MTF071 Computational Fluid Dynamics of Turbulent
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1 CHALMERS TEKNISKA HÖGSKOLA Termo- och Fluiddynamik Göteborg MTF07 Computational Fluid Dynamics of Turbulent Flow Task K2 Lars Davidson In Task K you worked with the diffusion equation Now we add the convective terms (left-hand side see Eq and put the diffusion terms on the right-hand side Equation is called the convection-diffusion equation or the transport equation for temperature You should write a computer program to solve Eq in two dimensions with finite volume methods Use the hybrid upwind/central differencing scheme see Eq 543 in [] Note that the continuity error It is recommended that you use Matlab The velocity field and the grid are given for each group and are available on the net; there is an m-file (pl_vectm by which you can read and plot the vector field of the velocity The task should be carried out in groups of two (or one and it should be presented both orally and written The two-dimensional transport equation for temperature reads #" $&%( The algebraic equation system should be solved using both Gauss-Seidel and TDMA ( How to proceed Fetch the grid and velocity field from the net Plot the velocity field using the m-file pl_vectm From the supplied velocity field you can find out the location and extent of the inlet and outlet Start to write you program to solve Eq
2 Use the coarse grid when you are writing/debugging your program When the program works you should also use the fine grid Note that when prescribing a heat flux you must divide by $ % since Eq is an equation for temperature rather than for energy Use ( denotes the coordinate normal to the wall at walls if no boundary condition is given (see tables Convergence It is very important to verify that a converged solution has been obtained At each iteration compute the residual as where " # $ % & ( is a temperature flux used to normalize the residual The temperature flux should be representative of the total flux in the domain In the present work it is suitable to take as the inlet mass flux multiplied by the difference of the temperature flux at inlet and outlet ie -/ 0 (3 where is the temperature difference between inlet and outlet The solution is consider as converged when 32 Presentation of the work The work should be presented both orally and in a form of a short report The oral presentation should be approximately 0 minutes (use some slides Present the problem how you carried out the work problems encountered and how you solved them The presentation and the report must include the following points: Sensitivity to boundary conditions Make (an interesting change of one boundary condition 2 Sensitivity to convergence Does the temperature change if you use another convergence criterium? Test and 3 Sensitivity to heat conductivity If you decrease (ie " the convective transport will increase relative to the diffusive one Increase and decrease by a factor of 00 Discuss how the results are affected 4 Plot the heat flux (along a Dirichlet boundary or the wall temperature (along a Neumann boundary as a function of the distance along the wall 5 Check if you have global conservation ie does the heat flux through all boundaries (inlet outlet and walls sum up to zero (as it should if you don t have any source term * + (2 2
3 GRID : to 5 Physical data: " $ % Boundary conditions: GRID 2: 6 to 0 Physical data: " $&% Boundary conditions: GRID 3: to 5 Physical data: " $ % Boundary conditions: ( GRID 4: 6 to 20 Physical data: " $ % Boundary conditions: ( ( GRID 5: 2 to 25 Physical data: " $&% Boundary conditions: ( References HK Versteegh and W Malalasekera An Introduction to Computational Fluid Dynamics The Finite Volume Method Longman Scientific & Technical Harlow England bc # " $ Table : Boundary conditions for to 5 3
4 0 0 ( ( bc $ $ Table 2: Boundary conditions for 6 to bc # " $ Table 3: Boundary conditions for to bc ( # 0( $ Table 4: Boundary conditions for 6 to bc ( $ Table 5: Boundary conditions for 2 to 25 4
5 PSfrag replacements Figure : Configuration The extent in the third coordinate direction is 5
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