Modeliranje sevalnega toplotnega toka v sistemu Navier-Stokesovih enačb z metodo robnih elementov - Crnjac, Peter (visokošolska dela, 2016)
In fluid dynamics all three physically different mechanisms of heat transfer can occur: conduction and convection described by the Navier-Stokes equations, and the radiation phenomenon, which is a complex nonlinear mode of heat transfer gaining importance at sufficiently high temperatures. The objective of this work is to develop a numerical model to solve coupled problems involving conductive, convective and radiative heat transfer in an absorbing, emitting, and isotropic scattering medium. The present development will be focused on the laminar flow of the isotropic Newtonian fluid at low Mach numbers, so the changes in mass density are negligible and the fluid is treated to be incompressible. The radiation impact on overall heat transfer is conveyed in the energy equation, where in the non-convective energy flux, besides the diffusion heat flux also the radiative flux needs to be taken into consideration. This is done in such a way that we include the term for the divergence of the radiative flux vector into the energy equation as the radiative energy source. The equation used to solve the temperature field is an integro-differential equation and still presents a serious issue in the computational fluid dynamics. In this study the contributions from radiant energy transfer are presented using two approaches for optical thick fluids: the Rosseland diffusion approximation with two different techniques for implementing boundary conditions and the P_1 approximation. In diffusion approximation we consider an absorbing and emitting medium with isotropic scattering. The impact of heat radiation is analysed by applying the Rosseland model, where the radiative flux is given in a gradient form with a temperature dependent coefficient of radiative conductivity. The coefficient of radiative conductivity is taken into account in the energy equation as a non-linear source term. However near a boundary the diffusion approximation may not be accurate as the radiation is not isotropic. To overcome this difficulty, the boundary condition at the edge of the medium is modified by applying the Robin radiative boundary condition. In the spherical harmonics approximation, the impact of radiation on the overall heat transfer is analysed by applying a more complicated P1 radiative model. In the model, we express the incident radiation at a given position in the radiation field by the nonlinear nonhomogeneous modified Helmholtz equation. Under the Marshak boundary condition, we solve the equation iteratively as a coupled system with the energy equation. The governing equations are in the next step transformed with the use of the velocity-vorticity variables formulation into transport equations for kinematics and kinetics. By applying the boundary domain integral method, we transform partial differential equations into integral equations. Green's function is chosen for the weighting function; with the boundary discretization we obtain the linear part of the solution, whereas by partitioning the domain into internal cells, we take into account the nonlinear contributions of the transport phenomena. We note down the obtained system of algebraic equations in the matrix form with the fully populated nonsymmetric matrix, with the column vector of unknown boundary field function and its normal derivative and with the vector of known values. To test the validity and accuracy of the proposed numerical scheme, we analyse the problem of the overall heat transfer in a closed square cavity filled with air, treated as the ideal gas. The velocity and temperature fields together with the total heat transfer are calculated for the Rayleigh numbers in the laminar regime, and the solutions compared to the published standard results.
Keywords: boundary-domain integral method; computational fluid dynamics; hitrostno-vrtinčna formulacija; naravna konvekcija; natural convection; prenosna sevalna enačba; računalniška dinamika tekočin; radiation; raiative transfer equation; robno-območna integralska metoda; sevanje; velocity-vorcity formulation