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HYPERGRID

Modular computational tools for high speed compressible flows

User guide and brief overview for a Python implementation of the method presented in Modular Method for Estimation of Velocity and Temperature Profiles in High-Speed Boundary Layers [AIAA JOURNAL, Vol. 60, No. 9, September 2022 DOI: https://doi.org/10.2514/1.J061735], and [J. Comput. Phys.: X, Volume 17, 2023, 100128, DOI: https://doi.org/10.1016/j.jcpx.2023.100128]

Questions on the velocity profile calculation method can be sent to the authors: Vedant Kumar (vkumar20@umd.edu) and Johan Larsson (jola@umd.edu)

Questions on the wall-normal grid stretching method can be sent to the authors: Alessandro Ceci (alessandro.ceci@uniroma1.it) and Sergio Pirozzoli (sergio.pirozzoli@uniroma1.it)

Update 30th October 2024: compute properties for a compressible boundary layer using the method described in Hasan et al. (https://doi.org/10.2514/1.J063335), but with additional modular features as seen in compressibleBLprofiles detailed further in Kumar & Larsson (AIAA JOURNAL) and Ceci & Pirozzoli (J. Comput. Phys.: X). The computational tools can now compute directly the compressible boundary layer profiles using boundaryLayerPropFunctions_direct.py in the source code.

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PREREQUISITES

Working install of python3 with: numpy, scipy, matplotlib

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HOW TO USE

Beginner users are recommended to look at the Python script methodDemo1.py. The script demonstrates an example of computing the skin friction coefficient, wall heat transfer along with velocity and temperature profiles for a compressible boundary layer at Mach number = 5, Reynolds number (ReDelta2) = 3000 with wall and freestream temperatures of 300 and 200 K, respectively. For the sake of simplification, the script uses default modeling choices which has been observed to return the most accurate predictions at the time of the publication of the above manuscript.

Next, an example for the same boundary layer case as above however with the specification of modeling choices is presented in the Python script methodDemo2.py. More information on the available modeling choices and how to change them is available below.

Regarding the wall-normal grid generation, the user is only required to set the parameters alf_plus (controlling the resolution requirements in terms of Kolmogorov scale in wall units), jb (defining the grid index at which transition between the near-wall and the outer mesh stretching) and ysp (setting the first grid point in the wall-normal direction in inner-units).

M2.py and M6.py are two examples for the boundary layer grid generation profile y^+(j), given the prescribed inputs.

Update 30th October 2024: Beginner users are recommended to look at the Python script methodDemodirect.py. The script demonstrates an example of computing the skin friction coefficient, wall heat transfer along with velocity and temperature profiles for a compressible boundary layer at Mach number = 5, Reynolds number (ReDelta2) = 3000 with wall and freestream temperatures of 300 and 200 K, respectively. The script now uses a direct approach computing directly the compressible boundary layer profiles. The grid generation process is unchanged, but two additional scripts M2direct.py and M6_direct.py are now released. The scripts generate the boundary layer grid generation profile y^+(j) using boundary layer properties of the computed compressible profiles.

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FUNCTION DEFINITION (indirect method)

Function: boundaryLayerProperties in the Python script boundaryLayerPropFunctions.py

boundaryLayerProperties(M, ReDelta2, Tw_Te, Te, N, **kwargs)

Computes boundary layer properties for the prescribed flow parameters. Subscripts _w (or w) and _e (or e) denote wall and edge (freestream) properties, respectively.

Parameters:

    M: float 
    Mach number (M = Ue/c_e, where c is the speed of sound)

    ReDelta2: float 
    Reynolds number (ReDelta2 = rho_e Ue theta / mu_w, where theta is the momentum thickness)

    Tw_Te: float 
    Ratio of wall (Tw) to freestream (Te) temperature

    Te: float 
    Freestream temperature  

    N: int 
    Number of grid points in the boundary layer 

    **kwargs: Optional inputs 


        Flags for different model choices

        flagViscTemp: int, optional 
        Flag for selecting viscosity-temperature relation. 

        flagIncCf: int, optional
        Flag for selecting an incompressible skin-friction relation

        flagTempVelocity: int, optional
        Flag for selecting a temperature-velocity relation

        flagVelocityTransform: int, optional
        Flag for selecting a velocity transform function


        Additional parameters

        gridStretchPar: float 
        Grid-stretching parameter for geometrical stretching.
        Hence, if gridStretchPar = 1, then we have a uniform grid and if
        gridStretchPar>1, then we have a grid stretching away from the wall

        underRelaxFactor: float 
        Extent of under-relaxation for the solution process such that
        x_new = (1 - underRelaxFactor)*x_old + underRelaxFactor*x_new

    Note that unspecified optional inputs switch to default values. 
    See below for more information.

