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    Validation Case: Compressible Flow In a de Laval Nozzle

    This validation case belongs to fluid dynamics. The aim of this test is to validate the following parameters for a compressible, steady-state turbulent flow through a de Laval Nozzle:

    • Mach number across the nozzle.
    • Pressure across the nozzle.

    The simulation results from SimScale were compared to analytical results obtained from methods presented in [1].

    Geometry

    A typical configuration of the de Laval nozzle with a non-smooth throat that was chosen as the geometry can be seen below:

    de laval nozzle geometry
    Figure 1: Geometry of the de Laval Nozzle

    The nozzle is axisymmetric and an angular slice of the complete geometry with a wedge angle of 18\(°\) was used. The following table contains the coordinates of the reference points:

    Point\(x\) \([mm]\)\(y\) \([mm]\)\(z\) \([mm]\)
    A000
    B05.543435
    C0-5.543435
    D68.6800
    E68.683.167720
    F68.68-3.167720
    G238.86.335440
    H238.8-6.335440
    Table 1:The coordinates of the reference points

    Analysis Type and Mesh

    Tool Type : OpenFOAM®

    Analysis type : Compressible with k-omega SST turbulence model

    Time dependency: Steady state

    Mesh and Element types:

    The hexahedral mesh was created externally, and then was imported on to the SimScale platform, with an inflation of \(y^+\) = 30 near the walls, as the ‘wall functions’ model was chosen. In order to keep the flow two-dimensional, the mesh was designed to have only one layer in the \(y\) direction.

    Mesh typeCells in the x-directionCells in the y-directionCells in the zdirectionNumber of nodesType
    blockMesh1501100150002D Hex
    Table 2: Mesh Metrics
    mesh details of the nozzle
    Figure 2: The mesh used for the SimScale case (zoomed in details at the bottom). It is only one cell thick in the third direction.

    Simulation Setup

    Fluid:

    • Custom He psi thermo type fluid:
      • Amount of substance: 1 mol
      • Molar mass: \(M_m\) = 28.9 \(kg \over \ kmol \)
      • Transport model: Sutherland
      • Reference viscosity: \(\nu_0\) = 1.716e-5 \(kg \over \ {s \times \ m}\)
      • Reference temperature: \(T_0\) = 273.15 \(K\)
      • \(T_s\) = 116 \(K\)
      • Thermo model: hconst
      • Specific heat: \(C_p\) = 1005 \(J \over\ {kg \times \ K}\)
      • Heat of formation: \(H_f\) = 0 \(J \over\ kg\)

    Boundary Conditions:

    ParameterInlet (ABC)Outlet (GHI)Wall (BEHIFC)Wedges (AGHEB + AGIFC)
    Velocity
    \([m/s]\)
    7.58Zero Gradient0Wedge
    Pressure
    \([Pa]\)
    Zero Gradient1.3×105Zero GradientWedge
    Temperature
    \([K]\)
    300 Zero GradientZero GradientWedge
    Turbulent kinetic energy \(k\)
    \([m^2/s^2]\)
    0.862 Zero GradientWall FunctionWedge
    Specific dissipation rate \(ω\)
    \([s^{-1}]\)
    484.269 Zero GradientWall FunctionWedge
    Turbulent thermal diffusivity
    \(α_t\)
    CalculatedCalculatedWall FunctionWedge
    Turbulent dynamic viscosity \(μ_t\)CalculatedCalculatedWall FunctionWedge
    Table 3: The boundary conditions

    The wedge boundary condition (BC) is used in computational fluid dynamics to define an axisymmetric situation. You can learn more about it here.

    Reference Solution

    The results are calculated from Bernoulli’s equation and the ideal gas equation. The speed of sound can be calculated as:

    $$c = \sqrt{\frac{\gamma RT}{m}} \tag{1}$$

    Where:

    • \(\gamma\) is the specific heat ratio
    • \(T\) is the temperature
    • \(m\) is the molecular weight of the fluid
    • \(R\) is the universal gas constant

    The Mach number of a cutting plane be calculated as:
    $$M = \frac{ U}{c} \tag{2}$$

    Where:

    • \(U\) is the average velocity magnitude of the plane

    Finally, the static pressure can be computed as:

    $$P_{stat} = P_{stag}\ – \frac{1}{2}\rho \times U^2 \ \tag{3}$$

    Where:

    • \(P_{stag}\) : the stagnation pressure, assumed to be 0.1301 \(MPa\)
    • \(\rho\) : the density of the fluid

    Result Comparison

    A comparison of the Mach number and pressure variation in the nozzle obtained with SimScale with analytical results is given below:

    the mach number across the de laval nozzle compared to the reference paper
    Figure 3: Visualization of the Mach number across the nozzle
    the pressure across the de laval nozzle compared to the reference paper
    Figure 4: Visualization of the pressure across the nozzle. Deviation in results is a result of 1D hypothesis assumed in the analytical method.

    A source of deviation between the calculated and analytical results exists because the latter is calculated with a one-dimensional hypothesis. This means that all parameters are assumed to be uniform in the radial direction, while in the following figures, it seems that there exists radial variation too in the flow variables.

    velocity contour on the de laval nozzle
    Figure 5: Velocity contour on the nozzle

    One can observe the radial variation causing results to slightly deviate:

    temperature distribution on the de laval nozzle shown within simscale
    Figure 6: Temperature distribution on the de Laval nozzle

    Last updated: January 7th, 2021

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