josie.euler package¶

Subpackages¶

josie.euler.schemes package

Submodules¶

josie.euler.eos module¶

This module contains the different Equation of State (EOS) implementations

class josie.euler.eos.EOS¶

Bases: object

An Abstract Base Class representing an EOS for an Euler System

abstract p(rho, e)¶

Return type: Union[ndarray, float]

abstract rho(p, e)¶

Return type: Union[ndarray, float]

abstract rhoe(rho, p)¶

Return type: Union[ndarray, float]

abstract sound_velocity(rho, p)¶

Return type: Union[ndarray, float]

class josie.euler.eos.PerfectGas(gamma=1.4)¶

Bases: josie.euler.eos.EOS

This class embeds methods to compute states for the Euler problem using an EOS (Equation of State) for perfect gases

$p = \rho \mathcal{R} T = \rho \left( \gamma - 1 \right)e$

gamma¶: The adiabatic coefficient

p(rho, e)¶

This returns the pressure from density and internal energy

Parameters

rho (Union[ndarray, float]) – A ArrayAndScalar containing the values of the density on the mesh cells
e (Union[ndarray, float]) – A ArrayAndScalar containing the values of the internal energy on the mesh cells

Returns

A :class:`ArrayAndScalar containing the values of the pressure multiplied by the density

Return type

p

rho(p, e)¶

This returns the sound velocity from density and pressure

Parameters

p (Union[ndarray, float]) – A ArrayAndScalar containing the values of the pressure on the mesh cells
e (Union[ndarray, float]) – A ArrayAndScalar containing the values of the internal energy on the mesh cells

Returns

A ArrayAndScalar containing the values of the density on the mesh cells

Return type

rho

rhoe(rho, p)¶

This returns the internal energy multiplied by the density

Parameters

rho (Union[ndarray, float]) – A ArrayAndScalar containing the values of the density on the mesh cells
p (Union[ndarray, float]) – A ArrayAndScalar containing the values of the pressure on the mesh cells

Returns

A ArrayAndScalar containing the values of the internal energy multiplied by the density

Return type

rhoe

sound_velocity(rho, p)¶

This returns the sound velocity from density and pressure

Parameters

rho (Union[ndarray, float]) – A ArrayAndScalar containing the values of the density on the mesh cells
p (Union[ndarray, float]) – A ArrayAndScalar containing the values of the pressure on the mesh cells

Returns

A ArrayAndScalar containing the values of the sound velocity multiplied by the density

Return type

c

class josie.euler.eos.StiffenedGas(gamma=3, p0=1000000000.0)¶

Bases: josie.euler.eos.PerfectGas

p(rho, e)¶

This returns the pressure from density and internal energy

Parameters

rho (Union[ndarray, float]) – A ArrayAndScalar containing the values of the density on the mesh cells
e (Union[ndarray, float]) – A ArrayAndScalar containing the values of the internal energy on the mesh cells

Returns

A :class:`ArrayAndScalar containing the values of the pressure multiplied by the density

Return type

p

rho(p, e)¶

This returns the sound velocity from density and pressure

Parameters

p (Union[ndarray, float]) – A ArrayAndScalar containing the values of the pressure on the mesh cells
e (Union[ndarray, float]) – A ArrayAndScalar containing the values of the internal energy on the mesh cells

Returns

A ArrayAndScalar containing the values of the density on the mesh cells

Return type

rho

rhoe(rho, p)¶

This returns the internal energy multiplied by the density

Parameters

rho (Union[ndarray, float]) – A ArrayAndScalar containing the values of the density on the mesh cells
p (Union[ndarray, float]) – A ArrayAndScalar containing the values of the pressure on the mesh cells

Returns

A ArrayAndScalar containing the values of the internal energy multiplied by the density

Return type

rhoe

sound_velocity(rho, p)¶

This returns the sound velocity from density and pressure

Parameters

rho (Union[ndarray, float]) – A ArrayAndScalar containing the values of the density on the mesh cells
p (Union[ndarray, float]) – A ArrayAndScalar containing the values of the pressure on the mesh cells

Returns

A ArrayAndScalar containing the values of the sound velocity multiplied by the density

Return type

c

josie.euler.exact module¶

class josie.euler.exact.Exact(eos, Q_L, Q_R)¶

Bases: object

This class implements the exact solution of the Riemann problem as scheme.

