R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 22

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 22)

If the first-orderderivatives of Φ are taken as the unknowns U\, u ,2Uz, then η = k = 3,and Eq. (18) may be replaced by the system of three equationsψΑιdx+,ψ ,ψ+Β \Αι+Αdydzdui _ dUz_ Q~dzdx~idUz\dydu \2+du _dU\ _ ^daT~dy ~~2'dz)As above, the additional equations result from the condition that theunknowns are derivatives of the same function Φ. This condition can alsobe expressed by curl u = 0; in (19) only the y- and z-components of thislast equation are written. T h e x-component, namely(19')Ρ ~ψdy= 0,dzis not a consequence of the other two.

In fact, from the vanishing of the yand 2-eomponents of curl u we can conclude that the z-component is inde­pendent of x, but not that it vanishes entirely. [That (curl u)* =du^/dy —9.5111FURTHER EXAMPLESdu /dz is independent of χ is seen by straightforward differentiation with2respect to χ and then substitution from the last t w o of Eqs. (19) t o showthat d(curl u) /dx vanishes.] Thus the problem is not completely stated byzEqs. (19), and eventually we shall have to take into account E q . ( 1 9 ' ) also.Let us first determine the characteristic surfaces of the system ( 1 9 ) .For this set of equations, the components of the coefficient vectors a areu1: (4i , Β,, B ), ( £ , A , BO, (ft , B , 4,);t =2L = 2:(0,I = 3:(0,-1,0,3211),(0,0,0),(-1,0,0);0),(1,0,0),(0,0).0,Thus the λ-equation ( 1 0 ) becomesA{\i+Β λ32+Β λ2Β λι +3+Αλ322Βχλ3Β λχ +2Βιλ2 +0(20)-λ=Α λ33-λι2λι[Αιλι2++Α\22Λ λ332+2£ιλ λ +22Β λ λι +322 Β λ ι λ ] = 0.332T h e expression in square brackets determines a cone of second order; againat a particular point Ρ the cone may be real, possibly degenerate, orimaginary, depending upon the values of the coefficients Ai and B at P.tIn particular, if we consider the case of steady three-dimensional potentialflow, i.e., E q .

(7.14) with d/dt = 0, and choose the x-direction parallel toq at P , the coefficients in (19) are Ai= BB23=1 — M,2A2= Az=1, Bx== 0. Then the cone is given by(21)(Μ2-1)λ! 2λ22-λ32= 0.This represents a cone of revolution about the x-axis, with semi vertex anglea satisfyingtan a = ^λ*\+^= ^W^lthe angle (and cone) being real only for Μ== cota,1. Thus a characteristic sur­face, normal to λ at P , makes the M a c h angle a with the x-axis, or direc­tion of q, and is tangent to the M a c h cone at P , as defined in Sec. 5.2.32W e see that E q . (20) has the additional solution λι = 0 for all points P ,so that any plane parallel to the x-axis is exceptional for the system ( 1 9 ) .These planes are, however, not exceptional for the original E q .

( 1 8 ) , asfollows from considering E q . ( 1 9 ' ) in addition to ( 1 9 ) . T a k e , for instance,the x,2-plane, and try to compute the derivatives with respect to y fromthe known x- and ^-derivatives. This cannot be done from (19) alone sincethe second of these equations does not involve any of the unknowns112II. GENERALTHEOREMSdui/dy, bu^/dy, duz/dy. From (19'), however, duz/dy is obtained at once,and then dui/dy and du /dy are determined b y the first and last of Eqs.(19).

Since the ^-derivatives can be determined, the #,z-plane is not excep­tional.* Thus, the cone (21) includes in fact all exceptional directions.!As a further example, let us briefly consider one-dimensional nonsteadymotion; the potential in this case satisfies Eq. (7.24), which is of the type(2) in the independent variables χ and t. Here A = a — q , Β =— q, C = — 1 , so that Eq. (13') gives22(22)_ ^ = - ^ V ^λα — q*22= _ ^=_L_q ± aq db axfor the slope of the characteristic directions in the :r,£-plane. T h e char­acteristic lines make with the /-axis an angle whose tangent is q ± a.This result plays an important role in the theory of one-dimensional mo­tion (see Chapter I I I ) , where the compatibility relations will be taken upfor Eq.

