Purpose
This paper aims to present a high-order algorithm implemented with the modal spectral element method and simulations of three-dimensional thermal convective flows by using the full viscous dissipation function in the energy equation. Three benchmark problems were solved to validate the algorithm with exact or theoretical solutions. The heated rotating sphere at different temperatures inside a cold planar Poiseuille flow was simulated parametrically at varied angular velocities with positive and negative rotations.
Design/methodology/approach
The fourth-order stiffly stable schemes were implemented and tested for time integration. To provide the hp-refinement and spatial resolution enhancement, a modal spectral element method using hierarchical basis functions was used to solve governing equations in a three-dimensional space.
Findings
It was found that the direction of rotation of the heated sphere has totally different effects on drag, lateral force and torque evaluated on surfaces of the sphere and walls. It was further concluded that the angular velocity of the heated sphere has more influence on the wall normal velocity gradient than on the wall normal temperature gradients and therefore, more influence on the viscous dissipation than on the thermal dissipation.
Research limitations/implications
This paper concerns incompressible fluid flow at constant properties with up to medium temperature variations in the absence of thermal radiation and ignoring the pressure work.
Practical implications
This paper contributes a viable high-order algorithm in time and space for modeling convective heat transfer involving an internal heated rotating sphere with the effect of viscous heating.
Social implications
Results of this paper could provide reference for related topics such as enhanced heat transfer forced convection involving rotating spheres and viscous thermal effect.
Originality/value
The merits include resolving viscous dissipation and thermal diffusion in stationary and rotating boundary layers with both h- and p-type refinements, visualizing the viscous heating effect with the full viscous dissipation function in the energy equation and modeling the forced advection around a rotating sphere with varied positive and negative angular velocities subject to a shear flow.
New collocation path-following approach for the optimal shape parameter using Kernel method
