R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 30
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T h e Eqs. (11.12) for steadyadiabatic one-dimensional flow reduce to a pair of relations between u andT, not involving x. T h e only possible solutions are therefore of the formu = constant and Τ = constant, from which follow also ρ = constant andρ = constant. I n other words, the only steady, strictly adiabatic onediniensional flow of an ideal fluid is a uniform flow with constant values ofu, ρ, ρ, T, etc. Thus, only nonsteady flow is of interest in this case.Returning to Eqs.
(11.2) and (11.3), and omitting the viscosity term,we obtain013mdudp*(2)dp+3P-o.dtdxdxT h e specifying equation which expresses the fact that the motion is adiabatic may be taken from Sec. 1.5. In the case of an ideal fluid it reads:p/p= constant for each particle, and this last qualification may beomitted if initially all particles have the same entropy. T h e mathematicalproblem remains unchanged if y is replaced, by any other constant greaterthan one, so we shall, in general, use as specifying equation the polytropicrelationy(3)— = constant = C,PKwhere κ is any constant greater than 1.If (3) holds, or more generally an arbitrary (p,p)-relationwe can introduce the square of the sound velocitya=with dp/dp > 0,—dpas a known function of ρ (see Sec.
5.2) and use this function to eliminateρ from ( 2 ) . Since dp/dx equals dp/dp times dp/dx, Eq. (2) takes the form(4)156I I I . ONE-DIMENSIONAL FLOWEquations (1) and (4) may be slightly rearranged to yielddxat(5)^ (pu) + u^- (pu) dtdxu ^ +dt(a-^ = 0,dxu)a system of two equations for the unknowns pu and p.Another convenient form of the equations may be obtained by using thepressure head P , defined in Sec. 2.5:(6)ρ — [ dp _fJJρ2adpdP _ a dpρ 'dxdP _ a dp2ρ dx'dtρ dtThen it is easily verified, by carrying out the indicated differentiations, that(4) and (1) are equivalent, respectively, todx\2(7)^^ dt'Alternatively, the first equations of (5) and (7) form a system equivalentto (1) and ( 2 ) .
I t may be noted that (5) and (7) are very similar: the variables pu and ρ in (5) parallel the variables (u /2 + P) and u in ( 7 ) .2In the particular case of the polytropic condition ( 3 ) , we have(8)a = kCP ~\2Ρ =kCp"'1="κ -1W i t h the velocities u and a as variables in place of u and p, the first equations of (5) and (7) become— \uadx(9)) = U,) + — (adtd (uadx\2\1/κ -+£=»·For adiabatic flow of a diatomic perfect gas (as assumed for air), we haveκ = y = 1.4 = 7/5, and thereforefdX<9>d( U(ua )s5+ % ( a ) = 0,5dt2\, du.12.2POTENTIAL AND PARTICLE FUNCTION157For a monatomic gas, κ = 5/3, we have 3 instead of 5 in the above equations.T h e t w o equations (9) are homogeneous, planar (but not linear) differential equations for the unknowns u and a with independent variables χand t, namely,κ — 1du2dx—τζ— a(10),Θα , dah u —+dx*>•da ,du— + u dx1 a dxκ -.—=0,dt, duh dt— =0.Of course, Eq.
(10) could have been derived directly from (1) and (4) byintroducing the variable a in place of p, using ( 8 ) .2. Potential and particle functionEither one of the t w o parallel sets of equations (5) and (7) can be reduced to a single differential equation of second order. Indeed, the firstE q .
(5) expresses the fact that ρ and — pu are, respectively, the x- and^-derivatives of one and the same functionψ(χ,ϊ):(11)- 7 7Ρ =ΤΓ"Ρ^ =,dx·dtT h e second Eq. (5) then supplies the following condition on ψ:<>Sn/+dt2^4- +(« -« )^=0.2/2dx dt2dx2In the same way, the system (7) is satisfied if a function Φ(χ,ί)is introduced for whichcm«=;*and which satisfies(12 )idx—dtF2+ 2u — — +dx dt+p-2-%dt(w-α) —dx= 0.2Equations ( I F ) and (12') appear to be the same; the coefficients 2u andu— a , however, do not bear the same relation to the unknown function,2Φ or ψ, in the two cases.T h e function Φ(χ,<) is none other than the potential introduced in Sec.7.1. Thus, one-dimensional isentropic flow of an ideal fluid is always a potential flow and irrotational.
Equation (12 ) is exactly the same as Eq.r(7.24), discussed in Sec. 7.4. Each function Φ(χ,ί)satisfying (12') deter-158III. O N E - D I M E N S I O N A LFLOWmines a particular one-dimensional flow of an ideal fluid and, conversely,for each continuous one-dimensional flow pattern there exists a potentialfunction satisfying (12').T h e function ψ(χ,(), which is known as the particle function, may be interpreted in the following way.
