R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 53
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Or, introducing a ,2s(43)ρ — ps= — a p2s28(-—-),\PPs/Β =a p.and hence, for the constants A, B,(430A = p + pa ,sss2282Von Karman and Tsien (Fig. 103) take for the point of tangency one withundisturbed stream conditions*, so that, using the subscript <*>y(43")A=p„+p^oo ,2Β=ajpj.For either choice of A and Β such a gas has some simple but unusual* W e may have in mind the flow past a profile with uniform stream conditions atinfinity.17.5 T H E C H A P L Y G I N - K A R M A N - T S I E NAPPROXIMATION281properties which are easily verified.
W e find from Bernoulli's equation*(44)a -q = a,2Μ2= q{a2+r/p.T h e relation between ρ and q is now18FIG.104a. a/a versus q/a for various values of κ (ellipses and hyperbolas).8sW e note that there is now no restriction as to the value of q: in fact, q —» <*>as Μ —> 1, as seen from the second Eq. (44), and in this case, as seen from(44'), ρ —> 0. Hence, as Μ —> 1, ρ —> 0, q —* oo, a —* ° o .
I t is thus seen thatin the case of the Chaplygin approximation, the sound velocity a/a increaseswith increasing rather than decreasing q, in contrast to the case κ > 1.Figure 104a shows a/a versus q/a for various values of κ, in particularfor κ = 1.4 and κ = — 1, whereas Fig. 104b shows Μ versus q/a, for variousvalues of κ. For κ = — 1 the curve has Μ = 1 as an asymptote. I t is thusseen that an initially subsonic flow will remain subsonic if κ = — 1; hencetransonic flow cannot be studied by this method.s8* We have in Kdrman-Tsien's ease as2s= aw2— q^.282IV.PLANESTEADYPOTENTIALFLOWFrom the first equation (42) we have, using (44),(45)dkdqa^λ = log8qV Wq+qvV +a, +29q2If the integration constant in the expression for λ is chosen as in (45), it isseen that λ —> — «>, as q —> 0, and λ —> 0 as g —> oo, Af —• 1; thus λincreases monotonically.
From the second formula (45) we see that forq-values small compared to a :8(45')x-tog-L + O ^ )hence2a *8(b) Relation to an incompressible flow problem. Eqs. (41) are CauchyRiemann equations in the Cartesian coordinates λ, — θ. It will however bemore convenient to introduce the new independent variable,19(46)=ν2^2a,e=1 +The choice of the multiplicative constant 2a is such that for a —» <», or q8F I G . 104b.
Μ versus q/a for various values of κ.t817.6 F L O W P A S Tsmall compared to a , the ν ~the form:8>7T=-%(47Aq [as in (45')]. T h e equations (41) then take?=%v283PROFILEvM < 1-dvΘΘdvd6These are Cauchy-Riemann equations in polar coordinates ν, — Θ, i.e.,exactly the form which Eqs. (16.31) take on as Μ —» 0. Consequentlyφ + ιψ is an analytic function of ve~ and the real and imaginary parts ofany analytic function of ve~ will be solutions of (47). These equationsmay be interpreted as the equations of an incompressible fluid with complex velocity ve~ = f where the speed ν varies between 0 and 2a . W estate: If we apply the Chaplygin approximation to the basic equations(16.31) and introduce in (41) instead of λ, a new variable ν by (46), theseequations take the form of incompressible flow equations in the polarcoordinates ν, — 0.If q and ρ are expressed in terms of ν we obtain, by (46) and (44'),x9x9%98W e see that as ν increases monotonically from 0 to 2a , q goes from 0 to <χ>,and ρ from 1 to 0.
T h e transformations (46), (46') are thus interpreted asrelations between an incompressible flow and a compressible Chaplyginflow.86. Continuation(a) Flow past a profile. L e t us try to construct the compressible Chaplygin flow past a given profile P o .
W e assume that this flow does not inv o l v e circulation. This restriction will be removed later in the section.W e assume as known the complex potential w(z) of the incompressibleflow around P (where ζ = χ + iy) and introduce the complex velocitydw/dz = f (z) = ve~ . W e use this last equation to express ζ in terms of ξand obtain w(z) = W o ( f ) = φο + ιψο where <p and ψ depend on ν and Θ,and ^ o = 0 along P . W e then define a stream function ψ(<?,0) of a compressible Chaplygin flow by setting0%e000(48)t(q,B)=*oM),where (46) holds between ν and q, and similarly for <p(qfi). These φ and ψthus defined are solutions of (16.31) under the Chaplygin approximation.W e shall however see that these ^(<?,0), <p(qfi) do not in general provide asolution of the given boundary-value problem of flow past P .
