R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 23

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(30), except that λ * must now be a vector which makes thesecond factor of Eq. (26) vanish.Article 10The Characteristics in the C a s e of Two Independent VariablesΊ . Characteristic directionsIn many problems of aerodynamics the number of independent variablesreduces to t w o : η = 2. Almost all that is known at present about com­pressible fluid flow refers to such cases. Examples (some of which will betreated in more detail in later chapters) include: (a) plane steady flow, inwhich the independent variables are χ and y (Chapters I V , V ) ; (b) nonsteady parallel flow, variables χ and t (Chapter I I I ) ; (c) nonsteady radialflow, variables r and t; and (d) steady axially symmetric flow, variables rand 2, where r is the distance from the axis of symmetry and ζ is measuredparallel to the axis, as in cylindrical coordinates.

In all these cases the char­acteristics are curves in the plane of the independent variables, as was seen inArt. 9. A study of these curves will be of great use in the task of integratingthe differential equations.If the specifying equation is a (p,p)-relation and if, in examples (a) and( d ) , the motion is further assumed to be irrotational, then the number ofunknowns is also t w o : k = 2. In (b) and (c) the unknowns are ρ (or p) anda velocity component; in cases (a) and (d) two velocity components maybe taken as the unknowns, since we have seen (Art.

8) that in a steady ir­rotational flow ρ (or ρ or a ) may be expressed as a function of the mag­nitude of the velocity vector.210.1CHARACTERISTIC117DIRECTIONSIf we let x\ = x , x = y for the independent variables, and U\ — u , u = νfor the dependent variables, the system ( 9 . 6 ) of planar differential equationscan be written out as22du.du.ai — + a — + α2dx(1),du.,3dydu.+oi — + o —2θ#θ 1/dv.dvτ - + «4 —dxdy, dv,6 — +dx3= α,, dv64 —=di/,b,where the vectors of Eq. ( 9 .

6 ) are an = (a\ , a ) , ai = (a^,a ), a i = (bi ,62),a = (63,64). All the coefficients and the right-hand members are, in gen­eral, functions of x , y, u , and i>; in the case of a linear problem the α; andbi depend on χ and y only, while a and 6 are linear functions of u and vwith coefficients depending on x y , if indeed they depend on u and t> at all.22242239yyN e x t we consider the vector X (Sec. 9.2) which is normal to a characteris­tic direction; the components λι and λ of λ in the x - and ^/-directions aredetermined by Eq.

( 9 . 1 0 ) , which here becomes2λιαι +λ α\ibi\b2λια2+\aλι&3 +\b324=(2)+2220.4Since the normal directions are in the same plane as the characteristicdirections, this condition may be so expressed as to determine the charac­teristic directions themselves, rather than the normals to these directions.If φ is the angle between the x-axis and a characteristic direction, then theslope of the characteristic at any point is-f-=tan φ =.λdx2If we take X of length 1 and suitably directed, we may write λι = — sin φ,λ = cos φ, and Eq.

(2) becomes2a cos φ — αϊ sin φα cos φ — α sin φ Ιb cos φ — bi sin φb cos φ — 6 sin φ |20=(202=Δ24 cos φ — ( Δ2Μ4343+ Δ23) cos φ sin φ + Δ ι sin φ,32where Δα- is used as an abbreviation for aib — aAi . All three coefficientsof this quadratic equation vanish in the following two cases: (a) if the lefthand member of one of the equations (1) is a multiple of the left-hand mem­ber of the other; ( b ) if the derivatives of u and ν appear only as linearcombinations μ du/dx+ ν dv/dxand μ du/dy+ ν dv/dy.W e assume thatneither of these occurs for any set of values^, y, u and ν considered.k118II.G E N E R A LTHEOREMSIn all other cases*, Eq. (2') has two, one, or no real solutions for tan φ;the system (1) is then called hyperbolic, parabolic, or elliptic, respectively.Thus, for example, we saw in Sec.

9.3 that the equations governing steadypotential flow in two dimensions are elliptic in a subsonic region, parabolicon a sonic line, and hyperbolic in a supersonic region. When the a, andbi are independent of u and v, the coefficients of (2') depend only on χand y and it is possible to state that the system (1) is hyperbolic in acertain region of the x,i/-plane.

In this case the two roots tan φ = dy/dxof Eq. (2') are functions of χ and y alone. B y integration of the resultingordinary differential equations, the t w o sets of characteristics can be foundindependently of u and v, thus, in the case of linear equations in two dimen­sions the characteristics form a network of curves determined once and for all,before any boundary conditions are given to single out a particular solution u, v.In the general case, however, where the coefficients a and 6. are func­tions of u and ν also, the system may be described as hyperbolic or ellipticat a point only in connection with a particular solution u, ν of the system.Here the characteristics are different for different solutions u, ν and mustbe determined progressively along with the solutions themselves, whichdepend on the boundary conditions.

Actual use of the characteristics willbe made only in a hyperbolic region and at its boundary, which is parabolic.tAs a simple example, consider the case of a small perturbation of a fluidat rest, discussed in Sec. 4.2. T h e equations corresponding to (1) are Eqs.(4.5), where u and ν are q and p, while χ and t stand for χ and y\ in thepresent notation these equations are given byxdudvpo — + — = 0,dxdyfo\(3)du2 dvpo — + a — = 0.dydx0Here all coefficients are constants: a\ = po, a =θ2 = a = bi = bi = 0. Eq.

