Nonlinear Bénard convection with rotation

The equations for nonlinear Bénard convection with rotation for a layer of fluid, thickness d, are derived using the Glansdorff & Prigogine (1964) evolutionary criterion as used by Roberts (1966) in his paper on non-rotational Bénard convection. The parameters of the problem in this case are the Rayleigh number R = αgΔθd/vK, the Taylor number T = 4d4Ω23/v2 and the Prandtl number Pr = v/K, where α is the coefficient of volume expansion, g the acceleration due to gravity, Δθ the temperature difference between the horizontal surfaces, v the kinematic viscosity, K the thermal diffusivity and Ω3 the rotation rate about the vertical direction. The asymptotic solution for two-dimensional cells (rolls) is investigated for large Rayleigh numbers and large Taylor numbers. For rolls the convection equations are found t o be independent of the Prandtl number. However, the solutions depend upon the Prandtl number for another reason. The rotational problem differs from the non-rotational one in that the Rayleigh number and the horizontal wavenumber a of the convection are now functions of the Taylor number. These are taken to be R ∼ ρTα′ and α ∼ ATβ, where α′ and β are positive numbers. Thermal layers develop as R becomes large with ρ or T becoming large. The order in which ρ and T are allowed to increase is important since the horizontal wavenumber a also increases with T and the convection equations can be reduced in this case. A liquid of large Prandtl number such as water has v [Gt ] K. Since R ∼ O (1/vK) and T ∼ O(1/v2), ρ will be greater than T for a given (large) Δθ and Ω3. Similarly, for a liquid of small Prandtl number such as mercury v [Lt ] K, and T is greater than ρ for a given Δθ and Ω3. For rigid-rigid horizontal boundaries with ρ large and then T large the ρ thermal layer has the same structure as for the non-rotating problem. As T → ∞ three types of thermal layers are possible: a linear Ekman layer, a nonlinear Ekman layer and a Blasius-type thermal layer. When the horizontal boundaries are both free the ρ thermal layer is again of the same structure as for non-rotating BBnard convection. As T → ∞ a nonlinear Ekman layer and a Blasius-type thermal layer are possible.When T is large and then ρ made large the differential equations governing the convection are reduced from eighth order to sixth order owing to a becoming large as T → ∞. There are Ekman layers as T → ∞, when the horizontal boundaries are both rigid. The ρ thermal layers now have a different structure from the non-rotating problem for both rigid-rigid and free-free horizontal boundaries. The equation for small amplitude convection near to the marginal case is derived and the solution for free-free horizontal boundaries is obtained.

A linearized theory for rotational supercavitating flow

In this paper methods are given for establishing qualitative and quantitative measures of the effects of rotation in supercavitating flows past slender bodies. A linearized theory is developed for steady, two-dimensional flow under the assumption that the flow has a constant rotation throughout. The stream function of the rotational flow satisfies Poisson's equation. By using a particular solution of this equation, the rotational problem is reduced to a problem involving Laplace's equation and harmonic perturbation velocities. The resulting boundary-value problem is solved by use of conformal mapping and singularities from thinairfoil theory. The theory is then applied to asymmetric shear flow past wedges and hydrofoils and to symmetric shear flow past wedges. The presence of rotation is shown to create significant changes in the forces acting on the slender bodies and in the shape and size of the trailing cavities.