R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 27
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Thus the specifying condition isEquations (2) through (5) are four equations for the four unknowns u,p, p, and T, provided that σ is expressed in terms of these variables. T h eusual assumption, made in the Navier-Stokes theory, is that σ is proportional to the rate of expansion du/dx, as mentioned at the end of Sec. 3.1:1χχ\v)σχ= Mo — ·dxHere μο is a function, presumed to be known, of some or all of the variablesu, p, p, and T. T h e valueΜ = f Mo,however, rather than μ itself, is usually called the (physical) viscosity, orcoefficient of viscosity, for the following reason.011.2EQUATIONS FOR S T E A D Y137MOTIONUnder the assumption of one-dimensional flow, the y- and z- directionsare interchangeable. Then a' and σ' must be equal, and the condition (3.7),yσ'χ + <r' + σ'ζ=ζ0, givesy(7)—2xaN o w consider an element of quadratic cross section dx = dy.
Equilibriumconsiderations of the half-cell (Fig. 50) with cross section ABCrequire·σ 'Μ-•ΧF I G . 50. T h e viscous stresses on a rectangular fluid element in one-dimensional flow.that on the face represented by i Cdirection AC of magnitude(<r' dx dz) +-dy dz) = y |(σ'va shearing force must result in theχ(σ* -σ)νdx dz,and since the area of this diagonal surface element is (dx y/2) dz, theshearing stress isT—2\ Χ ~σν)σ— ϊΧ 'σOn the other hand, the rate of shear η, i.e., the time rate of change of theangle CAD, where DA is perpendicular to A C in the x,y-p\&ne, is determined by the differences between the velocities at A, C, and D.
Point Cadvances against A by (du/dx) dx units of length per second, and D is leftbehind by the same amount. ThusV=AC\dxdxand, using (6), the ratio of shearing stress to rate of shear is |μ = μ.For its numerical value, see Sec. 5.02. Equations for steady motionT h e partial differential equations ( 2 ) , ( 3 ) , and ( 5 ) , in which the independent variables are χ and t, become ordinary differential equations in χwhen a state of steady motion, d/dt = 0, is considered.
In the Euler rule138III. ONE-DIMENSIONAL FLOWof differentiation, (1.4), d/dt reduces to u{d/dx)and the three differentialequations are nowd (\duη. d(p — σ' )χΛ"s(i+,4i')t>-"i-«-*here Κ represents the heatfluxk(dT/dx).T h e first equation shows that the mass flux(9)m = pu,or rate of flow of mass across a unit cross section normal to the .τ-axis is aconstant. Using this, the second equation gives(10)= constant =mu + ρ — σχdm,say, and the third equation yieldsm (f+ ^ΖΓΪΤ)+<V°z) ~ Κ = constant =-say.
From (10) we have ρ — σ' = m(dmay be written— u), so that the last equationχ(11)m("I+ γΖΓχT)"dm,K=m(C2"C i u )'Finally, ρ may be replaced in (10) by use of the equation of state ( 4 ) :=ρgRpT=mgRT/u.When Eqs. (10) and ( 1 1 ) are solved for σ'χ=μ du/dx and Κ = k dT/dx respectively, there results0Mo du.Τ- — = u + gRm dxuCi,m dx2Dn(12)7 — 1These are t w o simultaneous ordinary differential equations for u and T.T h e solutions of this system, depending upon the constants Ci, C , and malready introduced, and upon t w o additional constants of integration,represent all possible patterns of one-dimensional steady flow of a perfectgas with viscosity and heat conduction.22I t is convenient to replace u and Τ by dimensionless variables ν and Θ;since d has the dimensions of velocity [see Eq. (10)] and C the dimensionsof velocity squared, we introduce2. _C2„ _u2Λ_ gRTρ11.3STEADYFLOW W I T H O U THP]AT139CONDUCTIONThen the system (12) takes the form= 2 r - V2v+Qm dx(14)1 k dO-β--τ-Ο77=-« +—c.V 2y -gRmdxy — 1T h e solutions v and θ of this system will depend on four arbitrary constants: ra, c, and two constants of integration.
