boundary layer problem dependent on the downstream different powers, to show the difference with the heat flux conditions. This has been long recognized for the outward obtained when the pressure gradient is neglected. The ev bound flow below a horizontal heated surface, for which olution of small perturbations superposed on the stationary even approximate integral methods require a boundary flow is discussed, pointing out that upstream propagation condition at the edge of the surface, in addition to a regu is possible in the region of interest but the speed of the larity condition at the center. A variety of conditions at the upstream propagating waves decreases as the component of edge has been proposed. Wagner10 assumed that the buoyancy tangential to the solid surface begins to dominate boundary layer thickness is zero there, which is appropri the flow, or as the edge of the surface is approached. Spe ate only for the flow above a heated surface; Singh and cial attention is given to the flow under very flat surfaces, Birkebak11 adjusted the movable singularity appearing in for which the effect of the tangential buoyancy force the equations of their integral method to make it coincide changes from negligible to dominant within a very short with the edge; Clifton and Chapman12 used an adaptation distance. The analysis of this case demonstrates the tran of the idea of critical conditions from open channel hy sition from the flow below a curved body to that below a draulics. The structure of the boundary layer near the edge finite horizontal flat plate. has been analyzed elsewhere,13 showing that the boundary layer solution develops a singularity at the edge, where the II. ORDERS OF MAGNITUDE AND FORMULATION flow becomes critical, in the sense that upstream propaga tion of small perturbations ceases to be possible. Knowing Consider a two-dimensional body with a power law the nature of this singularity, an appropriate boundary shape in a region around its bottom, condition was devised to describe the flow upstream the edge. The same singularity was found independently by -aL (1) Daniels14 in the analysis of the horizontal boundary layers in a thermal cavity flow driven by lateral heating. where x is the distance along the surface from its lowest Pera and Gebhart15 and Jones16 analyzed the incipient point, zs is the height above this point, L is a characteristic effect of the tangential component of buoyancy on the pres length of the body, and «>l/2. The parameter a could be sure gradient-dominated boundary layer near the edge of absorbed in the definition of L, but it is kept for conve an upward facing heated semi-infinite plate slightly in nience. The surface of the body is maintained at a constant clined to the horizontal, and Ackroyd17 extended this anal temperature Tw above the temperature Tv of the sur ysis to include variable flow properties. Jones16 integrated rounding fluid. Assuming that the density of this fluid is numerically the boundary layer equations farther away P = Pa>[l—P(T—Ta>)], the boundary layer equations de scribing its motion at large Grashof numbers are, with the from the edge for both upward and downward inclined Boussinesq approximation, plates. In the first case, he described the transition from the pressure gradient dominated to the buoyancy dominated du dv (2) regime, while, in the second case, he found a regular sep dx dy ' aration at a certain distance from the edge, and was able to continue the numerical integration through the region of du du dp'/P. dz< d2u (3) reverse flow. Irrespective of the physical meaning to be dx dy dx +geTx+vW assigned to the reverse flow, the possibility of continuing the marching integration points out that some self- 3p'/Po = ±g0, (4) adjusting mechanism is at work, which would not occur in dy a parabolic boundary layer. Jones attributed this feature to de de v d2e the fact that the pressure gradient is not given in advance, (5) being determined by the flow itself. Thus the "elliptic" dx dy Pr dy character of the problem underlies this solution, even if it where y is the distance normal to the surface, u and v are can be obtained by a marching technique. the components of the velocity along and normal to the In this paper we analyze the boundary layer flows surface, g=g/3(Tw-TJ, 0={T-TJ/{TW-TJ, above and below the surface of a heated bowl-shaped body Pr is the Prandtl number, and a dzjdx in the region around the lowest point of the surface where =2na\x/L\¿n 'sgn(x)