(7.24), or Eq. (7.43') with ν = 0. T h e compatibility relations corre­sponding to this latter equation with ν = 0, 1, 2 may be derived directlyfrom Eq. ( 1 4 " ) . This is left as an exercise for the reader. T h e characteristicsare, in appropriate variables, the same for all three cases.x6. General case of fluid motionIn considering the general differential equations of nonsteady nonpotential flow in three dimensions at a point P, we again take the ζ-axis in thedirection of q.

Then the particle derivative d/dt of (1.4) is expressed byd/dt + q d/dx. W e use Eq. ( I ) of Art. 1, the equation of motion for aninviscid fluid, but neglect gravitational forces. A somewhat more generalcase than that of an elastic fluid may be included by supposing the exist­ence of a function S(p,p) (such as entropy), which has a constant valuefor each particle, although not necessarily the same constant everywhere,as in the case of an elastic fluid. T h e specifying condition is then dS/dt = 0.As pointed out in Sec. 5.2, the local sound velocity a can still be defined,using34«*>- i - -STI-* If in (19) we had used the x- and ^-components of curl u = 0, the factor Xwould have appeared in the analogue of (20), rather than the factor λι , andsimilar conclusions would apply.t If the general argument of Sec.

2 is applied to a single second-order planar differ­ential equation for Φ{χ\ ,x , · · · , x ), it is seen that the λ-equation is immediatelyobtained by replacing each factor β^Φ/βχ^ dx in the original equation by \ \ . Thiscan be proved in exactly the same way as E q . (20) is derived from E q . (18). Actuallythe theory of characteristics for second-order equations can be arrived at directly bymeans of a reasoning analogous to that of Sec.

1, and the result here explained is thenobtained.a2nK33tK9.6G E N E R A L C A S E OF F L U I D113MOTIONThenas dp _ dS[dpdpd§> _ dS dp(24)dtdp dtdp dt^_dpi2dp LdtIdtdtdty_so that, when dS/dp is different from zero, the specifying condition may beexpressed by setting the bracket in (24) equal to zero, with d/dt ex­panded as above. Then the system of five partial differential equations inthe five unknowns q , q , q , p, and ρ isxyzdq. dq.dx+ _dtdxdtx+ JJR10,ρ dxldpdq1(25)+dx+ Ρ dydq l_dpz+dtdq,dzdqxdxdp, dp* dxdt=0,=ρ dz+1 dp+ρ dx4 sρ dt=i ) -+0,ftThis is a system of the form ( 6 ) , with k = 5, η = 4.

There are A* =25 vector coefficients a , each having four components (in the directionsx, y z, and f), though many of these vanish because of the choice of thecoordinate axes. For example, the components of an are the coefficients ofthe derivatives of q in the first equation: q, 0, 0, 1. For a , a i , and a i ,the components are all zero, since derivatives of the unknowns q , q , andρ do not appear in the first equation, while a has components 1/p, 0, 0, 0.T h e entries in the first line of the λ-determinant are the scalar products ofthese vectors with λ = (λι , λ , λ , λ ) , namely, q\\ + λ , 0, 0, λι/ρ, 0.

T h eremaining terms can be found similarly, and the equation for λ is2t)Cyi 2x35yzJ42+(20)λ400?λι +00λ!λ00λ43440λι/ρ00λ /ρ0WP0q\i +2λ4(q\i +020Ρ<?λι ++λ )4λ24-λ )/ρ4— a(q\ia\\r +2+λ22λ )4+λ )]82114II. GENERAL THEOREMSThere are two types of characteristics, corresponding to the two factorsin Eq. (26). Before discussing the general case, let us consider the particularcase of steady motion, when the ^-component λdrops out. T h e second4factor in Eq. (26) then supplies(27)a [(M21)λ! --22λ22-λ ] = 0,32and the corresponding characteristic surfaces are again tangent to the M a c hcones (for Μ^ 1, of course). T h e first factor supplies λχ = 0, so that any3plane passing through the x-axis (or velocity direction) is exceptional.

