Rod W. Douglass

A modified tau spectral method that eliminated spurious eigenvalues

Abstract A modified tau spectral method is presented which eliminates the spurious eigenvalues produced by the usual tau method. The modified tau method essentially involves an appropriate factorization of the differential operators in the eigenvalue problem. It is developed for eigenvalue problems posed as single differential equations or as systems of such equations. Several eigenvalue problems are solved using both the usual and modified Chebyshev-tau methods, including the Orr-Sommerfeld stability equation for plane Poiseuille flow. The convergence of the modified tau method is shown to be at least as rapid as that of the usual tau method. The use of the tau coefficients in indicating convergence is also discussed.

Natural convection In narrow-gap, spherical annuli

Abstract The subject of this paper is natural convection of fluid contained within narrow-gap, spherical annuli. The flows are presumed to be steady and the fluid is assumed to follow the Oberbeck-Boussinesq model. With the gap being very small relative to the outer spheres radius, the dependent variables are solved for by using a regular perturbation method in powers of the relative gap width, e. Solutions were found for a heated outer sphere through terms of order ϵ 11 . The results include Nusselt numbers, and contours of streamlines and isotherms as functions of the Grashof and Prandtl numbers, e, and Q, the dimensionless uniform energy generation rate parameter. The value of ϵ ranged from 0.1 to 0.001, Pr from 0.01 to 10, and Gr from 7 × 10 6 to 5 × 10 12 .

Thermal convection in rotating spherical annuli—1. Forced convection

Abstract The steady forced convection of a viscous fluid contained between two concentric spheres which are maintained at different temperatures and rotate about a common axis with different angular velocities is considered. Approximate solutions to the governing equations are obtained in terms of a regular perturbation solution valid for small Reynolds numbers and a modified Galerkin solution for moderate Reynolds numbers. The resulting flow pattern, temperature distribution, and heat-transfer characteristics are presented for the various cases considered. The theoretical heat-transfer results for small and moderate Reynolds number flows within a spherical annulus with a stationary outer sphere are compared with previous experimental results for the large Reynolds number flow situation. The difference between conduction, Stokes flow, and boundary-layer convection is shown.

Prandtl number effects on the stability of natural convection between spherical shells

Abstract An analysis of the effect of the Prandtl number on the linear stability of axisymmetric ( m = 0) disturbances on steady natural convection contained between two concentric spherical shells when the gap is narrow are presented. The disturbance equations are solved using a truncated spectral series. Convergence of the series is examined. Prandtl numbers range from 0 to 100 while the relative gap-width is either 0.100, 0.075, or 0.050. Results confirm the hypothesis that experimentally observed changes in the basic motion for certain flow parameters are due to its instability and indicate that for any Prandtl number larger than a transition value, the unstable flows evolve to a steady pattern while for smaller Prandtl numbers the bifurcated flows are time periodic.

Linear stability of natural convection in spherical annuli

Natural convection in a Boussinesq fluid filling the narrow gap between two isothermal, concentric spheres at different temperatures depends strongly on radius ratio, Prandtl number, and Grashof number. When the inner sphere has a higher temperature than the outer sphere, and for fixed values of radius ratio and Prandtl number, experiments show the flow to be steady and axisymmetric for sufficiently small Grashof number and quasi-periodic and axisymmetric for Grashof numbers greater than a critical value. It is our hypothesis that the observed transition is a flow bifurcation. This hypothesis is examined by solving an appropriate eigenvalue problem. The critical Grashof number, critical eigenvalues, and corresponding eigenvectors are obtained as functions of the radius ratio, Prandtl number, and longitudinal wavenumber. Critical Grashof numbers range from 1.18 × 10 4 to 2.63 × 10 3 as Prandtl number Pr increases from zero to 0.7, for radius ratios of 0.900 and 0.950. A transitional Prandtl number Pr t exists such that for Pr Pr t the bifurcation is time-periodic and axisymmetric. For Pr > Pr t the bifurcation is steady and non-axisymmetric with wavenumber two A first approximation to the bifurcated flow is obtained using the critical eigenvectors. For Pr Pr t the bifurcation sets in as a cluster of relatively strong cells with alternating directions of rotation. The cells remain fixed in location, but pulsate with time. The cluster moves toward the top of the annulus as Pr increases toward Pr t . An important feature of the non-axisymmetric bifurcation for Pr > Pr t is a set of four cells located at each pole of the annulus in which the radial velocity alternates direction in moving from any one cell to an adjacent one. For fixed radius ratio, the average Nusselt number at criticality varies only slightly with Prandtl number.

