Enok Palm

On the tendency towards hexagonal cells in steady convection

This paper attempts to explain theoretically the observed results that (1) the cells in steady convection approach a hexagonal form, and (2) the occurrence of ascent or descent in the middle of the cell depends on how the kinematical viscosity varies with temperature. The theory is based on non-linear equations and, of course, a variable coefficient of viscosity. It is found that, due to the variation of viscosity with temperature, the non-linear terms contain a second-order term which is destabilizing. This second-order term regulates the development and leads to a final motion composed of regular hexagons with ascent or descent in the middle of the cell according as the viscosity decreases or increases with temperature. The influence of a variable viscosity on Rayleighs result concerning the initiation of convection is obtained as a by-product.

On steady convection in a porous medium

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcys law.

Wave forces on three-dimensional floating bodies with small forward speed

A boundary-integral method is developed for computing first-order and mean second-order wave forces on floating bodies with small forward speed in three dimensions. The method is based on applying Greens theorem and linearising the Green function and velocity potential in the forward speed. The velocity potential on the wetted body surface is then given as the solution of two sets of integral equations with unknowns only on the body. The equations contain no water-line integral, and the free-surface integral decays rapidly. The Timman-Newman symmetry relations for the added mass and damping coefficients are extended to the case when the double-body flow around the body is included in the free-surface condition. The linear wave exciting forces are found both by pressure integration and by a generalised far-field form of the Haskind relations. The mean drift force is found by far-field analysis. All the derivations are made for an arbitrary wave heading. A boundary-element program utilising the new method has been developed. Numerical results and convergence tests are presented for several body geometries. It is found that the wave exciting forces are presented for several body geometries. It is found that the wave exciting forces and the mean drift forces are most influenced by a small forward speed. Values of the wave drift damping coefficient are computed. It is found that interference phenomena may lead to negative wave drift damping for bodies of complicated shape.

WAVE RADIATION AND WAVE DIFFRACTION FROM A SUBMERGED BODY IN A UNIFORM CURRENT

Radiation and diffraction of free-surface waves due to a submerged body in a uniform current is considered. The fluid layer is infinitely deep and the motion is two-dimensional. Applying the method of integral equations, the radiation problem and the diffraction problem for a submerged circular cylinder are examined. For small speed U of the current a forced motion of a given frequency will give rise to four waves. It is shown, however, that, for a circular cylinder, an incoming harmonic wave gives rise to two waves only. Depending on the frequency, the new generated wave may be considered as a transmitted or a reflected wave. The mean second-order force is computed. For the radiation problem the first-order damping force is also obtained. It is shown that, for some values of the parameters, the damping force is negative. This result is closely related to the fact that a harmonic wave travelling upstream with a phase velocity less than U conveys negative energy downstream. The forces remain finite as U σ/ g (σ ≡ the frequency, g ≡ the acceleration due to gravity) approaches ¼.

CONVECTION DUE TO INTERNAL HEAT SOURCES

This paper is concerned with convection generated by uniformly distributed internal heat sources. By a numerical method it is found that the planform is down-hexagons for infinite Prandtl numbers and Rayleigh numbers up to at least 15 times the critical value. The motion is also studied for finite Prandtl numbers and small supercritical Rayleigh numbers by using an amplitude expansion. It turns out that a small subcritical regime exists. Moreover, it also emerges that for Prandtl numbers less than 0.25 the stable planform is up-hexagons. In §3 a necessary condition in order to obtain a hexagonal planform is derived when the coefficients in the differential equations are a function of the vertical co-ordinate z .

A method for computing unsteady fully nonlinear interfacial waves

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchys theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

PROPULSION OF A FOIL MOVING IN WATER WAVES

Propulsion of a foil moving in the water close to a free surface is examined. The foil moves with a forward speed U and is subjected to heaving and pitching motions in calm water, head waves or following waves. The model is two-dimensional and all equations are linearized. The fluid is assumed to be inviscid and the motion irrotational, except for the vortex wake. The fluid layer is infinitely deep. The problem is solved by applying a vortex distribution along the centreline of the foil and the wake. The local vortex strength is found by solving a singular integral equation of the first kind, which appropriately is transformed to a non-singular Fredholm equation of the second kind. The vortex wake, the forward thrust upon the foil and the power supplied to maintain the motion of the foil are investigated. The scattered free surface waves are computed. For moderate values of U σ/ g ( U is forward speed of the foil, σ is frequency of oscillation, g is acceleration due to gravity) it is found that the free surface strongly influences the vortex wake and the forces upon the foil. When the foil is moving in incoming waves it is found that a relatively large amount of the wave energy may be extracted for propulsion. As an application of the theory the propulsion of ships by a foil propeller is examined. The theory is compared with experiments.

The mean drift force and yaw moment on marine structures in waves and current

The effect of the steady second-order velocities on the drift forces and moments acting on marine structures in waves and a (small) current is considered. The second-order velocities are found to arise due to first-order evanescent modes and linear body responses. Their contributions to the horizontal drift forces and yaw moment, obtained by pressure integration at the body, and to the yaw drift moment, obtained by integrating the angular momentum flux in the far field, are expressed entirely in terms of the linear first-order solution. The second-order velocities may considerably increase the forward speed part of the mean yaw moment on realistic marine structures, with the most important contribution occurring where the wave spectrum often has its maximal value. The contribution to the horizontal forces obtained by pressure integration is, however, always found to be small. The horizontal drift forces obtained by the linear momentum flux in the far field are independent of the second-order velocities, provided that there is no velocity circulation in the fluid.

Convective momentum and mass transport in porous sloping layers

Natural convection, caused by nonhorizontal isolines for water density in sloping sandstone layers, is studied. The flow field is calculated for a fluid-filled three-layer model composed of two identical sandstone strata separated by a low-permeability layer (shale). The condition for recirculation within the sandstone is found. It is shown that for layers of sufficiently small aspect ratio, the shale layer has negligible effect on the flow. With the assumption that the fluid density is a function only of temperature, the effect of the flow on the change of porosity in the sandstone layers is examined. The spatial change of porosity is calculated for small values of time. It is shown that the maximum change takes place at the midpoints of the end walls if the angle between the isotherms and the slab is less than 36.5° and at the midpoints of the lateral boundaries if this angle is greater than 36.5°. This result is independent of the angle the slab makes with the gravity vector. Also discussed is how the angle the isotherms make with the horizontal depends on the thermal conductivities of the sandstone layers and the bounding rock.

On the Non-linear Stability of Plane Couette Flow

Plane Couette flow is defined as the flow which takes place between two parallel planes in the case of no pressure gradient in the flow direction. When no disturbance occurs, the velocity will be a linear function of the vertical coordinate z.