I tfollows that, in the general case of steady motion under the specifyingcondition dS/dt = 0, all surfaces composed of streamlines are also characteris­tics. These stream surfaces are real characteristics even for subsonic flowwhenever the flow is not assumed to be irrotational, and in fact even inrotational incompressibleflow.For this case, however, the system (25)must be modified, but the formula corresponding to (26) is-(^1+λ4)2( λ1+2λ22+λ3 ) = 0,2ρso that the same factor is present.35In the case of a nonsteady flow, the second factor in (26) representsa " c o n e " ofsecond-order in £,i/,2,£-space. This"cone"intersectsthe£,i/,z-space in the cone given by Eq.

(27). For the intersection with thex,£-plane—i.e., when λ and λ2XiV-3drop out—the bracket reduces toα) +22αλ!λ +4λ ,42and this last expression equated to zero leads to the same t w o directionsin the ζ,Ζ-plane as in the one-dimensional nonsteady case: Eq. (22), withλ in place of λ . T h e first factor in Eq. (26) leads to the surfaces in the42z,2/,z,£-space consisting of the world lines of the particles (see Sec.

1.2).Thus all surfaces composed of world lines are characteristic.36(See also Sec.15.2).I t is seen that the Mach cone keeps its role as the envelope of exceptionalplanes irrespective of whether or not the flow is irrotational.37Consider finally the compatibility relations, remembering in particularthe end of Sec. 3. Firstly, corresponding to the triple root, i.e., to the firstfactor in Eq. (26), there are r = 3 compatibility relations.38T h e y consistof the specifying equation, the last of Eqs. (25), and two components ofthe Newton equation which are perpendicular to the λ-direction.W i t h respect to the second factor in Eq. (26) there is, for each directionof the cone, one compatibility relation. Consider first the steady case:9.6G E N E R A L CASE OF F L U I D M O T I O NE q . (27) supplies (Μ1)λ? --2λ*22-2115λ ? = 0, where the λ? are only2determined up to a common factor.N e x t compute the a ? , a* , · · · , < * ? from ( 9 ' ) .

Omitting for the mo­ment the * in a and λ, we have the equationsaiq\i + α λι = 0,4«2<7λι + &4\ = 0,2«3<Ζλι +αιλι +ΟΊΟΛ2«4λ=3+3«X30,+«βρ^λι=0,«4^λι — ot apq\\ = 0.hFirst we note that λι = λ? ?± 0 since λ * is perpendicular to a direction thatmakes the M a c h angle a 5^ 0 with the flow direction, which is here thex-direction. Hence the first and last of these equations give2a\ = a p a 5 ,a\a\ =2aρ=Q«5qindependent of λ . Further, we have(l pλ2«4\—,2on — — — - =q λι——a« 3 =—«6«4 λ3— — Γ-qαρ=q λιbλιλ3q7-·λιSince the α are likewise determined up to a common factor only, we choosea6=— 1/ap and obtain, again using stars,*/oo\(28)α1=ιM,*a21 λ**=1 λ*_ _ «=a,.Λ.=-a,«,1- -=W e now choose λ? = 1/M and haveλ * + \?(29)22=1 -1/M ,or2λ? + λ * + λ?8222=1,thus making λ * a unit vector.

Using λ? = l/ilf, Eqs. (28) simplify toTOQ'\l^o J*«ι=T7, Qf1Μ**2 = λ ,λ2*α3=λ*λ3,*«4 =—α,*«6 —_1,αρwhere (29) holds. N o t e that the fourth of the equations for the a has nottbeen used; it is seen that it reduces identically to (29) if the values of (28')are introduced.116II. GENERALTHEOREMSThese a* determine a compatibility relation. Using Eqs. (25) and (28')we find according to ( 1 1 ) :(30)ι..(| 1^,)-(. ,,+Λ+Ι*)_οor, using λ * · ς = a:(30')λ*·(§ _qdiv q + - g r a d p - i q f )= 0.\atppa* at /Here λ * may be any one of the single infinity of unit vectors perpendicularto the tangent planes of the Mach cone.In the general, i.e., nonsteady case, the compatibility condition is quitesimilar to Eq.

Характеристики

Список файлов книги