Pressure gradient effect in natural convection boundary layers F. J. Higuera and A. Liñán
Citation: Physics of Fluids A: Fluid Dynamics 5, 2443 (1993); doi: 10.1063/1.858757 View online: http://dx.doi.org/10.1063/1.858757 View Table of Contents: http://aip.scitation.org/toc/pfa/5/10 Published by the American Institute of Physics Pressure gradient effect in natural convection boundary layers F. J. Higuera and A. LiRh E. T. S. Ingenieros Aeronciuticos, Pza. Curdenal Cisneros 3, 28040 Madrid, Spain (Received 8 December 1992; accepted 2 June 1993) The high Grashof number laminar natural convection flow around the lower stagnation point of a symmetric bowl-shaped heated body is analyzed. A region is identified where the direct effect on the flow of the component of the buoyancy force tangential to the body surface is comparable to the indirect effect of the component normal to the surface, which acts through the gradient of the nonuniform pressure that it induces in the boundary layer. Analysis of this region provides a description of the evolution of the flow from a pressure-gradient dominated regime to a buoyancy dominated regime. Numerical results are presented for the flows above and below heated power-law body shapes, and the upstream propagation of small perturbations to the stationary flow is discussed. An asymptotic analysis is carried out for the flow below nearly flat horizontal bodies, for which the change from pressure-gradient-driven to buoyancy-driven flow occurs very rapidly in a short region. The influence of body edges located in the region of interest is also discussed.
I. INTRODUCTION found in Gebhart et cd5 For bodies of finite radius of cur- vature, the neglect of the pressure gradient does not result The natural convection flow generated by heated bod- in an error near the lower stagnation point in the limit of ies is confined, for large Grashof numbers, to thin bound- infinite Grashof numbers, because the scaled boundary ary layers around the body and to ascending plumes. The layer thickness is finite and constant there and the effect of flow in the boundary layers is primarily induced by the the pressure gradient, evaluated with the buoyancy-driven component of the buoyancy force tangential to the surface. solution, is then negligible. On the other hand, while anal- The component of the buoyancy force normal to the sur- ogous solutions exist for the flow above a concave body, face gives rise to a nonuniform distribution of reduced these are seldom if ever observed, owing to their instability. pressure p’ =p-tpmgz (Z being upward vertical distance, Self-similar solutions exist for bodies with appropriate pm the density of the fluid far from the body, and p the shapes. Thus, Schuh6 considered boundary layers of con- static pressure), whose gradient along the surface is, how- stant thickness, while Merk and Prins’ investigated general ever, negligible compared with the tangential component of conditions for self-similarity below two-dimensional and the buoyancy force outside the bottom region of the body. axisymmetric bodies in the absence of the pressure gradient Here, if the angle of the surface to the horizontal is zero or effect. Under the same assumption, Braun et al8 found a small, the tangential component of the buoyancy force is family of self-similar solutions for body shapes ranging small and we may need to take into account the influence from sharp edged to finite curvature and to nearly horizon- of the reduced pressure gradient on the flow. tal surfaces. Their solutions lead to an infinite boundary A large amount of work, beginning with the classical layer thickness at the lowest point of surfaces with locally work of Hermarm’ for the flow around a heated circular infinite curvature radius, and to zero boundary layer thick- cylinder, has been devoted to the analysis of the natural ness for surfaces of vanishing local curvature radius. All convection flow around heated bodies, with the assumption these solutions are applicable only at a certain distance that the effect of the normal component of the buoyancy from the lowest point. force can be neglected. With this assumption, the boundary The nonuniform reduced pressure distribution across layer equations are parabolic and the integration can be the boundary layer results from a hydrostatic balance of carried out starting at the lower region, with the local the pressure gradient and the normal component of the self-similar solution of the equations. Since no self-similar buoyancy force, because the flow is nearly parallel to the solution of the boundary layer equations exists in general body surface and accelerations normal to the surface are for the whole body, several