Denote byX,Y(Z= X + iY) the coordinates in the physical plane of the compressi0284IV. P L A N E S T E A D Y P O T E N T I A LFLOWble flow in order to distinguish them from the x, y in the plane of incompressible flow. Then from ψ(ς,θ),Y(q 6)y(p(q,6) follow, in the usual way,X(q,6),(see Sec. 3 ) . Some curve Ρ in the Z-plane will correspond to thegiven profile Po in the 2-plane, along which ψ0=0. W e shall show thatthis contour in the Z-plane differs in shape from the given contourbut reduces to P0P,0as a —-> oo, Μ —> 0, ν —> q.
Thus, even in the presentesimplified situation, we do not solve an exact boundary-value problem.I t is, however, possible here (in contrast to the polytropic case which weshall study in following articles) to indicate a very simple formula for thedeviation between the two contours, the so-called shape correction.W e obtain from E q .
(25")dZ = dX +i dY= - elie(άφ + I d#j ,and substitute for ρ and q from (46') in terms of v. Then, denoting by w,f, etc., the conjugate complex functions to w, ξ, we havedZ = - (άφ +q \e— {άφ +- άψ) = -f^ (4aρ/4α* νie== - eν%9s-22ve "—(άφ -υ )άφ + i / ζ - (4a4αΛ2s2+ν )άψ2%9i άψ) -aw —4as02ve%θi άψ)aw.Therefore,d Z - ^ - ^ f d ® .(49)Replacing f byaw/άζ(49')and f byaw/az we obtain:dZ = ( f e - L ( ? Y d 2 .74o ^ \dz/sThenW e thus obtain for each w(z) the Ζ for the compressible flow which corresponds to a ζ in the incompressible flow, the respective speeds υ and qare linked by E q .
(46).*W e note that from E q . (49) φ + %φ is a nonanalytic function of X + z'F.* This simple formula compares, in the exact theory, with such involved resultsas (67) and (72) in [24] pp. 242, 243.17.6285CIRCULATION( b ) Circulation. For flow around a profile a difficulty arises if the incompressible flow around the profile involves circulation, since the function tothe right in (49") will be single-valued only if2dz = 0,for any path around P , and this is true only in the absence of circulation.In other words, when the flow has circulation the profile shapes in theZ-plane furnished by the above theory are not closed. Hence, our methodas explained here is valid only if in the incompressible flow circulationis absent.
Several authors have generalized the procedure to take careof this difficulty. W e indicate here a method due to C. C. L i n .020Denote by w(z) the complex potential of the incompressible flow withcirculation, and introduce instead of f, more generally,(50)ξ(ζ)=k{z)ve~ψ =(50')iefcdz(51) /^"έ/<Here k{z) is an analytic function of z, regular and without zeros in the exterior of the given profile P (including the point ζ = <*>)> and such that0=0for any closed contour enclosing the given profile P o . In addition(510II < I k(z) I <on Po.ooI t may then be proved in the same way as (49") was obtained that the Ζof the compressible flow is given by(50")Ζ = fk(z) d z - ± fedzk(z)to be compared to ( 4 9 " ) , and with an analogous interpretation.Thus to a complex potential which represents an incompressible flowwith circulation, a related compressible Chaplygin flow with circulationis given by (50), (50'), and ( 5 0 " ) , and we are still free in the choice of k(z).In these equations ζ can be considered a parameter; and after it is eliminated we obtain relations between X, F, g, 0, the compressible flow coordinates and the velocity.T h e essential point in the above method is that ve~ is equated to afunction of ζ which is not directly the complex velocity ξ{ζ) as in KarmanTsien's scheme, but equal to %(z)/k(z) where k(z) is still widely arbitrary.%e286IV.
P L A N E STEADY P O T E N T I A L FLOWFor k(z)=1 the condition (51) of the closure of Ρ in the Z-plane is notsatisfied if circulation is present. On the other hand, k(z) should not devpart too much from unity if the profiles Ρ and Po are not to differ too muchfrom each other.Finally, discussing the direct problem, Lin relates the choice of k(z)tothe mapping of Po onto a circle: with w(z) now the complex potential forincompressible flow (with same circulation) past the circle, the functionk(z),which must satisfy the above requirements, should " n o t differ toom u c h " from k (z), the derivative of the mapping function.
This idea is then0related to a well-known method of v . Mises21who has shown (in incompressible flow) how to transform a circle into an airfoil of very generalshape.(c) Additionalremarks. T h e influence of compressibility upon the pressure distribution may be easily estimated. W e refer the reader to the literature.22In order to demonstrate the change of contour by means of an example,Tsien has computed the compressible flow about an approximately ellipticprofile by starting with an ellipse in the 2-plane. T h e deviation of the new23contour from the ellipse tends to zero* as a —> <», or M8M—> 0; it is howevershown that for not too small M^-values, and if the given ellipse is nearlya circle, the deviation is quite appreciable.W e have presented here the Chaplygin approximation not only becauseit is widely and successfully used, but also because it can be consideredfrom various aspects: on the one hand, it constitutes an approximationwhich greatly simplifies the usual equations; on the other hand, it is aneasily understood exact theory.
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