School of Computer Science and Applied Mathematics (ETDs)

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Browsing School of Computer Science and Applied Mathematics (ETDs) by Keyword "Associated Lie point symmetry"

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    Two-dimensional turbulent classical and momentumless thermal wakes
    (University of the Witwatersrand, Johannesburg, 2023-07) Mubai, Erick; Mason, David Paul
    The two-dimensional classical turbulent thermal wake and the two-dimensional momentumless turbulent thermal wake are studied. The governing partial differential equations result from Reynolds averaging the Navier-Stokes, the continuity and energy balance equations. The averaged Navier-Stokes and energy balance equations are closed using the Boussinesq hypothesis and an analogy of Fourier’s law of heat conduction. They are further simplified using the boundary layer approximation. This leads to one momentum equation with the continuity equation for an incompressible fluid and one thermal energy equation. The partial differential equations are written in terms of a stream function for the mean velocity deficit that identically satisfies the continuity equation and the mean temperature difference which vanishes on the boundary of the wake. The mixing length model and a model that assumes that the eddy viscosity and eddy thermal conductivity depend on spatial variables only are analysed. We extend the von Kármán similarity hypothesis to thermal wakes and derive a new thermal mixing length. It is shown that the kinematic viscosity and thermal conductivity play an important role in the mathematical analysis of turbulent thermal wakes. We obtain and use conservation laws and associated Lie point symmetries to reduce the governing partial differential equations to ordinary differential equations. As a result we find new analytical solutions for the two-dimensional turbulent thermal classical wake and momentumless wake. When the ordinary differential equations cannot be solved analytically we use a numerical shooting method that uses the two conserved quantities as the targets.