approximate techniques have negligible. This balance leads to a high pressure in the been applied, including integral methods, series expan- regions below a heated surface where the fluid temperature sions, and numerical methods. Thus, Chiang and Kaye2 is high or the boundary layer is thick, whereas the opposite carried out an analysis in terms of Blasius series, and occurs above a heated surface. Thus the fluid is pushed Saville and Churchill3 used Goertler series that, because of tangentially to the surface away from (respectively to- its faster rate of convergence, allow the solution to be ex- ward) these regions. Under this driving force, the flow tended farther away from the stagnation point. Muntasser below a horizontal plate acquires a motion toward the and Mulligan4 obtained a solution of the boundary layer edge, and that above a horizontal plate moves from the equations by means of the so-called local nonsimilarity ap- edge (where the well-known Stewartson’ self-similar solu- proximation. A review and comparisons with other results tion holds locally) toward the center. The effect of the obtained without the boundary layer approximation can be self-generated pressure gradient renders the solution of the
2443 Phys. Fluids A 5 (IO); October 1993 0899-8213/93/5(10)/2443/11/$6.00 @ 1993 American institute of Physics 2443 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. Wagner” 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, Birkebak” 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 ChapmanI 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,r3 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 (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 Gebhartt5 and Jones16 analyzed the incipient point, z, 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 n>1/2. The parameter cz 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 Ackroyd” extended this anal- temperature T, above the temperature T, 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 = p, [ 1 -p( T - T, )], the boundary layer equations de- from the edge for both upward and downward inclined scribing its motion at large Grashof numbers are, with the plates. In the first case, he described the transition from the Boussinesq approximation, pressure gradient dominated to the buoyancy dominated au au -+-=o, (2) regime, while, in the second case, he found a regular sep- ax ay aration at a certain distance from the edge, and was able to continue the numerical integration through the region of au au apvp, _ dzs a% reverse flow. Irrespective of the physical meaning to be '&+"&'- ax +@;i;;+V@r (3) assigned to the reverse flow, the possibility of continuing the marching integration points out that some self- adjusting mechanism is at work, which would not occur in a parabolic boundary layer. Jones attributed this feature to ae a8 y a% the fact that the pressure gradient is not given in advance, uz+v;i;=gp (5) being determined by the flow itself. Thus the “elliptic” 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=gfi(T,--TT,), O=(T-T,)/(T,--T,), above and below the surface of a heated bowl-shaped body Pr is the Prandtl number, and a dg’dx in the region around the lowest point of the surface where =2nalx/Ly sgn(x) (1 is understood in (3). The f ’ the changes in surface height and the boundary layer thick- in (4) corresponds to the boundary layers above and below ness are of the same order and, therefore, the effects of the the solid surface, respectively. The balance of convection, pressure gradient and of the component of buoyancy tan- pressure gradient, and diffusion in these equations over a gential to the surface are equally important. Formally, for distance x, along the surface yields the well-known scales high Grashof numbers, this region has a size much smaller yc= xc/Gri’5, u - &gGr~““, u, = $%&r:“‘, and than the characteristic radius of curvature of the body, but Ap’/p m =gx&&c’5, with Grc=gX2/?, for the boundary it can be quite appreciable for values of the Grashof num- layer thickness, velocities, and variations of the reduced ber of practical interest, the more so for flat-bottomed bod- pressure, respectively (see, e.g., Gebhart et al. 5). The char- ies [cf. the estimate (6) below]. The shape of the body is acteristic length (x,) of the region where the pressure gra- approximated by a power law in the neighborhood of the dient and the tangential component of the buoyancy are lower stagnation point. Numerical results are presented for comparable is then obtained by equating the orders of mag-
2444 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. LiiSn 2444 TABLE I. Scaled heat flux and skin friction from the solution of (14) for several values of n.
