R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 46
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W i t h the dimensionless quantitytν = q/a Eqs. (7) and (10) become:8/ir2(Μ-(13)2* \ dvΘΘ1) - = ν —,dsdn_dv— =dn(« + l ) .2-(κ-1)ν2-θθν-,ds2'2which include no parameter except κ.2. Equations for the potential and stream functionsT h e differential equation for the potential of an irrotational inviscid flowhas been derived in its most general form in Sec. 7.2 and later consideredfor steady plane motion in Sec. 7.5. Writing now φ instead of Φ w e quoterEq. (7.31):(14)p(l- φ ) - 2 ^ ^ + Ρ ( ΐdx \a /2dxdy2ady \22-4)=0.a /2Hereα»,-««ι•-(£)• + (£)"Λand a is to be determined from ( 9 ) . I t follows from (15) thatψ=(15')% = 0.q,dsdnT o derive (14) directly and at the same time obtain its form for thenatural coordinate system introduced in Sec.
1, we obtain from the firstof Eqs. (7) and (15'),(16)=& ^ρ+Μ=Μ>ΡΑφ= Μ>%νdsdndsdsand since Αφ is the sum of second derivatives in any t w o directions orthogonal to each other, Eq. (16) gives2»'>-«·-.>§.%eIf, on the other hand, q dq/ds is replaced by q d(q/2)/dsin (16), q is taken2from (15), and q d/ds is replaced by q d/dx + q d/dy, we find immediatelyx1 /2which is identical with (14).d <p2yδφ22 d <p\2242IV.
P L A N E S T E A D Y P O T E N T I A LFLOWIn the case of two-dimensional steady motion another approach ispossible. Instead of satisfying Eq. (2) by introducing the function φ, wemay satisfy Eq. (1) by setting(17)« . =ρ*w=—*Hence,1^(18)^ = — 1 ^^ =dxρ dy'dyρ dx'In the natural coordinate system we have(18')^=^ = — 1 ^15^ρ dn'dnρ ds'T h e function ψ, the stream function, has the property that its value is constant along a streamline; in fact, using (15') and (18') we obtainT h e last equation states that άψ is the mass flux moving through thecross section of a stream tube of unit height between the lines ψ and ψ + άψ.Equations (18) and (18') show that the lines φ = constant, the equipotential lines, or potential lines, are orthogonal to the lines ψ = constant, thestreamlines.N e x t it is seen that ψ satisfies an equation of second order of exactly thesame form as (14). T o show this, we compute d yp/ds , and obtain2dfyd /\2dqnaccording to the definition of ψ, E q .
(17). T h e π-component of the N e w t o nequation then yields^(20)= \ψds2=q dn1 ψ ,q dnwhere, as before, a = άρ/άρ. Using the second expression for d \p/ds inthese last two equations, and the condition of irrotationality (7) in theform dq /ds = dq/dn, we have222n(20')(M - i)|V- ^ pfi-*M2?ds2dn+dndn=|Vdn2* If φ and ψ are to be of equal dimensions we have to set pq = po fy/dy p'q =—po θψ/dx, where p is some standard density. We set po = P« = 1, in accordance withthe remark at the end of Sec.
1.x0fv16.2 P O T E N T I A L A N D S T R E A M243FUNCTIONSThus ψ satisfies the same equation in natural coordinates as does φ [seeEq. (16')]·If now d(pq)/dn in Eq. (20') is replaced by (l/ q)d(y q )/dn,also p q by(θψ/dx) + (dt/dy) ,and q d/dn by q d/dy - q d/dx, we obtain2P22xθψ222dn-* =qMoreover, Αφ = d t/dsbecomes22q v+θψdfy2Μ "2™22dfy+qx= -θ φ/θχ22dxTy W 'ν3 ψ/θη2y222+ d^/dy . Hence, Eq. (20')2Although this is apparently the same equation as (14), it cannot be usedin the same way, for a is expressible in terms of the derivatives of φ by (9)and (15), while the relation between a and ψ is less direct.*A few comments may be added. While the existence of a potential depends on the assumption of irrotationality, the stream function is based onthe always-valid continuity equation in cases where this equation is essentially two-dimensional.
