R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 80
Текст из файла (страница 80)
Thus, realcharacteristic lines in the x,y-plane exist only for supersonic motion, Μ ^ 1,and then they are lines with inclinations θ db a, where a = arc sin (a/q) anda is the same function of ρ, ρ as before. Further conclusions analogous tothose of A r t .
16, however, cannot be drawn. T h e interchange of dependentand independent variables, the use of the hodograph, etc., are no longer ofavail, since (9) is nonhomogeneous.T h e above conclusions concerning the characteristics also follow by considering Eqs. (22.1), (22.2), and (1) as a system of four homogeneous* T h e corresponding temperature and internal energy functions T(p, p) and U(p, p)are solutions of Eqs.
(2.11) and (2.12).t Equation (18') shows that ρ and a are then functions of (θψ/dx) + (θψ/dy)alone, and the coefficients are in fact independent of ψ itself. (See N o t e I V . 1.)22430V. I N T E G R A T I O N T H E O R Y A N DSHOCKSfirst-order equations for q , q , p, and p. For this purpose Eq. (1) is writtenxyout as<>>'f -«' a= ·•see ( 8 ) . Following Sec. 9.6 we find that the characteristics of the systemare the streamlines and the two families of Mach lines. For the streamlinesthere are two compatibility relations, namely, (1') and the Bernoulliequation (10). Across the streamlines ρ and q may have discontinuities (seeN o t e I I .
31), as we have already seen in Sec. 23.3. T h e compatibility relations on the Mach lines aredp ± pq tan a dd — 0,along a2C.±However, we did not find the streamlines appearing as characteristics ofEq. (9). T h e reason is that we have assumed in the present section that Sand Η are prescribed functions of φ. This leads to tw o new equations forthe derivatives of the state variables normal to a streamline which, inconjunction with the above-mentioned system, are sufficient to determinethese derivatives from given values on the streamline.TI t was shown in Sec. 22.3 that the actual change in entropy across ashock is in most cases very small. Thus if the motion before the shock isisentropic with constant total head, if the shock is not too strong and if, atthe same time, the variation of slope along the shock line is not large, thederivative G' will be zero and F' insignificant.
Under these conditions onemay, as a rule, consider the flow after the shock to be of the same typeas that before the shock.453. Substitution principle. Modified stream functionFor the important case of flow behind a curved shock line with uniformincident stream, the entropy S varies from streamline to streamline butthe Bernoulli function Η is constant. W e shall now show that any strictlyadiabatic flow of a perfect gas can be replaced by one with this property.Moreover the substitute flow has the same streamline pattern and pressuredistribution as the original.Consider the equation of continuity and the two components of Newton'sequation in natural coordinates (Sec.
16.1):dq _d s~PT_ dpds'2p qd0 _dpds ~ ~dn'These equations are satisfied by the set of variables q> 0, ρ, ρ in the χ, ?/-plane24.3SUBSTITUTION431PRINCIPLEcorresponding to the adiabatic flow of a given gas. When these variablesare replaced by \q 0, ρ, ρ/λ , respectively, the equations are still satisfiedprovided that λ is a function which remains constant along streamlines:d\/ds = 0. For the second flow has the same streamlines, so that the s- andη-directions are the same as before. N o t e however that the second set ofvariables need not necessarily correspond to a strictly adiabatic flow of thesame gas.
For the fact that S(p, p) is constant along streamlines does notimply that S(p, ρ/λ ) has the same property. Of course, another entropyfunction can always be found which will have this property for the secondflow, i.e., the second flow may always be considered the strictly adiabaticflow of a suitable gas.2y2W e therefore have the following result: To each given strictly adiabaticflow in the x,y-plane there corresponds an infinity of such flows with thesame streamline pattern and pressure distribution. These flows are obtainedfrom the original by multiplying the velocity vector q at each point by λ andthe density ρ by 1 /λ , where λ is any function whose level lines are the streamlines. Moreover since a = dp/dp, where differentiation is along a streamline, the velocity of sound is also multiplied by λ.
The Mach number at apoint is therefore unchanged by this transformation, so that all the flowsare subsonic (or supersonic) in the same region. This is known as thesubstitutionprincipled22W e have just seen that in general the character of the gas changes underthis substitution principle; normally the entropy function cannot be thesame in the two cases.
A n exception is the perfect gas, for which S(p, p)is given by (11). For if p/p remains constant along streamlines then so alsodoes p\ /p . I n particular, by choosing λ proportional to p* lpwe makethe second flow isentropic, since then p\ /p takes the same constant valueon all streamlines. Similarly the function gH = yp/(y — l ) p + q/2 ismultiplied by λ in the substitute flow. Hence, in particular, if λ is choseninversely proportional to H\ the corresponding function in the substituteflow will be constant throughout.
