R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 81
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I t expresses the fact that the lines in the physicalplane on which φ is constant have the slope of the velocity vector, i.e.{dy/dp) / {dx/dp) = q /q . In order to interpret (31) we note that the lefthand side may be writtenyx2 dp \\δφ)τ\δφ)JThus the equation may be integrated to give<«\ Kl)' Gs)] «*.»>-««.++where Ρ(φ, ρ) = J dp/ρ is g times the pressure head introduced in Sec. 2.5,the integral being taken with φ held fixed, and 0(φ) is an arbitrary function.
N o w according to (29)p24.4A SECOND435APPROACHG(yf/)/gso that (33) is the Bernoulli equation withthe value of the total headΗ on each streamline. A s before we assume that this function is determined by the boundary conditions.Equation (33) may be written in the form(35)\f= <?(*) -Ρ(ψ,ρ),where the right member is a function of ψ and ρ determined by the boundary conditions. Thus the Bernoulli equation expresses the fact that the speedq is a known function of ψ and ρ for any given problem. Having determinedthis function we may replace (31) by (34).On differentiating (35) with respect to ρ we find(36)- - - « ? dpN o w a is the rate of change of ρ with respect to ρ as a particle moves alongΡa streamline, so thatdp=_1_dpa '2since this derivative is taken with φ constant.
Thus on differentiating (36)we obtainM ' - l = p V 0 .(37)T h e motion is therefore subsonic or supersonic according as d q/dp isnegative or positive. (Cf. Sec. 8.1)If (32) and (34) are solved simultaneously for θη/θψ and θ η/θρ , and ηis eliminated from the resulting equations by equating θ η/δρ θψ andθ*η/θψθρ , we obtain a nonplanar second-order differential equation for ξ.This equation is of Monge-Ampfere type, but unlike that in Sec.
15.7 it isvery complicated. Since (32) and (34) are symmetric in ξ and rj, the sameMonge-Amp&re equation is obtained for η when ξ is eliminated from them.22ζ2222A simpler equation is obtained if ξ and η are expressed in terms of thestream direction θ(ψ, ρ). According to (29) we may write(38)^ = q cos Θ,αψ£ = q sin Θ,οψdand then (34) is automatically satisfied. B y differentiating (32) twice withrespect to ψ we obtain three equations from which ξ and η may be eliminatedby means of (38). T h e resulting equation for θ isθψ2θψ dpdp2436V.
I N T E G R A T I O N T H E O R Y A N DSHOCKSwhereθθ ΘΘqF =C =θφ dp'2θθ_ Γ d q_ ag / d 0 V—dip [_ θψ dpθψ \dp/zd02-d0~dp θψ dpj2This equation, like (9) for ψ in the physical plane, is planar and nonhomogeneous. Its advantage in comparison with (9) is that the coefficientsA, B, C, and F are known explicitly as functions of ψ, ρ, θθ/θψ, θθ/dponce the distributions of total head and entropy from streamline to streamline are given by the boundary conditions. For, by (35), the functionq(yp,p) appearing in these coefficients is then known.
In Eq. (9) thereappear p, q , q which must be determined as functions of ψ and its derivatives by means of (5) and (10), so that in general they are not knownexplicitly.xy5. The sufficiency of the shock conditions49In Sec. 14.2 it was shown that the shock conditions (14.2) and (14.9)are necessary conditions relating the initial and final states of an abrupttransition which is the limit of a one-dimensional nonsteady viscous flow.In Sec. 22.2 the same was shown for the shock conditions (22.3) and (22.16)in the case of steady plane flow.
W e shall now show that these conditionsare, in a definite sense, also sufficient. T o this end we first discuss the onedimensional non steady case, and introduce a more convenient system ofcoordinates in the £,£-plane.W e consider a given line S: χ = f(t), which is nowhere parallel to thex-axis, but otherwise arbitrary, and introduce a curvilinear coordinate system whose coordinate lines consist of the parallels to the rr-axis and thecurves obtained by translating S in the ^-direction (see Fig.
164). For onecoordinate, CT, at a point Ρ we take the displacement in the x-direction fromS, and for the other, β, we take the time t:(39)a=X -Sit)β = t.Thus(40)d_dxd_da'θϊ'θβCWda'where c(fi) = ά/(β)/άβ is the slope of a = constant at Ρ (measured fromthe i-axis).T h e differential equations governing the simply adiabatic flow of a per-SUFFICIENCY24.5OFSHOCKCONDITIONS437tcSαt^constan tF I G . 164. Curvilinear coordinates in the z,£-plane.feet viscous fluid in the χ,ί-plane are (11.2), (11.3), and (11.4').