This is so in the problem of the present chapterand in steady three-dimensional flow with axial symmetry (see Sec. 7.7).If the axis of symmetry has the x-direction and y is the radial direction,the equation of continuity has the form22έi y p q z )+Tyi y p q v )=°'and a stream function may be defined byθψ ιθψ*pyq=dy~'m"=- T x -W e shall not consider this problem, although many results and procedures are very similar to corresponding ones for the plane (x,y) -problem(see Sec. 9.4). Stream functions cannot be defined in the corresponding unsteady problems, in contrast to the situation in the theory of incompressiblefluids.Various analogies exist between the (x,t)-problem of Chapter I I I and the(x,y) -problem considered in the present chapter. In both cases the equations of the problem form a system of two homogeneous, planar, nonlinearpartial differential equations of first order, whose coefficients contain onlythe dependent variables p, u and q , q , respectively; such systems aresometimes called "reducible".
Thus by an interchange of variables a linearxy* The present derivation was chosen in view of a generalization of E q . (21) (seeSec. 24.2) which will cover cases where irrotationality is not assumed.244IV. P L A N E S T E A D Y P O T E N T I A LFLOWF I G . 88. Determination of flow from data along «£.system may be obtained in the speedgraph or hodograph planes. T h e potential and particle functions of Sec. 12.2 correspond, of course, to the presentpotential and stream functions, and the apparently identical Eqs.
(12.11')and (12.12') to the Eqs. (14) and (21) of the present section. T h e (xjt)problem, which is always hyperbolic (so that there are always real characteristics), corresponds mathematically, as we shall see later in more detail,to the supersonic (x,y)-problem. Accordingly, we shall have to expectdifficulties not existing in the (x,t)-problem.23. Subsonic and supersonic flow. Characteristics3T h e general theory of characteristics developed in A r t .
9, when appliedto the potential equation (14), led to the following results. T h e equation iselliptic (no real characteristics) in a region where the velocity q is smallerthan the local sound velocity a (M < 1); when q > a (M > 1), the equation is hyperbolic and the lines crossing the streamlines at the angles ± a(where sin a = l/Af) are characteristics (Mach lines); the transitionoccurs for q = a (M = 1). A short derivation of these results, independentof the previous arguments and based on Eqs. (7), follows.Assume that a velocity distribution satisfying Eqs. (7) is known on oneside of a line <£ (Fig. 88). One may ask, to what extent is the flow on theother side of £ determined? Obviously the derivatives of q and Β in thedirection of £ are known.
If £ crosses the streamline at the point Ρ at theangle β, these given quantities are (if d/dl denotes the directional derivativein the indirection)/oo\(22)θθ3Θ_ . όθ . _— = — cos β + — sin β.dldsdndqdq_ . dq . ^-f = - 2 cos β +sin β,dldsdnWhen, by use of (7), derivatives of θ are replaced by derivatives of q,Eqs. (22) become3s °C(22')ψ{Μ*dsSβ+VnSmβ=1) i n 0 + ! - c o s 0 dnS9TVjg,dl16.3 S U B S O N I C A N D S U P E R S O N I C245FLOWtwo linear equations for dq/ds and dq/dn; they have a unique solution, except when the determinant of their coefficients vanishes, that is, exceptwhencos β- 1) sin β(Μ2sin βcos β=1 -Μ2sin 3 =20.T h e same condition is found if, in (22), one eliminates the derivatives of qby means of (7) and then tries to solve the two equations for θθ/ds andθθ/θη.T h e present result follows also from Eq.
(10.2') if we identify in turnx, y, u, v, and φ with s, n, 0, g, and β.T h e conclusion is that whereas, in general, the values of q and 0 givenalong a curve £ determine the first-order derivatives of both q and 0 ateach point of £, no such inference is possible if, in the case Μ > 1, £crosses the streamlines at an angle whose sine is ztl/M. Along such a linesegment any two solutions of the partial differential equations (7) withthe same values of q and 0 along that line can be patched together, sincethe derivatives of q and 0 in an " e x t e r i o r " direction, i.e., in a directiondifferent from that of £, are not determined by the data along £ and thedifferential equations, and may therefore be different on the two sides.
