Reflection of Oblique Shock Wave

This example solves a reflecting oblique shock wave, as shown in Figure 1. The system consists of two oblique shock waves, which separate the flow into three zones. The incident shock results from a wedge. The second reflects from a plane wall. Flow properties in all the three zones can be calculated with the following data:

  1. The upstream (zone 1) Mach number M_1 and the flow properties density, pressure, and temperature.
  2. The first oblique shock angle \beta_1 (between zone 1 and 2) or the flow deflection angle \theta (across zone 1/2 and zone 2/3). Only one of the angle is needed. The other one can be calculated from the given one and M_1. The calculation detail is in ObliqueShockRelation.calc_flow_angle() and ObliqueShockRelation.calc_shock_angle().
../_images/reflection.png

Figure 1: Oblique shock reflected from a wall

M_{1,2,3} are the Mach number in the corresponding zone 1, 2, and 3. \theta is the flow deflection angle. \beta_{1,2} are the oblique shock angle behind the first and the second zone, respectively.

SOLVCON will be set up to solve this problem, and the simulated results will be compared with the analytical solution.

../_images/obrf_fine_rho.png

Figure 2: Simulated density (non-dimensionalized).

Driving Script

SOLVCON uses a driving script to control the numerical simulation. Its general layout is:

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#!/usr/bin/env python2.7
# The shebang above directs the operating system to look for a correct
# program to run this script.
#
# We may provide additional information here.


# Import necessary modules.
import os # Python standard library
import numpy as np # http://www.numpy.org
import solvcon as sc # SOLVCON
from solvcon.parcel import gas # A specific SOLVCON solver package we'll use


# ...
# ... other code ...
# ...


# At the end of the file.
if __name__ == '__main__':
    sc.go()

Every driving script has the following lines at the end of the file:

if __name__ == '__main__':
    sc.go()

The if __name__ == '__main__': is a magical Python construct. It will detect that the file is run as a script, not imported as a library (module). Once the detection is evaluated as true, the script call a common execution flow defined in solvcon.go(), which uses the content of the driving script to perform the calculation.

Of course, the file has a lot of other code to set up and configure the calculation, as we’ll describe later. It’s important to note that a driving script is a valid Python program file. The Python language is good for specifying parameters the calculation needs, and as a platform to conduct useful operations much more complex than settings. Any Python module can be imported for use.

See Full Listing of the Driving Script for the driving script of this example: $SCSRC/examples/gas/obrf/go. SOLVCON separates apart the configuration and the execution of a simulation case. The separation is necessary for distributed-memory parallel computing (e.g., MPI). Everything run in the driving script is about the configuration. The execution is conducted by code hidden from users.

To run the simulation, go to the example directory and execute the driving script with the command run and the simulation arrangement name obrf:

$ ./go run obrf

The driving script will then run and print messages:

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********************************************************************************
*** Start init (level 0) obrf ... 
*** ****************************************************************************
*** ****************************************************************************
*** *** Start build_domain ... 
*** *** ************************************************************************
*** *** mesh file: None
*** *** ************************************************************************
*** *** *** Start create_block ... 
*** *** *** ********************************************************************
*** *** *** ********************************************************************
*** *** *** End create_block . Elapsed time (sec) = 0.0925829
*** *** ************************************************************************
*** *** ************************************************************************
*** *** End build_domain . Elapsed time (sec) = 0.092721
*** ****************************************************************************
*** ****************************************************************************
*** End init obrf . Elapsed time (sec) = 0.0942369
********************************************************************************
********************************************************************************
*** Start run ... 
*** ****************************************************************************
*** ****************************************************************************
*** *** Start run_provide ... 
*** *** ************************************************************************
*** *** ************************************************************************
*** *** End run_provide . Elapsed time (sec) = 0.000499964
*** ****************************************************************************
*** ****************************************************************************
*** *** Start run_preloop ... 
*** *** ************************************************************************
*** *** Relation of reflected oblique shock:
*** *** - theta = 10.00 deg (flow angle)
*** *** - beta1 = 27.38 deg (shock angle)
*** *** - beta1 = 31.80 deg (shock angle)
*** *** - mach, rho, p, T, a (1) =  3.000  1.000  1.000  1.000  1.183
*** *** - mach, rho, p, T, a (2) =  2.505  1.655  2.054  1.242  1.318
*** *** - mach, rho, p, T, a (3) =  2.090  2.565  3.833  1.494  1.446
*** *** Steps 0/600
*** *** Block information:
*** ***   [Block (2D/centroid): 565 nodes, 1592 faces (100 BC), 1028 cells]
*** *** ************************************************************************
*** *** End run_preloop . Elapsed time (sec) = 0.014756
*** ****************************************************************************
*** 
*** ****************************************************************************
*** *** Start run_march ... 
*** *** ************************************************************************
*** *** ##################################################
*** *** Step 200/600, 0.7s elapsed, 1.4s left
*** *** CFL = 0.11/0.95 - 0.11/0.95 adjusted: 0/0/0
*** *** Performance of obrf:
*** ***   0.696783 seconds in marching solver.
*** ***   0.00348392 seconds/step.
*** ***   3.38902 microseconds/cell.
*** ***   0.29507 Mcells/seconds.
*** ***   1.18028 Mvariables/seconds.
*** *** ##################################################
*** *** Step 400/600, 1.4s elapsed, 0.7s left
*** *** CFL = 0.42/0.95 - 0.11/0.95 adjusted: 0/0/0
*** *** Performance of obrf:
*** ***   1.35721 seconds in marching solver.
*** ***   0.00339303 seconds/step.
*** ***   3.30061 microseconds/cell.
*** ***   0.302974 Mcells/seconds.
*** ***   1.2119 Mvariables/seconds.
*** *** ##################################################
*** *** Step 600/600, 2.1s elapsed, 0.0s left
*** *** CFL = 0.47/0.95 - 0.11/0.95 adjusted: 0/0/0
*** *** ************************************************************************
*** *** End run_march . Elapsed time (sec) = 2.06248
*** ****************************************************************************
*** 
*** ****************************************************************************
*** *** Start run_postloop ... 
*** *** ************************************************************************
*** *** Probe result at Pt/poi#611(3.79306,0.358565,0)601:
*** *** - mach3 = 2.074/2.090 (error=%0.79)
*** *** - rho3  = 2.543/2.565 (error=%0.86)
*** *** - p3    = 3.824/3.833 (error=%0.23)
*** *** Performance of obrf:
*** ***   2.02795 seconds in marching solver.
*** ***   0.00337992 seconds/step.
*** ***   3.28786 microseconds/cell.
*** ***   0.30415 Mcells/seconds.
*** ***   1.2166 Mvariables/seconds.
*** *** Averaged maximum CFL = 0.945858.
*** *** Relation of reflected oblique shock:
*** *** - theta = 10.00 deg (flow angle)
*** *** - beta1 = 27.38 deg (shock angle)
*** *** - beta1 = 31.80 deg (shock angle)
*** *** - mach, rho, p, T, a (1) =  3.000  1.000  1.000  1.000  1.183
*** *** - mach, rho, p, T, a (2) =  2.505  1.655  2.054  1.242  1.318
*** *** - mach, rho, p, T, a (3) =  2.090  2.565  3.833  1.494  1.446
*** *** ************************************************************************
*** *** End run_postloop . Elapsed time (sec) = 0.00133896
*** ****************************************************************************
*** ****************************************************************************
*** *** Start run_exhaust ... 
*** *** ************************************************************************
*** *** ************************************************************************
*** *** End run_exhaust . Elapsed time (sec) = 7.51019e-05
*** ****************************************************************************
*** ****************************************************************************
*** *** Start run_final ... 
*** *** ************************************************************************
*** *** ************************************************************************
*** *** End run_final . Elapsed time (sec) = 9.20296e-05
*** ****************************************************************************
*** ****************************************************************************
*** End run obrf . Elapsed time (sec) = 2.07972
********************************************************************************

