R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 42

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Then the velocity has a jump, while ρ re­mains continuous. A more general case is that of a surface of discontinuityseparating two three-dimensional regions; the normal velocity vanishes oneither side of the surface, and the pressure, but not the tangential velocitycomponent, remains continuous across the surface. This type of discon­tinuity appears in the theory of wings of finite span.5455In our one-dimensional flow of an ideal perfect gas the following situa­tion can occur. Consider a fluid mass within a tube, which is initially (att = 0) situated between the points A and B, and whose particles move ac­cording to certain particle lines in the o:,i-plane (Fig.

83). This flow satisfiesthe differential equations with some constant d = pip . T h e pressure dis­tribution is determined throughout, and in particular along the particleline ΒΒ' which starts at B. This motion is compatible with, and uninfluencedby, the presence of a weightless piston which is initially at Β and whichmoves according to the space-time curve ΒΒ', subjected to a pressure fromthe right which at each moment just equals the computed p.

N o w we canimagine another fluid mass, initially between Β and C, which moves in sucha way that ΒΒ' is again the particle line of the particle initially at Β andthe pressure along ΒΒ' equal to that computed before, while in this section1FIG. 83. Contact discontinuity Β Β ' .15.3COLLISIONOF T W O221SHOCKSof the fluid p/p has a different constant value C .* This moving mass wouldsupply the required pressure on the piston from the right, while vice versathe mass initially between A and Β would supply the pressure from the leftnecessary to maintain the motion of the second mass.

Nothing would changeif the piston were omitted. W e would then have a flow pattern for the fluidmass initially between A and C, with a continuous set of particle lines andcontinuous p-distribution, but with a jump in the p-values along ΒΒ'. IfPi is the density on the left side of ΒΒ' and p the density on the right, wehavey22constant1for all points of ΒΒ'.T h e simplest example of this kind is the case of all particles between Aand C moving with constant velocity under the same pressure, as one rigideolumn, while the density has different constant values for the two massesto the right and to the left of B. I t is evident that the equation of continuityand Newton's equation (steady flow, no external forces) are satisfied.One may observe that the temperature, which is proportional to p/p,is not the same on both sides of ΒΒ'.

But since we do not admit heat con­duction, this has no influence on the motion. Thus we have seen that in thestrictly adiabatic flow of a perfect, inviscid fluid, flow patterns are possible inwhich the density jumps in a constant ratio along certain particle lines, whilevelocity and pressure remain continuous. This phenomenon is nowadayscalled a contact discontinuity.In the speedgraph plane the line ΒΒ' is mapped into two distinct curvesthe ordinates of which, for the same abscissa, have the constant ratio3. Example of a contact discontinuity:collision of two shocks*0A simple example of a contact discontinuity occurs in the case we shallconsider next, the head-on collision of two shock fronts. Assume that fluidalong the interval AB of the z-axis (Fig. 84) is initially in a uniform stateof rest: u = 0, ρ = p , ρ = po at t = 0.

Consider two shock fronts movingin opposite directions, one along AC causing the state 0,p ,Po to changeinto a state u ,pi,pi, and one along BC changing the initial state tou ,p ,p .A t the time t , represented by the ordinate of C, the particles00x222b70* Analytically, the required flow on the right is determined by thevalues of u and υ on ΒΒ', and can be constructed as in Sec. 10.3.prescribedIII. O N E - D I M E N S I O N A L222FLOWF I G . 84. Collision of two shocks AC, BC, production of a contact discontinuity.to the left of C have velocity u , pressure pi, and density p i , while thoseto the right have u ,p2,P2- In the speedgraph plane these two states arerepresented by two points 1 and 2 lying on the two branches of the shockcurve with its corner, 0, at.(0, vo), where v = Ayp /po(y — l ) .