0.5 0.7 1 1.5 2.5
-g'(O) (Pr=0.7) 0.3552 ' 0.3616 0.3722 0.3862 0.4007 0.4145 0.4269 0.4674 -g'(0) (Pr=7) 0.7456 0.7657 0.7928 0.8341 0.8691 0.9012 0.9304 1.0236 /"(0) (Pr=0.7) 0.9608 0.9135 0.8602 0.7963 0.7514 0.7169 0.6884 0.6140 /"(0) (Pr=7) 0.6375 0.6137 0.5853 0.5484 0.5215 0.4998 0.4820 0.4332
nitude of these two terms of (3), which yields l)/2 m2n l) 1/5(Zn x) x/;=x\x\^- f(V), e=g(V), xc/L={2na)- - Gr- - , or, in terms of the Grashof number based on the characteristic size of the with body (Gr^irLVvVl), y ¡TT= HJ71: (13) i v- (5/2)/(5«-l)Q l/2(5n-l) (6) (2«a) r where I¡J is the streamfunction {u = tl'y, V=—I/JX, and ^=0 at y=0). Carrying these expressions into (7)-(12) with The effects of variable flow properties are left out of the dp/dx left out, we find present formulation. These effects have been investigated by Ackroyd17 for the boundary layer above a horizontal or l+« , /' (14a) slightly inclined heated plate, and the conditions of appli '+-^ff"-nf'2+g=0, cability of the Boussinesq approximation are the same for 1+» this flow and for the problem at hand. Variable flow prop g"+^-Prfg'=0, (14b) erties effects are surely important when accurate quantita tive results are sought for in many real-world problems /=/'=*-1=0 at V=0, (14c) involving nonsmall relative temperature differences. We do f'=g=0 for Tj-oo, (14d) not believe, however, that they would introduce qualitative changes in the steady flows discussed in this paper, and where the primes mean derivatives with respect to 77. The therefore, mainly for the sake of brevity and definiteness, solutions of (14) for different n and Pr coincide with the we confine ourselves to the framework of Eqs. (2)-(5). family of exact self-similar solutions found by Braun et al.% From this point onward the variables are referred to for special body shapes (in the present context such solu the previous characteristic values (with the subscript c). tions apply for x> 1 only). The corresponding heat flux and Denoting the nondimensional variables with the same sym wall shear stress are given in Table I for reference. In what bols used before for their dimensional counterparts, and follows we shall look for solutions of (7)-(12) symmetric omitting the prime in the reduced pressure, the nondimen in x and with the asymptotic behavior (13). sional boundary layer equations become An interesting variation of this problem occurs for shallow bowl-shaped bodies, for which the region x=0( 1) du dv may cover the whole body surface and the ends of the body (7) dx+dy=0' are encountered before the flow takes on the asymptotic form (13). We shall consider the extreme case of a finite I In 2 du du dp d u surface with edges at | x \ =xe= L/xc and with (1) holding (8) u^-+v-^-——^-+e — -+ for J JC I
III. RESULTS Equations (7)-(13) have been integrated numerically for different values of n and Pr. A transient method was used for the flow below the surface, whereby time deriva tives of u and 9 are added to the left-hand sides of (8) and (c) (10) and the time evolution of the flow from a given initial condition (« = 0=0) is followed until it becomes station ary. For effectively infinite bodies, the computational do main extends to \x\
0.4
0.2 ¡ . / ._ „ , /_ ., .i;.. _
-0.2
' • n = 0.5 X -0.4 4 5
FIG. 4. Accumulated excess of the actual heat flux above that of the FIG. 2. Heat flux at the upper side of a solid surface as a function of the self-similar solution (13) as a function of the distance from the lowest distance from the lowest point of the surface for several values of n and point of a solid surface. Solid lines correspond to the flow below a surface Pr=0.7. The dashed lines represent the asymptotic solution (13), with and dashed lines to the flow above a surface. —g'(0) taken from Table I. The common asymptotic behavior for x-»0 is that of Stewartson's solution.
