## Two-dimensional and axisymmetric turbulent thermal jets

2019
Mubai, Erick
##### Abstract
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