Solution algorithms for assignment problems

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  • 1. Arvind Deshpande
  • 2. 1. Take the input from the user :Initial temp(Tinitial), Boundary condition information, material properties density, thermal conductivity, specific heat, length, height, no. of interior control volumes control volumes in x-direction and no. of control volumes in y- direction and time step (based on stability criterion).2. Calculate x and y coordinates for all points.3. Calculate aw, ae, an, as and ap for all cv’s. Formula changes for cv’s close to boundary.4. Implement all boundary conditions.
  • 3. 5. Set Told and Toldt= Tinitial for all cv’s and Tnew = 0 for all cv’s.6. Increase time by time step and calculate Tnew at all cv’s using Gauss-Seidal point by point or line by line TDMA method.7. Check for convergence for iterations within time step. Residual = Tnew – Told (max residual /average residual/rms residual <ε)8. If converged, goto step 9 otherwise assign Told = Tnew and go to step 6.
  • 4. 9. Check for “steady state” Residual = Tnew – Toldt (max residual /average residual/rms residual <ε)10. If converged, stop otherwise assign Toldt = Tnew and go to step 6.
  • 5. 1. Take the input from the user :Inlet temp of flow, Boundary condition information, material properties density, thermal conductivity, specific heat, length, height, no. of interior control volumes control volumes in x-direction and no. of control volumes in y-direction.2. Calculate x and y coordinates for all points.3. Calculate velocity u and v using given formula.4. Calculate Dw, De, Dn, and Ds for all cv’s. Formula changes for cv’s close to boundary.
  • 6. 5. Based on CDS/UDS/Hybrid calculate aw, ae, an, as ap for all cv’s. Formula changes for cv’s close to boundary.6. Implement all boundary conditions.7. Set Told and Tnew = 0 for all cv’s.8. Calculate Tnew at all cv’s using Gauss-Seidal point by point or line by line TDMA method.9. Check for convergence. Residual = Tnew – Told (max residual /average residual/rms residual <ε)
  • 7. 9. If converged, goto step 10 otherwise assign Told = Tnew and go to step 6.10. For each axial location, calculate bulk mean temp, heat transfer coefficient and Nusselt no.
  • 8. Solution algorithm – Lid driven cavity (SIMPLE)
  • 9. 1. Take the input from the user :Lid velocity, material properties density, dynamic viscosity, length, height, no. of interior control volumes control volumes in x-direction (j) and no. of control volumes in y-direction (i), under relaxation factor for pressure and velocity. (You can use recommended under relaxation factors)2. Calculate x and y coordinates for all points.3. Implement all boundary conditions.4. Set uold =unew = vold =vnew =Pold =Pnew = 0 for all interior cv’s.
  • 10. 5. Calculate Dw, De, Dn, and Ds for all cv’s.6. Calculate Fw, Fe, Fn, and Fs for all cv’s. You will have to use interpolation (average) for velocity.7. Based on CDS/UDS/Hybrid calculate aw, ae, an, as for all cv’s. Formula changes for cv’s near top and bottom boundary.8. Calculate ap , source term based on pressure gradient and d1 values for all cv’s.9. Solve X-momentum equation (modified with under-relaxation factors) to get new values of u using Gauss-Seidal method.
  • 11. 10. Calculate Dw, De, Dn, and Ds for all cv’s.11. Calculate Fw, Fe, Fn, and Fs for all cv’s. You will have to use interpolation (average) for velocity.12. Based on CDS/UDS/Hybrid, calculate aw, ae, an, as for all cv’s. Formula changes for cv’s near left and right boundary.13. Calculate ap , source term based on pressure gradient and d2 values for all cv’s.14. Solve Y-momentum equation (modified with under-relaxation factors) to get new values of u using Gauss-Seidal method.
  • 12. 15. Calculate aw, ae, an, as for all cv’s. Formula changes for cv’s near boundary. (Corresponding coefficient will be zero)16. Calculate ap , source term (mass source) for all cv’s.17. If mass source < ε , solve pressure correction to get new values of P’ using Gauss-Seidal method.18. Pressure correction at boundary points can be set based on zero gradient.
  • 13. 19. Correct pressure using under relaxation factor. Correct velocity without under relaxation.20. If mass source < ε, go to next step. Otherwise go to step 5 with uold =unew , vold =vnew , Pold =Pnew21. Calculate u-velocity profile at vertical centreline and v-velocity profile at horizontal centreline and compare with bench mark results. (GHIA et al. (1982) JOURNAL OF COMPUTATIONAL PHYSICS VOL. 48, pp.387-411)
  • 14. Convergence criteria No. of iterations 10-5 736 10-6 1531 10-7 3262 10-8 5924
  • 15. No. of Control volumes No. of iterations 10 X 10 202 20 X 20 710 40 X 40 2243 80 X 80 5924