n 0.5 0.7 1 1.5 2 2.5 3 5
--g’(O) (Pr=0.7) 0.3552 0.3616 0.3722 0.3862 0.4007 0.4145 0.4269 0.4674 --g’(O) (Pr=7) 0.7456 0.7657 0.7928 0.8341 0.8691 0.9012 0.9304 1.0236 s’(O) (Pr=0.7) 0.9608 0.9135 0.8602 0.7963 0.7514 0.7169 0.6884 0.6140 f’(O) (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 ~=XIXp-1)‘2f(~), 6=g(rl), &/L=(2na)- 1’(2n-‘)Gr~*‘5’2n-1), or, in terms of the Grashof number based on the characteristic size of the with body ( Gr=gL3/& 1) ,
XC 1 x= (2n(r)(5/2)/(5”-1)Gr1/2(5n-1) ’ (6) where 1F,is the streamfunction ( u = $5, v = -$?, and tF,=0 at y=O). Carrying these expressions into (7)-( 12) with The effects of variable flow properties are left out of the dp/& left out, we find present formulation. These effects have been investigated by Ackroyd17 for the boundary layer above a horizontal or yltil;n ffN-np+g=o, slightly inclined heated plate, and the conditions of appli- cability of the Boussinesq approximation are the same for l+n this flow and for the problem at hand. Variable flow prop- g”+ - 2 Pr fg’=O, erties e&&s are surely important when accurate quantita- tive results are sought for in many real-world problems f=f’=g-l=O at ?I=o, (14c) involving nonsmall relative temperature differences. We do f’=g=O for ~+co, not believe, however, that they would introduce qualitative (14d) changes in the steady flows discussed in this paper, and where the primes mean derivatives with respect to 7. 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) au au may cover the whole body surface and the ends of the body ~+-@A (7) are encountered before the flow takes on the asymptotic form ( 13). We shall consider the extreme case of a finite au au ap Ix12n a224 surface with edges at 1x 1=x,= L/x, and with ( 1) holding uaxfvay=-~+y-+av”’ (8) for 1x I 2445 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. LifSn 2445 toward the center for smaller 1x I, with a plume rising at the center. Which of these possibilities, if any, is realized in a given situation seems to depend not only of the boundary (4 layer equations but also on the nature of flow induced around the origin and outside the boundary layer and on stability considerations that lie beyond the scope of the present analysis. In particular, as for the problem of exist- ence, the second and third types of solutions mentioned before, as well as the solutions for x,-+ CO,involve bound- ary layers starting with zero thickness at one or more points on the surface. While this is possible as far as the boundary layer is concerned, small regions would also exist around the singular starting points where the boundary layer approximation is not applicable. Hence, the possibil- ity of having these two types of solutions depends on whether an acceptable structure exists for the flow in such small regions, which is unlikely for smooth surfaces with- out edges or holes anchoring the singularities. In this paper (b) we shall be concerned only with the boundary layer solu- tions sketched in Figs. 1 (b) and 1 (c) for the flow above a solid surface. We shall assume that the conditions at the center of the surface have been so arranged, if necessary (e.g., by applying suction through the solid surface in a small region around the center, or by suppressing half of the body), that the solution of Fig. 1 (c) can be realized. 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 6’ are added to the left-hand sides of (8) and ( 10) and the time evolution of the flow from a given initial condition (u=@=O) is followed until it becomes station- ary. For effectively infinite bodies, the computational do- main extends to Ix I 2446 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. LiH&n 2446 1.6 b Pr = 0.7 -%-dt 0.6 - -.n=2 1.4 “15’\ f ,-I-, ,.\ /,’ l- .- 2&, _ 0.4 - p ,” ;,-.------_-. ______---..;. g-z--<.------j’ / -=--- -______-1’/ 0.5_------_____ I’ ,- 0.2 4; n=2 : --~-- !F- 1.5 1 0.6 0, \ j\‘--?-.j I-- Q I2 3 45 6 78 FIG. 1. 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 ftlnction 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'(O) 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 1/2 0.8 - 1 Still negligible [so that (9) holds as it is]. And third, trans- port effects are confined to a thin Stokes layer near the 0.8 0.6 solid, which does not need be analyzed. Outside this layer, 0.6 the linearized conservation equations for the perturbations can be combined to yield the following second-order equa- 0.4 tion for the pressure perturbation: 0.2 I t . 