AdvancedTraining_8_Solver.pdf

Solver
-Discretized equations must be solved
-Coefficients in equations are coupled
-e.g. Velocities are influenced by pressure gradients
-e.g. Velocities are influenced by temperature gradients (buoyancy)
-Must therefore solve iteratively
-Pressure and Temperature are
evaluated at Cell Centres
-Velocities are evaluated at Cell Faces
-Staggered Grid
-Gives correct pressure gradient terms in momentum equations
-Velocities are evaluated where needed for convection-diffusion equations
-At start of solution, velocity field and pressure field are unknown
-FLOTHERM uses SIMPLEST algorithm to evaluate coupled velocity and pressure field
-SIMPLEST is based on the Semi-
Implicit Method for Pressure-Linked Equations (developed by Patankar and Spalding)

-Solution of the equations proceeds as follows:
-A pressure field p* is guessed
-The discretized momentum equation is solved, to give velocity components, u* and v*

-Our second constraint on the velocity field is that it must satisfy continuity
-We use this constraint to determine the pressure correction, p’

-Solver continues until the imbalances in the equations are less than that specified by the termination criteria
-There are two methods for controlling convergence:
-Linear relaxation
-False time step

-The central coefficient is adjusted using the false time step
-Mass of cell / false time step is added to central coefficient

False Time Step
-False time step should be based on physical time scale of problem
-Residence time in domain
-Residence time in average cell

Solver
-Residence time in domain = Mean domain size / average velocity
-Residence time in cell = Mean cell size/ average velocity
-Normally, will set false time step within these limits

-Because pressure correction comes from the continuity equation, no false time step for pressure
-Can use linear relaxation instead
-Or use false time steps for velocities
-Conduction only problems, no false time step is needed
-Should solve in 1 outer iteration
-More efficient to limit inner iterations and perform several outer iterations

Linear Solver
-The discretized equations for pressure, velocity and temperature must be solved
-Need an linear solver to do this
-FLOTHERM offers three options
-Segregrated Solver
-Segregrated Conjugate Residual
-Multigrid
-Segregated:
-Uses Gauss-Seidel solver
-Looks locally at flow field
-Can be slow to capture trends
-Segregrated Conjugate Residual
-Based on Conjugate Gradient method
-Considers whole-field interactions
-Fast at resolving trends through whole domain m.aji87.cn
-Multigrid
-Same as segregrated conjugate residual for velocities and pressure
-For temperature, solves on progressively coarser grid
-Potentially very fast to converge temperature field; removes need for block correction

Solver
-Block correction
-Splits domain into blocks
-Generates new discretized equations for each block
-Solves equations
-Should give fast convergence for systems with large conductivity variations

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