CMCC Centro EuroMediterraneo per i Cambiamenti Climatici COSMO

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici COSMO WG 2 Conservative Dynamic Core An implicit solver based on dual time stepping and finite volumes for meteorological applications Pier Luigi Vitagliano CIRA COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 1

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici OUTLINE • Motivation • Mathematical model • Numerical schemes • Test case • Future work COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 2

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici 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 General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 3

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici MATHEMATICAL FORMULATION W= Fx = Fy = Fz = B= EULER EQUATIONS IN CONSERVATIVE VARIABLES COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 4

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici 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 Q = fluxes R(W) = Q – B – D B = source terms COSMO General Meeting - September 8 th, 2009 D = k∆4 W COSMO WG 2 - CDC artificial dissipation 5

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici SPATIAL DISCRETISATION Example of flux evaluation Qm = Fmk = ½ (Fm + Fk) • • • k m Conservation laws applied to each sub-domain (cell) Variables stored at cell centers Fluxes approximated at cell face centers COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 6

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici DUAL TIME STEPPING (W)/ t + R(W) = 0 Wn+1/ t + ½(3 Wn+1 - 4 Wn + Wn-1)/Dt + R(Wn+1) = 0 add a pseudo-time t derivative to the unsteady equation 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 General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 7

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici DUAL TIME STEPPING Example of time integration with DTS: a norm of the residuals of mass transport equations is monitored COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 8

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici PRECONDITIONING Improve convergency in dual time for low Mach number flows Correct ill-behaved artificial viscosity fluxes at low Mach P· W/ t + R(W) = 0 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 General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 9

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici PRECONDITIONING Example of convergence to steady solution with and without Preconditioning COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 10

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici Discretisation of the gravity force term Field initialisation Effect of mesh skewness Flux – force unbalance COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 11

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici 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 General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 12

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici Mesh for test on complex orography with cold bubble COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 13

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici CONCLUSIONS • COMPUTER CODE FOR TEST RUN READY • INITIAL TESTS ON STEADY MOUNTAIN FLOW • STUDY ON COMPLEX OROGRAPHY STARTED COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 14

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici FUTURE WORK TEST CASES: 1) 2) 3) Atmosphere at rest with deformed mesh (Zaengl (2004)) Cold bubble (Straka et al. (1993)) Mountain test cases: • • • 4) (Schaer et al (2002) sect. 5 b) (Bonaventura(2000)) (Klemp, Wilhelmson) Linear gravity waves (Skamarock-Klemp (1994), Giraldo(2008)) COSMO General Meeting - September 8 th, 2009 COSMO WG 2 - CDC 15

CMCC Centro Euro-Mediterraneo per i Cambiamenti Climatici TEST CASE COLD BUBBLE Initial Field Density contour. Step Δρ/ρSL=0. 0001 COSMO General Meeting - September 8 th, 2009 Initial Field U-Velocity contour COSMO WG 2 - CDC 16