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.