Mixed mode convection in an inclined slot

Abstract

Nonlinear interaction between transverse disturbances and longitudinal rolls has been investigated for flow in an inclined slot with a heated lower wall when both modes of instability occur at nearly the same value of the control parameter. This condition is shown to be possible for a fluid with Prandtl number greater than 0.263 897, For slightly supercritical values of the Rayleigh number (R) when the critical Rayleigh number for longitudinal rolls RLC is somewhat less than that for transverse stationary rolls, RSC, and for transverse travelling waves, RTC, longitudinal rolls occur first and then remain stable as R is increased beyond RSC or RTC; no mixed mode state occurs. In contrast, if RSc or RTc is slightly below RLC, pure transverse modes exist for only a relatively small range of R beyond RSC or RTC. Thereafter, a three-dimensional mixed mode state occurs well before RLC is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches RLC the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than RLC. Mixed mode convection is also investigated for a special choice of parameters when three modes, namely transverse stationary rolls, transverse travelling waves and longitudinal rolls, become unstable simultaneously. Longitudinal rolls again emerge as the stable supercritical state.

Spatio-temporal patterns in inclined layer convection

“…In general, the codimension 2 point γ = γ c2 , where the critical Rayleigh numbers R l c (γ) of the longitudinal rolls and R t c (γ) of the transverse ones are equal, moves continuously towards γ = 0 for decreasing Pr . Below Pr < 0.264 (more precisely calculated in Fujimura & Kelly (1992)), only the primary bifurcation to transverse rolls prevails even at incremental inclinations from the flat γ = 0 state.…”

Pattern Formation in Inclined Layer Convection

“…For 0 • ≤ θ 77.5 • we observed longitudinal rolls, subharmonic oscillations, Busse oscillations, undulation chaos, and crawling rolls. In the neighborhood of the codimension two point [11] for thermal and shear driven instability (77.5 • θ 84 • ), we observed drifting bimodals, drifting transverse rolls, and localized longitudinal and transverse bursts. For inclinations θ 84 • we found drifting transverse rolls, switching diamondpanes, and longitudinal bursts.…”

“…Above a critical angle θ c ( 90 • ) the shear flow causes a primary instability to transverse rolls (TR) with roll-axes perpendicular to the shear flow [7]. The few prior experiments [8-10] on ILC showed reasonable agreement with the linear theory [7,[11][12][13]. These experiments also demonstrated that LR are unstable to some form of undulations [8][9][10], in qualitative agreement with theory [12,13], but the quantitative details of the state were inaccessible due to experimental limitations.Here we report the first experimental results on pattern formation in ILC for large aspectratio systems in a range of inclination angles 0 • ≤ θ ≤ 120 • , i.e., from horizontal (heated from below) to past vertical (heated from above).…”

“…Here ǫ ≡ (∆T − ∆T c (θ))/∆T c (θ) is the reduced control parameter. Theoretically, Fujimura and Kelly [11] predicted a forward bifurcation to transverse rolls, which lose stability to bimodals at ǫ ≈ 0.001 in a narrow angular region. We find good agreement with these predictions, but with the difference that the experimentally observed patterns are drifting.…”

On the oscillatory modes of compressible thermal convection in inclined differentially heated cavities

“…, the component of gravity tangential to the fluid layer will cause buoyant fluid to flow up along the warm wall and down along the cold wall. This shear flow, in turn, tends to break the in-plane isotropy of the usual horizontal layer heated from below, causing a departure of the dynamics from well-known RB mechanisms (Chen and Pearlstein[72]; Fujimura and Kelly[73]; Kaloni and Qiao[74]). For <0, the higher the absolute value of , the less important the contribution of the shear flow is, until for  = -90° classical RB convection isrecovered.…”