Relativistic magnetized Kelvin-Helmholtz problem. More...

#include "pluto.h"

Include dependency graph for init.c:

Functions
void	Init (double *v, double x1, double x2, double x3)

void	Analysis (const Data d, Grid grid)

void	UserDefBoundary (const Data d, RBox box, int side, Grid *grid)

Detailed Description

Relativistic magnetized Kelvin-Helmholtz problem.

Implements the initial condition for a relativistic KH flow in 2D Cartesian coordinates as in section 6.6 of Mignone et al. (ApJS, 2012). Density and pressure are initially constant and equal to $\rho = 1$ and $p = p_0$ where the latter value is retrieved from the definition of the Mach number $M = v/c_s$ . The sound speed is $c_s=\sqrt{\Gamma p/w}$ for the IDEAL equation of state and $c_s = \sqrt{\frac{p}{3w}\frac{5w - 8p}{w - p}}$ for the TAUB EoS (see Eq. [17], [31] and [32] of Mignone & McKinney [MNRAS, 2007]), where $w = \rho h$ is the gas enthalpy.

The velocity is parallel to the x axis with a hyperbolic tangent shear profile. Magnetic field has the form

$\vec{B} = \left(\sqrt{2\sigma_{\rm pol}p_0},\, 0,\, \sqrt{2\sigma_{\rm tor}p_0}\right)$

where $\sigma_{\rm pol}$ and $\sigma_{\rm tor}$ control the strength of the poloidal and toroidal component, respectively.

Periodic boundary conditions are assumed in the horizontal direction while zero-gradient (outflow) conditions hold at the lower and upper boundaries in the y direction.

The presence of a magnetic field gives rise to a filamentary structure and to the formation of a number of vortices arranged symmetrically with respect to the central point. The figure below shows density for configuration #03 using AMR with 4 levels of refinement at t = 5

The control parameters for this problem are:

g_inputParam[SIGMA_TOR]: control the strength of the toroidal (z) component of magnetic field;
g_inputParam[SIGMA_POL]: control the strength of the poloidal (x) component of magnetic field;
g_inputParam[VEL0]: the flow velocity (in units of the speed of light);
g_inputParam[MACH]: the flow Mach number used to recover the pressure value;

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References

"The PLUTO Code for Adaptive Mesh Computations in Astrophysical Fluid Dynamics", Mignone et al., ApJS (2012) 198, 7.
"Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index", Mignone & McKinney, MNRAS (2007) 378, 1118.

Author: A. Mignone (migno.nosp@m.ne@p.nosp@m.h.uni.nosp@m.to.i.nosp@m.t)

Date: Sep 22, 2014

Definition in file init.c.

Function Documentation

void Analysis	(	const Data *	d,
		Grid *	grid
	)

Perform runtime data analysis.

Parameters

[in]	d	the PLUTO Data structure
[in]	grid	pointer to array of Grid structures

Compute volume-integrated magnetic pressure, Maxwell and Reynolds stresses. Save them to "averages.dat"

Definition at line 131 of file init.c.

 {
 
 }

void Init	(	double *	v,
		double	x1,
		double	x2,
		double	x3
	)

The Init() function can be used to assign initial conditions as as a function of spatial position.

Parameters

[out]	v	a pointer to a vector of primitive variables
[in]	x1	coordinate point in the 1st dimension
[in]	x2	coordinate point in the 2nd dimension
[in]	x3	coordinate point in the 3rdt dimension

The meaning of x1, x2 and x3 depends on the geometry:

$\begin{array}{cccl} x_1 & x_2 & x_3 & \mathrm{Geometry} \\ \noalign{\medskip} \hline x & y & z & \mathrm{Cartesian} \\ \noalign{\medskip} R & z & - & \mathrm{cylindrical} \\ \noalign{\medskip} R & \phi & z & \mathrm{polar} \\ \noalign{\medskip} r & \theta & \phi & \mathrm{spherical} \end{array}$

Variable names are accessed by means of an index v[nv], where nv = RHO is density, nv = PRS is pressure, nv = (VX1, VX2, VX3) are the three components of velocity, and so forth.

Definition at line 64 of file init.c.

 {
   static int first_call = 1;
   double x,y, arg, eps, kx;
   double alpha, beta;
   static double pr0;
 
   if (first_call == 1){
     arg = g_inputParam[MACH]/g_inputParam[VEL0];
     arg = arg*arg;
     #if EOS == IDEAL
      g_gamma = 4./3.;
      pr0 = (g_gamma - 1.0)/((g_gamma - 1.0)*arg - 1.0)/g_gamma;
     #else
     { 
       double a,b,c;  /* See Eq. 32 of Mignone \& McKinney (MNRAS 2007) */
       a = 15.0 - 6.0*arg; b = 24.0 - 10.0*arg; c = 9.0;
       arg = 0.5*(- b - sqrt(b*b - 4.0*a*c))/a;
       pr0 = 2.0/3.0*sqrt(arg*arg/(1.0 - arg*arg));
     }
     #endif
     first_call = 0;
   }
 
   x = x1;
   y = x2;
 
   kx    = 2.0*CONST_PI*x1;
   alpha = 1.0/100.0;
   beta  = 1.0/10.0;
 
   arg = y*y/(beta*beta);
   eps = 0.01*g_inputParam[VEL0];
 
   v[RHO] = 1.0;
   v[VX1] = -g_inputParam[VEL0]*tanh(y/alpha);
   v[VX2] =  eps*0.5*(sin(kx) - sin(-kx))*exp(-arg);
   v[VX3] = 0.0;
 