7. /_.:.. - ._ ____,_ ” , , 5--y&:;:-;-:;;; P’ ’ L am..... * (u&)-cy - -P=O, 0 I 2 (154 -. 3 5 4 (6 ) 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’/C$, . Here the primes mean y The dotted lines represent the Richardson number at the wall. derivatives and the subscript b denotes the unperturbed 2447 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. Li7iBn 2447 flow at the location (x) where the perturbation is applied. 2 Equation (15a) is to be solved with the boundary condi- -G, tions I.5 P’=O at y=O and P-O for y+c~. (1%) (The first of these expresses the inviscid condition V=O 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 - ( Ix 1‘“ /ikx)P’ has been omitted in (15a); since kgl, such a term would matter only for very large values of Ix 1. Leaving it out, the de- pendence of the eigenvalue problem on x occurs through the functions ZQ,and 0; 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 flux and shear stress for the inbound Row above a solid with n= 1 and different values of x,. Excerpt: average heat flux as a function of the flow becomes critical. To investigate the existence of x,; the dashed part corresponds to boundary layers with recirculation. negative eigenvalues for x > 0 in the flow under a curved body, we consider the rather extreme case of the unper- turbed flow ( 13), which holds for x large. Equation (15a) rewritten in terms of q, f, and g takes the form above finite length parabolic bodies (n = 1) with boundary ~(~~-~)‘~(fr-e))~(P’/g’)‘--P=O, where the primes mean layers starting at the edges for several values of x, (revert- now q 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 x, ways possible for these prOtieS of ub and 6b [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( 1 )], However, as noted by .iones,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 x+ CO.A straightforward computation shows that, curves of Fig. 5 for x,=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 x,, can be observed 1 in the figure. This is due to the decrease of the tangential C- -fncOJX(3n-1)/2 for x> 1. (16) component of buoyancy, which opposes the boundary layer flow, as the surface inclination diminishes near the The conclusion to be drawn from this linear analysis is that center. The average Nusselt number, Nu/Grr’5 the change from elliptic to parabolic character of the sta- = - Jpe/dy)gdx/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-+ CO. boundary layers with reverse flow. A criterion on the max- The response of the boundary layer to a small station- imum value of x, 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 elsewhere’a that the influence Last, to demonstrate the elfect 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 (7 = -02~;~) is smaller than l/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= I) with X, finite. In this case, .Y> l/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 Y> l/4 $= mlo 1Cr2m+l(Y)X2m+‘, and Pr < 2 are satisfied simultaneously. The dotted lines of Fig. 3 give the value of 7 as a function of x. As can be m (17) seen, Y> l/4 in the central region of the surface. For P= m;, P2mwX2m, e= * 5 P;m(Y)x2mt large x (13) holds and then CF=x(1-sn)‘2 l?Z=D X[-g’(0)]/f”(O)2, which tends to zero for n> l/5 and and a hierarchy of problems is obtained by inserting ( 17) to infinity for II < l/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 2448 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. LiMn 2448 55 I are summarized in the following subsection as a prelimi- --01di 4.u nary to the discussion of the flow in the transition region. ..5 3 A. incoming critical flow Under the action of the large pressure gradient men- tioned before, transport effects become negligible in the .45 2 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 [ the distance to this origin, the flow in the inviscid region for 1/(2n .1 1 - 1) IV. ASYMPTOTIC SOLUTION BELOW A SOLID @,------BK/A2 K-Z/J SURFACE FOR n LARGE 1-2pY +*-* 3 When n is large, the buoyancy term 6JIx I 2n/x in Eq. (194 (8) goes from negligible to dominant in short transition u,= - A(;;;& y-‘+*** 3 regions, ( IX]- 1) =0[1/(2n- l)], where the character of the flow changes from pressure driven to buoyancy driven. A/A l--P vazE -- In this section we analyze the flow in the transition region 1-2pY +*-* 9 at the right