See [Tor09] for a detailed view on compressible schemes and [GR96] for a deep study on hyperbolic systems and the Riemann problem. This is also based on [CG85] and [Kam15]

Parameters

eos – The EOS to use
Q_L (EulerState) – The left state
state (Q_R The right) –

eos¶: The EOS to use

Q_L¶: The left state

Q_R The right state

Q_star_L¶: The left star state

Q_star_R¶: The right star state

left_wave¶

The type of the left wave (it’s populated after calling solve()

Type: josie.euler.exact.WaveType

right_wave¶

The type of the right wave (it’s populated after calling solve()

Type: josie.euler.exact.WaveType

full_state(p_star, U_star)¶: Compute the full Euler state from the value of $p^*$ and $u^*$

left_wave: josie.euler.exact.WaveType¶

rankine_hugoniot(rho, p, rho_k, p_k)¶: The Rankine-Hugoniot locus connecting the known state before the $k$ shock wave to the “star” state after the shock wave

rarefaction(p, Q_k, wave)¶

Non linear function for the rarefaction

Parameters

p (float) – The after rarefaction fan pressure value
rho – An initial estimate for the after rarefaction density (unused)
Q_k (EulerState) – The state before the rarefaction fan
wave (Wave) – The wave to consider

Return type

float

Returns

The value of the non linear function $f_k(p^*)$ which statisfies
$u^* = U_k \pm f_k(p^*)$

rarefaction_ode(p, y, wave)¶

Solve the rarefaction ODE for a vectorized set of states

Parameters

p – is the indipendent variable, i.e. the pressure
q – is the state variable. It’s an array of size [num_riemann x 2].

Returns

A tuple containing the derivatives of the state

Return type

derivatives

right_wave: josie.euler.exact.WaveType¶

sample(x, t, origin=0.5)¶

Sample the solution at given $(x, t)$

Returns: The state solution at the requested $(x, t)$
Return type: state

sample_rarefaction(U_c, V_k, wave)¶

Return the state within the rarefaction fan

Return type: EulerState

shock(p, Q_k, wave)¶

This function returns the after-shock velocity

Parameters

p (float) – The after shock pressure value
rho_guess – An initial estimate for the after shock density
Q_k (EulerState) – The state before the shock
wave (Wave) – The wave to consider

Returns

Return type

The value of the non linear function after the shock

solve()¶

class josie.euler.exact.RarefactionState(*args, **kwargs)¶

Bases: josie.state.State

fields: Type[josie.fields.RarefactionFields]¶

class josie.euler.exact.Wave(value)¶

Bases: enum.IntEnum

An enumeration.

LEFT = 1¶

RIGHT = -1¶

class josie.euler.exact.WaveType(value)¶

Bases: enum.Enum

An enumeration.

RAREFACTION = 2¶

SHOCK = 1¶

josie.euler.fields module¶

josie.euler.problem module¶

class josie.euler.problem.EulerProblem(eos, **kwargs)¶

Bases: josie.problem.Problem

A class representing an Euler system problem

eos¶: An instance of EOS, an equation of state for the fluid

F(cells)¶

This returns the tensor representing the flux for an Euler model

A general problem can be written in a compact way:

$\pdv{\vb{q}}{t} + \div{\vb{F\qty(\vb{q})}} + \vb{B}\qty(\vb{q}) \cdot \gradient{\vb{q}} = \vb{s\qty(\vb{q})}$

This function needs to return $\vb{F}\qty(\vb{q})$

Parameters

cells (CellSet) – A MeshCellSet that contains the cell data

Returns

An array of dimension $Nx \times Ny \times 4 \times 2$ , i.e. an array that of each $x$ cell and $y$ cell stores the $4 \times 2$ flux tensor

The flux tensor is:

$\pdeConvective = \begin{bmatrix} \rho u & \rho v \\ \rho u^2 + p & \rho uv \\ \rho vu & \rho v^ 2 + p \\ (\rho E + p)U & (\rho E + p)V \end{bmatrix}$

Return type

F

josie.euler.solver module¶

class josie.euler.solver.EulerSolver(mesh, scheme)¶

Bases: josie.solver.Solver

A solver for the Euler system

t: float¶

josie.euler.state module¶

We create one big state that contains the actual conservative variables that are used in the flux together with the “auxiliary” variables that are instead needed, for example, to compute the speed of sound.

$\eulerState$

rho: density $\rho$
rhoU: component along $x$ of the velocity $\vb{u}$ ,

multiplied by the density
rhoV: component along $y$ of the velocity $\vb{u}$ ,

multiplied by the density
rhoE: total energy multiplied by the density $\rho E$
rhoe: internal energy multiplied by the density $\rho e$
U: component along $x$ of the velocity $u$
V: component along $y$ of the velocity $v$
p: pressure $p$
c: sound velocity $c$