Lecture Objectives

1. Lecture Objectives • Correction & Review • Iterations vs. Unsteady-state calculation • Solution of CFD conservation equations • Link pressure and velocities in NS equations • SIMPLE Algorithm

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

3. Correction!General Transport Equation unsteady-state Fully explicit method: Or different notation: Implicit method For Vx>0 For Vx<0

4. Steady state vs. Unsteady state Steady state We use iterative solver to get solution Unsteady state We use iterative solver to get solution and We iterate for each time step • Make the difference between • - Calculation for different time step • - Calculation in iteration step

5. Advection diffusion equation 1-D, steady-state Dx Dx N N+1 N-1 Different notation: Dx General equation

6. General Iteration Procedure Point Jacobi Solverfor steady state problem Example: Advection diffusion equation, 1-D, steady-state, 4 nodes 1) Explicit format: 4 3 1 2 2) Guess initial values: 3) Substitute and calculate: Substitute and calculate: 4) ………………………….

7. Unsteady-state Iterations vs. Unsteady-state calculation For time step t or t+Dt ? Explicit or implicit method

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

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

10. Solution of CFD conservation equations How to solve pressure and velocities in NS equations?

11. 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

12. 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

13. Pressure and velocities in NS equations How to link pressure field with continuity equation? SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm Dx Dx P E W Dx Ae Aw 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

14. 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

15. 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

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