e.g., Refs. 2 and 3). The accumulated difference between in a way that does not depend on n because the tangential the actual heat flux and the one obtained when the solution buoyancy force is neglected in Stewartson's solution. On (13) is used down to x=0 is the contrary, the actual heat flux below the surface (Fig. 3) is finite at x=0 in all the cases (and has a peak there for * 39 l+ )/2 dx+—- g'(0)^ " |, l/2 (ub~cY[~\ -P=0, (15a) whereas the velocity perturbation normal to the boundary FIG. 3. Heat flux at the lower side of a solid surface as a function of the distance from the lowest point. The dashed lines are the same as in Fig. 2. layer is V = ik{ub — c)P'/9'h. Here the primes mean y The dotted lines represent the Richardson number at the wall. derivatives and the subscript b denotes the unperturbed flow at the location O) where the perturbation is applied. Equation (15a) is to be solved with the boundary condi tions P'=0 at y=0 and P—0 forj>^oo. (15b) (The first of these expresses the inviscid condition V=0 at the solid surface.) Equations (15a) and (15b) will have a nontrivial solution only for special values of c, and up stream perturbation propagation can occur if an eigenvalue c<0 exists for x>0 (the corresponding eigenfunction would then be regular). A term — (\x\ 2n/ikx)P' has been omitted in (15a); since £>1, such a term would matter only for very large values of \x\. Leaving it out, the de pendence of the eigenvalue problem on x occurs through the functions ub and 9'b only. For a horizontal plate, it has been found13 that upstream propagation is possible every where except in the immediate vicinity of the edge, where FIG. 5. Heat fluxan d shear stress for the inbound flowabov e a solid with the flow becomes critical. To investigate the existence of n= 1 and different values of xe. Excerpt: average heat fluxa s a function of x ; the dashed part corresponds to boundary layers with recirculation. negative eigenvalues for x > 0 in the flow under a curved e body, we consider the rather extreme case of the unper turbed flow (13), which holds for x large. Equation (15a) rewritten in terms of 17, /, and g takes the form above finite length parabolic bodies (« = 1) with boundary 5n 1)/2 2 x<- - (f'-c) (P'/g')'-P=0, where the primes mean layers starting at the edges for several values of xe (revert now 17 derivatives. The smallest eigenvalue of this problem, ing to dimensional variables this means a parabolic cylin numerically computed as a function of x for several values der of width much smaller than the curvature radius at its of n and Pr, is negative and decreases in modulus with nose). A region of reverse flow at the bottom of the bound increasing x. This means that upstream propagation is al ary layer appears on the central part of the surface when xe ways possible for these profiles of ub and 0b [and presum is greater than a certain critical value, function of n and Pr. ably the same is true of the actual profiles for x=0{\)], However, as noted by Jones,16 the numerical integration but the upstream speed of the perturbations tends to zero can be continued through this region (the end points of the when JC->OO. A straightforward computation shows that, curves of Fig. 5 for xe=4 correspond to the failure of the asymptotically, the smallest eigenvalue is numerical method). A tendency to reattachment, which actually occurs for moderate values of xe, can be observed 1 in the figure. This is due to the decrease of the tangential c~ — f"(0)x°n~l)n f°r X**' (16) component of buoyancy, which opposes the boundary layer flow, as the surface inclination diminishes near the /5 The conclusion to be drawn from this linear analysis is that center. The average Nusselt number, Nu/Gr' the change from elliptic to parabolic character of the sta = - r0<(d9/dy)0dx/xe, is given in the excerpt of this tionary problem does not occur at any definite value of x figure; the dashed part of the curve