Of these, m occurs in theequations only as a factor of dx and therefore in the general solution onlyas a factor of x, and one constant of integration can be absorbed by translating the origin χ = 0 (since Eqs. (14) are unchanged in form if, insteadof χ, x' = χ + C is used as the independent variable). Thus, except forsimilarity transformations: x" = mx + C", the solution depends on onlytwo parameters.Before discussing the general equations (14), we turn to a special case.3. Steady flow without heat conductionG.
I . Taylor has shown that the system (14) can be integrated in closedform when k is set equal to zero.* This is not a realistic assumption, since itis known that the ratio μο/k varies over a small finite range (see Sec. 5 ) .I t will be seen later, however, that some principle features of the flow canbe found in the solution of (14) under the assumption k = 0.Eliminating θ from Eqs. (14), with k = 0, we find3(15)=(y+l)„-yV 2 U + C (7-1).m axExcept for the scale factor m and translation of x, the solution of thisequation depends on only one parameter, c.Suppose that c lies within the limits° 2W^T)=%(16)<C<(Ύ =1· ·4)Then there exist two real positive values V\ and v (v\ > v , say) for whichthe right-hand member of (15) vanishes:22(7 + l)v - yV2v + (7 - l)c = 0,(17)and it can be written as(170(7 +yV2~v +l)v -c(y -1) = (7 + 1 ) ( V £ -VviKVv-Vv ).For constant μ and for ν between v\ and v , the solution of (15) for χ as02* Equation (3.23) shows that this amounts to strictly adiabaticflow.2140III.
O N E - D I M E N S I O N A LFLOWΥ (PROPORTIONOL TOVV)TX"TU8F I G . 5 1 . Variation of velocity u (or y/υ) with position x, for k = 0.a function of ν takes the form(18)27 +Vv~iμlm0log (ΛΛΙ λ / ί )—log (Λ/ν —+\V )/2constant.H e r e * χ decreases from + <χ> to — co as ν increases from v tovi . Forz; > v\the argument of the first logarithm in (18) must be changed in sign, andthen χ increases from — °o to + <*> as ν increases from v\ to + <*>. For0 < ν < v the argument of the second logarithm must be changed, andthen χ increases from a finite value to +as ν increases from 0 to v .
Ifwe now restrict our attention to flows for which the state variables tendto finite limits as χ —> ± oo then these last two branches of the solutionmay be neglected and attention focused on the function v(x) defined by(18). Figure 51 shows V ^ a s a function of x\ since yj ν is proportional tothe velocity u, we also obtain the graph of u as a function of χ merelyby taking a different scale on the vertical axis.
Since pu = constant, \/vis also inversely proportional to p. T w o considerations are of major interestto us: the relation between the initial and final values of u , and the steepness of the descent from Ui to u .22002y2Since Λ ΑΙ and s/vin \/Ί), we have2are roots of (17), considered as a quadratic equation(19)7 + 1or, in terms of u, using (13),(190Ui + u227 + 1which gives an interpretation of the constant Ci . When k = 0, the second* For the remainder of this article it is assumed that m is positive, i.e., χ is chosento increase along the direction of the flow.11.3STEADY FLOW WITHOUT HEATCONDUCTION141of Eqs.
(14) shows that ν and θ satisfy(20)—^—7 - 1V2v -ν +-c = 0.Eliminating c from (20) and ( 1 7 ) , which is valid however only for ν = Vior v , and then eliminating \/2v from the same equations, we find2(21)Oi + 2vi - Λ/2^ = 0and— 2 — O + v> = c7 - 1{i = 1, 2 ) ,tso thatOly,(210+V2Λ1Vl77 — 1vV2th^θι +=027 7 = +V2v/V2^2,2νχ = — ^ — 07 — 1+2v .2When ν and θ are replaced by their values from ( 1 3 ) , ν =0 = p/Cip,the first equation multiplied by m = pu and d(22)pi + mill = pu/2Ciandgives+ mM ,22and the second, multiplied by C i , gives2(23)^+2^L_P}=^7 — 1 Pi+^L_PJ.27 - IP2Equation (23) is the same as the Bernoulli equation (see Eq.
