
Quick search Euler equations¶ All the necessary routines that implement the 2D Euler Equations (given in conservative variables) are defined in `GoverningEquations/EulerEquations.py` Governing equations Equations of state Initial conditions Governing equations ¶ class `3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.EulerEquations.Equations`(FluidProp, EOs)¶ Class furnishing all the methods that are required to define numerical schemes and evolve the solution according to the Euler Equation Fields It contains the following initialisation parameters (later available as well as attributes) FluidProp (list of FluidModel) – the list of fluid property instances (NbFluids) EOS (list of callbacks) – the list of Equation of state functions (NbFluids) It further contains the following attributes, filled upon initialisation NbVariables (integer) – number of variables of the model VariablesNames (list of strings) – ordered name of the variables VariablesUnits (list of strings) – symbols of the units of the variables VariablesLatexNames (list of strings) – ordered name of the variables, latex encoding VariablesLatexUnits (list of strings) – symbols of the units of the variables, latex encoding XYLatexUnits (list of strings) – units of the x-y coordinates in latex encoding Parameters FluidProp (list of FluidModel) – the list of fluid property instances (NbFluids) EOs (list of EOS) – the list of Equation of state instances (NbFluids) Methods `ConservativeToPrimary`(Var, FluidIndex, args)¶ Converts the conservative variables to the primary ones Parameters Var* (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) Returns (NbVars x NbGivenPoints) the corresponding primary variables Return type Var (numpy array) Note args is there only for compatibility reason at call time `PrimaryToConservative`(PrimVar, FluidIndex, args)¶ Converts the primary variables to the conservative ones Parameters Var (2D numpy array) – the primary variables of the problem (NbVars x NbGivenPoints) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) Returns (NbVars x NbGivenPoints) the corresponding conservative variables Return type Var (numpy array) Note args is there only for compatibility reason at call time `Flux`(Var, args)¶ Flux function of the (conservative) EulerEquations Parameters Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints) Optional argument x (float numpy array), – (optional, not impacting here) x-y coordinates of the given points (NbGivenPoints x 2) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) Returns fills direclty the Parameters data structure Return type None Note This function is Vectorised Fluid index is the fluid index in the initially given list, not the fluid type FluidProp is the list of all the fluid properties for each fluid (given the list, not the type) args is there only for compatibility reason at call time `GetUV`(Var, args)¶ Function giving back the x-y velocity at the considered point, given the variational values given for the the conservative Euler equation’s variables. Parameters Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints) x (float numpy array) – (generally optional, required for this problem), x-y coordinates of the given points (NbGivenPoints x 2) Optional argument FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) Returns UV (float numpy array) – the velocities values at the points (2 x NbGivenPoints) Note This function is Vectorised args is there only for compatibility reason at call time `Jacobian`(Var, x, FluidIndex, args)¶ Computes the Jacobian of the flux for the Euler equations Parameters Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints) x (float numpy array) – (generally optional, required for this problem), x-y coordinates of the given points (NbGivenPoints x 2) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) Returns J[:,:,i] gives the jacobian of the flux taking care of the dynamic of the ith spatial coordinate. Return type J (3D numpy array) Note For each flux fi = (fi1,…, fin), the returned Jacobian reads: J[:,:] = [dfi1/dx1, ….., dfi1/dxn …. df in/dx1, ….., dfin/dxn] args is there only for compatibility reason at call time `EigenValues`(Var, FluidIndex, n, x, args)¶ Computes the eigenvalues associated to the flux. Parameters Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2) x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2) Returns (NbGivenPoints x 4) the four eigenvalues at each given point Return type lbd (numpy array) Note args is there only for compatibility reason at call time `RightEigenVectors`(Var, FluidIndex, n, x, args)¶ Computes the right eigenvectors associated to the eigenvalues. Parameters Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2) x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2) Returns (NbVars x MbVars x NbGivenPoints) the matrix of eigenvectors for each given point Return type reg (numpy array) Note args is there only for compatibility reason at call time `LeftEigenVectors`(Var, FluidIndex, n, x, args)¶ Computes the left eigenvectors associated to the eigenvalues. Parameters Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2) x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2) Returns (NbVars x MbVars x NbGivenPoints) the matrix of eigenvectors for each given point Return type reg (numpy array) Note args is there only for compatibility reason at call time `SpectralRadius`(Var, FluidIndex, n, x, args)¶ Computes the spectral radius