Data will be output in directory result/.

Arrangement

An arrangement sits at the center of a driving script. It’s nothing more than a decorated Python function with a specific signature. The following function obrf() is the main arrangement we’ll use for the shock reflection problem:

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def obrf(casename, **kw):
    return obrf_base(
        # Required positional argument for the name of the simulation case.
        casename,
        # Arguments to the base configuration.
        ssteps=200, psteps=4, edgelength=0.1,
        gamma=1.4, density=1.0, pressure=1.0, mach=3.0, theta=10.0/180*np.pi,
        # Arguments to GasCase.
        time_increment=7.e-3, steps_run=600, **kw)

It’s typical for the arrangement function obrf() to be a thin wrapper which calls another function (in this case, obrf_base()). It should be noted that an arrangement function must take one and only one positional argument: casename. All the other arguments need to be keyword.

To make the function obrf() discoverable by SOLVCON, it needs to be registered with the decorator gas.register_arrangement (gas was imported at the beginning of the driving script):

@gas.register_arrangement
def obrf(casename, **kw):
    # ... contents ...

The function obrf_base() does the real work of configuration:

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def obrf_base(
    casename=None, psteps=None, ssteps=None, edgelength=None,
    gamma=None, density=None, pressure=None, mach=None, theta=None, **kw):
    """
    Base configuration of the simulation and return the case object.

    :return: The created Case object.
    :rtype: solvcon.parcel.gas.GasCase
    """

    ############################################################################
    # Step 1: Obtain the analytical solution.
    ############################################################################

    # Calculate the flow properties in all zones separated by the shock.
    relation = ObliqueShockReflection(gamma=gamma, theta=theta, mach1=mach,
                                      rho1=density, p1=pressure, T1=1)

    ############################################################################
    # Step 2: Instantiate the simulation case.
    ############################################################################

    # Create the mesh generator.  Keep it for later use.
    mesher = RectangleMesher(lowerleft=(0,0), upperright=(4,1),
                             edgelength=edgelength)
    # Set up case.
    cse = gas.GasCase(
        # Mesh generator.
        mesher=mesher,
        # Mapping boundary-condition treatments.
        bcmap=generate_bcmap(relation),
        # Use the case name to be the basename for all generated files.
        basefn=casename,
        # Use `cwd`/result to store all generated files.
        basedir=os.path.abspath(os.path.join(os.getcwd(), 'result')),
        # Debug and capture-all.
        debug=False, **kw)

    ############################################################################
    # Step 3: Set up delayed callbacks.
    ############################################################################

    # Field initialization and derived calculations.
    cse.defer(gas.FillAnchor, mappers={'soln': gas.GasSolver.ALMOST_ZERO,
                                       'dsoln': 0.0, 'amsca': gamma})
    cse.defer(gas.DensityInitAnchor, rho=density)
    cse.defer(gas.PhysicsAnchor, rsteps=ssteps)
    # Report information while calculating.
    cse.defer(relation.hookcls)
    cse.defer(gas.ProgressHook, linewidth=ssteps/psteps, psteps=psteps)
    cse.defer(gas.CflHook, fullstop=False, cflmax=10.0, psteps=ssteps)
    cse.defer(gas.MeshInfoHook, psteps=ssteps)
    cse.defer(ReflectionProbe, rect=mesher, relation=relation, psteps=ssteps)
    # Store data.
    cse.defer(gas.PMarchSave,
              anames=[('soln', False, -4),
                      ('rho', True, 0),
                      ('p', True, 0),
                      ('T', True, 0),
                      ('ke', True, 0),
                      ('M', True, 0),
                      ('sch', True, 0),
                      ('v', True, 0.5)],
              psteps=ssteps)

