Thermal+and+gravity+driven+probs

=Warm bubbles and density current problems solved with a FE Variational Multiscale Compressible solver (VMS-C):=


 * 3D warm bubble:
 * Domain: [0:1] x [0:1] x [0:1] km3
 * Resolution: dx=dy=dz=100 meters

Initial steps:




 * **Warm bubble** **__Type 1__**: as defined in Ahmad and Lindeman 2007 (after Wiker and Skamarok 1998)
 * Domain: [0:20] x [0:10] km2
 * Resolution: dx=dz=50 meters

From top to bottom : TOP LEFT: Variational Multiscale Compressible (FE, Ref. 3) TOP RIGHT: WRF-V2, 5th order solution (FD, Ref 1) BOTTOM LEFT: WRF-V2, 2nd order solution (FD, Ref 1) BOTTOM RIGHT: f-wave solution (FV, Ref 1)


 * [[image:warm_ahmad_1020s_contourPseudoDetail.jpg width="406" height="321" caption="ALYA - CVMS - bilinear elements"]] || [[image:5th_wrf.jpg width="428" height="333" caption="WRF - 5th order FD solution"]] ||
 * [[image:warm-ahmad_WRF2ndorder_1020s.jpg align="right" caption="WRF - 2nd order FD solution"]] || [[image:warm-ahmad_1020s.jpg caption="Godunov f-wave (FV). See ref 1"]] ||
 * [[image:warm-ahmad_WRF2ndorder_1020s.jpg align="right" caption="WRF - 2nd order FD solution"]] || [[image:warm-ahmad_1020s.jpg caption="Godunov f-wave (FV). See ref 1"]] ||

[1] Amhad and Lindeman, 2007 "Euler solutions using flix-based wave decomposition" [2] Wiker and Skamarock, 1998 "A time-splitting scheme for the elastic equations incorporating second-order Runge-Kutta time differencing" [3] Marras et al., 2010 "Application of a Galerkin Finite Element Scheme to Atmospheric Buoyant and Gravity Driven Flows" AIAA 48th Aerospace Sciences Meeting, ORlando, FL 4-7 Jan 2010


 * **Warm bubble** **__Type 2__**: as defined in Robert 1993
 * Domain: [0:1.0] x [0:1.5] km2
 * Resolution: dx=dz = 5 meters


 * [[image:robert_smooth_dxdz2_5m_1070s_color.jpg width="359" height="420" caption="ALYA 18min: Robert case: VMS solution"]] || [[image:robert_18min_2dot5m.jpg width="266" height="390" caption="18min: Robert case: Robert reference"]] ||
 * [[image:robert_smooth_dxdz2_5m_720s_color.jpg width="361" height="421" caption="12 min: Robert case: VMS solution"]] || [[image:robert_12min_2dot5m.jpg width="266" height="395" caption="12 min: Rober case: Reference"]] ||

Ref: [1] Robert 1993, "Bubble convection experiements with a semi-implicit formulation of the Euler equations" [2] Marras et al. "Application of a Variational ...", in preparation


 * **Density current**: Euler solution __without__ any eddy viscosity added.
 * Domain: [-25.0:25.0] x [0.0:6.0] km2
 * Resolution: dx=dz=50 meters

Top Pair of images: Contour-line images: Variational multiscale [2] Color images: f-wave solution [1]


 * [[image:ahmad-densityCurrent_600.jpg width="454" height="180" caption="Ahmad f-wave solution: 600s"]] || [[image:DENSITY_CURRENT_50m_600s_BW_min10punto39_max26punto5_15contorni.jpg width="388" height="163" caption="VMS-C solution: 600s"]] ||
 * [[image:ahmad-densityCurrent_900.jpg width="478" height="165" caption="Godunov f-wave (FV): at 900 s Ref. 1"]] ||  ||
 * [[image:DENSITY_CURRENT_50mAND100m_4images_legend.jpg width="879" height="341" caption="ALYA - CVMS - Bilinear elements, 50m (left ) and 100 m (right)"]] ||  ||
 * [[image:DENSITY_CURRENT_50mAND100m_4images_legend.jpg width="879" height="341" caption="ALYA - CVMS - Bilinear elements, 50m (left ) and 100 m (right)"]] ||  ||


 * [[image:WRF-densityCurrent-5thorder.jpg width="432" height="515" caption="WRF-ARW, 5th order solution"]] ||  || [[image:WRF-densityCurrent-2ndorder.jpg width="430" height="508" caption="WRF-ARW, 2nd order solution"]] ||

Ref: [1] Amhad and Lindeman, 2007 "Euler solutions using flix-based wave decomposition" [2] Marras et al., 2010 "Application of a Galerkin Finite Element Scheme to Atmospheric Buoyant and Gravity Driven Flows" AIAA 48th Aerospace Sciences Meeting, ORlando, FL 4-7 Jan 2010 [3] Straka et al., 1993 "Numerical solution of a nonlinear density current: a benchmark solution and comparisons"


 * **Double bubble (Robert 1993): to study the sensitivity to sharp discontinuities and interaction**



Ref: [1] Robert 1993, "Bubble convection experiements with a semi-implicit formulation of the Euler equations" [2] Marras et al. "Application of a Variational ...", in preparation