T h e rectilinear motion of any material particle is completely determined when its position χ is given as a function oft. T h e curve in the x,2-plane representing this function for any particle hasbeen called a particle line (Sec. 1.2).
T h e slope of the particle line, measured by the tangent of its angle with the <-axis, is dx/dt or the instantaneous velocity u of the particle. N o w , along the lines ψ = constant we have,using (11),(13)0 = ^ dx + ^ dt = ρ dx — pu dt,dxdt^ = u.dtThus the family of curves φ = constant (Fig. 57) consists of the particlelines for all elements of the fluid mass under consideration. T h e differencebetween the ψ-values on two distinct particle lines may be found by integrating the first E q . (11) along any parallel to the x-axis:(14)Φ Β - * Λ =fJA3T *dΓ=pdx,JAOXand is therefore equal to the quantity of mass enclosed in the flow by acylinder of unit cross section extending between the two planes χ = χ , andΛX—XB·Both (11') and (12') are second-order differential equations of the type(9.2), withA = 1,Β = u,C = u-a,F = 0,2where the present independent variables χ and t are to be identified withthe former y and x, respectively.
From (9.13') [see end of Sec. 9.5] theslopes of the characteristics are given by( 1 5 )^= j!( £ ± VB*-AC)=u±a.ore ton uF I G . 57. Particle lines ψ = constant.12.2159POTENTIAL A N D PARTICLE FUNCTIONF I G . 58. Orientation of the characteristics with respect to the £-axis for velocityu > 0.(a) u supersonic,( b ) u subsonic.Since both u and a depend on derivatives of the solution, Φ or ψ, theseslopes vary with the solution being considered, in agreement with the factthat the equations (11') and (12') are nonlinear. T h e slopes are real in allcases: whether the flow is supersonic or subsonic, the problem is hyperbolic.
I nthe first case u + a and u — a have the same sign, so that the characteristicdirections lie on the same side of the vertical in the £,£-plane (Fig. 58a, ifu > 0 ) , while in subsonic flow the characteristic directions fall on oppositesides of the vertical line (Fig. 58b, if u > 0 ) .T h e same conclusions can be obtained b y applying to (5) or (7) the general theory of characteristics (Sec. 9.2) or the discussion of the two-dimensional case ( A r t . 10).Some examples of one-dimensional flow were given in Sec.
7.4. Thesewere all such that Φ was a quadratic, and u a linear function in χ withcoefficients depending on t. W e arrive at essentially the same solutions ifwe start from the assumption(16)φ(χ,1) = (ax +β)\where a and β depend only on t, and η is constant. From* (11) we have then(ΛΚ'\(lb )W(ρ = — = ηα(αχj _+ β),u==dxθψ /3ψ- ^ 7 / ^ Z=~dt / dxαχ + β'>aand, with the polytropic relation ( 3 ) ,(16")a = KC ~kp1= κΟ(ηα) -\αχκ+β)^- ^.1When these expressions are substituted into (11') or ( 4 ) , the equation re-160III. ONE-DIMENSIONALFLOWduces t o(2α'2- aa)x+ {2α'β' - οβ") + κ(η -+ β*»-™"-»-\)a Cn -\axK+2K1= 0.This equation can be satisfied identically in χ only if all coefficients vanish,or if the equation is independent of x, or if the last term is a suitable linearfunction of x.
T h e three values of η making these results possible, and thecorresponding conditions on the coefficients, are:(a) η = 1;(i7)(b)()Cn2a;=-2η = i ± ] ;2= 0,aa-a'2«' -2a"a°>=™22<*'0' -=Ka = constant · Γ2 / (+ 1 ),K^K-+=2;Κ α β,2κ+2+ 1)* (κ - 1 ) " .-12*=*«'β'where K = -C[K/(K- 1)] ,K= -2CK(KA particular solution of (17c) isxαβ" = 0;2a β' -β0 = constant • « "u1 ) / u + 1 ),which leads towhere A is a simple function of C and κ, and c is arbitrary. This is the particle function* corresponding to the example ( a ) , Eqs. (7.28), of Sec. 7.4with c = 0, except for different meaning of the constant A. I n the sameway a simple solution of (17b) supplies2/φ(χ,ί)= Β (jV/U-i)Ύ+t-Cll K+cJ2where Β and c are arbitrary and Ci is a simple function of B, C, and κ.For c = 0 this is the particle function for example ( b ) of the same section[see Eqs.
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