Natural convection in a fluid layer periodically heated from above

Natural convection in a horizontal layer subject to periodic heating from above has been studied. It is shown that the primary convection leads to the cooling of the bulk of the fluid below the mean temperature of the upper wall. The secondary convection may lead either to longitudinal rolls, transverse rolls, or oblique rolls. The global flow properties (e.g., the average Nusselt number for the primary convection and the critical conditions for the secondary convection) are identical to those of the layer heated from below. However, the flow and temperature patterns exhibit phase shifts in the horizontal directions.

Figure 1

Fluid layer with sinusoidal heating applied at the upper wall.

Figure 2

The flow topology and the isotherms for ${\mathrm{Ra}}_p$ = 400, α = 5 for the upper (a) and lower (b) heatings. Thick (thin) lines identify streamlines (isotherms); solid (dashed) lines identify positive (negative) temperature.

Figure 3

Variations of the location of the roll center $y_{\mathrm{c}}$ as a function of α (a) and as a function of ${\mathrm{Ra}}_p$ (b), variations of the maximum of the stream function as a function of α and ${\mathrm{Ra}}_p$ (c) and variations of Nusselt number Nu as function of α and ${\mathrm{Ra}}_p$ (d). Solid (dotted) lines correspond to heating applied at the upper (lower [6]) wall in these figures.

Figure 4

The mean temperature component for ${\mathrm{Ra}}_p$ = 400. Solid (dashed) lines correspond to the upper (lower [6]) heating.

Figure 5

Variations of the critical periodic Rayleigh number ${\mathrm{Ra}}_{p,\mathrm{cr}}$ as a function of α. Curves 1, 2, 3 correspond to the longitudinal, transverse, and oblique rolls, respectively. See [6] for derivation of the form of the asymptote of ${\mathrm{Ra}}_{p,\mathrm{cr}}$ for $\alpha \to \infty$.

Figure 6

Variations of the critical wave numbers ${\delta }_{\mathrm{cr}}$ and ${\beta }_{\mathrm{cr}}$ as functions of α. Curves 1, 2, 3 correspond to the longitudinal, transverse, and oblique rolls, respectively.

Figure 7

The disturbance flow field for the upper (a) and lower (b) [6] heatings for the longitudinal rolls with α = 4, δ = 2, and ${\mathrm{Ra}}_p$ = 3042. Solid (dotted) lines correspond to the secondary (primary) convection. The actual pattern consists of a superposition of the primary convection and the secondary convection multiplied by an arbitrary constant.

Figure 8

The disturbance isotherms for the upper (a) and lower (b) [6] heatings for conditions from Fig. 7.

Figure 9

The disturbance flow field at $y$ = 0 for the upper (a) and lower (b) heatings for the transverse rolls for α = 5, β = 1.62, and ${\mathrm{Ra}}_p$ = 3326. Path lines are illustrated using solid lines.

Figure 10

The disturbance isotherms for the upper (a) and lower (b) heatings for conditions from Fig. 9. Solid (dotted) lines identify the positive (negative) temperatures.

Figure 11

The disturbance flow field at $y$ = 0 for the upper (a) and lower (b) heatings for the oblique rolls for α = 2.7, β = 0.8, δ = 1.35, and ${\mathrm{Ra}}_p$ = 2411.

Figure 12

The disturbance isotherms for the upper (a) and lower (b) heatings. Conditions are the same as in Fig. 11. Solid (dotted) lines identify the positive (negative) temperatures.