corresponding to but only asymptotically, as x —• 00. boundary layers with reverse flow. A criterion on the max The response of the boundary layer to a small station imum value of xe for which these solutions can be realized ary perturbation introduced at the solid wall is also of is lacking at the present time. some interest. It was found elsewhere18 that the influence Last, to demonstrate the effect of the pressure gradient of such perturbation is mainly confined to a thin transport below finite size bodies, which, as mentioned before, en sublayer if the local Richardson number at the wall ables upstream propagation of perturbations generated at (JT = —Q'j/u'^) is smaller than 1/4, whereas the per some downstream location, we consider the boundary layer turbation influences the bulk of the boundary layer if below a parabolic body {n— 1) with xe finite. In this case, S~> 1/4. In addition, the perturbation propagates up (13) is not valid anywhere, as can be seen by noting that, stream through the outer low velocity region of the bound owing to the symmetry, the solution would be of the form ary layer if Pr < 2, being felt ahead of the location on the wall where it was generated if the two conditions ST> 1/4 2m+i 4>= X hm+dy)x , and Pr < 2 are satisfied simultaneously. The dotted lines of m=0 Fig. 3 give the value of ST as a function of x. As can be (17) 2m lm seen, 3~>\/4r in the central region of the surface. For p= I P2m(yU , e=± I P'2m{y)x , large x (13) holds and then jr=;c(1-5")/2 m=0 X[-g'(0)]/f"(0)2, which tends to zero for n> 1/5 and and a hierarchy of problems is obtained by inserting (17) to infinity for n < 1/5. into (7)-(12) and separating terms of like powers of x. Figure 5 shows the heat flux and wall shear stress The leading terms of (17) satisfy equations analogous to are summarized in the following subsection as a prelimi nary to the discussion of the flow in the transition region. A. Incoming critical flow Under the action of the large pressure gradient men tioned before, transport effects become negligible in the bulk of the boundary layer approaching the transition re gion, being confined to a sublayer whose thickness tends to zero at the apparent origin. Denoting by t, the distance to this origin, the flow in the in viscid region for l/(2« — 1)<£< 1 is of the form l u-ul{y)=£ U{y) + - p-p,(y)=£xP(y) + - 0-e¡(y)=^e(y) + - (18) where (u¡,pi,0¡) are the limiting velocity, pressure, and FIG. 6. Heat flux and shear stress for the outbound flow below a solid temperature distributions at the apparent origin, and the with n=l and different values of xe. The dashed lines correspond to the similarity solution (13) for an infinite parabolic cylinder. right-hand sides of (18) represent the departures from this limiting flow. The limiting distributions depend on the ev olution of the flow for (1— x)=0{\), but satisfy certain (14), with f—$\, g = ±p'o, and n = \, but with the criticality conditions. Thus, the limiting velocity and tem additional term — 2p2 appearing in (14a). This^2» which is perature are of the form u^Ay^ and 6¡=* 1 — By* for y4,1, different from zero for xe finite, would be determined by with yi¿=0.364 and /c=/c(Pr) determined from the match the next higher order problem in the hierarchy. But this ing with the viscous sublayer, and matching also implies problem, in turn, involves the unknown p4, and the cou k=2\I/{\L+2) =* 0.308. Since [X < 1, the shear stress on the pling persists for all higher orders, so that, ultimately, the solid tends to infinity at the apparent origin, in keeping location of the edges influences the solution everywhere. with the divergent pressure gradient there. The right-hand Numerical results for the heat flux and shear stress are sides of (18) satisfy the conservation equations (7)-(10) presented in Fig. 6 for Pr=0.7, together with those of the linearized about the limiting distributions and with the self-similar solution (13) with n=\, holding