associated to the flux. Parameters Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints) FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2) x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2) Returns the spectral radius computed at each given point Return type Lmb (numpy array) Note args is there only for compatibility reason at call time `RoeMatrix`(Var, n, FluidIndex, args)¶ A Roe matrix corresponding to the Euler Equations, accross one given edge Parameters n (float numpy array) – the outward normal of the considered element (nx, ny) Var (float numpy array) – the value of the variables (NbVariables x 2). Var[:,1] average value of the considered element, Var[:,2] average value of the neigborhing element. FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints) x (float numpy array) – (optional, not impacting here) x-y coordinates of the given points (NbGivenPoints x 2) Returns a Roe matrix of the system Return type RoeMatrix (float numpy array) Note This function is NOT vectorised Fluid index is the fluid index in the initially given list, not the fluid type FluidProp is the list of all the fluid properties for each fluid (given the list, not the type) Equations of state ¶ class `3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.EulerEquations.EOS`(Id)¶ Class furnishing the methods giving the Equation of State that can be used in combination with the `SimpleFluid` definition. Upon creation, fills the field `EOS` with the function callback corresponding to the wished initialisation function and the field `EOSName` with the associated label. The implemented EoS are linked as follows. See their respective documentation for more information. Stiff fluid Ideal Parameters Id (integer) – the index corresponding to the equation of fluid the user wants when considering the governing equations given in this module and the associated fluid. Methods `Stiff`(Var, FluidProp, Type)¶ Equation of State for stiff fluids, giving back either the internal energy or pressure. Parameters Type (string) – desired output: “e”: internal energy, “p”: pressure Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints). If type “e”, the variables should be [rho, rhoU, rhoV, p].T. If type “p”, the variables should be [rho, rhoU, rhoV, E]. FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point Returns the pressure values at the given points (NbGivenPoints) Return type p (float numpy array) `dEoSRhoStiff`(Var, FluidProp)¶ Derivative of the Equation of State for stiff fluids along the density variable at the given points. Parameters Type (string) – desired output: “e”: internal energy, “p”: pressure Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints). FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point Returns derivative of the Equation of State for stiff fluids along the density variable (NbGivenPoints) Return type deos (float numpy array) `dEoSEStiff`(Var, FluidProp)¶ Derivative of the Equation of State for stiff fluids along the Energy variable at the given points. Parameters Type (string) – desired output: “e”: internal energy, “p”: pressure Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints). FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point Returns derivative of the Equation of State for stiff fluids along the Energy variable (NbGivenPoints) Return type deos (float numpy array) `Ideal`(Var, FluidProp, Type)¶ Equation of State for ideal fluids, giving back either the internal energy or pressure. Parameters Type (string) – desired output: “e”: internal energy, “p”: pressure Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints). If type “e”, the variables should be [rho, rhoU, rhoV, p].T. If type “p”, the variables should be [rho, rhoU, rhoV, E]. FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point Returns the pressure values at the given points (NbGivenPoints) Return type p (float numpy array) `dEoSRhoIdeal`(Var, FluidProp)¶ Derivative of the Equation of State for ideal fluids along the density variable at the given points. Parameters Type (string) – desired output: “e”: internal energy, “p”: pressure Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints). FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point Returns derivative of the Equation of State for stiff fluids along the density variable (NbGivenPoints) Return type deos (float numpy array) `dEoSEIdeal`(Var, FluidProp)¶ Derivative of the Equation of State for ideal fluids along the Energy variable at the given points. Parameters Type (string) – desired output: “e”: internal energy, “p”: pressure Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints). FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point Returns derivative of the Equation of State for stiff fluids along the Energy variable (NbGivenPoints) Return type deos (float numpy array) Initial conditions ¶ class `3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.EulerEquations.InitialConditions`(Id, params)¶ Class furnishing the solution initialisation routines that are suitable to study the the Euler Equations, defined for working on a subdomain. Access the routine computing the initial conditions through the field `IC`, filled upon creation with the function callback corresponding to the index of the wished initialisation function. The implemented initialisation method are linked as follows. See their respective documentation for more information. ConstantState ConstantState2 