    ############################################################################
    # Final: Return the configured simulation case.
    ############################################################################
    return cse

There are three steps:

  1. Obtain the Analytical Solution to set up all quantities for the simulation.
  2. Instantiate the simulation case object (of type GasCase). The GasCase object needs to know how to set up the mesh (see Mesh Generation) and the boundary-condition (BC) treatment (see BC Treatment Mapping). Section Case Instantiation will explain the details.
  3. Configure callbacks for delayed operations by calling defer() of the constructed simulation GasCase object. Section Callback Configuration will explain these callbacks.

At the end of the base function, the constructed and configured GasCase object is returned.

Although the example has only one arrangement, it’s actually encouraged to have multiple arrangements in a script. In this way one driving script can perform simulations of different parameters or different kinds. Conventionally we place the arrangement functions near the end of the driving script, and the decorated functions (e.g., obrf()) are placed after the base (e.g., obrf_base()). The ordering will make the file easier to read.

Analytical Solution

To set up the numerical simulation for the shock-reflection problem, we’ll use class ObliqueShockRelation to calculate necessary parameters by creating a subclass of it:

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class ObliqueShockReflection(gas.ObliqueShockRelation):
    def __init__(self, gamma, theta, mach1, rho1, p1, T1):
        super(ObliqueShockReflection, self).__init__(gamma=gamma)
        # Angles and Mach numbers.
        self.theta = theta
        self.mach1 = mach1
        self.beta1 = beta1 = self.calc_shock_angle(mach1, theta)
        self.mach2 = mach2 = self.calc_dmach(mach1, beta1)
        self.beta2 = beta2 = self.calc_shock_angle(mach2, theta)
        self.mach3 = mach3 = self.calc_dmach(mach2, beta2)
        # Flow properties in the first zone.
        self.rho1 = rho1
        self.p1 = p1
        self.T1 = T1
        self.a1 = np.sqrt(gamma*p1/rho1)
        # Flow properties in the second zone.
        self.rho2 = rho2 = rho1 * self.calc_density_ratio(mach1, beta1)
        self.p2 = p2 = p1 * self.calc_pressure_ratio(mach1, beta1)
        self.T2 = T2 = T1 * self.calc_temperature_ratio(mach1, beta1)
        self.a2 = np.sqrt(gamma*p2/rho2)
        # Flow properties in the third zone.
        self.rho3 = rho3 = rho2 * self.calc_density_ratio(mach2, beta2)
        self.p3 = p3 = p2 * self.calc_pressure_ratio(mach2, beta2)
        self.T3 = T3 = T2 * self.calc_temperature_ratio(mach2, beta2)
        self.a3 = np.sqrt(gamma*p3/rho3)

    def __str__(self):
        msg = 'Relation of reflected oblique shock:\n'
        msg += '- theta = %5.2f deg (flow angle)\n' % (self.theta/np.pi*180)
        msg += '- beta1 = %5.2f deg (shock angle)\n' % (self.beta1/np.pi*180)
        msg += '- beta1 = %5.2f deg (shock angle)\n' % (self.beta2/np.pi*180)
        def property_string(zone):
            values = [getattr(self, '%s%d' % (key, zone))
                      for key in 'mach', 'rho', 'p', 'T', 'a']
            messages = [' %6.3f' % val for val in values]
            return ''.join(messages)
        msg += '- mach, rho, p, T, a (1) =' + property_string(1) + '\n'
        msg += '- mach, rho, p, T, a (2) =' + property_string(2) + '\n'
        msg += '- mach, rho, p, T, a (3) =' + property_string(3)
        return msg

    @property
    def hookcls(self):
        relation = self
        class _ShowRelation(sc.MeshHook):
            def preloop(self):
                for msg in str(relation).split('\n'):
                    self.info(msg + '\n')
            postloop = preloop
        return _ShowRelation

For the detail of ObliqueShockRelation, see Oblique Shock Relation.

Case Instantiation

An instance of GasCase represents a numerical simulation using the gas module. In addition to Mesh Generation and BC Treatment Mapping, other miscellaneous settings can be supplied through the GasCase constructor.

Mesh Generation

An unstructured mesh is required for a SOLVCON simulation. A mesh file can be created beforehand or on-the-fly with the simulation. The example uses the latter approach. The following is an example of mesh generating function that calls Gmsh:

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class RectangleMesher(object):
    """
    Representation of a rectangle and the Gmsh meshing helper.

    :ivar lowerleft: (x0, y0) of the rectangle.
    :type lowerleft: tuple
    :ivar upperright: (x1, y1) of the rectangle.
    :type upperright: tuple
    :ivar edgelength: Length of each mesh cell edge.
    :type edgelength: float
    """