for an infinite transport and buoyancy terms omitted. Two linearly inde parabolic cylinder. As can be seen, the influence of the pendent solutions of these equations, denoted by the sub edges extends to the whole boundary layer when xe is suf scripts a and b in what follows, have the following asymp ficiently small, while, as xe increases, the solution tends to totic forms for y 4,1: that of the infinite parabolic cylinder except in the neigh BK/A1 borhood of the edges. :1- + l-2ft 0c+l-2fO(l-2iO f. + ' 1 IV. ASYMPTOTIC SOLUTION BELOW A SOLID BK/A SURFACE FOR n LARGE (19a) When n is large, the buoyancy term 0\x\2n/x in Eq. (8) goes from negligible to dominant in short transition regions, ( |JC| — l)=0[l/(2«—1)], where the character of the flow changes from pressure driven to buoyancy driven. X/A . In this section we analyze the flow in the transition region 1-2/i- at the right half of the body for l<(2«-l)< d_ ldPx/dy\ Px 6/uj6/uj (19a) alone must satisfy the boundary condition at infinity dy~ [dO/dy)~^+(2n-l)1-c =0. (23) imposes a condition on the limiting distributions them selves, which is the criticality condition alluded to before. This equation does not contain |" derivatives. A particular solution accounting for the nonhomogeneous buoyancy term, and its associated velocity and temperature, are B. Transition from pressure driven to buoyancy 0^ -Uié'£ driven flow Pp 1 T, U,p ~ d\j>2 u (28b) 0.2 wall \ n 0.2 0A 0.6 0.-8 1 1.2 1.4 1-M t { Ay Ír(§)%;' -Ané%¿Vi- for y2-> oo. dy2 ^ A{1 2/0 (28c) FIG. 8. Heat flux, shear stress, and pressure below a surface with n= 15 (solid lines), and below a flat horizontal strip whose edges coincide with The first term in (28c) matches the limiting velocity u¡; the inflection points of the previous pressure distribution (dashed lines). the second matches the outer velocity perturbation associ ated with the pressure perturbation a{g)Pa{y); and the third, which represents the outward displacement of the side of (28c) are comparable (region 110 isy2=0{e^)>l. L limiting distributions, matches Up. The unknown function The velocity is of order j%=0{é $), and the viscous term is if(£) is determined by the condition that the outward dis negligible in the momentum equation [transport effects be placement be that given by the third term of (28c). ing confined to a thinner sub-sublayer II2 of thickness Upstream of the region of nonlinear response, for 0(e~té/2)]. The balance of inertia and pressure force yields (— £)>1, the outward displacement becomes negligible Tr=0{e2^), so that the second term of (28c) remains im and (28) has a solution of the form portant in IIX. The solution outside the viscous subsublayer is of the form (,í+I)/(íí+2) 2 + 2) : t/-2=(-^) /20(^0). 7F=(-^) ^ í 2o, +l 2 (29) i>2=e^ ^f2l(V2l), Z=e ^2l, with ^i^M 1/(ii+2) (31) where i720=JV'(— £) , TT2Q is a constant, and where f2i{f]2\) and the constant 7?21 satisfy (fi+2)f%+nf&- (M+1)/2o/20+2^20=0, (30a) M/ÍÍ- (/*+ D/2l/21 + 2^21=0, (32a) at /20 = /20 = ° VlO = 0, (30b) /2i=0 at 1721=0, (32b) A *W20 /:20 ' ^20 vlo*+0v%o+--- f A „«+l ^21 ^l-ft M , .. V+i ¿(1-2/0 V21 i7JT/21 Ai~M+1'72i -A{l_2fM) ' for r¡2o—> oo. (30c) for -Í721-» oo- (32c) Here the constant A can be absorbed by a scale transfor 1/i,l+2) The solution of (32a) and (32b) with the asymptotic be mation, rewriting (30) in terms of A r]20, i/iil+2 4/(fl+2) havior given by the leading term of (32c) can be written as fis/A \ and Tf20/A . This asymptotic solution, and the perturbation to the limiting flow that it induces in the outer inviscid region, matches the incoming flow con V2i= i/ 2 t/( i+i (33) Jo ^(íí+i)i "/] ' ' '-2í2r sidered in the previous subsection. In particular, the con dition that the coefficient of 17^0 in (30c) be zero is fulfilled The value of Jr21 is determined by expanding this integral for ^==-0.364, as was advanced before, and then for /2i>l, substituting fn from the three-terms expansion v(fl+2) Tf20/A -0.355. (32c), and