StationaryVortex SOD SOD_Fluid1 SOD_Fluid2 Karni_Fluid1(bublle) Karni_Fluid2(complement to the bubble) Parameters Id (integer) – the index corresponding to the initialisation method the user wants when considering the governing equations given in this module. params* (list of arguments) – the (fixed arguments to pass to the selected function) Methods `SOD_Fluid1`(PointsID, Mesh, EoS, FluidProp, args)¶ Initialising the solution to a constant state that matches the SOD-inner value. (The given points should all belonging to a same subdomain). Parameters PointsID* (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – (optional), the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – (optional, not used here) the the properties of the fluid present where the given points are Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Note args is here only for compatibility on call There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution. For running smothly the inner circle diameter of the given subdomain should be at least 2 `SOD_Fluid2`(PointsID, Mesh, EoS, FluidProp, args)¶ Initialising the solution to a constant state that matches the SOD-outer value. (The given points should all belonging to a same subdomain). Parameters PointsID (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Note args is here only for compatibility on call There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution. For running smothly the inner circle diameter of the given subdomain should be at least 2 `ConstantState`(PointsID, Mesh, EoS, FluidProp, args)¶ Initialising the solution to a constant state on the given points, all belonging to a same subdomain. Parameters PointsID (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Note args is here only for compatibility on call There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution. `ConstantState2`(PointsID, Mesh, EoS, FluidProp, args)¶ Initialising the solution to a constant state on the given points, all belonging to a same subdomain. Parameters PointsID (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – (optional), the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – (optional, not used here) the the properties of the fluid present where the given points are Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Note args is here only for compatibility on call There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution. `StationaryVortex`(PointsID, Mesh, EoS, FluidProp, subdomain, args)¶ Initialising the solution to a StationaryVortex centred at the center of mass of the subdmain. (The given points should all belonging to a same subdomain). Parameters PointsID (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are subdomain (shapely multipolyon) – (optional, not used here) the shapely polygon to which the given points belong Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Note args is here only for compatibility on call There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution. For running smothly the inner circle diameter of the given subdomain should be at least 2 `SOD`(PointsID, Mesh, EoS, FluidProp, subdomain)¶ Initialising the solution to a SOD centred at the center of mass, of width 0.5 of the subdmain. (The given points should all belonging to a same subdomain). Parameters PointsID* (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are subdomain (shapely multipolyon) – (optional, not used here) the shapely polygon to which the given points belong Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Note There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution. For running smothly the inner circle diameter of the given subdomain should be at least 2 `Bubble`(PointsID, Mesh, EoS, FluidProp, subdomain, args)¶ Initialising the solution to a constant state corresponding to the bubble in the Karni test, where the density is considered as Rconsidered/Rair fluid. Parameters PointsID* (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – the properties of the fluid present where the given points are subdomain (shapely multipolyon) – (optional, not used here) the shapely polygon to which the given points belong Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Note args is here only for compatibility on call There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution. `PlanarShock`(PointsID, Mesh, EoS, FluidProp, subdomain, MachNum=1.19, args)¶ Initialising the solution to two constante state distributed on either side of a shock for the Planar shock test. The shock is set at 25% of the total x width of the domain, starting from the right. The (left) rested part is set to be such that (p, u, v, p) = [1., 0., 0., 1]. The shock is set to move from right to left. Parameters PointsID (integer array-like) – the index of the points to consider Mesh (MeshStructure) – the considered mesh EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points FluidProp (FluidSpecs) – the properties of the fluid present where the given points are subdomain (shapely multipolyon) – the shapely polygon to which the given points belong MachNum (float) – the desired Mach number (given on the still fluid) Returns The initialised values at the considered points (NbVariables x NbGivenPoints) Return type Init (float numpy array) Warning The computation of the state assumes air as fluid component Note *args is here only for compatibility on call There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.