Forexample, we shall see (Art. 18) that a uniform flow on parallel straightstreamlines can be continued as a flow along curved streamlines if and onlyif the transition takes place along a line that intersects the streamlines atthe Mach angle a = arc sin l/M (see Fig.
44). (Here the transition line £is a straight line, since in the uniform flow region q, a, and consequently aare constant along the line.) T h e line which we obtain by turning q at eachpoint in the positive (negative) sense through the angle a is called the pluscharacteristic C , or plus M a c h line (minus characteristic C~, or minusMach line).+Denoting by φ (φ~) the angle which a C(C~~) makes with a fixeddirection, say with the x-axis, and measuring 0 likewise from the x-axis,we arrive at the characteristic conditions or direction conditions++φ(23)sin a =+= θ + α,,orφ~ =0 -acot a = Λ/Μ2— 1.Here a depends only on q or, by Eq.
(10), on M. T h e formulas showclearly how the characteristic directions depend on the solution g, 0 underconsideration.Since sin a is smaller than 1, except at a sonic point, and is nonnegative,the positive direction on each characteristic may be defined as the one thatmakes an acute angle with the velocity vector. I t is then obvious that pre4246IV.P L A N ESTEADYPOTENTIALFLOWscribing the two positive characteristic directions at a point, except whereΜ = 1, is equivalent to prescribing q. In other words, the velocity distribution in a region is uniquely determined by the two sets of directedcharacteristics.W e know from the general theory that, in contrast to noncharacteristiccurves, along a characteristic the values of q and θ cannot be chosen arbitrarily.
In fact, from Eqs. (22') we find~ cos a — q ^ sin a = ~ [cos β cos a — (Μds2— 1) sin β sin a]+ ~~~ [sin β cos a — cos β sin a].dnSince M — 1 = cot a, both brackets vanish if β = α, that is, if d/dl refersto differentiation along a plus characteristic. In the same way we see thatcos a dq/dl + q sin a 36/dl = 0 if d/dl refers to a minus characteristic.Hence one has, with an obvious notation, the following relations betweenderivatives of q and Θ along a Mach line:22(24)_= , t a n a ^ ,- g t a n * ^ ,or [see also Eqs. (9.17)](24 );du= gtana,along a C ;^+do= — gtan a ,In rectangular coordinates, letting q = u and q(9.15) or by direct computationxyalong a C~.= v, we obtain from Eq.along a C*.Equations (24) are the compatibility relations, introduced in a general formin Sec.
9.3 and discussed in detail for k = η = 2 in Sec. 10.2, which holdbetween the derivatives of the dependent variables along a C and alonga C~, respectively. They show that along a characteristic, θ is a functionof q, and uniquely determined by specifying q and 0 at a single point.W e see that here in contrast to the linear case a characteristic may beany geometrically given' curve, with compatible values of the dependentvariables prescribed along it such as to make it "characteristic \ Althoughit follows from the general theory of Arts.
9 and 10, as well as from thederivation at the beginning of this section, we state explicitly: For a steadyplane potential flow of an elastic fluid, the streamlines are not exceptionalor characteristic; the only characteristics are the Mach lines.+,16.4 B A S I C B O U N D A R Y - V A L U E P R O B L E M S2474. Basic boundary-value problemsT h e role played by the characteristics in the solution of boundary-valueproblems (see Sees. 10.2 and 10.3) can now be explained in our presentcase as follows. Along an arc AB values of q and θ are given (Cauchy problem)in such a way that the curve A Β has at no point the direction of either ofthe characteristics; this means that at no point is the angle between thecurve and the z-axis equal to 0 + a(q) or 0 — a(q), where a(q) is a knownfunction of q whose form depends on the (p,p)-relation.* Consider theneighboring points 1 and 2 on A Β (Fig.
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