Returns:

    cf: float 
    Skin friction coefficient

    Bq: float 
    Wall heat transfer rate

    ch: float 
    Stanton number, undefined for adiabatic walls

    yPlus: array
    Array of y-plus coordinates inside the boundary layer

    uPlus: array
    Array of scaled velocity (uPlus = u/u_tau) values in the boundary layer

    Tplus: array
    Array of T-plus values (Tplus = T/Tw) values in the boundary layer

    mu_muw: array
    Arrray of wall-scaled viscosity values in the boundary layer

    [Refer to the manuscript for the definition of the output parameters above]

---

Function: gridProperties in the Python script boundaryLayerPropFunctions.py

gridProperties(yPlus,TTw,mumuw,alf_plus,jb,ysw)

Computes natural wall-normal grid distribution given the prescribed resolution requirements Subscripts _w (or w) denote wall properties, respectively.

Parameters:

    yPlus, array
    Wall-normal points for the compressible boundary layer (in plus units)

    T_Tw, array
    Boundary layer temperature profile, normalized by the wall temperature Tw

    mu_muw, array
    Boundary layer viscosity profile, normalized by the wall viscosity muw

    alf_plus, float
    Target resolution in wall Kolmogorov units

    jb, integer
    Transition node from viscous to outer stretching

    ysp, float
    Target first point wall distance in inner units

    **kwargs: Optional inputs:

            myNy, integer
            Number of points to use to build the wall normal grid stretching.
            By default the method with run with a number of points based on
            an asyptotic estimation for the given Reynolds number

Returns:

    yStar_j, array
    semilocal scaled wall distance function of the number of points in the wall normal direction

    yPlus_j, array
    wall scaled  wall distance function of the number of points in the wall normal direction

    jj, array 
    index array of points in the wall normal direction

    Ny, integer
    number of points in the wall normal direction

    alf_opt, float
    optimal alf star to respect threshold resolution in wall Kolmogorov units

    etaPlus_j, array
    estimated etaPlus profile, useful to chek resolution requirements

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FUNCTION DEFINITION (direct method)

Function: boundaryLayerProperties in the Python script boundaryLayerPropFunctions_direct.py

boundaryLayerProperties(M, ReDelta2, Tw_Te, Te, N, **kwargs)

Computes boundary layer properties for the prescribed flow parameters. Subscripts _w (or w) and _e (or e) denote wall and edge (freestream) properties, respectively.

Parameters:

    M: float 
    Mach number (M = Ue/c_e, where c is the speed of sound)

    ReDelta2: float 
    Reynolds number (ReDelta2 = rho_e Ue theta / mu_w, where theta is the momentum thickness)

    Tw_Te: float 
    Ratio of wall (Tw) to freestream (Te) temperature

    Te: float 
    Freestream temperature  

    N: int 
    Number of grid points in the boundary layer 

    **kwargs: Optional inputs 


        Flags for different model choices

        flagViscTemp: int, optional 
        Flag for selecting viscosity-temperature relation. 

        flagIncCf: int, optional
        Flag for selecting an incompressible skin-friction relation

        flagTempVelocity: int, optional
        Flag for selecting a temperature-velocity relation

        flagVelocityTransform: int, optional
        Flag for selecting a velocity transform function


        Additional parameters

        gridStretchPar: float 
        Grid-stretching parameter for geometrical stretching.
        Hence, if gridStretchPar = 1, then we have a uniform grid and if
        gridStretchPar>1, then we have a grid stretching away from the wall

        underRelaxFactor: float 
        Extent of under-relaxation for the solution process such that
        x_new = (1 - underRelaxFactor)*x_old + underRelaxFactor*x_new   

        calibrateWakePar: string ('Yes' or 'No', bool)
        Compute the equivalent incompressible boundary layer velocity profile
        in order to calibrate the wake parameter

    Note that unspecified optional inputs switch to default values. 
    See below for more information.

