Objective Numerical methods Finite volume

Finite Volume Method - Conservation of f for the finite volume Divide the whole computation domain into sub-domains One dimension: n h W P dx E dx w e s Dx l e w - Finite volume is a fixed space in the flow domain with imaginary boundaries that allow the fluid to flow in and out. - Integral conservation of the quantities such as mass, momentum and energy. f

Convection term dxw P dxe W E Dx – Central difference scheme: - Upwind-scheme: If Vx>0 and If Vx<0 and

Diffusion term W dxw P dxe E Dx w e

Summary: Steady–state 1D I) X direction If Vx > 0, If Vx < 0, Convection term - Upwind-scheme: W P dxw dxe E and a) and Dx w e Diffusion term: b) When mesh is uniform: DX = dxe = dxw Assumption: Source is constant over the control volume c) Source term:

1D example - uniform mesh After substitution a), b) and c) into I): We started with partial differential equation: same and developed algebraic equation: We can write this equation in general format: Unknowns Equation coefficients

1D example multiple (N) volumes N unknowns 1 2 3 i N-1 N Equation for volume 1 N equations Equation for volume 2 …………………………… Equation matrix: For 1D problem 3-diagonal matrix

3D problem Equation in the general format: H N W P E S L Wright this equation for each discretization volume of your discretization domain A F 60,000 elements 60,000 cells (nodes) N=60,000 x = 60,000 elements 7-diagonal matrix This is the system for only one variable ( ) When we need to solve p, u, v, w, T, k, e, C system of equation is larger

Iteration method Alternative to use matrix solver tool is to use iterations You can use excel if you are not familiar with matrix solver tools General Iteration Procedure: 1) Express equation in explicit form 2) Guess initial values 3) Substitute initial values and calculate new values 4) Substitute new values and calculate newer values 6) Repeat step 4) until convergence is achieved example Iterations -residual Value: T1 Residual initial guess 22 iteration 1 23 1.00000 iteration 2 23.25 0.25000 iteration 3 23.390625 0.14063 iteration 4 23.483459 0.09283 iteration 5 …… --- 23.96441 Iteration 98 23.96444 0.00003 2 2 2 Difference of value between two iteration 2 2

Numerical instability divergency divergence variable solution convergence iteration

Navier Stokes Equations Continuity equation This velocities that constitute advection coefficients: F=rV Momentum x Momentum y Momentum z Pressure is in momentum equations which already has one unknown In order to use linear equation solver we need to solve two problems: find velocities that constitute in advection coefficients 2) link pressure field with continuity equation

Pressure and velocities in NS equations How to find velocities that constitute in advection coefficients? For the first step use Initial guess And for next iterative steps use the values from previous iteration

Pressure and velocities in NS equations How to link pressure field with continuity equation? SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm W Dx P Dx E Dx Aw Ae Aw=Ae=Aside We have two additional equations for y and x directions The momentum equations can be solved only when the pressure field is given or is somehow estimated. Use * for estimated pressure and the corresponding velocities

SIMPLE algorithm Guess pressure field: P*W, P*P, P*E, P*N , P*S, P*H, P*L 1) For this pressure field solve system of equations: x: y: ……………….. ……………….. z: Solution is: 2) The pressure and velocity correction P = P* + P’ P’ – pressure correction For all nodes E,W,N,S,… V = V* + V’ V’ – velocity correction Substitute P=P* + P’ into momentum equations (simplify equation) and obtain V’=f(P’) V = V* + f(P’) 3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V 4) Solve T , k , e equations

SIMPLE algorithm start Guess p* p=p* Step1: solve V* from momentum equations Step2: introduce correction P’ and express V = V* + f(P’) Step3: substitute V into continuity equation solve P’ and then V Step4: Solve T , k , e equations no Converged (residual check) yes end

Other methods SIMPLER SIMPLEC variation of SIMPLE PISO COUPLED - use Jacobeans of nonlinear velocity functions to form linear matrix ( and avoid iteration )

Relaxation Relaxation with iterative solvers: When the equations are nonlinear it can happen that you get divergency in iterative procedure for solving considered time step divergence variable solution convergence Solution is Under-Relaxation: Y*=f·Y(n)+(1-f)·Y(n-1) Y – considered parameter , n –iteration , f – relaxation factor For our example Y*in iteration 101=f·Y(100)+(1-f) ·Y(99) f = [0-1] – under-relaxation -stabilize the iteration f = [1-2] – over-relaxation - speed-up the convergence iteration Value which is should be used for the next iteration Under-Relaxation is often required when you have nonlinear equations!

Example of relaxation (example from homework assignment) Example: Advection diffusion equation, 1-D, steady-state, 4 nodes 1) Explicit format: 1 2 3 4 2) Guess initial values: 3) Substitute and calculate: 4) Substitute and calculate: Substitute and calculate: ………………………….