TY - JOUR
T1 - A bifurcation study of convective heat transfer in porous media. Part II
T2 - Effect of tilt on stationary and nonstationary solutions
AU - Ryland, D. K.
AU - Nandakumar, K.
PY - 1992
Y1 - 1992
N2 - In the present work the investigation of Weinitschke et al. [Phys. Fluids A 2, 912 (1990)] on the bifurcation structure of stationary, two-dimensional solutions of the Darcy-Oberbeck-Boussinesq model equations, which governs the convection heat transfer in a porous medium, is extended. The effect of imposing a symmetry breaking geometrical perturbation, viz., a tilt, on the unfolding of the bifurcation structure, is investigated first. The symmetry breaking bifurcation points that are found at zero tilt are structurally unstable to even a slight degree of tilt, and they unfold into limit points that coalesce with the neighboring limit points as the degree of tilt is increased. Two such limit points disappear through the formation of a double limit point at very small angles of tilt. On the fold curve of such limit points are found origins of paths of Hopf points, also known as the B-point singularity. Several such B points are located on the fold curves of limit points. This is helpful in pointing to regions of parameter space, where interesting dynamical behavior with oscillatory and chaotic flow structures are possible. The dynamical behavior is explored through simulation of the governing time-dependent equations after suitable spatial discretization through the Arakawa scheme. Linear stability analysis of stationary solutions indicate that such solutions remain stable over an increasing range of Rayleigh number (Ra) as the degree of tilt is increased. The first limit point (L1), below which there is a unique solution, tends toward infinite Rayleigh number as the tilt approaches 45°. At 45° tilt, a reflective symmetry about the diagonal of a square cell is restored and the stationary solutions computed by the continuation method remain stable for Ra as high as 200 000. Results from the dynamic simulation confirm this behavior. For tilts of 5° and 10° stationary solutions lose stability at Rayleigh numbers in the range of 4000-5000. Periodic solutions interspersed with regions of chaotic behavior are observed as Ra is increased continuously. For 20°, however, periodic solutions begin only above Ra≈10 000.
AB - In the present work the investigation of Weinitschke et al. [Phys. Fluids A 2, 912 (1990)] on the bifurcation structure of stationary, two-dimensional solutions of the Darcy-Oberbeck-Boussinesq model equations, which governs the convection heat transfer in a porous medium, is extended. The effect of imposing a symmetry breaking geometrical perturbation, viz., a tilt, on the unfolding of the bifurcation structure, is investigated first. The symmetry breaking bifurcation points that are found at zero tilt are structurally unstable to even a slight degree of tilt, and they unfold into limit points that coalesce with the neighboring limit points as the degree of tilt is increased. Two such limit points disappear through the formation of a double limit point at very small angles of tilt. On the fold curve of such limit points are found origins of paths of Hopf points, also known as the B-point singularity. Several such B points are located on the fold curves of limit points. This is helpful in pointing to regions of parameter space, where interesting dynamical behavior with oscillatory and chaotic flow structures are possible. The dynamical behavior is explored through simulation of the governing time-dependent equations after suitable spatial discretization through the Arakawa scheme. Linear stability analysis of stationary solutions indicate that such solutions remain stable over an increasing range of Rayleigh number (Ra) as the degree of tilt is increased. The first limit point (L1), below which there is a unique solution, tends toward infinite Rayleigh number as the tilt approaches 45°. At 45° tilt, a reflective symmetry about the diagonal of a square cell is restored and the stationary solutions computed by the continuation method remain stable for Ra as high as 200 000. Results from the dynamic simulation confirm this behavior. For tilts of 5° and 10° stationary solutions lose stability at Rayleigh numbers in the range of 4000-5000. Periodic solutions interspersed with regions of chaotic behavior are observed as Ra is increased continuously. For 20°, however, periodic solutions begin only above Ra≈10 000.
UR - http://www.scopus.com/inward/record.url?scp=36449006540&partnerID=8YFLogxK
U2 - 10.1063/1.858364
DO - 10.1063/1.858364
M3 - 文章
AN - SCOPUS:36449006540
SN - 0899-8213
VL - 4
SP - 1945
EP - 1958
JO - Physics of fluids. A, Fluid dynamics
JF - Physics of fluids. A, Fluid dynamics
IS - 9
ER -