12).
T h e adjoint equation for (43) is_UA)^θξση;.2.f + nW4+ v)(f2N o w the general solution of (44) is(45)Ω =v)[F(i)(£ ++(?(„)] -i(i+„) [F'(£) +2G'd,)],where F and (? are arbitrary functions of a single variable.* T h e particularsolution, or Riemann function, used in Sec. 10.5 was the one satisfying theboundary conditionsΩ(£ι,= 1,^= 6Ω for η = ηι,= αΩ for ξ =ξι,where ξ ι , 771 were a pair of parameters. I t can be verified by differentiationthat the Riemann function corresponding to Eq. (43) is22* + \ , [2^!Ω(ξ, η; ξι , ηι) =(46)( ξ ΐ++ 2ξτ, +(ξ -η)&-m)]7 7 0^1 /22 , 22\,T h e general initial value problem can be solved explicitly once the Riemannfunction is known, as shown in Sec. 10.5 (see also Sec.
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