half of the body for lg(2n- 1) (Gri’5 [the and boundary layer approximation breaks down in this region when (2n- 1) =U(GI-~‘~)]. As will be seen, the pressure &,=&f-j--*-, @,=-BK~~-~+-*, gradient attains very large values in this region, 0( 2n - 1 ), u,=A,UJ+f-* , vb=‘&if+--’ , (1%) before its erect is finally overcome by that of the buoyancy. Hence, matching of the transition region and the upstream and, in general, only a certain linear combination of these boundary layer would not be possible without a pressure solutions satisfies the conditions ( LJ,P,O) -0 for y--+ CU.As gradient in the incoming flow that, with the scales used in can be seen, the velocity associated with the solution ( 19b) the formulation, diverges at some apparent origin located diverges at the solid wall faster than the velocity associated inside the transition region in the asymptotic limit (2n with the solution (19a), and would therefore dominate the - 1) -+ 00. It turns out that this single condition determines flow in the viscous sublayer. It turns out, however, that the the structure of the incoming pressure driven flow indepen- problem for the viscous sublayer has no solution matching dently of its subsequent evolution. This flow has been an- (19b) and, thus, the more rapidly divergent branch must alyzed elsewherei and the main features of this analysis not appear in the solution of the problem. That the solution 2449 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. LWm 2449 t=- 1)= & [Wn - 1,’ + O(l)] (=- 11= Gii [h(2”- I)‘+ O(l)] ao, de, (2n-l)uzz+ Vl du’o, (22d) FIG. 7. Sketch of the transition region for large n. which can be combined into a single equation for P, : (19a) alone must satisfy the boundary condition at in&&y (23) imposes a condition on the limiting distributions them- selves, which is the criticality condition alluded to before. This equation does not contain c derivatives. A particular solution accounting for the nonhomogeneous buoyancy term, and its associated velocity and temperature, are B. Transition from pressure driven to buoyancy f@ - de5 driven flow cJ=[~n-l)l-O’ (I,‘(&‘-” Figure 7 is a sketch of the different parts of the tran- sition region. The rapid increase of the buoyancy term in P-,=(2?+ l)“U&, $7 (2n~‘;:vo, this region leads first to a splitting of the boundary layer, with viscous effects confined to the thin sublayer II. More- where the primes mean y derivatives. As can be seen, the over, owing to the critical character of the limiting flow, incipient effect of buoyancy on the pressure gradient- buoyancy manifest itself in a rather peculiar way: it in- dominated flow is to increase the pressure in this part of duces a velocity perturbation in the inviscid bulk of the the transition region. The increase of pressure is achieved layer I that is much smaller than the right-hand sides of by an outward displacement of the boundary layer profiles ( 18) but increases near the wall at the faster rate ( 19b), [see (22~) and the expressions of UP and OP above], lead- instead of ( 19a). In this way, the buoyancy induced veloc- ing to a deceleration of the flow in the inner part of the ity perturbation forces a nonlinear response of the viscous layer and to an acceleration in the outer part. sublayer II, which, in its turn, requires a driving pressure The general solution of (23) is (built up in the inviscid region I) much larger than that induced directly by the buoyancy, Then, farther down- P1=4OPa(Y) +&ml(Y) +pp, (25) stream, region II splits into an inviscid nonlinear sublayer II1 and a viscous subsublayer II,. The first of these grows and the associated velocity perturbation is until it covers the whole boundary layer (region III), at which point the buoyancy ceases to be a small Gerturba- tion. The buoyancy force overcomes the pressure gradient shortly afterwards, leading to a decrease of the boundary Here a(c) and b(c) are arbitrary functions and P,(y) and layer thickness, which comes to an end when transport P,(i) *are the two functions introduced in the previous effects become again important (in region IV). The differ- subsection. On account of the critical character of the lim- ent stages of this transition are successively analyzed in the iting flow, the boundary condition at inlinity implies 6(g) remainder of this section. ==O. The part UP of the velocity has the same behavior Let near the wall as the solution (19b) (i.e., I$“-‘), being more singular than the only acceptable solution x=l+--- 2nll M2~--lY+~l,- [ - a(ul’Pp3~)‘=O(ay- “)I of the homogeneous problem. Let us consider now the flow in the viscous sublayer II. with 5=0( 1) and u still undetermined, denote the region The thickness of this sublayer, determined from the bal- of nonlinear response of the viscous sublayer II. Then ance of convection (with a velocity of order f) and diffu- sion, is of order S = 0[ 1/(2n - 1) “(!