R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 70
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A particular example of such a flow of a viscous fluidcan be constructed from the steady one-dimensional flow presented inSees. 11.3 and 11.4. Suppose the latter flow is viewed from a coordinatesystem moving in the negative ^-direction with constant speed v. Thisamounts to superimposing on the original flow a constant speed ν in the^/-direction. Then along any line parallel to the z-axis in the resulting motion a rapid change from the state q = U\, q = vi, pi, pi to another stateQx = u , q = v , pi, p takes place, in which these eight variables satisfythe three relations (11.22), (11.23) and (11.24):x2y(3a)p\U\ = p u2(3b)(3c)y22pi +2= m,2mux = p +mu ,27 — 1 pi2+ ^ L * "7 — 1p22as well as the extra condition(3d)If qi is written for Ui + vvi = v2 =2and q2for uv.2+ v2then (3c) may be replacedby(3c')27 — 1 pi27 — 1 p2by virtue of (3d).
I n the limit μ —» 0 the thickness of the transition regiontends to zero [see (11.47)] and Eqs. (3) then relate the initial and finalvalues of an abrupt transition.Thus Eqs. (3) are the shock conditions for a particular kind of steadyplane motion. I t will now be shown that these equations are the transitionconditions for the most general case of steady plane flow.022.2OBLIQUE SHOCK C O N D I T I O N S FOR P E R F E C T GAS3772. The oblique shock conditions for α perfect gasFirst it is necessary to obtain the equations governing the steady planeflow of a viscous fluid.
T h e equation of continuity is the same as for an inviscid fluid, namely Eq. ( 2 ) . For a viscous fluid the three components ofN e w t o n ' s equation are given b y Eqs. (3.8), and these must be specializedto the case of steady plane motion with gravity neglected.For a volume element dx dy dz (see Fig.
6 of A r t . 3) in such a motion, noshearing stresses due to viscosity can exist on the two faces parallel to the£,?/-plane, since adjacent particles on opposite sides of these faces movewith the same velocity. Consequently(4)τ=ζχT=zy= 0,σ2and hence also(5)TyZ== 0.TXgThere remain only the normal stresses σ , σ and the shearing stressesχT(6)υ= Tyx — τXV(say).Thus with gravity neglected Eqs.
(3.8) reduce to,dqxdt/o\\o)ρdqy—j-_dtdpda, drdxdxdydp— —dyx-τdrday—"Tdx—dyywhere d/dt = q d/dx + q d/dy.A s the specifying condition we suppose that the motion is adiabatic except for heat conduction, i.e. simply adiabatic (see Sec. 1.5). Under theseconditions the energy equation is (3.24), and on restricting the latter to thecase of steady plane flow we havexywhen gravity is omitted. Here w is defined by (2.5) and w' by (3.10), whichin the case of steady plane flow reduce todxdxby virtue of Eqs. ( 4 ) , (5) and ( 6 ) ; hencedydyV. I N T E G R A T I O N T H E O R Y A N D378w + W dx= ^-[q (px-σ)χ-q r]+ ^ayySHOCKS[-q r+xq (py-σ )].νFinally, if it is assumed that the fluid is a perfect gas, so that ( 1 . 6 ) :ρ =(9)holds and U = c Tcondition isvΡat \_2= p/(ygRpT— l ) p according to (2.13), then the specifyingox7 — 1 pj(10)Equations (2) and (7) through (10) areknowns q , q , p, p, and T , provided σ , σthese variables (see last paragraph of Sec.and the result added to ( 7 ) , then the latterxχy(Π)^<r'*l + ^[pq? + V -five equations for the five un"and r are expressed in terms of3.3).
If (2) is multiplied by qxis replaced byνIpQxQv -τ ] = 0;similarly if ( 2 ) , multiplied by q , is added to ( 8 ) , it becomesy(12)£[qqPxv-τ] + ±[pq+ ρ -2va' ] = 0.yAgain, on adding ( 2 ) , this time multiplied by [q/2 +specifying condition (10) we obtain(13)dx~'"["*(! ν^ιί)++^Wl+ qx{p+σ χ)7^1p)"i l T9vTp/(y— l ) p ] to the~ Sfc~ S] '+ ?i(p-<7i)fc=0Finally if Γ is expressed in terms of ρ and ρ from the equation of state ( 9 ) ,then ( 2 ) , (11), (12), and (13) become four equations for q , q , p, and ρ asfunctions of χ and y.Each equation is of the formxdAdB=y0and when integrated over the interval X\ to x supplies a relation of the form2Jr 2X3 D22.2O B L I Q U E SHOCK C O N D I T I O N S F O R P E R F E C T GAS379W e now consider solutions for which the ^-derivatives remain bounded asμο and k tend to zero. Then (14) holds also for the limit flow, with the integral tending to zero as x approaches X\.
