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    Validation Case: Natural Convection Between Horizontal Concentric Cylinders

    This validation case belongs to CFD, more specifically to the Convective Heat Transfer solver. The reference experiment by Kuehn and Goldstein\(^1\) studies the natural convection between two horizontal concentric cylinders.

    The test case aims to validate the following parameters:

    • Convective heat transfer
    • Natural convection

    The simulation project containing the CFD model and the results can be accessed by using the following link:

    Geometry

    The geometrical model captures the air portion between the two concentric cylinders, to mimic the physical setup performed in the experiment of reference\(^1\). Figure 1 shows a translucent view of the model where the geometry can be appreciated:

    geometry concentric cylinders convection simscale
    Figure 1: Geometrical model for the fluid region between the concentric cylinders

    The relevant dimensions used for the geometry and simulation are as follows:

    • Length: \(L=\) 500 \(mm\)
    • Radius of inner cylinder: \(R_i=\) 17.8 \(mm\)
    • Radius of outer cylinder: \(R_o=\) 46.25 \(mm\)

    Analysis Type and Mesh

    Tool Type: OpenFOAM®

    Analysis Type: Steady-state, incompressible, laminar Convective heat transfer

    Mesh and Element Types:

    SimScale’s Standard meshing algorithm was used for the creation of the mesh. This algorithm implements a strategy with a surface triangulation, followed by a transition layer of tetrahedrals and prisms, and finally a hexahedral core for the gross region of the flow field.

    mesh concentric cylinders convection simscale
    Figure 2: General view of the mesh created by SimScale’s Standard algorithm

    The mesh has a total of 1387582 volumetric elements, which are tetrahedrals, prisms, and hexahedrals. The triangular surface mesh can be appreciated in Figure 2, while Figure 3 shows a mesh clip to appreciate the hex core and the transition layer. It can be seen also that, in spite of implementing a laminar turbulence model, boundary layers are also included in the mesh.

    mesh clip concentric cylinders convection simscale
    Figure 3: Mesh clip showing the different types of volumes used to construct the mesh

    Simulation Setup

    Fluid Material Properties:

    • Material: Air
    • Kinematic viscosity (\(\nu\)): 1.529e-5 \(m^2/s\)
    • Density (\(\rho\)): 1.196 \(kg/m^3\)
    • Thermal expansion coefficient (\(\alpha\)): 3.43e-3 \(1/K\)
    • Laminar Prandtl number: 0.713
    • Specific heat: 1004 \(J/(kg \cdot K)\)

    Boundary Conditions:

    • Inner, Hot Wall: Fixed temperature of 347 \(K\)
    • Outer, Cold Wall: Fixed temperature of 327 \(K\)
    • End walls: Adiabatic

    Initial Conditions:

    • Uniform temperature of 293 \(K\) in the fluid region

    Model:

    • Gravity in the \(-g_z\) direction with magnitude 9.81 \(m/s^2\)

    Results

    For the comparison of results, the radial distribution of the temperature at different polar angles is presented in the reference work. The radial coordinates and temperature are normalized to be dimensionless according to the following relations:

    $$\hat{R} = \frac{R – R_i}{R_o – R_i} $$

    $$\hat{T} = \frac{T – T_o}{T_i – T_o} $$

    For comparison, the dimensionless radius and temperature are measured at polar angles of \(\theta = 0°\) and \(\theta = 180°\), with the direction of gravity coinciding with the latter set. Figure 4 shows a comparison of the results with respect to the reference work.

    temperature comparison concentric cylinders convection simscale
    Figure 4: Comparison of temperature distribution, reference vs SimScale

    A good agreement is found between the results of the Convective Heat Transfer solver and the results reported in the experimental work for the convection between the concentric cylinders. The temperature distribution in the cross-section of the air volume can be visualized in Figure 5:

    temperature distribution concentric cylinders convection simscale
    Figure 5: Temperature distribution in the cross-section of the volume between the concentric cylinders.

    References

    • Kuehn T.H and Goldstein R.J, “An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders”, Journal of Fluid Mechanics, vol 74, part 4, 1976.

    Last updated: September 26th, 2022

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