Returns:

    cf: float 
    Skin friction coefficient

    Bq: float 
    Wall heat transfer rate

    ch: float 
    Stanton number, undefined for adiabatic walls

    yPlus: array
    Array of y-plus coordinates inside the boundary layer

    uPlus: array
    Array of scaled velocity (uPlus = u/u_tau) values in the boundary layer

    Tplus: array
    Array of T-plus values (Tplus = T/Tw) values in the boundary layer

    mu_muw: array
    Arrray of wall-scaled viscosity values in the boundary layer

    kappa_VK: float
    Value of the Von Karman constant used in the boundary layer profile generation,
    that must be also given as input for the wall-normal grid generation process

    [Refer to the manuscript for the definition of the output parameters above]

---

Function: gridProperties in the Python script boundaryLayerPropFunctions.py

gridProperties(yPlus,TTw,mumuw,alf_plus,jb,ysw)

Computes natural wall-normal grid distribution given the prescribed resolution requirements Subscripts _w (or w) denote wall properties, respectively.

Parameters:

    yPlus, array
    Wall-normal points for the compressible boundary layer (in plus units)

    T_Tw, array
    Boundary layer temperature profile, normalized by the wall temperature Tw

    mu_muw, array
    Boundary layer viscosity profile, normalized by the wall viscosity muw

    alf_plus, float
    Target resolution in wall Kolmogorov units

    jb, integer
    Transition node from viscous to outer stretching

    ysp, float
    Target first point wall distance in inner units

    kappa, float
    Von Karman constant
    (REMARK: 
    FOR CONSISTENCY IT MUST BE THE OUTPUT OF THE 
    FUNCTION GENERATING THE BOUNDARY LAYER PROFILES)

    **kwargs: Optional inputs:

            myNy, integer
            Number of points to use to build the wall normal grid stretching.
            By default the method with run with a number of points based on
            an asyptotic estimation for the given Reynolds number

Returns:

    yStar_j, array
    semilocal scaled wall distance function of the number of points in the wall normal direction

    yPlus_j, array
    wall scaled  wall distance function of the number of points in the wall normal direction

    jj, array 
    index array of points in the wall normal direction

    Ny, integer
    number of points in the wall normal direction

    alf_opt, float
    optimal alf star to respect threshold resolution in wall Kolmogorov units

    etaPlus_j, array
    estimated etaPlus profile, useful to chek resolution requirements

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METHOD DESCRIPTION

A detailed description of the method can be found in the papers referenced above.

In brief, the properties for the compressible boundary layer are computed by relating it with an equivalent incompressible state.

In the indirect method, an incompressible boundary layer velocity profile is defined using a known model followed by transforming it to the desired compressible state (defined using the function input parameters).

These transformation functions assume that the only effect of non-zero Mach numbers comes from the boundary layer density profile as well as viscosity profile for some functions.

The density and viscosity profiles are hence computed using models relating them with the boundary layer temperature which in turn is related to the boundary layer velocity profile using a temperature-velocity modeling relation.

The method follows an iterative process which, on convergence, returns the outputs described above.

In the direct method, the compressible boundary layer velocity profile is directly computed in the desired compressible state (defined using the function input parameters) using a known model

For the generation of the natural wall-normal grid stretching, first a universal scaling for the Kolmodorov length-scale in semilocal units is assumed, i.e. eta^* = ( k y^* )^1/4; then the y^* profile is created. Finally the generated semilocal wall-normal grid ( y^* ) is converted into the wall scaled one ( y^+ ).

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MODELING CHOICES

In the indirect method, the current work models the incompressible velocity profile using the definition presented in Huang et al. [AIAA Journal, Vol. 31, No. 9, 1993, pp. 1600– 1604]. It can be changed by editing the function universalVelocityProfile in boundaryLayerPropFunctions.py

In the direct method, the target compressible velocity profile are directly computed by integrating an ODE as described in Hasan et al. [PRF, (https://doi.org/10.2514/1.J063335)] work.

