On the boundary-layer structure of cavity flow in a porous medium driven by differential heating

On the boundary layer structure of differentially heated cavity flow in a stably stratified porous medium

“…However, since contributions to the left-hand side of (2.8) from the horizontal boundary layer are of order R 1/3 , this change in H has no impact on the leading-order solution there, which therefore satisfies at the upper surface, as in the insulating case. Numerical and asymptotic solutions of (4.2)-(4.5) and (4.7)-(4.11) reported by Daniels & Punpocha (2005) show that the temperature profile in (4.3) drives a twoway flow in the horizontal boundary layer which is entrained and detrained by the vertical boundary layer. As Z → ∞, the solutions in the horizontal and vertical boundary layers assume the algebraically decaying forms…”

“…Apart from being a monotonic function, the precise form of S is not expected to have a significant impact on the large-Darcy-Rayleigh-number structure of the solution; the form (2.7) is chosen to enable comparison with previous results (Daniels & Punpocha 2005). Note that because the sidewalls of the cavity are thermally insulated, it follows that the heat-flux integral…”

“…This constant represents the possibility of a correction (of order R −1/4 ) to b but would only be present if a correction to the temperature of this magnitude were generated within the inner layer. Substitution of (5.19) into (5.3) gives (5.20) and the solution found by Daniels & Punpocha (2005) is…”

“…The problem is formulated in § 2 and some numerical solutions are presented in § 3. Daniels & Punpocha (2005) have shown that, for the case where the lower surface is thermally insulated, the main circulation takes place within horizontal and vertical boundary layers of depth order R −1/3 near the upper surface, where R 1 is the Darcy-Rayleigh number based on the cavity depth and the temperature difference along the upper surface. In that case, fluid moves along the upper surface to the colder end where it descends within a vertical boundary layer and is detrained back into the horizontal layer.…”

“…In addition, a new double horizontal/vertical boundary layer structure occurs near the upper surface. The inner horizontal layer of depth order R −1/3 is similar to that of the insulated problem analysed by Daniels & Punpocha (2005), and is considered in § 4. The outer horizontal layer, which has depth of order R −1/4 , is considered in § 5 and is needed to allow the temperature field to adjust to its stably stratified form in the core region below.…”