R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 66
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4. However, we do not enter into an explanation of the general pro1cedure recommended by Lighthill in the case of supersonic velocities.2. Replacement of Chaplygin's factor [ ψ η ( τ ι ) ]1W e refer to Eqs. (20.13) and (20.14) with ψ (τ)given by Eq. (20.8).ηFor r = η , the right side of Eq.
(20.14) coincides with that of Eq. (20.13)with q =1, where q = qi =1 is the velocity at infinity and η =l/q .2mW e have indicated earlier that if we start with one branch of the compressible flow ^ ( r , 0 ) , the main problem is to find its continuation over thedesired region. Lighthill noticed that for this purpose Chaplygin's factor[^ (ri)]n_ 1is inappropriate (leading to avoidable complications) and replaced it by another factor which retains the essential properties of theformer and is better adapted to the problem of continuation.
W h a t arethese properties? W e have seen thatlim ψηΜ/ψηΜ=q,nand require accordingly, /(n, n) denoting the factor in question, that/(1)lim ψ (τ)/(η,τι)η=\ n/2lim ( - )i.e., t h a t / ( n , τι) should behave like τ ι ~η / 2=as qmq,n—> oo. A second requirement is that(2)φ =g [Σ^ » ( r ) / ( n , τι)β- *] ,< ηwhich replaces E q . (20.14), should have the same circle of convergence asthe original f-series expansion (20.13) about f=0; and in fact shouldexhibit for τ = τ\ , a behavior similar to that of (20.13) for q = 1.T o arrive at an appropriate normalizing factor / ( η , τι) we introduce in-V. INTEGRATION THEORY AND SHOCKS354stead of r, the new variable, λ, the same as that in (17.17), defined bydX/dg=Λ/1 ~ M /q]it leads t o a normal form of the second-order2equation for ^ ( g , 0). In terms of τ we have(sVdX)=dr;dX/dg=V l-dr/dgMqj2g-V l=2gΜ2=Jl_2r/l2 r f-r/nl - τ'W e chose the limits of the integral so that X = 0 for r = r* :(3')tanh"W e shall use the variable8=(4)λ +σ,where σ is a constant defined by the requirement that slog (q/q )mforsmall r, i.e.e*lim — = 1e2(5)τ-*·0orΤΤThis gives with κ = y = 1.4 and ft = r "2σ =- f t tanh" J -e~ .2a= 6,1J log (ft -12Xlim — =log 2 =1) +2-1.17.ΔftI t is seen that s —> — oo, as r —> 0, Μ —> 0.
A s r increases from 0 tos increases from — oo to σ =r,t—1.17.W i t h s (or λ ) as an independent variable the stream-function equation(20.5) is transformed to(6)^tdsW+2^^t =ΘΘT2^oVwhere Τ is a function of s (or r or g ) , namely,Τ =-dΡV l-M22/ V l-ρds \M\2/'If we apply to Eq.
(6) the separation of variables ψ = ^ (s)e \n(6' ,λ_^_γnwe obtain*an equation which will be used later.I n order to obtain an equation with a term containing φ rather than21.3FLOW AROUND A CIRCULARCYLINDER355d\p/ds as in E q . (6), the dependent variable is changed by putting(7)ν(τ)ψ*.φ =A n elementary computation gives»>+%+where, with a prime denoting differentiation with respect to s,2V'—(9)V"F = —= Τ ,2V'——2-' - 0 - 4γ - [ ^ Τ - ( - 5 Γ · ·«T h e equation corresponding to E q .
(6') is thenwd ψ2, *η,. *ΓΛds2.*which suggests that asymptotically for large | η |, ψη~e .n8This, togetherwith E q . ( 7 ) , makes somewhat plausible the following result (which we givewithout proof). T h e function ψ (τ)is asymptoticallyηequal to e V{r),nsasI η I —> oo, for subsonic r (negative integers η excluded); or, more precisely,(10)φ (τ)η=[• (0]+oV(r)enas I η I —> oo , uniformly for 0 ^ r ^ rt— e, for complex η and | η + m | ^ δfor all positive integers m (δ, € arbitrarily small positive numbers).W e can now see that for η <τ< a simple and appropriate normalizingfactor is(11)/(n, n ) = e "In fact, from the asymptotic equality of en s i28in E q .,where=β(η).3and r for small r, as expressed( 5 ) , we see that e~ behaves like n *-8β ιin the limitη—> 0, aswas intended. Also it follows from E q .
(10) that the general term ofthe series in E q . (2) will behave for large η like F ( r ) c enr =η , s =n ( s - S l - t 0 )and forSi , this is indeed a behavior similar to that of the generalterm in E q . (20.13) for q =1.3. Flow around a circular cylinderT o gain insight into the problem of flow around an obstacle, we considerhere what is probably the simplest profile, the circle. W e shall give the4* N o t e that VA= A " , where Κ was introduced in (17.24')·-1356V. I N T E G R A T I O N T H E O R Y A N DSHOCKSy4F I G . 138. Flow past circular cylinder.main line of procedure and refer the reader for details to the literature thatwill be quoted.*L e t the radius of the circle be equal to one, let, as before, q„ = q = 1,and the corresponding η be subsonic.