Further, the following modeling choices are incorporated in the current implementation:\ 1. Viscosity-Temperature relation - Chosen by specifying value to flagViscTemp\ [For implementation details, see function viscosityFromTemperature in boundaryLayerPropFunctions.py]\ flagViscTemp = 1 : Keyes' viscosity law\ flagViscTemp = 2 : Sutherland's law\ flagViscTemp = 3 : Curve-fit to the CoolProp dataset [DEFAULT]\ flagViscTemp = 4 : Power law, (mu/muw) = (T/Tw)^n, where n = 2/3

1. Incompressible skin-friction relation - Chosen by specifying value to flagIncCf\ [For implementation details, see function skinFrictionIncompressible in boundaryLayerPropFunctions.py]\ flagIncCf = 1 : Karman-Schoenherr's relation\ flagIncCf = 2 : Blasius relation\ flagIncCf = 3 : Smits' relation [DEFAULT]

2. Temperature-Velocity relation - Chosen by specifying value to flagTempVelocity\ [For implementation details, see function temperatureFromVelocity in boundaryLayerPropFunctions.py]\ flagTempVelocity = 1 : Zhang's relation [DEFAULT]\ flagTempVelocity = 2 : Walz's relation

3. Velocity transform function (indirect method) - Chosen by specifying value to flagVelocityTransform\ [For implementation details, see function inverseVelocityTransform in boundaryLayerPropFunctions.py]\ flagVelocityTransform = 1 : Inverse of the Van Driest velocity transform\ flagVelocityTransform = 2 : Inverse of the Trettel-Larsson velocity transform\ flagVelocityTransform = 3 : Inverse of the Volpiani velocity transform function [DEFAULT]

4. Velocity transform function (direct method), used as a compressible eddy viscosity model - Chosen by specifying value to flagVelocityTransform\ [For implementation details, see function inverseVelocityTransform in boundaryLayerPropFunctions_direct.py]\ flagVelocityTransform = 1 : The Van Driest velocity transform\ flagVelocityTransform = 2 : The Trettel-Larsson velocity transform\ flagVelocityTransform = 3 : The HLPP velocity transform function [DEFAULT]

5. Wall normal grid stretching\ The semilocal y^* profile is created according to the proposed stretching of Pirozzoli & Orlandi [J. Comput. Phys. Volume 439,2021, 110408, https://doi.org/10.1016/j.jcp.2021.110408 ], the final y^+ grid is then obtained using the tranformation from semilocal to wall-scaled units.

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POINTS TO NOTE

1. For grid-converged results, it is recommended to use grids with near-wall resolution < 0.2 plus units. The distribution of grid points can be controlled using the number of grid points "N" and, the grid-stretching parameter "gridStretchPar". By default, gridStretchPar = 1.016 and can be changed by specifying it as an input parameter to the function boundaryLayerProperties. The following warning messages are displayed if the grid is coarser than the above threshold:\ "WARNING: Grid spacing at the wall = ___ is GREATER than threshold value = 0.2 (in plus units)"\ "RUN AGAIN WITH A FINER GRID AT THE WALL"\ The threshold for grid convergence check is hardcoded and can be altered by modifying the variable wallResolutionThreshold in the function boundaryLayerProperties.

2. The indirect method implementation has been found to be stable with the use of the Van Driest and Volpiani velocity transform functions. Hence, these cases are run without any under-relaxation (underRelaxFactor = 1). However, using the Trettel-Larsson transform typically needs moderate to strong under-relaxation. Therefore, by default, underRelaxFactor = 0.5 for this case to ensure stability of the solution process. These default values can be over-written by specifying the underRelaxFactor as an input parameter to the function boundaryLayerProperties.

3. If the resolution requirement in terms of Kolomogorov length-scale are not very restrictive, the number of computed points in the wall normal direction might be lower than the parameter jb. The script will output the message:\ "WARNING: Predicted Ny points in natural stretching are less than jb"\ "Bounding min(Ny) to jb"\ "PLEASE VERIFY YOUR RESOLUTION TRESHOLD"

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ADDITIONAL DOCUMENTATION

A detailed description of each function implemented in can be found in the files boundaryLayerPropFunctions.py and boundaryLayerPropFunctions_direct.py