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An implicit solver based on dual time stepping and finite volumes for meteorological applications Pier Luigi Vitagliano

An implicit solver based on dual time stepping and finite volumes for meteorological applications Pier Luigi Vitagliano CIRA. COSMO WG2 Conservative Dynamic Core. OUTLINE. Motivation Mathematical model Numerical schemes Test case Future work. MOTIVATIONS AND GOALS.

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An implicit solver based on dual time stepping and finite volumes for meteorological applications Pier Luigi Vitagliano

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  1. An implicit solver based on dual time stepping and finite volumes for meteorological applicationsPier Luigi VitaglianoCIRA COSMO WG2 Conservative Dynamic Core COSMO WG 2 - CDC

  2. OUTLINE • Motivation • Mathematical model • Numerical schemes • Test case • Future work COSMO WG 2 - CDC

  3. MOTIVATIONS AND GOALS • Improve numerical efficiency • Improve conservation properties • Improve capability to deal with steeper orography • Test a time integration scheme for meteorological applications • Test spatial schemes based on finite volumes • Issue recommendations on future implementation in COSMO COSMO WG 2 - CDC

  4. MATHEMATICAL FORMULATION W= Fx = Fy = Fz = B= EULER EQUATIONS IN CONSERVATIVE VARIABLES COSMO WG 2 - CDC

  5. SPATIAL DISCRETISATION • Finite Volumes approach • Integral form allows discontinuities in the flow field • Conservation laws applied to each sub-domain (cell) • Variables stored at cell centers • Fluxes approximated at cell face centers (W)/t + R(W) = 0 R(W) = Q – B – D Q = fluxes B = source terms D = k∆4Wartificial dissipation COSMO WG 2 - CDC

  6. SPATIAL DISCRETISATION Example of flux evaluation Qm = k m Fmk = ½ (Fm+ Fk) • Conservation laws applied to each sub-domain (cell) • Variables stored at cell centers • Fluxes approximated at cell face centers COSMO WG 2 - CDC

  7. DUAL TIME STEPPING (W)/t + R(W) = 0 Wn+1/t + ½(3Wn+1- 4Wn + Wn-1)/Dt + R(Wn+1) = 0 add a pseudo-time tderivative to the unsteadyequation advance the solution in t until the residual of the unsteady equation is negligible iterations in t are performed by explicit Runge-Kutte scheme convergence acceleration techniques can be adopted without loss of time accuracy: residual averaging, local time stepping, multigrid formulation is A-stable and damps the highest frequency very large physical time step Dt can be used Jameson, A., 1991: Time Dependent Calculations Using Multigrid,with Applications to Unsteady Flows Past Airfoils and Wings. AIAA Paper 91–1596 COSMO WG 2 - CDC

  8. DUAL TIME STEPPING Example of time integration with DTS: a norm of the residuals of mass transport equations is monitored COSMO WG 2 - CDC

  9. P·W/t + R(W) = 0 PRECONDITIONING Improve convergency in dual time for low Mach number flows Correct ill-behaved artificial viscosity fluxes at low Mach Difficulties rise from large ratio between acoustic wave speed and fluid speed Premultiplying the time derivative changes the eigenvalues of the system and accelerates the convergence to steady state. Turkel, E., 1999: Preconditioning techniques in computational fluid dynamics. Annu.Rev.Fluid Mech. 1999,31:385-416. Venkateswaran, S., P. E. O. Buelow, C. L. Merkle, 1997: Development of linearized preconditioning methods for enhancing robustness and efficiency of Euler and Navier-Stokes Computations, AIAA Paper 97-2030. COSMO WG 2 - CDC

  10. PRECONDITIONING Example of convergence to steady solution with and without Preconditioning COSMO WG 2 - CDC

  11. Discretisation of the gravity force term Field initialisation Effect of mesh skewness Flux – force unbalance COSMO WG 2 - CDC

  12. TEST CASE MOUNTAIN FLOW Flow over a gaussian mountain simulated with a test code based on finite volumes conservative schemes. Vertical velocity component. The dashed line shows the lower boundary of the Rayleigh damping layer, which prevents the wave reflection. COSMO WG 2 - CDC

  13. Mesh for test on complex orography with cold bubble COSMO WG 2 - CDC

  14. CONCLUSIONS • COMPUTER CODE FOR TEST RUN READY • INITIAL TESTS ON STEADY MOUNTAIN FLOW • STUDY ON COMPLEX OROGRAPHY STARTED COSMO WG 2 - CDC

  15. FUTURE WORK TEST CASES: • Atmosphere at rest with deformed mesh (Zaengl (2004)) • Cold bubble (Straka et al. (1993)) • Mountain test cases: • (Schaer et al (2002) sect. 5b) • (Bonaventura(2000)) • (Klemp,Wilhelmson) • Linear gravity waves (Skamarock-Klemp (1994),Giraldo(2008)) COSMO WG 2 - CDC

  16. TEST CASE COLD BUBBLE Initial Field Density contour. Step Δρ/ρSL=0.0001 Initial Field U-Velocity contour COSMO WG 2 - CDC

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