R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 34
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(65) determine Z\ and Z , and finally any three Eqs. (64) serve2(Β', (Β'. T h e results areto determinea'(x)(69)= \ (v 42(B'fc)β= * Y -{χ)=UU3u ) '2' +— —— Uu+υ4-(xu-V[ (vx2-u) +22uV)v',V)v',— - XV .44T h e right-hand members are expressed entirely in terms of the given functions u(x) and v(x), since the function V(x) is defined by (67). Then /(£)and ^(T?) are given throughout A'B'Cby (62), witha'(x) dx,α[χ(η)] = / a'(χ) dx, etc;the lower limits in the integrals are arbitrary, but must be the same in eachpair of integrals.Examples(a) L e t a and β be distinct constants, and suppose that for t = 0u = αχ,ν = βχ,for all x.Thenχ/Xdu = ^x\ ξ = (β + a)x, η = (β - a)12.7GIVENINITIALVALUES179Equations (69) reada'(x)= | (20 + 3« )(/J 22=G'(x)«'(*)a ),22- | (2/3 +=+3a )«,23« ).22Then, if Eqs. (70) are evaluated, with lower limit 0, and substituted into(62), we find/(«)9(v)== <Vwithr=11 8β + 9αβ + 3α240(0 + α )'22„=31 8ff - 9αβ + 3α240(0 - α )23Together with (47) these solutions give t, χ —tions of u and v.and therefore z, as func(b) A particularly simple formula results if u(x) = 0, so that the flowstarts from rest.
Then V(x) in (67) is 0, Eqs. (68) give Y = 7 = 0, andfinally from (69) we get30a'(x)2x= |i>V,e'(x)(B'(x) = 0,=-\vv.Here we may find/' and g' directly from Eqs. (63), rather than follow allthe steps used in the general case:/'(*) = 2£C = -|JWdz,g(v) = -2τ76 = I Jxvv dx.In this case, £ = v(x) + w(:r) reduces to ξ = v(x), so that if χ = α (ν) denotes the inverse function of ν = v(x), we may write/'(*) = - I /*α(υ) υ dv,g'( )v=\fa(v) ν dv.I t can be verified that this result agrees with that obtained from the formulas that give the solutions in the general case.180I I I .
O N E - D I M E N S I O N A L FLOWArticle 13Simple W a v e s . Examples1. Simple waves; definition and basic relationsIn the preceding article we treated some problems of one-dimensionalnonsteady flow by transforming from the χ,/-plane onto the speedgraphplane of the variables u,v.
In general, any finite region of the x,/-plane ismapped onto a finite area in the w,v-plane, and any element of area dx dtis transformed into an element of area in the speedgraph. T h e extreme degenerate case is that of uniform flow: u = constant, ρ = constant, wherethe entire χ,/-plane corresponds to a single point of the speedgraph plane.In between lies the case, now to be considered, in which the whole £,/-areais mapped into a single arc Γ in the speedgraph plane and each element ofarea dx dt is transformed into an element of arc on Γ. This type of flowhas been given the name simple wave, for reasons which will appear below.31As was mentioned in Sec. 10.6, this situation can occur only when the Jacobian of the transformation mapping the physical plane into the Μ,ν-planevanishes at all points; consequently some of the results of the precedingarticle are not applicable, e.g., Eq.
(12.23).T h e curve Γ in the speedgraph plane cannot be an arbitrary curve. Indeed, we have seen that the (u ±a)-characteristics in the .r,/-plane arealways mapped into the =F45° straight lines in the u,v-plane. A t a point Ρwith coordinates x,t one of the two line elements dx/dt = u ±a may bemapped into the point P ' , but certainly not both of them, for then thewhole element of area dx dt at Ρ would map into a single point. Thus eachelement of arc on Γ must be the image of an element of arc of a characteristic and have either the + 4 5 ° or —45° direction.
Accordingly we can distinguish two kinds of simple waves, which will be called forward and backward waves, respectively. In the first case the complete speedgraph consistsof a single line in the + 4 5 ° direction, with equation ν =v +0u\ in thewhole flow pattern (that is, for all χ and all t) the difference ν — u has aconstant value. In a backward wave the speedgraph is a line ν — v — u0and the sum ν + u is constant for all points of the flow.In both kinds of simple wave each curve of the second set of characteristics is mapped into a single point of the line Γ, since the image must lieon the line Γ and on a line having the other of the ± 4 5 ° directions.
Thismeans that along these characteristics u and ν are constant, and with ν alsoρ and a. Since the slope of these characteristics, measured from the /-axis,is either u +a or u — a, it follows that the curves of this second set ofcharacteristics must be straight lines. Thus we arrive at the following defi-13.1SIMPLE181WAVEScrosscharacteristicsF I G . 65. Construction of a forward rarefaction w a v e . Assumed g i v e n : T h e envelope Ε of straight (u + a)-characteristics in the z^-plane, and the -f 45° line Γ :ν = VQ + u forming the speedgraph.+nition :* A simple wave is a flow pattern in which one set of characteristics inthe x,t-plane consists of straight lines along which u and ν (and also ρ and a)have constant values.