(2') therefore reads41, b2=po, bz=ao ,23—po cos φ + ροαο sin φ = 0,222tan φ = db —.OoT h e differential equationsdy _1dxao1dy _1dxoVare satisfied by y = x/oo + constant and y = — X/OQ + constant, respec­tively, so that the characteristics consist of two sets of parallel straight* Δ24 = 0 and Δ η — 0 imply that there are linear combinations of equations (1)with left-hand sides μι du/dx + PI dv/dx and μ du/dy -f- vi dv/dy, respectively.

If inaddition ΔΜ + Δ23 = 0, then μιν — μιν\ = 0. From this it follows that (a) and (b)are the only exceptional cases.2210.2COMPATIBILITY119RELATIONSlines meeting the x-axis at angles arc cot do and arc cot (—do) respectively.If, however, the one-dimensional flow were not treated as a small pertur­bation, i.e., with terms of higher order neglected, the constant po in ( 3 )would be replaced b y ρ = ν, and other terms would be added.

Then theleft-hand members of (3) would be nonlinear, and the characteristics wouldchange with the boundary conditions to which the flow is subjected.2. Compatibility relationsFrom the discussion in Sec. 9.3, it follows that there is for each character­istic a suitable linear combination (9.11) of the left-hand members of thesystem ( 1 ) , involving only the components of grad u and grad ν parallel t othat characteristic, that is, du/da and dv/da, if d/da means differentiationin that characteristic direction. W e called the equation so derived, whichlinks the changes of u and ν along a characteristic, the compatibility relation.T h e compatibility relation in the present case, where η = k = 2, willnow be derived directly and discussed in some detail.

W e first introduce theabbreviations Αι, A , Βι, and B for the four elements of the determinantin ( 2 ' ) , so that E q . ( 2 ' ) becomes AiB — A Bi = 0. Then, with the aidof the relation ( 2 ' ) between tan φ and the A / s occurring there, it can beseen b y direct computation that the following identities hold:2222t(4a)aiBi — biA\αιΒι — 62Λ.1cos φsin φ(4b)CL3B2(4c)α Βι — b Ai _ CLABI — b Aisin φcos φ223sin φcos φζaj$ — 6 Δ4 ^42,— bAzΔ12,zAΔ3 2+ Δ13 tan φΔ ι + Δ42 cot φ = Κ,4a\B — b\AaiB — b Acos φsin φ2(4d)2222ΔΜΔ3 2+ Δ ι tan φ3+ Δ24 cot φ = L.Equations (4c) and (4d) are definitions of Κ and L.If the first and second of Eqs.

(1) are multiplied by Bi and A (or B andA ) respectively, and subtracted, the results, in view of the identities (4)arex22(5)A p+ Kf= Β -Αφ,dada12ια+ A ^= Βχι daddauA b,2where for Κ and L either one of the expressions in Eqs. (4c) and (4d) maybe taken. I n general, any one of the four alternatives can be used as the40120II. GENERALTHEOREMScompatibility relation since they are equivalent. In certain special cases,however, some of the alternatives may fail. For example, if in (1) all co­efficients except αϊ and 6 vanish, the only nonvanishing Δ is A = α ι 6 ,and it follows from Eq. (2') that the characteristic directions are φ = 0°and φ = 90°.

For φ = 0° the second expression for Κ and the same for Lbecome indefinite (0· o o ) . If the alternative expressions for Κ and L areused, the first Eq. (5) has only zero terms, but the second gives Δ du/θσ =B a = b a or a\ du/βσ = a, which is identical with the first Eq. ( 1 ) .Hence one of the given equations is the compatibility relation in this case.In all cases in which E q . (2') has two distinct real solutions, at least one ofthe four alternative forms of (5) supplies the compatibility relation.4i 44Χ 42AW e may now return to the compatibility relations derived in Sec.

9.4 forparticular equations and check the results obtained against the presenttheory.3. Two important theoremsW e are now in a position to formulate, for the hyperbolic case, two prin­cipal theorems concerning the existence and the uniqueness of solutions ofthe system (1) which satisfy certain boundary (or initial) conditions. T h ereal characteristic curves, which first appeared as curves across which asolution could not be extended by means of the system ( 1 ) , can now be usedas curves along which a solution can be extended by means of the com­patibility relations ( 5 ) .Theorem A.

Let the values of u and ν be given along a curve A Β in thex,y-plane (see Fig. 45a), in such a way that at no point does the directionof A Β coincide with either of the t w o characteristic directions determinedby x, y, u, and ν (Cauchy's problem). Then a solution, assuming the givenvalues at the points of AB, exists in the neighborhood of AB on both sides,lying within a region bounded (partly) by the four curves ACi, ADi , BC2CIID(a)(bb))F I G .

45. Illustration of the Cauchy problem.10.3TWO IMPORTANT121THEOREMSand BD that are characteristics for this solution. As far out as the solutionexists, that is, at most in a characteristic quadrangle ACBD (Fig. 45a), itis uniquely determined by the given values on AB. In the linear case (seeSec. 4) the existence (and the uniqueness) of the solution can be proved inthe whole characteristic quadrangle ACBD, whose boundaries are knownindependently of the given values of u and v.T o understand this theorem let us consider on the curve A Β two pointsP i and Ρ 2 an infinitesimal distance apart.

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