
Quick search Linear advection (Rotation)¶ All the necessary routines that implement a simple 2D translation, linear advection are defined in `GoverningEquations/LinearAdvectionRotation.py` Governing equations Equations of state Initial conditions Governing equations ¶ class `3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.LinearAdvectionRotation.Equations`(FluidProp, EOs)¶ Class furnishing all the methods that are required to define numerical schemes and evolve the solution according to the Linear Advection (translation) 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 EOS) – the list of Equation of state instances (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, x, args)¶ Flux function of the LinearAdvection (rotation case) 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 the obtained flux (spatialdim x NbVars x NbPoints) Return type Flux (3D numpy array) 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, x, argv)¶ Function giving back the x-y velocity at the considered point, given the variational values given for the advection. 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, argv)¶ Computes the Jacobian of the flux for the LinearAdvection (rotation case) 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 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 `SpectralRadius`(Var, FluidIndex, n, x, argv)¶ 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 `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 NbEigs) the 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 Equations of state ¶ class* `3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.LinearAdvectionRotation.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. Dummy Note After having been initialised the instance, the EOS is accessible from the field EOS within this class. Warning For advection, this class is void, just here for compatibility properties in the code 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 `Dummy`(Var, FluidProp, Type)¶ Dummy EOS returning the identity for the sake of the code compatibility. Parameters Type (string) – desired output (here inactive) Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints) FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point Returns the variable retrieved from the EoS (here identity as dummy function) Return type val Initial conditions ¶ class `3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.LinearAdvectionRotation.InitialConditions`(Id, params)¶ Class furnishing the solution initialisation routines that are suitable to study the the Linear Advection Rotation 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 Not implemented Bump 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 `ConstantState`(PointsID, Mesh, EOS=[], FluidProp=[], subdomain=[], argv)¶ 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 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 argv 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. `Bump`(PointsID, Mesh, EOS=[], FluidProp=[], subdomain=[], args)¶ IInitialising the solution to a bump 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) – (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 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

Linear advection (Rotation)¶

All the necessary routines that implement a simple 2D translation, linear advection are defined in GoverningEquations/LinearAdvectionRotation.py

Governing equations
Equations of state
Initial conditions

Governing equations ¶

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

Class furnishing all the methods that are required to define numerical schemes and evolve the solution according to the Linear Advection (translation)

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 EOS) – the list of Equation of state instances (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, x, *args)¶

Flux function of the LinearAdvection (rotation case)

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: the obtained flux (spatialdim x NbVars x NbPoints)
Return type: Flux (3D numpy array)

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, x, *argv)¶

Function giving back the x-y velocity at the considered point, given the variational values given for the advection.

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, *argv)¶

Computes the Jacobian of the flux for the LinearAdvection (rotation case)

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: 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

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

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

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 NbEigs) the 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

Equations of state ¶

class 3_LESARD.SourceCode.ProblemDefinition.GoverningEquations.LinearAdvectionRotation.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.

Dummy

Note

After having been initialised the instance, the EOS is accessible from the field EOS within this class.

Warning

For advection, this class is void, just here for compatibility properties in the code

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

Dummy(Var, FluidProp, Type)¶

Dummy EOS returning the identity for the sake of the code compatibility.

Parameters

Type (string) – desired output (here inactive)
Var (float numpy array) – the value of the variables (NbVariables x NbGivenPoints)
FluidProp (FluidModel list) – the list of the fluid properties associated to each given Point

Returns

the variable retrieved from the EoS (here identity as dummy function)

Return type

val

Initial conditions ¶

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

Class furnishing the solution initialisation routines that are suitable to study the the Linear Advection Rotation 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

Not implemented

Bump

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

ConstantState(PointsID, Mesh, EOS=[], FluidProp=[], subdomain=[], *argv)¶

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
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

*argv 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.

Bump(PointsID, Mesh, EOS=[], FluidProp=[], subdomain=[], *args)¶

IInitialising the solution to a bump 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) – (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
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

Linear advection (Rotation)¶

Governing equations¶

Equations of state¶

Initial conditions¶

Governing equations ¶

Equations of state ¶

Initial conditions ¶