Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 102
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Real-gas solutions for air obtained in this mannerare given in Ref. 17.Nonuniform Surface Temperature. Transformations (Eq. 6.76) are applicable to flowswith nonuniform surface temperature provided a linear viscosity law is assumed (gp = constant). The fiat-plate results given previously for constant Prr may be applied to a cone withthe requirement that the surface boundary conditions be the same in terms of ~. For a flatplate, ~ - x, and for a cone, ~ - x 3.
Therefore, the flat-plate results must be modified in such away that lengths in the x direction are replaced by x 3 to obtain the cone results. For example,Eq. 6.66, which expresses the effect of a stepwise surface temperature for a fiat plate, becomesfor a conelst( ,s)St (x, 0) lcone= [ 1 -(xS_)9/41-1/3(6.82)Similar expressions may be derived for the effect of an arbitrary heat flux distribution.Mass Transfer, Uniform Surface Temperature.
The uniform-surface-temperature resultsabove may be extended to include mass transfer. Similarity requires that f(0) be constant,6.28CHAPTER SIXwhich determines the blowing distribution along the surface. The normal velocity componentfrom Eq. 6.99 isVw / ( 2 k + 1)~twUe----~/ 2pwUeX f(O)(6.83)For a cone, Vw X-1/2as for a flat plate, but is larger by the factor ~/3 for given values of x andf(0) (thus, the blowing parameter PwVw/peUeSt0 remains unchanged). With this difference, thefiat-plate results may be applied to a cone according to the equations above.For a nonsimilar blowing distribution, for example, Vw= constant, Eq. 6.83 is not applicable.
Solutions to this problem may be found in Ref. 40.-Surface With Streamwise Pressure GradientGas With Uniform Elemental Composition in Chemical Equilibrium. Except for a fewconfigurations such as a fiat plate and wedges or cones in supersonic flow, the pressure variesover the body surface as determined by inviscid flow theory. The influence of pressure gradients on forced convection in laminar boundary layers is presented in this section. Axisymmetric bodies at zero angle of attack and yawed cylinders of infinite length will be treatedtogether to illustrate their close relationship (see Fig. 6.18).
Because boundary layer theoryrequires negligible pressure gradients across the boundary layer, the techniques presentedhere apply only to those bodies whose local surface radius of curvature (1, in Fig. 6.18) is largecompared with the boundary layer thickness, thereby minimizing centrifugal forces.Governing Differential Equations. In the absence of foreign gas injection, the boundarylayer can be considered to have uniform elemental composition and be in chemical equilibrium, and is governed by the following equations:3 (purk) +axOu -~x + ov ay -a7ydx +-~ypu -~x + pv igy - igy l-tpu -~x + pv ~ y = -ffy-y(pvr k) = 0(6.84)~t(6.85)(for k = 0 only)ffy-y+ g 1 --~y(6.86)2where k = 1 for the axisymmetric body and k = 0 for the yawed cylinder.VUXVooUoo(a) k= 1i/(b)k:OFIGURE 6.18 Sketch of coordinates employed for related two-dimensional flows.
(a) axisymmetric body; (b) yawed cylinder of infinite length.(6.87)FORCED CONVECTION, EXTERNAL FLOWS6.29The boundary conditions arex = O, y > 0u = Ue(O)I=IeforW=We=V=sinAx > O, y ---) ook=0u ---) Ue(X)I~IeforW~We=V=sinAy=0>k=0(6.88)u=w=0v = 0 for an impervious surface or vw(x) with transpirationI=iw(X)forq"¢0bIby-0orforI=iaw(X)The yaw angle A is the complement of the angle between the free-stream direction and thecylinder axis (see Fig. 6.18).Transformation o f Variables and Equations. The extensions of transformations Eq. 6.36to include the effects of pressure gradients are1 fox{ r ~ 2k; = ~t--~r pw~twUe~-~}Ue(F/L)k foxTi = ~trV/2(;_~ )pdx = ;(x)(6.89)dywhere Br is a reference viscosity and_ L is a reference length introduced to make ~ and 11dimensionless [16, 41].
The function ~(~) is yet to be determined and is a key element in theextension of similarity solutions to flows where the inviscid boundary conditions do not permit boundary layer similarity. (Note the change in symbols employed here from those of Refs.16 and 41.)The dependent variables are defined asufn - Ue(X)-II - lew-~ -- We(for k = 0 only)(6.90)Additional parameters that enter the equations areit = -7- = I - (1 - ts)~ 2 - (ts - te)f~(6.91)le2Wewherets = 1 - ~andte = 1 -21e2U 2 d- W e2/e(6.92)(6.93)The pressure gradient parameter is defined as2 ( ~ - ~) t, due~p=-Uete d~Both te and I]e are functions of x through their dependence on Ue(X).(6.94)6.30CHAPTER SiXThe transformed partial differential equations of momentum and energy are+ 2(~ (Cw~)~ + a - - ~f~= 2(~ - ~ ) ( f ~ -~)(fnfq;-f;fnq)f~)(6.95)(6.96)-- - )fln--{Cw(k-1)[(ts-te)(f2)rldt'(l-ts)(-W2)Tl]lr I+ 2(~--~)(f,a[;-f;La)(6.97)The boundary conditions for zero mass transfer arerl=0f(¢.
