NewtonianEuler::Solutions::IsentropicVortex< Dim > Class Template Reference

Newtonian isentropic vortex in Cartesian coordinates. More...

#include <IsentropicVortex.hpp>

Classes
struct	AdiabaticIndex
	The adiabatic index of the fluid. More...
struct	Center
	The position of the center of the vortex at $t=0$. More...
struct	MeanVelocity
	The mean flow velocity. More...
struct	PerturbAmplitude
	The amplitude of the perturbation generating a source term. More...
struct	Strength
	The strength of the vortex. More...

Public Types
using	equation_of_state_type = EquationsOfState::PolytropicFluid< false >
using	options = implementation defined

Public Member Functions
	IsentropicVortex (const IsentropicVortex &)=default
IsentropicVortex &	operator= (const IsentropicVortex &)=default
	IsentropicVortex (IsentropicVortex &&)=default
IsentropicVortex &	operator= (IsentropicVortex &&)=default
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
	IsentropicVortex (double adiabatic_index, const std::array< double, Dim > &center, const std::array< double, Dim > &mean_velocity, double strength, double perturbation_amplitude=0.0)
template<typename DataType , typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, Dim, Frame::Inertial > &x, const double t, tmpl::list< Tags... >) const
	Retrieve a collection of hydrodynamic variables at position x and time t.
template<typename DataType >
DataType	perturbation_profile (const DataType &z) const
	Function of z coordinate to compute the perturbation generating a source term. Public so the corresponding source class can also use it.
template<typename DataType >
DataType	deriv_of_perturbation_profile (const DataType &z) const
double	perturbation_amplitude () const
const EquationsOfState::PolytropicFluid< false > &	equation_of_state () const
void	pup (PUP::er &) override
virtual auto	get_clone () const -> std::unique_ptr< InitialData >=0

Static Public Attributes
static constexpr Options::String	help

Friends
template<size_t SpatialDim>
bool	operator== (const IsentropicVortex< SpatialDim > &lhs, const IsentropicVortex< SpatialDim > &rhs)

Detailed Description

template<size_t Dim>
class NewtonianEuler::Solutions::IsentropicVortex< Dim >

Newtonian isentropic vortex in Cartesian coordinates.

The analytic solution to the 2-D Newtonian Euler system representing the slow advection of an incompressible, isentropic vortex [208]. The initial condition is the superposition of a mean uniform flow with a gaussian-profile vortex. When embedded in 3-D space, the isentropic vortex is still a solution to the corresponding 3-D system if the velocity along the third axis is a constant. In Cartesian coordinates $(x,y,z)$, and using dimensionless units, the primitive quantities at a given time $t$ are then

$$\begin{array}{rl}\rho & ={[1-{\displaystyle \frac{(\gamma -1){\beta }^2}{8\gamma {\pi }^2}}\exp (1-r^2)]}^{1/(\gamma -1)},\\ v_x& =U-{\displaystyle \frac{\beta \tilde{y}}{2\pi }}\exp \left({\displaystyle \frac{1-r^2}{2}}\right),\\ v_y& =V+{\displaystyle \frac{\beta \tilde{x}}{2\pi }}\exp \left({\displaystyle \frac{1-r^2}{2}}\right),\\ v_z& =W,\\ ϵ& =\frac{{\rho }^{\gamma -1}}{\gamma -1},\end{array}$$

with

$$\begin{array}{rl}{r}^2& ={\tilde{x}}^2+{\tilde{y}}^2,\\ \tilde{x}& =x-X_0-Ut,\\ \tilde{y}& =y-Y_0-Vt,\end{array}$$

where $(X_0,Y_0)$ is the position of the vortex on the $(x,y)$ plane at $t=0$, $(U,V,W)$ are the components of the mean flow velocity, $\beta$ is the vortex strength, and $\gamma$ is the adiabatic index. The pressure $p$ is then obtained from the dimensionless polytropic relation

$$\begin{array}{r}p={\rho }^{\gamma }.\end{array}$$

On the other hand, if the velocity along the $z-$axis is not a constant but a function of the $z$ coordinate, the resulting modified isentropic vortex is still a solution to the Newtonian Euler system, but with source terms that are proportional to $d{v}_z/dz$. (See NewtonianEuler::Sources::VortexPerturbation.) For testing purposes, we choose to write the velocity as a uniform field plus a periodic perturbation,

$$\begin{array}{r}{v}_z(z)=W+ϵ\sin z,\end{array}$$

where $ϵ$ is the amplitude of the perturbation. The resulting source for the Newtonian Euler system will then be proportional to $ϵ\cos z$.

Member Function Documentation

get_clone()

template<size_t Dim>

auto NewtonianEuler::Solutions::IsentropicVortex< Dim >::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

help

template<size_t Dim>

constexpr Options::String NewtonianEuler::Solutions::IsentropicVortex< Dim >::help

staticconstexpr

Initial value:

= {
      "Newtonian Isentropic Vortex. Works in 2 and 3 dimensions."}

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The documentation for this class was generated from the following file:

• src/PointwiseFunctions/AnalyticSolutions/NewtonianEuler/IsentropicVortex.hpp