    GMSH_SCRIPT = """
    // vertices.
    Point(1) = {%(x1)g,%(y1)g,0,%(edgelength)g};
    Point(2) = {%(x0)g,%(y1)g,0,%(edgelength)g};
    Point(3) = {%(x0)g,%(y0)g,0,%(edgelength)g};
    Point(4) = {%(x1)g,%(y0)g,0,%(edgelength)g};
    // lines.
    Line(1) = {1,2};
    Line(2) = {2,3};
    Line(3) = {3,4};
    Line(4) = {4,1};
    // surface.
    Line Loop(1) = {1,2,3,4};
    Plane Surface(1) = {1};
    // physics.
    Physical Line("upper") = {1};
    Physical Line("left") = {2};
    Physical Line("lower") = {3};
    Physical Line("right") = {4};
    Physical Surface("domain") = {1};
    """.strip()

    def __init__(self, lowerleft, upperright, edgelength):
        assert 2 == len(lowerleft)
        self.lowerleft = tuple(float(val) for val in lowerleft)
        assert 2 == len(upperright)
        self.upperright = tuple(float(val) for val in upperright)
        self.edgelength = float(edgelength)

    def __call__(self, cse):
        x0, y0 = self.lowerleft
        x1, y1 = self.upperright
        script_arguments = dict(
            edgelength=self.edgelength, x0=x0, y0=y0, x1=x1, y1=y1)
        cmds = self.GMSH_SCRIPT % script_arguments
        cmds = [cmd.rstrip() for cmd in cmds.strip().split('\n')]
        gmh = sc.Gmsh(cmds)()
        return gmh.toblock(bcname_mapper=cse.condition.bcmap)

BC Treatment Mapping

Boundary-condition treatments are specified by creating a dict to map the name of the boundary to a specific BC class.

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def generate_bcmap(relation):
    # Flow properties (fp).
    fpb = {
        'gamma': relation.gamma, 'rho': relation.rho1,
        'v2': 0.0, 'v3': 0.0, 'p': relation.p1,
    }
    fpb['v1'] = relation.mach1*np.sqrt(relation.gamma*fpb['p']/fpb['rho'])
    fpt = fpb.copy()
    fpt['rho'] = relation.rho2
    fpt['p'] = relation.p2
    V2 = relation.mach2 * relation.a2
    fpt['v1'] = V2 * np.cos(relation.theta)
    fpt['v2'] = -V2 * np.sin(relation.theta)
    fpi = fpb.copy()
    # BC map.
    bcmap = {
        'upper': (sc.bctregy.GasInlet, fpt,),
        'left': (sc.bctregy.GasInlet, fpb,),
        'right': (sc.bctregy.GasNonrefl, {},),
        'lower': (sc.bctregy.GasWall, {},),
    }
    return bcmap

Callback Configuration

SOLVCON provides general-purpose, application-agnostic solving facilities. To describe the problem to SOLVCON, we need to provide both data (numbers) and logic (computer code) in the driving script. The supplied code will be called back at proper points while the simulation is running.

Classes MeshHook and MeshAnchor are the fundamental constructs to make callbacks in the sequential and parallel runtime environment, respectively. The module gas includes useful callbacks, but we still need to write a couple of them in the driving script.

The shock reflection problem uses three categories of callbacks.

  1. Initialization and calculation:
  1. Reporting:
  1. Output:

The order of these callbacks is important. Dependency between callbacks is allowed.

View Results

After simulation, the results are stored in directory result/ as VTK unstructured data files that can be opened and processed by using ParaView. The result in Figure 2 was produced in this way. Other quantities can also be visualized, e.g., the Mach number shown in Figure 3.

../_images/obrf_fine_mach.png

Figure 3: Mach number at the final time step of the arrangement obrf_fine.

Both of Figures 2 and 3 are obtained with the arrangement obrf_fine.

Oblique Shock Relation

An oblique shock results from a sudden change of direction of supersonic flow. The relations of density (\rho), pressure (p), and temprature (T) across the shock can be obtained analytically [Anderson03]. In addition, two angles are defined:

  1. The angle of the oblique shock wave deflected from the upstream is \beta; the shock angle.
  2. The angle of the flow behind the shock wave deflected from the upstream is \theta; the flow angle.

See Figure 4 for the illustration of the two angles.

../_images/oblique_shock.png

Figure 4: Oblique shock wave by a wedge

M is Mach number. \theta is the flow deflection angle. \beta is the oblique shock angle.

Methods of calculating the shock relations are organized in the class ObliqueShockRelation. To obtain the relations of density (\rho), pressure (p), and temprature (T), the control volume across the shock is emplyed, as shown in Figure 5. In the figure and in ObliqueShockRelation, subscript 1 denotes upstream properties and subscript 2 denotes downstream properties. Derivation of the relation uses a rotated coordinate system (n, t) defined by the oblique shock, where \hat{n} is the unit vector normal to the shock, and \hat{t} is the unit vector tangential to the shock. But in this document we won’t go into the detail.

../_images/oblique_relation.png

Figure 5: Properties across an oblique shock

The flow properties in the upstream zone of the oblique shock are v_1, M_1, \rho_1, p_1, T_1. Those in the downstream zone of the shock are v_2, M_2, \rho_2, p_2, T_2.
class solvcon.parcel.gas.oblique_shock.ObliqueShockRelation(gamma)

Calculators of oblique shock relations.

The constructor must take the ratio of specific heat:

>>> ObliqueShockRelation()
Traceback (most recent call last):
    ...
TypeError: __init__() takes exactly 2 arguments (1 given)
>>> ob = ObliqueShockRelation(gamma=1.4)

The ratio of specific heat can be accessed through the gamma attribute:

>>> ob.gamma
1.4

The object can be used to calculate shock relations. For example, calc_density_ratio() returns the \rho_2/\rho_1:

>>> round(ob.calc_density_ratio(mach1=3, beta=37.8/180*np.pi), 10)
2.4204302545

The solution changes as gamma changes:

>>> ob.gamma = 1.2
>>> round(ob.calc_density_ratio(mach1=3, beta=37.8/180*np.pi), 10)
2.7793244902
gamma

Ratio of specific heat \gamma, dimensionless.