o)= y,(;. o)= ~(~. o)= o[(~, O) = iw(~) = i~(~)andrl~oo--In(~,0)=0for q : , 0(6.98)ppforqw =0fn(~, oo) = ~(~, oo) = [(~, oo) ~ 1With mass transfer, f(~, 0) depends onvw(x) as follows:Ue(X ) ---- ~l,r \ L ]Lv~(6.99)+Similar SolutionsSimilarity Criteria and Reduced Equations.The partial differential equations (Eqs. 6.956.97) are not amenable to solution except by numerical methods utilizing high-speed computers. Considerable simplifications can result, as in the case of the flat plate, if these equations are reduced to ordinary differential equations through the similarity concept where thedependent variables f, ~, and [ are assumed functions of rl alone.
Equations 6.95-6.97become, for ~ = constant(c~f,,),+Tf,,=~ '~-~+ ~,-~-~(6.100)(CwW)' +fW= 0(-~rrf')'+ff '(6.101)l)[(t,-t~)(f'2)'+(l-t,)(-~2) "]{ C ~ ( ~ rr---_(6.102)IpConsistent with the similarity assumption, none of the terms that appear in these equations or in_the related boundary conditions can be dependent on x or ~; that is, ~p, t~, and Iw,as well as ~, must be constant.
The parameter t~, defined by Eq. 6.93, however, violates thisrequirement when Uevaries with x. The similarity assumption is also violated by the terms thatcontain t~ explicitly in Eqs. 6.100 and 6.102 and by the gas properties Cw, Prr, pe/p, which canbe expressed in terms of t, which in turn depends on t~ through Eq. 6.91. Consequently, exactsimilar solutions are not possible under general stagnation region flow conditions.Exact similar solutions are possible for stagnation regions where Ue = 0 and t~ is a constantand equal to unity for an axisymmetric body and to t, for a yawed cylinder. The terms involving t~ drop out of Eqs. 6.100-6.102, and similarity occurs for constant iw and ~p.FORCED CONVECTION, EXTERNAL FLOWS6.31For similar flows, the pressure gradient parameter expressed as follows must be constant:{ r ~2k2 -isfox/~ P - Ue te dxr ~2k(6.103)pw~l,wtle~---~)In a stagnation region, the fluid properties are nearly constant, and Ue- X; also r --- x.
Thus,13p= 1A for an axisymmetric body, and 13p= 1 for a yawed cylinder.The skin friction coefficient and Stanton number are defined under the conditions of similarity on axisymmetric and two-dimensional bodies as follows. The components of the shearstress in the xi direction are given bygwPwtsUedue () ui"r'wi=dx ~~pte(6.104)wwhere the subscript i = 1 or 3 represents the x or z direction, respectively. The skin frictioncoefficient is defined asCfn mT,vn2Cfl~pwUeX_with2(6.105)pwUeUie~tsxdue~1.w(6.106)~ptel, le dx f~vtCf3 /pwUeX ~ ts X dUe-~P2 v~tw(6..107)~deUe dxAlternative forms of these equations that are sometimes more convenient areCf1 .t / pwUeXeff2 Vktw1- V ~ f'wCf34/ pwUeXeff2 Vgw(6.108)1- ~/2 ~'w(6.109)F ~2kfoXpw~l.wUe~)wherexeff=dx{ r ~2k(6.110)pw~l,wUe~-~)The corresponding Stanton number expressions for a surface at constant temperature arest~pwUeX ~ -is x due 1~l'wSt--pwUeXeff~tw["~pte Ue dx Prow f ~ - I o w(6.111)w11I"- V ~ Pr~w L - [ ~ w(6.112)Uniform Surface Temperature, Ideal Gas With Viscosity-Density Product and PrandtlNumber Equal to Unity.
For the case of an ideal gas with Cw = Pr = 1, similarity is possibleaway from the stagnation region of a body. Equations 6.100 and 6.102 for an axisymmetricbody or a cylinder normal to the free stream reduce to6.32CHAPTER SIXwhenUef,,, +ff,,= ~jp(f,2_ I )(6.113)[" + fl'= 0(6.114)satisfiesUe~e(6.115)- A~f~d2These equations are equivalent to those solved in Ref. 42 for a uniform surface temperature.Examples of the extensive solutions in Refs.
16 and 42 are presented in this section.Figure 6.19 shows the velocity distribution in the boundary layer of an axisymmetric bodyor an unswept cylinder for a cold wall (Iw = 0). (Note that the value of 11 used here is a factorof V ~ smaller than the one employed in sections on the flat plate.) An accelerating freestream (13p > 0, Ue increasing) reduces the flow boundary layer thickness and increases thevelocity gradient rather uniformly through the boundary layer. A decelerating free stream(13p< 0) thickens the flow boundary layer rather severely and causes the velocity distributionto acquire an S shape. Eventually, the boundary layer will separate; that is, f g = 0.
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