Browsing by Author "Mubai, Erick"

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    Two-dimensional and axisymmetric turbulent thermal jets
    (2019) Mubai, Erick
    In this dissertation, the two-dimensional and axisymmetric turbulent thermal jets arestudied. TheclosureproblemintheReynoldsaveragedNavier-Stokesequations is resolved by introducing the eddy viscosity while the closure problem in the averaged energy balance equation is resolved by introducing the turbulent thermal conductivity. The kinematic eddy viscosity and turbulent thermal conductivity are modelled using the mixing lengths formulation. The governing equations are then written in terms of the stream function to give a non-linear system of a third order partial differential equation coupled with a second order partial differential equationforthetemperaturedifference. Thegoverningsystemoftwopartialdifferential equations is reduced to two ordinary differential equations using the associated Lie point symmetry approach. The associated Lie-point symmetry approach is very powerful when dealing with partial differential equations. If a system of partial differential equations has a conserved vector, the Lie point symmetry associated with that conserved vector can be used to reduce the system of partial differential equations to a system of ordinary differential equations. This system of ordinary differential equations can be integrated at least once, according to the double reduction theorem [10,11]. TheassociatedLiepointsymmetryisderivedusingconservedvectorsfromconservationlawsofthegoverningpartialdifferentialequations. Theconservedvectors are obtained from the multiplier method. The conservation laws are used to obtain conserved quantities by integrating the conservation laws along the axis perpendicular to the centreline of the jet from the centreline to the boundary of the jet. The reduced ordinary differential equations for the invariant solutions can only be solved analytically when the kinematic viscosity and the thermal conductivity arebothzeroforthetwo-dimensionaljetandforothercasestheyaresolvednumerically. Theconservedquantitiesareusedtosupplementthehomogeneousboundary conditions to fully solve the jet problem. The shooting method is used to obtain the numerical solution
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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.