ObliqueShockRelation provides three methods to calculate the ratio of flow properties across the shock. M_1 and \beta are required arguments:

With M_1 available, the shock angle \beta can be calculated from the flow angle \theta, or vice versa, by using the following two methods:

The following method calculates the downstream Mach number, with the upstream Mach number M_1 and either of \beta or \theta supplied:

Listing of all methods:

ObliqueShockRelation.calc_density_ratio(mach1, beta)

Calculate the ratio of density \rho across an oblique shock wave of which the angle deflected from the upstream flow is \beta and the upstream Mach number is M_1:

\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1) M_{n1}^2} {(\gamma - 1) M_{n1}^2 + 2}

where M_{n1} = M_1\sin\beta.

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> round(ob.calc_density_ratio(mach1=3, beta=37.8/180*np.pi), 10)
2.4204302545

as well as numpy.ndarray:

>>> angle = 37.8/180*np.pi; angle = np.array([angle, angle])
>>> np.round(ob.calc_density_ratio(mach1=3, beta=angle), 10).tolist()
[2.4204302545, 2.4204302545]
Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • beta – Oblique shock angle \beta deflected from the upstream flow, in radian.
ObliqueShockRelation.calc_pressure_ratio(mach1, beta)

Calculate the ratio of pressure p across an oblique shock wave of which the angle deflected from the upstream flow is \beta and the upstream Mach number is M_1:

\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}(M_{n1}^2 - 1)

where M_{n1} = M_1\sin\beta.

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> round(ob.calc_pressure_ratio(mach1=3, beta=37.8/180*np.pi), 10)
3.7777114257

as well as numpy.ndarray:

>>> angle = 37.8/180*np.pi; angle = np.array([angle, angle])
>>> np.round(ob.calc_pressure_ratio(mach1=3, beta=angle), 10).tolist()
[3.7777114257, 3.7777114257]
Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • beta – Oblique shock angle \beta deflected from the upstream flow, in radian.
ObliqueShockRelation.calc_temperature_ratio(mach1, beta)

Calculate the ratio of temperature T across an oblique shock wave of which the angle deflected from the upstream flow is \beta and the upstream Mach number is M_1:

\frac{T_2}{T_1} = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2}

where both p_2/p_1 and \rho_1/\rho_2 are functions of \gamma, M_1, and \beta. See also calc_pressure_ratio() and calc_density_ratio().

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> round(ob.calc_temperature_ratio(mach1=3, beta=37.8/180*np.pi), 10)
1.5607602899

as well as numpy.ndarray:

>>> angle = 37.8/180*np.pi; angle = np.array([angle, angle])
>>> np.round(ob.calc_temperature_ratio(mach1=3, beta=angle), 10).tolist()
[1.5607602899, 1.5607602899]
Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • beta – Oblique shock angle \beta deflected from the upstream flow, in radian.
ObliqueShockRelation.calc_dmach(mach1, beta=None, theta=None, delta=1)

Calculate the downstream Mach number from the upstream Mach number M_1 and either of the shock or flow deflection angles:

M_2 = \frac{M_{n2}}{\sin(\beta-\theta)}

where M_{n2} is calculated from calc_normal_dmach() with M_{n1} = M_1\sin\beta.

The method can be invoked with only either \beta or \theta:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> ob.calc_dmach(3, beta=0.2, theta=0.1)
Traceback (most recent call last):
    ...
ValueError: got (beta=0.2, theta=0.1), but I need to take either beta or theta
>>> ob.calc_dmach(3)
Traceback (most recent call last):
    ...
ValueError: got (beta=None, theta=None), but I need to take either beta or theta

This method accepts scalar:

>>> round(ob.calc_dmach(3, beta=37.8/180*np.pi), 10)
1.9924827009
>>> round(ob.calc_dmach(3, theta=20./180*np.pi), 10)
1.9941316656

as well as numpy.ndarray:

>>> angle = 37.8/180*np.pi; angle = np.array([angle, angle])
>>> np.round(ob.calc_dmach(3, beta=angle), 10).tolist()
[1.9924827009, 1.9924827009]
>>> angle = 20./180*np.pi; angle = np.array([angle, angle])
>>> np.round(ob.calc_dmach(3, theta=angle), 10).tolist()
[1.9941316656, 1.9941316656]
Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • beta – Oblique shock angle \beta deflected from the upstream flow, in radian.
  • theta – Downstream flow angle \theta deflected from the upstream flow, in radian.
  • delta – A switching integer \delta. For \delta = 0, the function gives the solution of strong shock, while for \delta = 1, it gives the solution of weak shock. This keyword argument is only valid when theta is given. The default value is 1.
ObliqueShockRelation.calc_normal_dmach(mach_n1)

Calculate the downstream Mach number from the given upstream Mach number M_{n1}, in the direction normal to the shock:

M_{n2} = \sqrt{\frac{(\gamma-1)M_{n1}^2 + 2} {2\gamma M_{n1}^2 - (\gamma-1)}}

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> round(ob.calc_normal_dmach(mach_n1=3), 10)
0.4751909633

as well as numpy.ndarray:

>>> np.round(ob.calc_normal_dmach(mach_n1=np.array([3, 3])), 10).tolist()
[0.4751909633, 0.4751909633]
Parameters:mach_n1 – Upstream Mach number M_{n1} normal to the shock wave, dimensionless.
ObliqueShockRelation.calc_flow_angle(mach1, beta)

Calculate the downstream flow angle \theta deflected from the upstream flow by using calc_flow_tangent(), in radian.

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> angle = 48.25848/180*np.pi
>>> round(ob.calc_flow_angle(mach1=4, beta=angle)/np.pi*180, 10)
32.0000000807

as well as numpy.ndarray:

>>> angle = 48.25848/180*np.pi; angle = np.array([angle, angle])
>>> np.round((ob.calc_flow_angle(mach1=4, beta=angle)/np.pi*180), 10).tolist()
[32.0000000807, 32.0000000807]

See Example 4.6 in [Anderson03] for the forward analysis. The above is the inverse analysis.

Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • beta – Oblique shock angle \beta deflected from the upstream flow, in radian.
ObliqueShockRelation.calc_flow_tangent(mach1, beta)

Calculate the trigonometric tangent function \tan\beta of the downstream flow angle \theta deflected from the upstream flow by using the \theta-\beta-M relation:

\tan\theta = 2\cot\beta \frac{M_1^2\sin^2\beta - 1} {M_1^2(\gamma+\cos2\beta) + 2}

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> angle = 48.25848/180*np.pi
>>> round(ob.calc_flow_tangent(mach1=4, beta=angle), 10)
0.6248693539

as well as numpy.ndarray:

>>> angle = 48.25848/180*np.pi; angle = np.array([angle, angle])
>>> np.round(ob.calc_flow_tangent(mach1=4, beta=angle), 10).tolist()
[0.6248693539, 0.6248693539]

See Example 4.6 in [Anderson03] for the forward analysis. The above is the inverse analysis.

Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • beta – Oblique shock angle \beta deflected from the upstream flow, in radian.
ObliqueShockRelation.calc_shock_angle(mach1, theta, delta=1)

Calculate the downstream shock angle \beta deflected from the upstream flow by using calc_shock_tangent(), in radian.

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> angle = 32./180*np.pi
>>> round(ob.calc_shock_angle(mach1=4, theta=angle, delta=1)/np.pi*180, 10)
48.2584798722

as well as numpy.ndarray:

>>> angle = np.array([angle, angle])
>>> np.round(ob.calc_shock_angle(mach1=4, theta=angle, delta=1)/np.pi*180,
...          10).tolist()
[48.2584798722, 48.2584798722]

See Example 4.6 in [Anderson03] for the analysis.

Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • theta – Downstream flow angle \theta deflected from the upstream flow, in radian.
  • delta – A switching integer \delta. For \delta = 0, the function gives the solution of strong shock, while for \delta = 1, it gives the solution of weak shock. The default value is 1.
ObliqueShockRelation.calc_shock_tangent(mach1, theta, delta)

Calculate the downstream shock angle \beta deflected from the upstream flow by using the alternative \beta-\theta-M relation:

\tan\beta = \frac{M_1^2 - 1 + 2\lambda\cos\left(\frac{4\pi\delta + \cos^{-1}\chi}{3}\right)} {3\left(1 + \frac{\gamma-1}{2}M_1^2\right)\tan\theta}

where \lambda and \chi are obtained internally by calling calc_shock_tangent_aux().

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> angle = 32./180*np.pi
>>> round(ob.calc_shock_tangent(mach1=4, theta=angle, delta=1), 10)
1.1207391858

as well as numpy.ndarray:

>>> angle = np.array([angle, angle])
>>> np.round(ob.calc_shock_tangent(mach1=4, theta=angle, delta=1),
...          10).tolist()
[1.1207391858, 1.1207391858]

See Example 4.6 in [Anderson03] for the analysis.

Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • theta – Downstream flow angle \theta deflected from the upstream flow, in radian.
  • delta – A switching integer \delta. For \delta = 0, the function gives the solution of strong shock, while for \delta = 1, it gives the solution of weak shock.
ObliqueShockRelation.calc_shock_tangent_aux(mach1, theta)

Calculate the \lambda and \chi functions used by calc_shock_tangent():

\lambda = \sqrt{(M_1^2-1)^2 - 3\left(1+\frac{\gamma-1}{2}M_1^2\right) \left(1+\frac{\gamma+1}{2}M_1^2\right)\tan^2\theta}

\chi = \frac{(M_1^2-1)^3 - 9\left(1+\frac{\gamma-1}{2}M_1^2\right) \left(1+\frac{\gamma-1}{2}M_1^2+\frac{\gamma+1}{4}M_1^4\right) \tan^2\theta} {\lambda^3}

This method accepts scalar:

>>> ob = ObliqueShockRelation(gamma=1.4)
>>> lmbd, chi = ob.calc_shock_tangent_aux(mach1=4, theta=32./180*np.pi)
>>> round(lmbd, 10), round(chi, 10)
(11.2080188412, 0.7428957121)

as well as numpy.ndarray:

>>> angle = 32./180*np.pi; angle = np.array([angle, angle])
>>> lmbd, chi = ob.calc_shock_tangent_aux(mach1=4, theta=angle)
>>> np.round(lmbd, 10).tolist()
[11.2080188412, 11.2080188412]
>>> np.round(chi, 10).tolist()
[0.7428957121, 0.7428957121]

See Example 4.6 in [Anderson03] for the analysis.

Parameters:
  • mach1 – Upstream Mach number M_1, dimensionless.
  • theta – Downstream flow angle \theta deflected from the upstream flow, in radian.

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#!/usr/bin/env python2.7
# -*- coding: UTF-8 -*-
#
# Copyright (c) 2014, Yung-Yu Chen <yyc@solvcon.net>
#
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are met:
#
# - Redistributions of source code must retain the above copyright notice, this
#   list of conditions and the following disclaimer.
# - Redistributions in binary form must reproduce the above copyright notice,
#   this list of conditions and the following disclaimer in the documentation
#   and/or other materials provided with the distribution.
# - Neither the name of the SOLVCON nor the names of its contributors may be
#   used to endorse or promote products derived from this software without
#   specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
# POSSIBILITY OF SUCH DAMAGE.