W h a t happensfor t > U?x22002W e can find a flow pattern for t > to which satisfies all conditions if weassume that two new shock fronts (reflected shocks) form at C and movealong appropriate lines CD and CE. T h e two states of particles after pass­ing through the second shock fronts will be represented by two points 3 and4, with 3 lying on the shock curve starting at 1, and 4 on the shock curvestarting at 2. T h e two points 3 and 4 must fulfill two conditions: theymust have the same abscissa u in order that the final particle velocity in thewedge between CD and CE be one and the same for all particles, and second,the pressure value ρ must be equal for all particles.

N o w it is possible, ingeneral, to satisfy two conditions by choosing the slopes c and c of theshock lines CD and CE appropriately. T h e values of the density after thereflected shocks, however, will not necessarily be the same for particles tothe left of C as for those to the right. Thus the final particle line throughC (broken line CF in Fig. 84) will, in general, be a discontinuity line of thetype described in the preceding section.I t is easy, in principle, to find the numerical solution, that is, the valuesof u, p, and ρ after the second shocks, when the two states Ui,p piand^2,2?2,P2 are given.

W e simply apply to the transitions across CD and CEthe equation (14.29) which gives the relation between the pressure and34h15.3223C O L L I S I O N OF T W O S H O C K Svelocity changes across a shock. Using u and ρ to denote the final values ofvelocity and pressure, we have(p - P i ) = § ( « - «ι) Κγ + D P + ( 7 2DPI],2(7)(Ρ -p)2= I2(«« ) [(7 +-DP +22(7 -DPJ.Eliminating ρ between these two equations, we obtain one equation deter­mining u; one root of this equation is the common abscissa of 3 and 4 andtherefore determines the slope of the final particle lines between CD andCE.

Using the equation (14.32) of the shock curve, applied to the arcs 24and 13? we can then find the ordinates, corresponding to the abscissa u, ofthe points 3 and 4, and from these we know the values of p/pz and p / p .4On the other hand, the common value of ρ is determined by ( 7 ) , now thatu is known, so that finally the t w o densities p and p , prevailing to the left34and to the right of the discontinuity line CF, can also be computed.T h e occurrence of such contact discontinuities after a shock collision hasbeen observed, although in most cases the effect is rather weak unless theoriginal shocks are of great, but very different, strengths.T o carry out the computation, we may suppose that the pressure ratiosVh V2 of the initial shocks are given.

Then the density ratios ξι, £ are deter­2mined from the Hugoniot equation (14.26). Finally, the second of Eqs.(14.32') yields, with u= 0,0φ)Wi27 -_21 (?n h7l)+2m_ 7 -u221 'vo221 (m 7h2V2l)+21*Thus Ui/vo and u /vo are also expressed in terms of the compression ratios.2If we write χ and y for the unknowns u/vo and p/po, the equations (7) be­come\2/(ν - να = (7ττ^-τν,ίι (j - *)* K7 + Dv + (7 - Dml,- D22(70o& " ri* = 2 & ( * {y21(7 + Dy + (7 - DnJ.iyOnce y has been found, the density ratios follow from the Hugoniot equa­tion:/ \QP3POthy +2171y + η?ηιTaking, for example, ηι = 8, η2p4po= 4, andKy +η2y + Λ ^2> 0, u2< 0, with y =i,224III.

ONE-DIMP]NSIONAL FLOWwe find ξι =(70 givesξ = f,2=Wi/i>o1/λ/7, W^O =s = 0.1405,— 3/5λ/7· T h e solution ofy = 21.87.58When these values are substituted in ( 9 ) , we obtain for the density ratiosPz/po = 6.97, pt/po = 7.37, so that p /p = 1.057. These values were used in43the example in Fig. 84. T h e reader might compute for himself, from thegiven data 771,772, the four shock velocities chc , c , c , all as multiples of23v.4I t is seen from the figure, and from the value of pjp%,0that the disconti­nuity is insignificant, even though we have started with high pressure ratios77 .24.

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