-‘+‘)I. The velocity per- x2”-1=exp[(2n-l)lnx]~(2n-l)ue~. (21) turbation Up is of order @“-‘/(2ti- l)lWU in the viscous The solution in region I, outside the viscous sublayer, is of sublayer, and the response of the sublayer becomes nonlin- the form u=ul+Ul, p=prfPl, 8=81+0,, and v=v,, ear when this velocity is of order P, which occurs for with U,, PI and O1 small. These variables satisfy the lin- U= (p+ 1 )/(,u+2>. A pressure of order S2P is then re- earized equations quired to influence the flow in the sublayer; this is much larger than Pp above, and therefore the pressure gradient au, av, must come from the homogeneous solution a(c in un-- 1) af;+ay=o, W-4 (25), requiring an a(E) of order 1/(2n-l)2p’(“+2) (the 2450 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. Li716n 2450 resulting pressure is uniform across the sublayer). Thus 2 r 5 appropriate variables of order one in the viscous sublayer -Cd, 4,,, Ad are E and I.6 4 yz2= (2n- l)m+2’ y, &gjg = (2n-l)@+‘)‘(@)$, 1.4 ~(~>=(2n--)*/1’(‘1+2)a(~). (27) I.2 3 PT = 0.7 In terms of these variables, the inner problem becomes I 0.8 2 a& a2& a&a$ diT a3& (284 2g~-xg~=-z+2p 0.6 -a& _ 0.4 I 7&=~=0 at y2=0, (28b) 0.2 0 0 0 0.2 II.4 0.6 0.d I 1.2 1.4 ~(g)jTg~-Ape~j$-l for yZ- co. 2 (28~) FIG. 8. Heat flux, shear stress, and pressure below a surface with n= 15 The first term in (28~) matches the limiting velocity uz; (solid lines), and below a flat horizontal strip whose edges coincide with the inflection points of the previous pressure distribution (dashed lines). the second matches the outer velocity perturbation associ- ated with the pressure perturbation a( g)P,b>; and the third, which represents the outward displacement of the side of (28~) are comparable (region Iii) is FZ= O(.&) > 1. limiting distributions, matches v{. The unknown function The velocity is of order j$= 0( PC), and the viscous term is ii(<) is determined by the condttton that the outward dis- negligible in the momentum equation [transport effects be- placement be that given by the third term of (28~). ing confined to a thinner sub-sublayer II2 of thickness Upstream of the region of nonlinear response, for O(e-PY2)]. The balance of inertia and pressure force yields ( -c) > 1, the outward displacement becomes negligible Z=O(e2pf), so that the second term of (28~) remains im- and (28) has a solution of the form portant in II,. The solution outside the viscous subsublayer is of the form &= (-~)(~+*)‘(~+2)f20(r/20), & ( ~g-)w(P+~)ij20 i29) &=e(~f1)~~Zl(r]21), ij=e2%?Z1, with r]21=y2/ej, where r]20=j$/( -6) i’(pL+‘), ?YZOis a constant, and (31) where f21(r/21) and the constant ZZl satisfy (p+2>f~6+pf$- (p+ 1>f20.&+$&=0, (304 .- ~~~f;:-(~+l)f2lf;;+2~~~iT21=0, (!+I fi~=f&=O at 7720=0, (30b) fil=O at ~i=0, (32b) A 77,J rl:;p+o&+ * * * A f20-- P-1+1 V~(+.&2~) p+l- ;r,l f21--- jJ+1 r/Z1 ‘4(1-2p) d7--A&+*** for 77zo-+CO. (3Oc) for r]21-+co. (32~) Here the constant A can be absorbed by a scale transfor- mation, rewriting (30) in terms of A 1’(P+2)r]2a, The solution of (32a) and (32b) with the asymptotic be- fdA WG-2) ) and F2dA “(P+2). This asymptotic solution, havior given by the leading term of (32~) can be written as and the perturbation to the limiting flow that it induces in f21 d.f the outer inviscid region, matches the incoming flow con- 1121= (33) 0 sidered in the previous subsection. In particular, the con- J dition that the coefficient of vgo in (30~) be zero is fulfilled The value of SZiizlis determined by expanding this integral for ,u=O.364, as was advanced before, and then for f2i) 1, substituting f2, from the three-terms expansion &/A4’(“+‘) ~0.355. (32c), and identifying like powers of q2,. The result is It is worth noticing that a favorable pressure gradient /.