"""
This is an example for solving the problem of oblique shock reflection.
"""


import os # Python standard library
import numpy as np # http://www.numpy.org
import solvcon as sc # SOLVCON
from solvcon.parcel import gas # A specific SOLVCON solver package we'll use


class ObliqueShockReflection(gas.ObliqueShockRelation):
    def __init__(self, gamma, theta, mach1, rho1, p1, T1):
        super(ObliqueShockReflection, self).__init__(gamma=gamma)
        # Angles and Mach numbers.
        self.theta = theta
        self.mach1 = mach1
        self.beta1 = beta1 = self.calc_shock_angle(mach1, theta)
        self.mach2 = mach2 = self.calc_dmach(mach1, beta1)
        self.beta2 = beta2 = self.calc_shock_angle(mach2, theta)
        self.mach3 = mach3 = self.calc_dmach(mach2, beta2)
        # Flow properties in the first zone.
        self.rho1 = rho1
        self.p1 = p1
        self.T1 = T1
        self.a1 = np.sqrt(gamma*p1/rho1)
        # Flow properties in the second zone.
        self.rho2 = rho2 = rho1 * self.calc_density_ratio(mach1, beta1)
        self.p2 = p2 = p1 * self.calc_pressure_ratio(mach1, beta1)
        self.T2 = T2 = T1 * self.calc_temperature_ratio(mach1, beta1)
        self.a2 = np.sqrt(gamma*p2/rho2)
        # Flow properties in the third zone.
        self.rho3 = rho3 = rho2 * self.calc_density_ratio(mach2, beta2)
        self.p3 = p3 = p2 * self.calc_pressure_ratio(mach2, beta2)
        self.T3 = T3 = T2 * self.calc_temperature_ratio(mach2, beta2)
        self.a3 = np.sqrt(gamma*p3/rho3)

    def __str__(self):
        msg = 'Relation of reflected oblique shock:\n'
        msg += '- theta = %5.2f deg (flow angle)\n' % (self.theta/np.pi*180)
        msg += '- beta1 = %5.2f deg (shock angle)\n' % (self.beta1/np.pi*180)
        msg += '- beta1 = %5.2f deg (shock angle)\n' % (self.beta2/np.pi*180)
        def property_string(zone):
            values = [getattr(self, '%s%d' % (key, zone))
                      for key in 'mach', 'rho', 'p', 'T', 'a']
            messages = [' %6.3f' % val for val in values]
            return ''.join(messages)
        msg += '- mach, rho, p, T, a (1) =' + property_string(1) + '\n'
        msg += '- mach, rho, p, T, a (2) =' + property_string(2) + '\n'
        msg += '- mach, rho, p, T, a (3) =' + property_string(3)
        return msg

    @property
    def hookcls(self):
        relation = self
        class _ShowRelation(sc.MeshHook):
            def preloop(self):
                for msg in str(relation).split('\n'):
                    self.info(msg + '\n')
            postloop = preloop
        return _ShowRelation


class RectangleMesher(object):
    """
    Representation of a rectangle and the Gmsh meshing helper.

    :ivar lowerleft: (x0, y0) of the rectangle.
    :type lowerleft: tuple
    :ivar upperright: (x1, y1) of the rectangle.
    :type upperright: tuple
    :ivar edgelength: Length of each mesh cell edge.
    :type edgelength: float
    """

    GMSH_SCRIPT = """
    // vertices.
    Point(1) = {%(x1)g,%(y1)g,0,%(edgelength)g};
    Point(2) = {%(x0)g,%(y1)g,0,%(edgelength)g};
    Point(3) = {%(x0)g,%(y0)g,0,%(edgelength)g};
    Point(4) = {%(x1)g,%(y0)g,0,%(edgelength)g};
    // lines.
    Line(1) = {1,2};
    Line(2) = {2,3};
    Line(3) = {3,4};
    Line(4) = {4,1};
    // surface.
    Line Loop(1) = {1,2,3,4};
    Plane Surface(1) = {1};
    // physics.
    Physical Line("upper") = {1};
    Physical Line("left") = {2};
    Physical Line("lower") = {3};
    Physical Line("right") = {4};
    Physical Surface("domain") = {1};
    """.strip()

    def __init__(self, lowerleft, upperright, edgelength):
        assert 2 == len(lowerleft)
        self.lowerleft = tuple(float(val) for val in lowerleft)
        assert 2 == len(upperright)
        self.upperright = tuple(float(val) for val in upperright)
        self.edgelength = float(edgelength)

    def __call__(self, cse):
        x0, y0 = self.lowerleft
        x1, y1 = self.upperright
        script_arguments = dict(
            edgelength=self.edgelength, x0=x0, y0=y0, x1=x1, y1=y1)
        cmds = self.GMSH_SCRIPT % script_arguments
        cmds = [cmd.rstrip() for cmd in cmds.strip().split('\n')]
        gmh = sc.Gmsh(cmds)()
        return gmh.toblock(bcname_mapper=cse.condition.bcmap)


def generate_bcmap(relation):
    # Flow properties (fp).
    fpb = {
        'gamma': relation.gamma, 'rho': relation.rho1,
        'v2': 0.0, 'v3': 0.0, 'p': relation.p1,
    }
    fpb['v1'] = relation.mach1*np.sqrt(relation.gamma*fpb['p']/fpb['rho'])
    fpt = fpb.copy()
    fpt['rho'] = relation.rho2
    fpt['p'] = relation.p2
    V2 = relation.mach2 * relation.a2
    fpt['v1'] = V2 * np.cos(relation.theta)
    fpt['v2'] = -V2 * np.sin(relation.theta)
    fpi = fpb.copy()
    # BC map.
    bcmap = {
        'upper': (sc.bctregy.GasInlet, fpt,),
        'left': (sc.bctregy.GasInlet, fpb,),
        'right': (sc.bctregy.GasNonrefl, {},),
        'lower': (sc.bctregy.GasWall, {},),
    }
    return bcmap