Euler equations¶

All the necessary routines that implement the 2D Euler Equations (given in conservative variables) are defined in GoverningEquations/EulerEquations.py

Governing equations
Equations of state
Initial conditions

Governing equations ¶

class 3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.EulerEquations.Equations(FluidProp, EOs)¶

Class furnishing all the methods that are required to define numerical schemes and evolve the solution according to the Euler Equation

Fields

It contains the following initialisation parameters (later available as well as attributes)

FluidProp (list of FluidModel) – the list of fluid property instances (NbFluids)

EOS (list of callbacks) – the list of Equation of state functions (NbFluids)

It further contains the following attributes, filled upon initialisation

NbVariables (integer) – number of variables of the model

VariablesNames (list of strings) – ordered name of the variables

VariablesUnits (list of strings) – symbols of the units of the variables

VariablesLatexNames (list of strings) – ordered name of the variables, latex encoding

VariablesLatexUnits (list of strings) – symbols of the units of the variables, latex encoding

XYLatexUnits (list of strings) – units of the x-y coordinates in latex encoding

Parameters

FluidProp (list of FluidModel) – the list of fluid property instances (NbFluids)
EOs (list of EOS) – the list of Equation of state instances (NbFluids)

Methods

ConservativeToPrimary(Var, FluidIndex, *args)¶

Converts the conservative variables to the primary ones

Parameters

Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)

Returns

(NbVars x NbGivenPoints) the corresponding primary variables

Return type

Var (numpy array)

Note

*args is there only for compatibility reason at call time

PrimaryToConservative(PrimVar, FluidIndex, *args)¶

Converts the primary variables to the conservative ones

Parameters

Var (2D numpy array) – the primary variables of the problem (NbVars x NbGivenPoints)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)

Returns

(NbVars x NbGivenPoints) the corresponding conservative variables

Return type

Var (numpy array)

Note

*args is there only for compatibility reason at call time

Flux(Var, *args)¶

Flux function of the (conservative) EulerEquations

Parameters: Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints)

Optional argument

x (float numpy array), – (optional, not impacting here) x-y coordinates of the given points (NbGivenPoints x 2)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)

Returns: fills direclty the Parameters data structure
Return type: None

Note

This function is Vectorised
Fluid index is the fluid index in the initially given list, not the fluid type
FluidProp is the list of all the fluid properties for each fluid (given the list, not the type)
*args is there only for compatibility reason at call time

GetUV(Var, *args)¶

Function giving back the x-y velocity at the considered point, given the variational values given for the the conservative Euler equation’s variables.

Parameters

Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints)
x (float numpy array) – (generally optional, required for this problem), x-y coordinates of the given points (NbGivenPoints x 2)

Optional argument

FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)

Returns: UV (float numpy array) – the velocities values at the points (2 x NbGivenPoints)

Note

This function is Vectorised
*args is there only for compatibility reason at call time

Jacobian(Var, x, FluidIndex, *args)¶

Computes the Jacobian of the flux for the Euler equations

Parameters

Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints)
x (float numpy array) – (generally optional, required for this problem), x-y coordinates of the given points (NbGivenPoints x 2)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)

Returns

J[:,:,i] gives the jacobian of the flux taking care of the dynamic of the ith spatial coordinate.

Return type

J (3D numpy array)

Note

For each flux fi = (fi1,…, fin), the returned Jacobian reads:

J[:,:] = [dfi1/dx1, ….., dfi1/dxn

….

df in/dx1, ….., dfin/dxn]
*args is there only for compatibility reason at call time

EigenValues(Var, FluidIndex, n, x, *args)¶

Computes the eigenvalues associated to the flux.

Parameters

Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)
n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2)
x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2)

Returns

(NbGivenPoints x 4) the four eigenvalues at each given point

Return type

lbd (numpy array)

Note

*args is there only for compatibility reason at call time

RightEigenVectors(Var, FluidIndex, n, x, *args)¶

Computes the right eigenvectors associated to the eigenvalues.

Parameters

Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)
n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2)
x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2)

Returns

(NbVars x MbVars x NbGivenPoints) the matrix of eigenvectors for each given point

Return type

reg (numpy array)

Note

*args is there only for compatibility reason at call time

LeftEigenVectors(Var, FluidIndex, n, x, *args)¶

Computes the left eigenvectors associated to the eigenvalues.

Parameters

Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)
n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2)
x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2)

Returns

(NbVars x MbVars x NbGivenPoints) the matrix of eigenvectors for each given point

Return type

reg (numpy array)

Note

*args is there only for compatibility reason at call time

SpectralRadius(Var, FluidIndex, n, x, *args)¶

Computes the spectral radius associated to the flux.