Thus in particular, if the x-direc2tion is taken to be the normal to the shock line at the point under consideration, and 1, 2 refer to adjacent points on opposite sides of the curve,see Fig. 142, the difference A2— Ai must vanish. Introducing successivelyfor A the four expressions from ( 2 ) , (11), (12), and (13) we obtain the fourconditions[pgxli=[pq0,+2xρ-σ ]\χ=0,[pq qxy-τ]\=0,ni+Th*h*-*-»->*l-*(15)In order to compare these with Eqs.
(3a) through (3d) we first replacethe values of q and q on the t w o sides 1 and 2 of the shock by U\, Vi andxu,2yv , respectively. Then the first of E q s . (15) is exactly ( 3 a ) . N o w since2we assume that the fluid behaves like an ideal fluid on either side of theshock, the viscous stresses σ , r and the heat flux k(dT/dx)χmust vanish onthe two sides of the shock. Thus the second of Eqs. (15) gives (3b) immediately, while the third givespuv222— piUiVi=m(v2— v{) =0.Therefore, since m ^ 0 (i.e., particles cross the shock line), (3d) must hold.Under the same condition the fourth of Eqs.
(15) yields (3c').Thus we have shown that the four equations (3a) through (3d) representnecessary conditions relating the initial and final values of an abrupt transition in steady plane flow. One restriction must still be added to the conditions. T h e flow studied in A r t . 11, which in the limit supplies a specialcase, at least, of such a transition (namely one with ν constant), is not re-F I G . 142.
Choice of axes at shock.380V. I N T E G R A T I O N T H E O R Y A N D S H O C K Sversible: it always goes from lower to higher values of θ [see (11.13)] andtherefore of Τ or p/p.Thus the program outlined in the last section m a y b e22formulated more precisely as follows.We consider flow patterns in the x,y-plane which satisfy the differentialequations of ideal fluid theory everywhere except on certain curves (of unknownshape); across these "shock lines"the tangential componentof velocity ν iscontinuous, while p, p, and the normal component u have discontinuitieswhichsatisfy the three conditions (3a) through (3c) and the inequality(16)P2Piwhere for any particle state 1 precedes state 2.I t is not correct to refer to these flow patterns as discontinuous solutions4of the equations of ideal fluid flow", see Sec.
15.2. Rather they should becalled asymptotic solutions of the equations of viscous flow.3. Analysis of the shock conditionsT h e shock conditions consist of the four equations (3a) through (3d) andthe inequality (16). T h e .main features of the conditions are that the tangential components of velocity V i , v enter only into ( 3 d ) , and that the2normal components u ,u,xthe pressures pi,2p,2and the densities p i , psatisfy exactly the same equations as the relative velocities u[ , u ,pressures pi,p,22the2and the densities p i , p do in the case of nonsteady one2dimensional flow, see Eqs. (14.2).
Moreover the inequalities (14.9) and (16)are the same. Thus to every result which is a consequence of Eqs. (14.2) andthe inequality (14.9) there corresponds a result in the present case, obtained from it by changing u[, u t o U\, u . This remark saves us a certain22amount of work, since results corresponding to those obtained in Sees.
14.3and 14.4 may be given immediately.For this purpose it is convenient to introduce the M a c h numbers Μ,ίηM2ncorresponding to the normal velocities U\, u \ these normal Mach2bers are related to M ,ΜxΜin2numby the equations= Mi sin σ ι ,M2n= M2sin σ ,2where σι and σ are the angles (between 0° and 180°) at which the shockline is inclined to the stream direction for states 1 and 2 respectively, seeFig. 143a. T h e y replace the Μ[ , M of Sees. 14.3 and 14.4.22(a) Consider first some limiting cases which satisfy the shock conditions(3a) through ( 3 d ) . If u = u = 0, then m = 0, and Eq.
(3b) gives p = p .For this case the third condition (3c) is satisfied for an arbitrary value ofP i = P 2 . N 0 particle crosses the line of discontinuity and hence this case isnot usually included in the concept of shock. Another limiting case is U\ =x2x222.3ANALYSISOFT H ESHOCK381CONDITIONS(a)(b)F I G . 143. Deflection of streamline at shock, (a) T h e angles σι and σ , ( b ) V e l o c i t y2components.u τ* 0. Again from (3b) it follows that pi = p , while the third conditionleads to pi = p .
W i t h the fourth condition (3d) it follows that no actualdiscontinuity occurs, and this case will be referred to as zero shock. T h esame conclusion follows if we know only that p = p , or that pi = p ,provided the particles actually cross the shock line.222x22(b) A physically possible shock (i.e., a rapid transition governed by viscousfluid theory) is always a compression shock"; pressure, density and temperature increase, while the normal component of the velocity decreases and thetangential component remains unchanged.N o t e that this implies that the speed q decreases and that the streamline isdeflected towards the shock line on passing through the shock (Fig. 143).(c) T h e normal M a c h numbers M , Mare given in terms of the pressureratio p /pi by the analogue of (14.13), namelyuin2n22MΜin_ 7+ 1——P2 , 7 ~ 1-T—~,2MΜ2n_ 7 + 1 Vi , 7 ~ 1ο—75I·2ypi2y2yp2yIn the preceding section we assigned to the state in front of the shock thesubscript 1, and to that behind the shock the subscript 2.
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