@A2 (diF/dc < 0) is required to maintain the zero outward dis- ;r,, = -w” -0.295 A’, (34) placement of the solution of (30). Hence, the displacement becomes positive when the pressure gradient is less nega- where J= sc[( 1+~~)-~‘~--~-~+~-~/2]~~‘~d~~O.7497. tive than for this solution, but no adverse pressure gradient Nonlinearity affects the whole of the boundary layer is required to generate the outward displacement. Indeed, when FZ= O[ (2n - 1) 1’(p+2)], which, according to the re- the numerical results (see Fig. 8) show that positive values sults above, occurs for (2n-l)(x-l)=ln(2n-l)+{3, of dF/dc never occur. with c3=O( 1) (region III of Fig. 7). Appropriate inde- For (~1, the thickness of the region of nonlinear re- pendent variables to describe this region are g3 and the sponse where the first and third terms on the right-hand streamfunction $, which takes the place of y. Leaving out 2451 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. Lii%n 2451 TABLE II. Scaled heat flux and skin friction from Eq. (38) for several values of Pr. Pr 0.5 0.7 1 2 3 5 7 10 --g;(o) 0.2161 0.2466 0.2823 0362 23 0.4158 0.4911 0.5457 0.6083 f;(o) 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 f4= fi=g4- 1 =O at r/4=0, (39c) resulting equations, with transport effects neglected, are for r/4-+63. fi=g4=0 (39d) au ap ap 24 -= --+u3 -+&h, (354 This problem is the limiting form of ( 14) for (2n - 1) + w 86, ac3 w with f=(2n-l)-3’4f4, g=g4, and ~=(2n--l)-“~r/~. -=ap _-9 Results of the numerical solution of (39) for several values a* d of Pr are given in Table II. Figure 8 shows the distributions of heat flux, shear stress, and pressure on the solid below a surface with II = 15 (35c) (solid lines), and below a flat strip (n infinite; dashed where q=u/(2n-- 1) = u(dy/dc,)+. For c3 large, the ve- lines) whose edge coincides with the position of minimum locity grows exponentially according to the law u pressure gradient in the solution for n = 15. As can be seen, - [28( qb)]“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 decreases as e-c3’2, and the transport terms, evaluated with V. SUMMARY this solution, are of order ec3’2/( 2n - 1) relative to the terms displayed in (35a) and (35~). 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- layer when c3=ln(2n-- 1)2+&, with &=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 V where this pressure gradient is comparable to the compo- y4= (2n- l)y, f.Q=~ 2n-1’ 04=5g7 2 8, (36) 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 au4 au, relevant boundary layer equations leads to the distribution -+-=or (374 ac4 a4 of heat flux, which, along with the boundary layer thick- ness, is finite at any point below a power law surface. Even au4 au4 aG4 though the pressure gradient effect is thought to be notice- u4~-k~4ay4=~ec4+-g. (376) able under realistic conditions, no detailed experiment ex- ists, to our knowledge, with which the present results could ae de i a28 be compared. In the very important case of body surfaces 24 -+u4-=-7, (37c) 4 ac4 aY4 Pr ay, of finite nonzero curvature, to which the greatest amount of work is devoted in the literature, the pressure gradient u4=u4=0- 1=0 at yi=O, (37d) effect is known to be weaker than for zero or infinite cur- td4=e=o for y4+ co, We) vature radius, because it automatically disappears at the leading order in an expansion of the boundary Iayer solu- which describe a purely buoyancy-driven flow. Asymptot- tion around the lower stagnation point. However, the stan- ically, for g4k, 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. III in $4=e54’4f4(vd, e=gdrlJ, dme r/4=e%4 relation with the influence of the edges of a finite parabolic ’ (38) cylinder, because the region where the pressure gradient with effect may appear represents a large fraction of the curva- ture radius (about 40% for Gr= 105). f’“+f4fY A2 4 --y+g4=0, (394 4 ACKNOWLEDGMENTS Pr This work was partially supported by the CICYT un- & +z f&=09 (39b) der projects ESP 187/90 and PB92-1075-A. 2452 Phys. Fluids A, Vol. 5, No. 10, October 1993 F. J. Higuera and A. Lit%n 2452 ‘R. Hermann, “Heat transfer by free convection from horizontal cylin- ‘K. Stewartson, “On the free convection from a horizontal plate,” 2. ders in diatomic gases,” NACA Tech. Memo. 1366 (1954). Angew. Math. Phys. 9a, 276 (1958). 2T. Chiang and J. Kaye, “On laminar free convection from a horizontal “C. Wagner, “Discussion of integral methods in natural convection Rows cylinder,” in Proceedings of the 4th National Congress of Applied Me- by S. Levy,” J. Appl. 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