(2.200 withgravity omitted) found in the case of a steady, strictly adiabatic, inviscidflow, and implies conservation of energy. Equation (22) may be interpretedas expressing the conservation of momentum. Finally, the continuity equation (9) yields(24)pitti =pu,22or conservation of mass. Equations ( 2 2 ) , ( 2 3 ) , and (24) also follow directlyfrom ( 9 ) , ( 1 0 ) , and (11) on setting σχ= Κ= 0 for χ = ± o o .In studying the transition from U\ to u , we use Eq.
( 1 8 ) , in which y/v,2y/vi,and χ/v2can be replaced by u, U\ , and u without further change,2except in the additive constant. L e t e be any number satisfying 0 < e < \(Fig. 51) satisfyingand consider the t w o intermediate values u' and u"(25)wi -Then also U\ — u"=u' = t(ui-u)2= u" -u(0 <2€ <J).(1 — €)(^i — u ), and the change from u' to u" is2(1 — 2e) times the total velocity change from Ui to u .2T h e difference be-142III.
ONE-DIMENSIONAL FLOWtween the abscissas χ corresponding to these values u is/ofi\Τ(26)L'"= χ0-χ2/1\ μ= — — - log ( - 7 +1\e0UI+U1)/ mUI—U22.For given values of the flux m and the ratio u /ui,2the right-hand sidetends to zero as μ decreases, no matter how small e may be. L e t p* denote0the value of the density at that point of flow where u = (ui + u ) /2 and u*2the velocity at the point where ρ = (pi + P2)/2; then the last factor in (26)may be written in two ways:^27)MOU\ +muiU2_μο P2 +Pi _m p2 — pi— u22μp*(ui_02μ— u)u*(p220— pi) 'If, for example, we take e = 0.05 and use the standard values (see Sec. 5)μο ~ 5 Χ 1 0 " slug/ft sec and ρ* ~ 0.0025 slug/ft , then (26) and (27) give73χ" — x' ~ 0.001/(ui — u ). Thus, if the total velocity drop UI — u amounts22to 10 ft/sec, then 90 per cent of this drop is effected within a distance of0.0001 ft, or about 0.03 mm. This is a significant result: the thickness of thelayer within which occurs the major part of the transition from U\, pi, pi toP2 tends toward zero with μ, and is actually extremely small in airu>2, p ,2under normalconditions.If we use a = yp/p as the expression for the sound velocity, as is usual2in discussing the adiabatic flow of an inviscid fluid, the M a c h number isgiven in terms of our dimensionless variables by(28)M = l = 4-=4r = ^227Θayp/pygRTUsing the first equation ( 2 1 ) , we obtain2(OQ)^y1==2»,yM?;1_V2^11'22Since v > v , Eqs.
(29) imply Μι > Μ .x=yM221_ 1V2v~2Moreover, vi and v satisfy ( 1 7 ) .2If the right-hand member of ( 1 5 ) , (7 + l)v — y\/2v + (7 — l ) c , is considered as a function of\/2vits derivative is (7 +l)\/2v —7, whichvanishes only for \/2v = 7 / ( 7 + 1 ) . Since the zero of the derivative mustlie between the zerosΛ/2νι and y/2v2of the function,\/2v~ < 7 / ( 7+2\/2v[ follows. If these inequalities are introduced into ( 2 9 ) , we find thatM2>1 and M22< 1. The transition flow represented by (18) begins in asupersonic and ends in a subsonic state. T h e inflection point of the curve ofν against χ occurs when \/2v = 7 / ( 7 +1 ) . This value of \/2Λ) wouldcorrespond to Μ = 1 if (29) held for all the flow; for υ between ν and v ,λ2however, the left-hand member of (17) is negative [see ( 1 7 ' ) ] and we findΜ> 1 at the inflection point.1) <11.4THE COMPLETE143PROBLEMA n interpretation of c may be found by combining Eqs.
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