class ReflectionProbe(gas.ProbeHook):
    """
    Place a probe for the flow properties in the reflected region.
    """

    def __init__(self, cse, **kw):
        """
        :param relation: Instruct shock relations.
        :type relation: ObliqueShockReflection
        :param rect: Instruct the mesh rectangle.
        :type rect: RectangleMesher
        """
        rect = kw.pop('rect')
        self.relation = relation = kw.pop('relation')
        factor = kw.pop('factor', 0.9)
        # Detemine location.
        theta = relation.theta
        beta1 = relation.beta1
        beta2 = relation.beta2
        x0, y0 = rect.lowerleft
        x1, y1 = rect.upperright
        lgh = (y1-y0) / np.tan(beta1)
        hgt = factor * (x1-x0-lgh) * np.tan((beta2-theta)/2)
        lgh = factor * (x1-x0-lgh) + lgh
        poi = ('poi', x0+lgh, y0+hgt, 0.0)
        # Call super.
        kw['coords'] = (poi,)
        kw['speclst'] = ['M', 'rho', 'p']
        super(ReflectionProbe, self).__init__(cse, **kw)

    def postloop(self):
        super(ReflectionProbe, self).postloop()
        rel = self.relation
        self.info('Probe result at %s:\n' % self.points[0])
        M, rho, p = self.points[0].vals[-1][1:]
        self.info('- mach3 = %.3f/%.3f (error=%%%.2f)\n' % (
            M, rel.mach3, abs((M-rel.mach3)/rel.mach3)*100))
        self.info('- rho3  = %.3f/%.3f (error=%%%.2f)\n' % (
            rho, rel.rho3, abs((rho-rel.rho3)/rel.rho3)*100))
        self.info('- p3    = %.3f/%.3f (error=%%%.2f)\n' % (
            p, rel.p3, abs((p-rel.p3)/rel.p3)*100))


def obrf_base(
    casename=None, psteps=None, ssteps=None, edgelength=None,
    gamma=None, density=None, pressure=None, mach=None, theta=None, **kw):
    """
    Base configuration of the simulation and return the case object.

    :return: The created Case object.
    :rtype: solvcon.parcel.gas.GasCase
    """

    ############################################################################
    # Step 1: Obtain the analytical solution.
    ############################################################################

    # Calculate the flow properties in all zones separated by the shock.
    relation = ObliqueShockReflection(gamma=gamma, theta=theta, mach1=mach,
                                      rho1=density, p1=pressure, T1=1)

    ############################################################################
    # Step 2: Instantiate the simulation case.
    ############################################################################

    # Create the mesh generator.  Keep it for later use.
    mesher = RectangleMesher(lowerleft=(0,0), upperright=(4,1),
                             edgelength=edgelength)
    # Set up case.
    cse = gas.GasCase(
        # Mesh generator.
        mesher=mesher,
        # Mapping boundary-condition treatments.
        bcmap=generate_bcmap(relation),
        # Use the case name to be the basename for all generated files.
        basefn=casename,
        # Use `cwd`/result to store all generated files.
        basedir=os.path.abspath(os.path.join(os.getcwd(), 'result')),
        # Debug and capture-all.
        debug=False, **kw)

    ############################################################################
    # Step 3: Set up delayed callbacks.
    ############################################################################

    # Field initialization and derived calculations.
    cse.defer(gas.FillAnchor, mappers={'soln': gas.GasSolver.ALMOST_ZERO,
                                       'dsoln': 0.0, 'amsca': gamma})
    cse.defer(gas.DensityInitAnchor, rho=density)
    cse.defer(gas.PhysicsAnchor, rsteps=ssteps)
    # Report information while calculating.
    cse.defer(relation.hookcls)
    cse.defer(gas.ProgressHook, linewidth=ssteps/psteps, psteps=psteps)
    cse.defer(gas.CflHook, fullstop=False, cflmax=10.0, psteps=ssteps)
    cse.defer(gas.MeshInfoHook, psteps=ssteps)
    cse.defer(ReflectionProbe, rect=mesher, relation=relation, psteps=ssteps)
    # Store data.
    cse.defer(gas.PMarchSave,
              anames=[('soln', False, -4),
                      ('rho', True, 0),
                      ('p', True, 0),
                      ('T', True, 0),
                      ('ke', True, 0),
                      ('M', True, 0),
                      ('sch', True, 0),
                      ('v', True, 0.5)],
              psteps=ssteps)

    ############################################################################
    # Final: Return the configured simulation case.
    ############################################################################
    return cse


@gas.register_arrangement
def obrf(casename, **kw):
    return obrf_base(
        # Required positional argument for the name of the simulation case.
        casename,
        # Arguments to the base configuration.
        ssteps=200, psteps=4, edgelength=0.1,
        gamma=1.4, density=1.0, pressure=1.0, mach=3.0, theta=10.0/180*np.pi,
        # Arguments to GasCase.
        time_increment=7.e-3, steps_run=600, **kw)


@gas.register_arrangement
def obrf_fine(casename, **kw):
    return obrf_base(
        casename,
        ssteps=200, psteps=4, edgelength=0.02,
        gamma=1.4, density=1.0, pressure=1.0, mach=3.0, theta=10.0/180*np.pi,
        time_increment=1.e-3, steps_run=4000, **kw)


if __name__ == '__main__':
    sc.go()

# vim: set ff=unix fenc=utf8 ft=python ai et sw=4 ts=4 tw=79: