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SpECTRE
v2024.04.12
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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 = tmpl::conditional_t< Dim==3, tmpl::list< AdiabaticIndex, Center, MeanVelocity, Strength, PerturbAmplitude >, tmpl::list< AdiabaticIndex, Center, MeanVelocity, Strength > > |
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 > ¢er, 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) |
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 [191]. 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{align*} \rho &= \left[1 - \dfrac{(\gamma - 1)\beta^2}{8\gamma\pi^2}\exp\left( 1 - r^2\right)\right]^{1/(\gamma - 1)}, \\ v_x &= U - \dfrac{\beta\tilde y}{2\pi}\exp\left(\dfrac{1 - r^2}{2}\right),\\ v_y &= V + \dfrac{\beta\tilde x}{2\pi}\exp\left(\dfrac{1 - r^2}{2}\right),\\ v_z &= W,\\ \epsilon &= \frac{\rho^{\gamma - 1}}{\gamma - 1}, \end{align*}
with
\begin{align*} r^2 &= {\tilde x}^2 + {\tilde y}^2,\\ \tilde x &= x - X_0 - U t,\\ \tilde y &= y - Y_0 - V t, \end{align*}
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{align*} p = \rho^\gamma. \end{align*}
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 \(dv_z/dz\). (See NewtonianEuler::Sources::VortexPerturbation.) For testing purposes, we choose to write the velocity as a uniform field plus a periodic perturbation,
\begin{align*} v_z(z) = W + \epsilon \sin{z}, \end{align*}
where \(\epsilon\) is the amplitude of the perturbation. The resulting source for the Newtonian Euler system will then be proportional to \(\epsilon \cos{z}\).
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overridevirtual |
Implements evolution::initial_data::InitialData.
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staticconstexpr |