This validation case belongs to fluid dynamics and the aim of this test case is to validate the multiphase solver implemented in SimScale with the rising bubble case. Specifically, the following parameters are of interest:
Bubble vertical velocity
Bubble center of mass
Bubble profile
The simulation results from SimScale were compared to the results presented in the study “Bubble Benchmark“\(^1\) done by TU Dortmund.
As explained before, the mesh was created with the blockMesh tool. This is a uniform mesh with only one cell-layer in the z-axis, this is done to maintain the two-dimensional flow.
Mesh Type
Number of cells
Type
snappyHexMesh
28800
2D hexahedral
Table 1: Mesh settings
Figure 2: Uniform hexahedral mesh of fluid domain with 1 layer in the z-axis created with blockMesh
Simulation Setup
Fluid:
Case 1:
Gravity \((g)\) : 0.98 \(m/s^2\)
Surface tension \((\sigma)\): 24.5 \(N/m\)
Material 1:
Kinematic viscosity \((\nu)\): 0.01 \(m^2/s\)
Density \((\rho)\): 1000 \(kg/m^3\)
Material 2:
Kinematic viscosity \((\nu)\): 0.01 \(m^2/s\)
Density \((\rho)\): 100 \(kg/m^3\)
Case 2
Gravity \((g)\) : 0.98 \(m/s^2\)
Surface tension \((\sigma)\): 1.96 \(N/m\)
Material 1:
Kinematic viscosity \((\nu)\): 0.01 \(m^2/s\)
Density \((\rho)\): 1000 \(kg/m^3\)
Material 2:
Kinematic viscosity \((\nu)\): 0.1 \(m^2/s\)
Density \((\rho)\): 1 \(kg/m^3\)
The location of the bubble at t = 0 \(s\) is at y = 0.5 \(m\).
Initial and Boundary Conditions:
Initial conditions
Only the global phase fraction was initialized and was set to the value of 1, which is the fluid surrounding the bubble.
Boundary conditions
To simulate the rising bubble phenomenon, custom boundary conditions were used. The specific settings of the boundary conditions can be seen in the table below:
Parameter
Top and Bottom
Left and Right
Front and Back
Velocity
Fixed Value – 0 \(m/s\)
Slip
Empty 2D
Pressure
Fixed flux pressure – 0 \(Pa\)
Fixed flux pressure – 0 \(Pa\)
Empty 2D
Phase fraction
Zero gradient
Zero gradient
Empty 2D
Table 2: Custom boundary conditions for each parallel face
Reference Solution
The reference solution for the center of mass and the rising velocity is given by the following equations:
\(X_c\): x- and y-coordinates of the center of the bubble
\(\Omega_2\): the region the bubble resides\(^1\)
\(U_c\): rising velocity of the bubble
\(u\): velocity of the center of the bubble
Result Comparison
The comparison for the center of mass and the rising velocity of the bubble obtained from SimScale against the reference results obtained from the “Bubble Benchmark“\(^1\) is shown in figures below:
Figure 3: Temporal center of mass comparison for each case between results from SimScale and reference study
Figure 4: Temporal rising velocity comparison for each case between results from SimScale and reference study
The movement of the bubble for each case can be seen in the animation below:
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