Η Eqs. (40)are introduced into them we obtain(41a)A[ («-Pc)] +| | = 0,θβda(41b)(u-Pc)p + ΑdaP(« -[A +2«0 Ada Ι 2σ' ) + ρ(ρ -χda:-J *7 — 1ψ= 0,d/3r l+JA[m(p_ο_K]dawhere Κ = k dT/da and, according to (11.6), σχ= μ du/da. These equa0tions are similar to (14.4), (14.5), and (14.6). T h e derivative d'/dt appearing in the latter is equivalent to d/dβ.
I t was not necessary to introducethere the new coordinate system explicitly.W e now magnify the α-coordinate in the ratio 1 : μ , i.e., we introduce a0new independent variable s =α / μ , assuming for convenience that μο is0constant. Under this change of variable Eqs. (41) become(42a)/.ΟΙ \(42b)« L/p(u -W„ _du . d ,c) — + — ( ρ dsdsλe) ]/νσ«) =-Mopdu^dp,438V.,(42c)λINTEGRATIONθ Γη2,THEORYgRS\7 — 1ds |_2A N DSHOCKS.
d+fJdsiθβL—Γ γ - J'[«(p -Κ]σί) -4-|_2gRwhere/(43)duT h e right-hand side of each of Eqs. (42) has the factor μ which, as we saw inSec. 11.5, is extremely small for air (and other gases). L e t now the righthand sides be replaced by zeros, and consider a solution of the resultingequations in which u, p, p, and Τ tend to finite values at both s = — ooand s = + oo (each state depending on β).
W e shall use subscripts 1 and 2for,these states, assigning 1 to s = — oo and 2 to s = + oo if u — c ispositive, and vice versa if u — c is negative. Such a solution will be denotedby T(s, β) where the single symbol stands for the set of state variables.Then T(s, β) is called a shock transition solution of the original Eqs. (42),with respect to the line Sand the states 1 and 2.
I t depends implicitly on theviscosity coefficient μ through the variable s which is 1/μ times the displacement from S.000If the right-hand members of Eqs. (42) are replaced by zeros, the resultingequations no longer contain derivatives with respect to β, and are in factequivalent to Eqs. (11.8), governing the one-dimensional steady flow of aperfect viscous gas, when μ « = a stands for x, u — u — c for u, and partial for ordinary derivatives. This is easily checked once it is rememberedthat c does not depend on s. Hence the solution of these equations followsthe same lines as for Eqs.
(11.8), except of course that now the integrationconstants m, C i , C must be considered functions of β.102W i t h reference to Eqs. (11.8) we found that when the system possessesa solution in which the particles pass from a state 1 at χ = — oo to a state2 at χ = + oo, these states satisfy the shock conditionspiUi = p ^2 = m,(44a)2(44b)(44c)= p +pi + piu2\ u?2+- J L -£!7 — 1 piwith m > 0, and also T2^p ^2=J2222«,»Cm,=+£? =_ J L ^7 — 1pC ,227\ .W e shall now verify the converse: if Ui, pi, pi and u , p , P2 are two setsof values satisfying (44), with m > 0, and the condition T 7> T , then there222x1Τ24.5439SUFFICIENCY OF SHOCK CONDITIONSexist solutions of (11.8) in which the particles pass from state 1 at χ = — »to state 2 at a; = + ° ° .
For let (44) now be the definitions of the constantstn, C i , C%, and rewrite (44b) and (44c) in terms of ν = u/2Ci2« +ϊ ±Ί*=«+ν? Γ -1and θ=* - % ·These last equations are precisely Eqs. (11.21). I n the notation of Sec. 11.4they express the fact that (vi, θ ι ) and (v , θ ) are the points of intersection22of the parabolas λ = ± oo and λ = — 1 in the v, θ-plane, with the point 1to the right of the point 2 b y virtue of the condition T2=T\.
Suppose nowthat υ{χ) θ(χ) is any solution of Eqs. (11.14). Then u(x), T(x) as defined}by Eqs. (11.13) will satisfy Eqs. (11.12). If now we define p{x) =and p{x) = gRp(x)T(x),m/u(x)then the four functions u(x), p(x), p(x), T{x) willsatisfy Eqs. (11.9), (11.10), and (11.11) and hence Eqs. (11.8). But we havealready seen in Sec. 11.4 that for ra > 0 there are solutions of Eqs. (11.14)in which ν, θ pass from state 1 to state 2 as χ increases from — oo to + o o .A n y t w o such solutions are obtained from each other b y translating theorigin of x. Hence there are solutions of the system (11.8) in which u, ρ, ρpass from their values in the state 1 at χ = — oo to their values in the state2 at χ = + o o .
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