Μoreover, in a forward wave υ — u, and in a backwardwave ν + u is constant for all χ and t; in the first case the straight lines arethe (u + a)-characteristics, while in the second they are the (u — a)-characteristics. T h e characteristics of the other set, e.g.
the (u — a)-characteristicsin the case of a forward wave, are usually called cross-characteristics.T h e expression " w a v e " refers to the fact that the state u, p, i.e., thegeometrical point at which u and ρ have a given pair of values, "progresses"at constant speed, u + a or u — a, while, of course, any material particlemoves with velocity u. Thus, a forward wave propagates forwards relativeto the moving particles, while a backward wave propagates backwards.
Obviously a uniform flow, u = constant, ρ = constant, with two sets ofstraight (and parallel) characteristics, is a limiting case of a simple wave.In Fig. 65 is given an example of a forward wave. I t is assumed 1) thatthe 45° line Γ in the speedgraph plane has equation ν = v + u, whereis arbitrary; and 2) that the straight-line characteristics are an arbitrary0* T h e proof that such a flow pattern does in fact satisfy the equations of motionfollows the same lines as the proof for simple waves in the x,y-plane given in Sec. 18.1.182III. ONE-DIMENSIONALFLOWgiven set of lines in the :r,£-plane, tangent to an envelope E. From thesedata the flow pattern can be derived in the following way. A s in Sec. 12.5,the three directions u — a,u + a corresponding to any point on Γ (suchas point 2 in the figure) may be found by means of the vertical line 2-2°and the lines with slopes ± 5 , 2-2' and 2-2".
T h e straight characteristicthat maps into the point 2 on Γ is the tangent to Ε which is parallel to theline M2". A t any point on this characteristic the direction of the particleline is that of M2° and the direction of the cross-characteristic is that ofM2'.
Thus these two sets of curves, particle lines and cross-characteristics,are determined respectively by two direction fields; analytically, each setis determined by a single ordinary differential equation of first order. T h eisoclines for either set are the straight-line characteristics. If both parts ofFig. 65 are reflected on a vertical axis, they picture a backward wave. Inthis example the particles move (in both cases) from points of higher pressure (greater v) to points of lower pressure. This type of flow is called ararefaction wave.
One may, however, reverse simultaneously the signs of uand t in any flow pattern, as may be seen from Eq. (12.20). Geometrically,this means that the first of Fig. 65 is reflected in a horizontal axis and theother in a vertical one. Then we have a compression wave (backward or forward), and the particles move so that pressure increases. I t is seen thatthe density changes in the same sense as the velocity in a forward wave(both diminish in Fig. 65), and in the opposite sense in a backward wave,i.e., if u increases, then ν decreases.
T h e two solid lines with slopes ± 5 inthe second part of Fig. 65 are the boundaries between subsonic and supersonic flow, as mentioned in Sec. 12.3. A t a sonic point one of the two characteristics in the a:,2-plane is vertical.A physically possible flow is represented only by portions of the x,i-planein which the straight characteristics do not cross each other. Intersectionsoccur necessarily in the neighborhood of the envelope E. Moreover, theparticle lines approach each other, which means that the density increasesindefinitely, as they come closer to E.
Thus the following difference betweenrarefaction and compression waves can be stated: For given initial conditions at t = 0, the rarefaction wave can extend to t = QO , while the compression wave is restricted to a finite interval of time.Having constructed the flow geometrically, let us now consider theanalytic solution, which can be found starting from the same data: theconstantand the chosen set of straight characteristics. L e t the latter begiven in terms of a parameter β in the form(1)X =Χο(β)+ fit.Obviously, β is the slope of the characteristic as measured from the i-axis.Then, using the upper signs for a forward (and the lower for a backward)wave, we have13.1β = u(2)zfc α,SIMPLE183WAVESν = v ± u,±β = a + ν — v .00Since ν is a known function of a (equal t o 5a in the case of a diatomic gas),the last equation serves t o express a in terms of β.
W e can then define another function F of β b y\(3)ogF=^f^..JaDifferentiation of (1) gives(4)dx = (xo + t)dfi + β dt.N o w particle lines are defined b y the condition dx = udt. When the differentiation in (4) takes place in the direction of a particle line, i.e., if dx isreplaced b y u dt w e find%*'-¥-<>•(5)where β — u has been replaced b y ± α from ( 2 ) . Integration of (5) nowyields(6)Equation (6) gives t as a function of β (and a constant of integration C)along a particle line; when (6) is substituted in ( 1 ) , we obtain χ also as afunction of β and thus have a parametric representation of the particlelines.On the cross-characteristics the condition is dx = (u Τ a) dt, and whenthis is used t o replace dx/άβ in ( 4 ) , we obtain E q .
(5) with the denominatora replaced b y 2a, and E q . (6) is replaced b y(7)In the polytropic case ν = 2a/(κ — 1) [Eq. (12.19')] so that from (2)and (3)a =v±±l,hF = (vo ± βΓ>\where h =2l-±^;κ—1Kand therefore from (6) and (7)βΓ" [ τ f χΌ(υο ± β? '- άβ + c],κ1)(60t(70t = \ h\vo ± / Τ * " * [ = F / " x'oivo ± β ) ^ ^ .
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