G. de Vahl Davis

Three-dimensional natural convection in a box: a numerical study

The solution of the steady-state Navier–Stokes equations in three dimensions has been obtained by a numerical method for the problem of natural convection in a rectangular cavity as a result of differential side heating. In the past, this problem has generally been treated as though it were two-dimensional. The solutions explore the three-dimensional motion generated by the presence of no-slip adiabatic end walls. For Ra = 10 4 , the three-dimensional motion is shown to be the result of the inertial interaction of the rotating flow with the stationary walls together with a contribution arising from buoyancy forces generated by longitudinal temperature gradients. The inertial effect is inversely dependent on the Prandtl number, whereas the thermal effect is nearly constant. For higher values of Ra , multiple longitudinal flows develop which are a delicate function of Ra, Pr and the cavity aspect ratios.

An evaluation of upwind and central difference approximations by a study of recirculating flow

Abstract The method of “upwind differencing” to approximate the convection terms in numerical studies of fluid motion was introduced to overcome stability restrictions believed to be imposed by the use of central differencing. Here, both methods are applied to a two dimensional model of the recirculating flow in a cavity with a sliding top. It is shown that the false diffusion associated with first order upwind difference approximations can cause the numerical solution to severely misrepresent the physical transport processes. The instability exhibited by solutions based upon second order approximations is elucidated by examining the conservation laws implied by the governing equations. Finally, by calculating three dimensional solutions, it is shown that the assumption of two dimensionality is of questionable validity at high values of the Reynolds number.

Laminar natural convection in an enclosed rectangular cavity

Abstract A study is described of the steady laminar two dimensional motion of a fluid in an enclosed cavity, the motion being generated by a temperature gradient normal to the direction of the body force. The governing equations have been solved numerically. The results are compatible with, and form an extension of, some previous theoretical and experimental results. Maximum values considered of the Rayleigh number (based on cavity height) were 2 × 10 5 for a square cavity and 1.25 × 10 6 for one of height/thickness ratio = 5. Some new details of the flow at high Rayleigh numbers have been revealed. It has been found that high Prandtl numbers exert a stabilizing influence on the numerical solution, while they have only a small effect on the final results (over the range 10 −1 ⩽ Pr ⩽ 10 3 ).

The method of the false transient for the solution of coupled elliptic equations

Abstract A method for the numerical solution of a system of coupled, nonlinear elliptic partial differential equations is described, and the application of the method to the equations governing steady, laminar natural convection is presented. The essential feature of the method is the conversion of the equations to a parabolic form by the addition of false time derivatives, thus, enabling a marching solution, equivalent to a single iterative procedure, to be used. The method is evaluated by applying it to a well known two-dimensional problem and some examples of its use in three dimensions are given.

A numerical method for calculating temperature distributions in machining, from force and shear angle measurements

Abstract The finite element method is applied to calculate the temperatures in orthogonal machining with account being taken of the finite plastic zones, (a) in which the chip is formed and (b) in which further plastic flow occurs at the tool-chip interface, and also of the shape and thermal properties of the cutting tool. Mathematical models of both primary and secondary deformation are described. These allow complete temperature distributions to be obtained, given only the experimental values of tool force and chip thickness, and the thermal properties of the work and tool. The method has potential use in a predictive theory where only the fundamental properties of the work and tool materials are known.

Natural Convection between Concentric Vertical Cylinders

The motion of a fluid in the closed annular cavity formed by two concentric vertical cylinders and two horizontal planes has been analyzed by a numerical solution of the equations of motion and energy using a high‐speed digital computer. The motion is generated by a radial density gradient caused by the thermal boundary conditions which are, typically: inner cylinder at a (dimensionless) temperature of unity; outer cylinder at a temperature of zero; horizontal boundaries adiabatic. The fluid is assumed to have constant thermodynamic and transport properties except for the density, which is temperature‐dependent in the buoyancy term of the equation of vertical motion (the Boussinesq approximation); the flow is assumed to be axisymmetric. The equations of time‐dependent motion have been solved, so that both transient and steady‐state solutions are obtained. The parameters of the problem, and the respective ranges of values which have been considered, are: Rayleigh number (based on gap width) up to 2 × 105; ...

A computational study of transient plane front solidification of alloys in a Bridgman apparatus under microgravity conditions

A mathematical model of heat, momentum and solute transfer during directional solidification of binary alloys in a Bridgman furnace has been developed. A fixed grid single domain approach (enthalpy method) is used. The effects of coupling with the phase diagram (a concentration-dependent melting temperature) and of thermal and solutal convection on segregation of solute, shape and position of the solid/liquid interface are investigated. A vorticity–stream function formulation is used for calculation of the velocity fields. The results presented include calculations at 1 and 10 μg, both neglecting and including the dependence of melting temperature on concentration.

Mixed convection in vertical, cylindrical annuli

Abstract The laminar flow patterns and heat transfer for air contained in the enclosure formed between two vertical, concentric cylinders and two horizontal planes have been studied numerically. The inner cylinder and one of the horizontal planes are heated and rotated about the vertical axis; the other horizontal plane and outer cylinder are cooled and kept stationary. This geometry simulates the gaps at the ends of the rotor of a small, air-cooled, vertically mounted electric motor. The results facilitate the thermal design of such a motor. The influences of geometry (described by the radius ratio R and aspect ratio A), Ra and Re on temperature and velocity distributions have been investigated. Solutions have been obtained for 0.25 ⩽ A ⩽ 4.0, 1.2 ⩽ R ⩽ 8.0, 10 ⩽ Re ⩽ 300 and 103 ⩽ Ra ⩽ 105. It has been found that for low values of R and high values of Re the flow is dominated by centrifugal forces, whereas for high A and Ra buoyancy effects determine the flow patterns and, therefore, the heat transfer. Monocellular flow patterns have been found for the cases where one of these forces is dominant; otherwise two- or three-cell structures have been obtained.

A numerical and experimental study of natural convection and interface shape in crystal growth

Abstract A numerical and experimental study has been conducted on the crystal growth of succinonitrile in a horizontal Bridgman apparatus. The shape of the solid—liquid interface was significantly influenced by three-dimensional natural convection in the liquid adjacent to the interface. The interface profile observed during experiments was compared with predictions from a two-dimensional (2D) finite element analysis and a three-dimensional (3D) finite difference approach. Good agreement was achieved between the experimental and predicted results. The computed velocities in the vicinity of the interface were found also to be in good agreement with the measured experimental velocities.

Numerical solutions and experimental results for three dimensional buoyancy driven flows in tilted cylinders

Abstract Finite difference solutions and experiments are presented for free convection in differentially heated cylinders when the inclination varies with respect to the gravity vector. The results are compared with asymptotic solutions in horizontal cylinders for the core- and boundary layer-driven regimes. Over a range of 60° from the horizontal, a good agreement is obtained between the numerical and experimental results for large aspect ratio (A=10) and Rayleigh number (Ra≈20,000).