T h e complex potential w(z) of theincompressible flow is well known:x(12)/\w(z) =z,1+ ->.dw-χβ1f = - τ - = qe= 1 - - ,dzz12zζ = (1 -f)l= wo(t)w(z)= (1 -f)* +(1 -iT*.Consider the upper half of the z-plane. I t is seen (Fig. 138) that the #-axisfrom point 5 to oo and from - co to 1 is mapped onto the cut betweenf = 0 and f = 1; the image of the profile streamline 1 2 3 4 5 appears inthe hodograph as the circle of center q = 1, q = 0 through the origin,and a few more streamlines are sketched roughly indicating how the flowregion in the upper z-plane outside the obstacle is mapped onto the insideof the circle in the g ,g -plane.xxyyT h e expansion of w ( f ) must be made separately for | ξ \ < 1 and | f | > 1.T h e line | f | = 1 is an arc of circle in the hodograph plane, and its image11 — 1/z 1 = 1, or x — y = |, separates the flow region above the x-axis0222* T h e example of this section requires only some idea of analytic continuation andthe residue theorem.21.3FLOW AROUND A CIRCULAR C Y L I N D E Rinto three regions Ri,R,with corresponding images in the hodoRz,2357graph.1 we must take ζ =Expanding for | f | <+ (1 — f )— (1 — f ) ~ * in Rz (then for f = 0 we obtain ζ =Rz); for I Π1, ζ = i Γ * (1 ->F o r the regions Ri,w22in Ri and ζand 2 =x=— 1 in.β , ft we then obtain the expansions2Ml =(13)1 / f ) " * in β_ i1 in RΣ( »-1)Γ(η -I )^Σ(η +Ί)Γ(» -I)I (i)n=o»_oΓ(5)=ζη !n!If we consider also the lower part of the 2-plane, namely, y < 0, the regionsRi and Rz are continued symmetrically below the x-axis, and we denoteby Ri the region symmetric to R .T h e hodograph image of this lower2part covers a second time the same hodograph circle as in Fig.
138 and isnot shown. T h e formulas (13) remain correct and we have to add the formula(130=-u>2.One may verify that each of the four series is the analytic continuation ofits t w o neighbors. This expansion into series of w(z)= tA>(f) is the firststep.N e x t we seek a corresponding compressible flow and begin b y constructcorresponding to Wi and w , according to Eqs.ing the series W\ and W ,22(2) and ( 1 1 ) ,(14)WX=*Σ(-N1)F1 ν2/ = °(15)W2='r(J)ΣT(N"Η"^Ur)e-^ °\+i^'N(N"+),Γ(h )t U r ) e(n- ^+m.nln=oW e shall see immediately that W2is not the analytic continuation of Wiand that it can never be so, no matter h o w / ( n , n ) might have been chosen.T o find the continuation of Wi for r >ri is a mathematical problem whichcan be approached in various ways. A comparatively simple solution (seeN o t e 4 ) is based on a generalization of the representation of the hypergeometric function F(a, b, c; x) b y a "Barnes Integral".T o explain this idea we first return for a moment to the incompressibleflow problem.
Denote by Β the integrale- ffa C 'S!(1)r<5'-* -' - '*)r<)(f)358V. 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 SlcF I G . 139. Integration paths used in connection with Barnes integral.along the path C in the complex y-plane indicated in Fig. 139. W e thenapply the residue theorem to the above integral taken around an appropriate closed circuit (/) to the right of the imaginary axis, such as the oneindicated in the figure. W e use the fact thatν =Γ(*>) has simple poles at0, 1, 2, · · · with residues ( — l) /n\.—n, ft =nThen the poles of(ν — 1)Γ(—J >), which are at ft = 0, 2, 3, · · · are inside (/) and those ofwhich are at J — η, η = 0, 1, 2, · · · are outside (/).
I t can be—shown that, as | ν \ —> <χ>, q <1, the integral around the semicircle to theright converges to zero and therefore —B equals the limit of the sum of theresidues at the poles which are inside (/), or(13"}=Bf k ) £in-1)Ti-nh)% 'Ifl<1-T h e series to the right in E q .
( 1 3 " ) is identical with the expansion (13) ofWi . In this way W\ is now expressed by the integral B, and we can use thisrepresentation to find the continuation of w for | ξ \ >x1. W e shall nowexplain this main point directly for the compressible flow problem.If, with fψη(τ)β~ ,η81qe~ \ we replace, according to Eqs. ( 2 ) and ( 1 1 ) , q=l^(16)nbywe are led to consider, instead of B, the integral100Λ +Β' = ^-±φ(„ -1)Γ(ν-£)r(-„)(-l)V,(r)e-' "(+ W )dvalong the same path C as in Fig.
139. W e now apply the residue theorem tothe integrand in (16) and the circuit (/). W e find that for r < r i , (corresponding to q < 1) — B equals the limit of the sum of the residues of thef21.3FLOW AROUND A CIRCULARCYLINDER350integrand at ν = 0 , 2 , 3 , · · · and that this sum is exactly the series W\ . T ofind then the analytic continuation we integrate around the closed circuit( i 7 ) (indicated in the figure) to the left, where for | ^ ) —> oo and τ >τ ι , theintegral around the left semicircle tends to zero.
Inside this circuit are nownot only the poles of T(v — \), but also those of ψΛτ),of ν for fixed τ (0 <r <considered as a function1). T h e former are at \ — η (η= 0, 1, 2, · · · )and the limit of the corresponding sum of the residues equals W2negative and — Wif θ is positive.* T h e (simple) poles of φ (τ)29(η = 2, 3, · · ·)> with residues p=nwhere C—ηΟ ψ (τ)ηηndepends on theconstants a , b , introduced in E q .
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