1. Lecture Objectives: Explicit vs. Implicit Residual, Stability, Relaxation Simple algorithm

2. General Transport Equation unsteady-state 1-D Fully explicit method: Implicit method: Value form previous time step (known value) Make the difference between - Calculation for different time step - Calculation in iteration step

3. Explicit: Unsteady state Advection diffusion equation, 1-D Rarely used due to the problem with stability of calculation To achieve stable calculation  should be very small

4. Unsteady state Advection diffusion equation, 1-D Implicit: Iterative method: Explicit format (to solve system of equations) 2) Guess initial values: 3) Substitute and calculate: 4….) Iterate for considered time step In iteration substitute only these values Make the difference between iteration and calculation for next time step

5. Residual Example: x-exp(1/x)-2=0 Find x using iteration Explicit form 1: x=exp(1/x)+2 Explicit form 2: x=1/(ln(x)-ln(2)) Solution process: Guess x 0 Iteration : x 1 =exp(1/x 0 )+2, R 1 =x 1 -x 0 X 2 =exp(1/x 1 )+2, R 2 =x 2 -x 1 …….. ……. Not all iteration process converge! See the example for the same equation

6. Convergence example Explicit form 2: x=1/(ln(x)-ln(2)

7. Residual calculation for CFD Residual for the cell R  ijk =  k ijk -  k-1 ijk Total residual for the simulation domain R  total =  R  ijk | Scaled (normalized) residual R  =  R  ijk |/F  iteration cell position Variable: p,V,T,… For all cells Flux of variable  used for normalization Vary for different CFD software

8. 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 Under-Relaxation is often required when you have nonlinear equations! iteration convergence variable divergence solution 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 Value which is should be used for the next iteration

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

10. Navier Stokes Equations In order to use linear equation solver we need to solve two problems: 1)find velocities that constitute in advection coefficients 2) link pressure field with continuity equation This velocities that constitute advection coefficients: F=  V Pressure is in momentum equations which already has one unknown Continuity equation Momentum x Momentum y Momentum z

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

12. Pressure and velocities in NS equations How to link pressure field with continuity equation? SIMPLE ( Semi-Implicit Method for Pressure-Linked Equations ) algorithm 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 P E W xx xx xx Ae Aw Aw=Ae=A side We have two additional equations for y and x directions

13. 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: Solution is: x: y: z: ……………….. P = P* + P’ 2) The pressure and velocity correction P’ – pressure correction V = V* + f(P’) For all nodes E,W,N,S,… V’ – velocity correction Substitute P=P* + P’ into momentum equations (simplify equation) and obtain 3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V V’=f(P’) V = V* + V’ 4) Solve T, k,  equations

14. SIMPLE algorithm 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 Guess p* start end Converged (residual check) yes no p=p*

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