Parameters

Var (2D numpy array) – the variables of the problem (NbVars x NbGivenPoints)
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)
n (2D numpy array) – the x-y values of the normal at each given point (NbGivenPoints x 2)
x (2D numpy array) – (generally optional, required for this problem) the x-y locations at which the variables are given (NbGivenPoints x 2)

Returns

the spectral radius computed at each given point

Return type

Lmb (numpy array)

Note

*args is there only for compatibility reason at call time

RoeMatrix(Var, n, FluidIndex, *args)¶

A Roe matrix corresponding to the Euler Equations, accross one given edge

Parameters

n (float numpy array) – the outward normal of the considered element (nx, ny)
Var (float numpy array) – the value of the variables (NbVariables x 2). Var[:,1] average value of the considered element, Var[:,2] average value of the neigborhing element.
FluidIndex (integer numpy array) – the fluid present at each given point (NbGivenPoints)
x (float numpy array) – (optional, not impacting here) x-y coordinates of the given points (NbGivenPoints x 2)

Returns

a Roe matrix of the system

Return type

RoeMatrix (float numpy array)

Note

This function is NOT vectorised
Fluid index is the fluid index in the initially given list, not the fluid type
FluidProp is the list of all the fluid properties for each fluid (given the list, not the type)

Equations of state ¶

class 3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.EulerEquations.EOS(Id)¶

Class furnishing the methods giving the Equation of State that can be used in combination with the SimpleFluid definition. Upon creation, fills the field EOS with the function callback corresponding to the wished initialisation function and the field EOSName with the associated label. The implemented EoS are linked as follows. See their respective documentation for more information.

Stiff fluid

Ideal

Parameters: Id (integer) – the index corresponding to the equation of fluid the user wants when considering the governing equations given in this module and the associated fluid.

Methods

Stiff(Var, FluidProp, Type)¶

Equation of State for stiff fluids, giving back either the internal energy or pressure.

Parameters

Type (string) – desired output: “e”: internal energy, “p”: pressure
Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints). If type “e”, the variables should be [rho, rhoU, rhoV, p].T. If type “p”, the variables should be [rho, rhoU, rhoV, E].
FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point

Returns

the pressure values at the given points (NbGivenPoints)

Return type

p (float numpy array)

dEoSRhoStiff(Var, FluidProp)¶

Derivative of the Equation of State for stiff fluids along the density variable at the given points.

Parameters

Type (string) – desired output: “e”: internal energy, “p”: pressure
Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints).
FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point

Returns

derivative of the Equation of State for stiff fluids along the density variable (NbGivenPoints)

Return type

deos (float numpy array)

dEoSEStiff(Var, FluidProp)¶

Derivative of the Equation of State for stiff fluids along the Energy variable at the given points.

Parameters

Type (string) – desired output: “e”: internal energy, “p”: pressure
Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints).
FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point

Returns

derivative of the Equation of State for stiff fluids along the Energy variable (NbGivenPoints)

Return type

deos (float numpy array)

Ideal(Var, FluidProp, Type)¶

Equation of State for ideal fluids, giving back either the internal energy or pressure.

Parameters

Type (string) – desired output: “e”: internal energy, “p”: pressure
Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints). If type “e”, the variables should be [rho, rhoU, rhoV, p].T. If type “p”, the variables should be [rho, rhoU, rhoV, E].
FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point

Returns

the pressure values at the given points (NbGivenPoints)

Return type

p (float numpy array)

dEoSRhoIdeal(Var, FluidProp)¶

Derivative of the Equation of State for ideal fluids along the density variable at the given points.

Parameters

Type (string) – desired output: “e”: internal energy, “p”: pressure
Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints).
FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point

Returns

derivative of the Equation of State for stiff fluids along the density variable (NbGivenPoints)

Return type

deos (float numpy array)

dEoSEIdeal(Var, FluidProp)¶

Derivative of the Equation of State for ideal fluids along the Energy variable at the given points.

Parameters

Type (string) – desired output: “e”: internal energy, “p”: pressure
Var (float numpy array) – the value of the conservative variables (NbVariables x NbGivenPoints).
FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point

Returns

derivative of the Equation of State for stiff fluids along the Energy variable (NbGivenPoints)

Return type

deos (float numpy array)

Initial conditions ¶

class 3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.EulerEquations.InitialConditions(Id, *params)¶

Class furnishing the solution initialisation routines that are suitable to study the the Euler Equations, defined for working on a subdomain. Access the routine computing the initial conditions through the field IC, filled upon creation with the function callback corresponding to the index of the wished initialisation function. The implemented initialisation method are linked as follows. See their respective documentation for more information.