Partial Spectral Expansions for Problems in Thermal Convection

The use of spectral expansions for solving nonlinear partial differential equations is explained, and two examples drawn from convective heat transfer are presented. For both problems the results agree well with regular perturbation solutions at parameter values for which the latter remain valid. Evidence is given to indicate that the spectral solutions are valid for considerably larger parameter values than can be reached with the perturbation methods.

Thermal convection in rotating spherical annuli—2. Stratified flows

Abstract The steady combined thermal convection of a viscous Boussinesq fluid contained between two concentric spheres is considered. The spheres are maintained at different temperatures and rotate about a common axis with different angular velocities. A uniform radial gravitational field acts on the fluid. Approximate solutions to the governing equations are obtained with a modified Galerkin technique for moderate Reynolds numbers. The resulting flow patterns, temperature distributions, and heat-transfer and torque characteristics are presented for several angular velocity ratios and degrees of stratification. It is shown that increasing the buoyancy forces alters the primary and secondary flow patterns as well as the temperature distributions. The total rate of heat transfer and torque are subsequently enhanced.

The effect of stable stratification on the motion in a rotating spherical annulus

Abstract The effects of stable stratification on the steady laminar flow of a viscous fluid in a rotating spherical annulus are investigated. Three rotational configurations are discussed: inner sphere rotating, outer sphere at rest; inner sphere at rest, outer sphere rotating; and both spheres rotating in opposite directions. The calculations include the primary and secondary circulations, the temperature distribution, and the heat-transfer characteristics of the flow. It is shown that the buoyancy forces are effective in reducing the intensity of the secondary circulation, and in one case cause a new circulation pattern to appear.

Convection in a rotating spherical annulus with a uniform axial gravitational field

Abstract The steady combined convection of a Boussinesq fluid enclosed between two concentric rotating spheres is analytically investigated. The spheres rotate at constant rates and are maintained at uniform, but unequal temperatures. A uniform gravity field acts parallel to the rotation axis. The governing equations are solved using a partial spectral expansion method. This method provides solutions for Reynolds and Prandtl numbers significantly larger than currently available. The general nature of the flow field is shown to depend on the Reynolds number, the Prandtl number, and the Grashof number (presented in the ratio Gr Re 2 ). Increasing any or all of these parameters causes enhanced convective heat transfer and an increase in the torque required to rotate the spheres. The secondary flow field is strongly dependent on the ratio Gr Re 2 , approaching the single-eddy pattern of natural convection as Gr Re 2 becomes large.

A numerical simulation of acrylic grout curing in a femoral implant

Abstract A numerical simulation of the thermal response of a femoral prosthesis-acrylic grout-bone composite during and after polymerization of the grout is presented. It is intended to model total hip replacement just after implantation of the prosthesis. The problem is formulated such that the governing energy equation depends only on the radial coordinate and time. The equations for the composite are solved numerically using finite differences. Time differencing is by the Crank-Nicolsen method. Results include thermal histories of the composite, emphasizing the grout-bone interface, as altered by various initial composite temperatures and the size and location of the grout. It is found that the grout-bone interface temperature can be dangerously large (50–90% above body temperature) and that this can be overcome by chilling of the metal prosthesis. Also, the amount of heat generated, thus the grout-bone interface temperature, is dependent on the volume of the grout and its location.