Thus: If the original flow is that of a certain perfect gas then the substitute flows are also; in addition one of them isisentropic and another has constant total head.y2yyl2y2yy2For strictly adiabatic flow of a perfect gas in which Η is constantthroughout, the equations of the preceding section can be simplified. Thereis no loss in generality when we take this constant value of Η to be l/2g(this is equivalent to choosing units so that q = 1). Then (1G) becomesm(19)where(20)/ ' exp [F(t)/gR]άφ.432V. I N T E G R A T I O N T H E O R Y A N DThus q is a function of (ΘΨ/dx)+ (dV/dy)22SHOCKSalone, and it then follows from(10) that the same is true for a:a =(21)- g ).(122From (5) and (15) we find99-1/(7-1)(22)N o t e that the derivatives of Φ, unlike those of φ, are determined by thevelocity components alone.
T h e modified stream function Ψ was introduced by L. Crocco. I t has the property of remaining constant on streamlines, since it is a function of φ alone; unlike φ, however, it is discontinuousacross shock lines.47T o find the equation satisfied by Ψ we compute the left member of (9)with Φ replaced by Ψ. Thus/\_ g, \2θ*qq2xyaa / dx2a*= ΓΛ --W+/2ΓΛL\-22ya J dy22^ J J L + ( i - rf\2a J dx2θ*q \dxdy ^ \22a22* * \ /^Ydxdy\- 2 Μ 2.
^ ^W W/a=- ^ R[aa J dy ]2^ι_^Λ+(-y1ί^Υ]^a / Vdy/ J άφ\+άφ22)q2 9d t]+pqW 'where in the second step we have used Eqs. ( 5 ) , ( 9 ) , and (12). N o w from(20) we have ά Ψ/άφ = (F'/gR) άΨ/άφ, so that this last expression becomes22Finally from (15) and (19) we findso that Ψ satisfies the equationα / dy*2=(q2-ygRα),2K2x2/(7-0_q>dFd9'24.4A SECONDAPPROACH433All coefficients on the left-hand side of this equation depend on d^/dxand ΘΨ/dy alone [see (21) and (22)]. Apart from dF/άΨ, which is a givenfunction of Ψ, the right member depends only on (d^/dx) + (d^f/dy) .24. A second approach248I t is possible to give a different approach to the problem of strictlyadiabatic rotational flow. This parallels, to some extent, the discussion ofadiabatic one-dimensional flow given in Sec.
15.7.If q and q times the equation of continuity (22.2) are added to the firstand second components (22.1) of Newton's equation respectively, we obtainxy^+ P?* ) +(P~-2(23)(pq*Qy)0,=yThese two equations allow us to introduce two new functions l(x,y)v(x,y) such that(24)dl =-pq qdrj =~ (p + pq ) dx + pq qxydx +and(p + pq ) dy,2x2yxydy,just as the equation of continuity permits the introduction of \l/(x,y) suchthat(25)άψ =— pq dx + pq dyyx[compare (5)]. Substituting from this into Eq. (24), we have(26)d£ = q dψx+ ρ dy,drj = q άψ — ρydx*W e have seen that in strictly adiabatic flow the entropy S(p, p) is a function of ψ alone. This function, F ( ^ ) say, is supposed to be determined by theboundary conditions.
Prescription of F therefore provides a relation between ρ, ρ, ψ throughout the flow. If any two of these three variables areselected as new independent variables in place of χ and y, then the thirdmay be considered a known function of these two for any given problem.Moreover, (42) can be rewritten as(27)where ξ =d£ = q dψx-y dp,άη = q άψ + χ dp,yζ — py and η = η + px.
W e are thus led to select ψ, ρ as new* For an element of streamline άψ = 0 , and d\, άη are the x- and ?/-components ofthe pressure force across the element. Hence the total changes in | and ή along afixed boundary give the components of the force exerted by the fluid.434V. I N T E G R A T I O NT H E O R YA N DSHOCKSindependent variables in place of x, y and to replace f, rj by ξ, η. I n the newvariables, we have(28)y =χ = δη/dp,(29)q= δξ/δφ,xq-dt/dp,=yδη/δφ,and Eq.
(25) yields(30)dydx _ 1dyyd\f/ρdp opdydxSubstitution from (28) in (30) gives^ _ ^ L + ^ J j L(31)d\f/ d\f/dp< >d\p d\//dp5 7 ^ + 5 7 ^82dp2+iρ=0,= 0,dpas simultaneous equations for £, η as functionsof ψ, p. In these equations ρis considered to be a known function of φ and ρ; once a suitable solution ofthem has been determined, the position coordinates x y and the velocitycomponents q , q are given as functions of φ, ρ by (28) and (29), respectively.2}xyEquation (32) has an immediate interpretation since it is equivalent tothe second of Eqs. (30).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
