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    Validation Case: Rising Bubble

    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.

    Geometry

    The domain of interest is a 2-dimensional rectangular space with dimensions as below:

    top view of mesh used for rising bubble validation case
    Figure 1: 2-dimensional rectangular fluid domain

    The width AB, CD is 1 \(m\) while the height AD, BC is 2 \(m\).

    For simulation purposes, a mesh was created with the blockMesh tool in OpenFOAM® whose details can be found later in the following section.

    Analysis Type and Mesh

    Tool Type: OpenFOAM®

    Analysis Type: Multiphase

    Mesh and Element Types:

    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 TypeNumber of cellsType
    snappyHexMesh288002D hexahedral
    Table 1: Mesh settings
    uniform hexahedral mesh created with blockmesh for rising bubble validation case
    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:

    ParameterTop and BottomLeft and RightFront and Back
    Velocity Fixed Value – 0 \(m/s\)SlipEmpty 2D
    Pressure Fixed flux pressure – 0 \(Pa\)Fixed flux pressure – 0 \(Pa\)Empty 2D
    Phase fractionZero gradientZero gradientEmpty 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:

    $$X_c = (x_c,y_c) = \frac{\int_{\Omega_2}x\,dx}{\int_{\Omega_2}1\,dx} \tag{1}$$

    $$U_c = \frac{\int_{\Omega_2}u\,dx}{\int_{\Omega_2}1\,dx} \tag{2}$$

    where:

    • \(X_c\): 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:

    comparison of the bubble's center of mass for each case between simscale and reference study
    Figure 3: Temporal center of mass comparison for each case between results from SimScale and reference study
    comparison of the bubble's rising velocity for each case between 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:

    animation  of rising bubble for case 1
    Animation 1: Case 1 rising bubble deformation
    animation of rising bubble validation case for case 2
    Animation 2: Case 2 rising bubble deformation

    Last updated: May 20th, 2021

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