identifying like powers of 7721. The result is It is worth noticing that a favorable pressure gradient {dTr/d£, Pr 0.5 0.7 1 2 3 5 7 10 -ÍÍC0) 0.2161 0.2466 0.2823 0362 23 0.4158 0.4911 0.5457 0.6083 f'i(0) 1.1503 1.1119 1.0681 0.9754 0.9181 0.8450 0.7970 0.7467 a thin transport sublayer that need not be analyzed, the /4=/4==g4_l=0 at ^74=0, (39c) resulting equations, with transport effects neglected, are A=g4=0 for ?74— •» • (39d) du dp dp (35a) This problem is the limiting form of (14) for (2K — 1) -> oo dip 3/4 1/4 with /=(2«-l)- /4. g=g*, and 77=(2«-1)- T?4- dp e Results of the numerical solution of (39) for several values dljr~~u' (35b) of Pr are given in Table II. Figure 8 shows the distributions of heat flux, shear de (35c) stress, and pressure on the solid below a surface with «=15 (solid lines), and below a flat strip {n infinite; dashed lines) whose edge coincides with the position of minimum where u3=u/(2n— l)=u(dy/d¡¡3)^. For J"3 large, the ve locity grows exponentially according to the law u pressure gradient in the solution for n—15. As can be seen, ~ [20(^)]1/2e&/2, and the pressure gradient in (35a) be the agreement is good over most of the surface, the only comes negligible. This marks the beginning of the buoy differences arising in the transition region. ancy dominated flow. The thickness of the boundary layer n decreases as e~^ , and the transport terms, evaluated with V. SUMMARY this solution, are of order e&/2/(2n — 1) relative to the terms displayed in (35a) and (35c). These transport terms An analysis has been carried out of the effect of the become important to the flow in the bulk of the boundary pressure gradient induced by the component of the buoy 2 layer when |'3=ln(2«—1) +|4, with f4=0(l) (region ancy force normal to the surface of a curved body on the IV). Then, in terms of the variables natural convection boundary layers above and below the body. The size of the region around the bottom of the body where this pressure gradient is comparable to the compo y4=(2n-\)y, u4= 9, (36) 2n-l' v4= 2n-r nent of the buoyancy force tangential to the surface scales with a low power of the inverse of the Grashof number [cf. the conservation equations and boundary conditions be (6)], being therefore reasonably wide for values of this come parameter of practical interest. Numerical solution of the du4 dv4 relevant boundary layer equations leads to the distribution (37a) of heat flux, which, along with the boundary layer thick d§4 dy4 ness, is finite at any point below a power law surface. Even 2 du4 du4 d u4 though the pressure gradient effect is thought to be notice • 6e^+ u*dJ+v*dj: (37b) able under realistic conditions, no detailed experiment ex 2 ists, to our knowledge, with which the present results could de de i d e be compared. In the very important case of body surfaces U +V (37c) *di4 *dy4-Prdyl' of finite nonzero curvature, to which the greatest amount of work is devoted in the literature, the pressure gradient u4=v4=6—1=0 at y4=0, (37d) effect is known to be weaker than for zero or infinite cur vature radius, because it automatically disappears at the u4=6=0 for_i>4->oo, (37e) leading order in an expansion of the boundary layer solu which describe a purely buoyancy-driven flow. Asymptot tion around the lower stagnation point. However, the stan ically, for |"4>1, the solution in terms of the streamfunction dard expansion itself may require modification for moder takes the form ate Grashof numbers, as discussed at the end of Sec. Ill in ^4==el4/4y4(?74)) Q=gi(v^t where t]4=e^y4, relation with the influence of the edges of a finite parabolic (38) cylinder, because the region where the pressure gradient effect may appear represents a large fraction of the curva with ture radius (about 40% for Gr=105). fji n2 J7+ -2"+*4= =0, (39a) ACKNOWLEDGMENTS Pr This work was partially supported by the CICYT un- ár '+^/"4¿f4=o, (39b) 4 der projects ESP 187/90 and PB92-1075-A.