 /* --------------------------------------------
     find pressure values given the Mach number
    -------------------------------------------- */
 
   v[PRS] = pr0;
 
 /* v[PRS] = 20.0;  */
 
   v[TRC] = (y < 0.0 ? 1.0:-1.0);
 
   #if PHYSICS == MHD || PHYSICS == RMHD
 
    v[BX1] = sqrt(2.0*v[PRS]*g_inputParam[SIGMA_POL]);
    v[BX2] = 0.0;
    v[BX3] = sqrt(2.0*v[PRS]*g_inputParam[SIGMA_TOR]);
 
    v[AX1] = 0.0;
    v[AX2] = 0.0;
    v[AX3] = y*v[BX1];
 
   #endif
 }

void UserDefBoundary	(	const Data *	d,
		RBox *	box,
		int	side,
		Grid *	grid
	)

Assign user-defined boundary conditions.

Parameters

[in,out]	d	pointer to the PLUTO data structure containing cell-centered primitive quantities (d->Vc) and staggered magnetic fields (d->Vs, when used) to be filled.
[in]	box	pointer to a RBox structure containing the lower and upper indices of the ghost zone-centers/nodes or edges at which data values should be assigned.
[in]	side	specifies the boundary side where ghost zones need to be filled. It can assume the following pre-definite values: X1_BEG, X1_END, X2_BEG, X2_END, X3_BEG, X3_END. The special value side == 0 is used to control a region inside the computational domain.
[in]	grid	pointer to an array of Grid structures.

Assign user-defined boundary conditions in the lower boundary ghost zones. The profile is top-hat:

$V_{ij} = \left\{\begin{array}{ll} V_{\rm jet} & \quad\mathrm{for}\quad r_i < 1 \\ \noalign{\medskip} \mathrm{Reflect}(V) & \quad\mathrm{otherwise} \end{array}\right.$

where $V_{\rm jet} = (\rho,v,p)_{\rm jet} = (1,M,1/\Gamma)$ and M is the flow Mach number (the unit velocity is the jet sound speed, so $ v = M$ ).

Assign user-defined boundary conditions:

left side (x beg): constant shocked values
bottom side (y beg): constant shocked values for x < 1/6 and reflective boundary otherwise.
top side (y end): time-dependent boundary: for $x < x_s(t) = 10 t/\sin\alpha + 1/6 + 1.0/\tan\alpha$ we use fixed (post-shock) values. Unperturbed values otherwise.

Assign user-defined boundary conditions at inner and outer radial boundaries. Reflective conditions are applied except for the azimuthal velocity which is fixed.

Assign user-defined boundary conditions.

Parameters

	[in/out]	d pointer to the PLUTO data structure containing cell-centered primitive quantities (d->Vc) and staggered magnetic fields (d->Vs, when used) to be filled.
[in]	box	pointer to a RBox structure containing the lower and upper indices of the ghost zone-centers/nodes or edges at which data values should be assigned.
[in]	side	specifies on which side boundary conditions need to be assigned. side can assume the following pre-definite values: X1_BEG, X1_END, X2_BEG, X2_END, X3_BEG, X3_END. The special value side == 0 is used to control a region inside the computational domain.
[in]	grid	pointer to an array of Grid structures.

Set the injection boundary condition at the lower z-boundary (X2-beg must be set to userdef in pluto.ini). For $R \le 1$ we set constant input values (given by the GetJetValues() function while for $ R > 1 $ the solution has equatorial symmetry with respect to the z=0 plane. To avoid numerical problems with a "top-hat" discontinuous jump, we smoothly merge the inlet and reflected value using a profile function Profile().

Provide inner radial boundary condition in polar geometry. Zero gradient is prescribed on density, pressure and magnetic field. For the velocity, zero gradient is imposed on v/r (v = vr, vphi).

The user-defined boundary is used to impose stress-free boundary and purely vertical magnetic field at the top and bottom boundaries, as done in Bodo et al. (2012). In addition, constant temperature and hydrostatic balance are imposed. For instance, at the bottom boundary, one has:

$\left\{\begin{array}{lcl} \DS \frac{dp}{dz} &=& \rho g_z \\ \noalign{\medskip} p &=& \rho c_s^2 \end{array}\right. \qquad\Longrightarrow\qquad \frac{p_{k+1}-p_k}{\Delta z} = \frac{p_{k} + p_{k+1}}{2c_s^2}g_z$

where $g_z$ is the value of gravity at the lower boundary. Solving for $p_k$ at the bottom boundary where $k=k_b-1$ gives:

$\left\{\begin{array}{lcl} p_k &=& \DS p_{k+1} \frac{1-a}{1+a} \\ \noalign{\medskip} \rho_k &=& \DS \frac{p_k}{c_s^2} \end{array}\right. \qquad\mathrm{where}\qquad a = \frac{\Delta z g_z}{2c_s^2} > 0$

where, for simplicity, we keep constant temperature in the ghost zones rather than at the boundary interface (this seems to give a more stable behavior and avoids negative densities). A similar treatment holds at the top boundary.

Assign user-defined boundary conditions. At the inner boundary we use outflow conditions, except for velocity which we reset to zero when there's an inflow

Definition at line 140 of file init.c.

145 {

146 }

Functions

Detailed Description

Function Documentation