ConstantState

ConstantState2

StationaryVortex

SOD

SOD_Fluid1

SOD_Fluid2

Karni_Fluid1(bublle)

Karni_Fluid2(complement to the bubble)

Parameters

Id (integer) – the index corresponding to the initialisation method the user wants when considering the governing equations given in this module.
params (list of arguments) – the (fixed arguments to pass to the selected function)

Methods

SOD_Fluid1(PointsID, Mesh, EoS, FluidProp, *args)¶

Initialising the solution to a constant state that matches the SOD-inner value. (The given points should all belonging to a same subdomain).

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – (optional), the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – (optional, not used here) the the properties of the fluid present where the given points are

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Note

*args is here only for compatibility on call
There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.
For running smothly the inner circle diameter of the given subdomain should be at least 2

SOD_Fluid2(PointsID, Mesh, EoS, FluidProp, *args)¶

Initialising the solution to a constant state that matches the SOD-outer value. (The given points should all belonging to a same subdomain).

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Note

*args is here only for compatibility on call
There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.
For running smothly the inner circle diameter of the given subdomain should be at least 2

ConstantState(PointsID, Mesh, EoS, FluidProp, *args)¶

Initialising the solution to a constant state on the given points, all belonging to a same subdomain.

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Note

*args is here only for compatibility on call
There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.

ConstantState2(PointsID, Mesh, EoS, FluidProp, *args)¶

Initialising the solution to a constant state on the given points, all belonging to a same subdomain.

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – (optional), the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – (optional, not used here) the the properties of the fluid present where the given points are

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Note

*args is here only for compatibility on call
There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.

StationaryVortex(PointsID, Mesh, EoS, FluidProp, subdomain, *args)¶

Initialising the solution to a StationaryVortex centred at the center of mass of the subdmain. (The given points should all belonging to a same subdomain).

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are
subdomain (shapely multipolyon) – (optional, not used here) the shapely polygon to which the given points belong

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Note

*args is here only for compatibility on call
There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.
For running smothly the inner circle diameter of the given subdomain should be at least 2

SOD(PointsID, Mesh, EoS, FluidProp, subdomain)¶

Initialising the solution to a SOD centred at the center of mass, of width 0.5 of the subdmain. (The given points should all belonging to a same subdomain).

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – the the properties of the fluid present where the given points are
subdomain (shapely multipolyon) – (optional, not used here) the shapely polygon to which the given points belong

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Note

There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.
For running smothly the inner circle diameter of the given subdomain should be at least 2

Bubble(PointsID, Mesh, EoS, FluidProp, subdomain, *args)¶

Initialising the solution to a constant state corresponding to the bubble in the Karni test, where the density is considered as Rconsidered/Rair fluid.

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – the properties of the fluid present where the given points are
subdomain (shapely multipolyon) – (optional, not used here) the shapely polygon to which the given points belong

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Note

*args is here only for compatibility on call
There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.

PlanarShock(PointsID, Mesh, EoS, FluidProp, subdomain, MachNum=1.19, *args)¶

Initialising the solution to two constante state distributed on either side of a shock for the Planar shock test. The shock is set at 25% of the total x width of the domain, starting from the right. The (left) rested part is set to be such that (p, u, v, p) = [1., 0., 0., 1]. The shock is set to move from right to left.

Parameters

PointsID (integer array-like) – the index of the points to consider
Mesh (MeshStructure) – the considered mesh
EOS (function callback) – the equation of state given by the model of the fluid that is present at the given points
FluidProp (FluidSpecs) – the properties of the fluid present where the given points are
subdomain (shapely multipolyon) – the shapely polygon to which the given points belong
MachNum (float) – the desired Mach number (given on the still fluid)

Returns

The initialised values at the considered points (NbVariables x NbGivenPoints)

Return type

Init (float numpy array)

Warning

The computation of the state assumes air as fluid component

Note

*args is here only for compatibility on call
There is usually one initialisation routine per subdomain, and the PointsID are not necessarily contigous, be careful when assigning the values back in the regular solution.

Euler equations¶

Governing equations¶

Equations of state¶

Initial conditions¶

Governing equations ¶

Equations of state ¶

Initial conditions ¶