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:

3D warm bubble

Warm bubbleType 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)

ALYA - CVMS - bilinear elements

WRF - 5th order FD solution

WRF - 2nd order FD solution

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 bubbleType 2: as defined in Robert 1993

Domain: [0:1.0] x [0:1.5] km2

Resolution: dx=dz = 5 meters

ALYA 18min: Robert case: VMS solution

18min: Robert case: Robert reference

12 min: Robert case: VMS solution

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]

Ahmad f-wave solution: 600s

VMS-C solution: 600s

Godunov f-wave (FV): at 900 s Ref. 1

ALYA - CVMS - Bilinear elements, 50m (left ) and 100 m (right)

WRF-ARW, 5th order solution

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

ALYA - CVMS - Bilinear elements 10m res.

Robert, FD scheme, 10m resolution

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

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

Initial steps:

Warm bubble: as defined in Ahmad and Lindeman 2007 (after Wiker and Skamarok 1998)Type 1From 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)

[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: as defined in Robert 1993Type 2Ref:

[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 solutionwithoutany eddy viscosity added.Top Pair of images:

Contour-line images: Variational multiscale [2]

Color images: f-wave solution [1]

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 interactionRef:

[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