RMHD rotor test problem in 2 or 3 dimensions. 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

RMHD rotor test problem in 2 or 3 dimensions.

The 2D rotor problem [dZBL03][Mig12] consists of a rapidly spinning disk embedded in a uniform background medium treaded by a costant magnetic field. The initial conditin reads as

$\left(\rho, p, v_x, v_y, B_x\right) = \left\{\begin{array}{ll} \left(10, 1,-\omega y,\,\omega x,\, 1\right) & \quad\mathrm{for}\quad r < 0.1 \\ \noalign{\medskip} \left(1,\, 1,\, 0,\, 0,\, 1\right) & \quad\mathrm{otherwise} \end{array}\right.$

where $\omega = 9.95$ is the angular frequency of rotation. The computational domain is the unit square and outflow boundary conditions are imposed everywhere.

As the disk rotates, strong torsional Alfven waves form and propagate outward carrying angular momentum from the disk to the ambient. The emerging flow structure is enclosed by a circular fast forward shock traveling into the surrounding medium. An inward fast shock bounds the innermost oval region where density has been depleted to lower values. The presence of the magnetic field slows down the rotor, and the maximum Lorentz factor decreases from the nominal value of 10 to 2.2 (approx).

A list of tested configurations is given in the following table:

Conf.	GEOMETRY	DIM	T. STEPPING	RECONSTRUCTION	divB	AMR
#01	CARTESIAN	2	RK3	LINEAR	CT	NO
#02	CARTESIAN	2	RK2	LINEAR	CT	NO
#03	CARTESIAN	2	HANCOCK	LINEAR	8W	NO
#04	CARTESIAN	2	HANCOCK	LINEAR	CT	NO
#05	CARTESIAN	3	RK2	LINEAR	GLM	NO
#06	CARTESIAN	3	RK2	LINEAR	GLM	NO
#07	CARTESIAN	3	HANCOCK	LINEAR	CT	NO
#08	CARTESIAN	3	RK3	LimO3	GLM	NO

The final solutin at t = 0.4 on a grid of 400x400 grid points is shown below.

Density map (in log scale) at t = 0.4 in Cartesian coordinates using 400 x 400 grid zones (configuration #03). Field lines are super-imposed.

The three dimensional version extends the current configuration to a spinning sphere and it is described in [MUB09].

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

Date: Aug 6, 2015

Reference:

[dZBL03] "An efficient shock-capturing central-type scheme for multidimensional relativistic flows" Del Zanna, Bucciantini, Londrillo, A&A (2003) 400, 397
[MUB09] "A five-wave Harten-Lax-van Leer Riemann solver for relativistic magnetohydrodynamics" Mignone, Ugliano & Bodo, MNRAS (2009) 393, 1141.
[Mig12] "The PLUTO Code for Adaptive Mesh Computations in Astrophysical Fluid Dynamics" Mignone et al., ApJS (2012) 198, 7 (see Sect. 6.3)

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 126 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 68 of file init.c.

 {
   double r, r0, Bx, omega;
 
   g_gamma   = 5./3.;
   omega = 0.995;
   Bx    = 1.0;
   r0    = 0.1;
 
   #if GEOMETRY == CARTESIAN
 
    r = D_EXPAND(x1*x1, + x2*x2, + x3*x3);
    r = sqrt(r);
 
    v[RHO] = 1.0;
    v[VX1] = 0.0;
    v[VX2] = 0.0;
    v[PRS] = 1.0;
    v[BX1] = Bx;
    v[BX2] = 0.0;
 
    if (r <= r0) {
      v[RHO] = 10.0;
      v[VX1] = -omega*x2/r0;
      v[VX2] =  omega*x1/r0;
    }
 
    v[AX1] = v[AX2] = 0.0;
    v[AX3] = v[BX1]*x2;
 
   #elif GEOMETRY == POLAR
 
    r = x1;
 
    v[RHO] = 1.0;
    v[VX2] = 0.0;
    v[VX1] = 0.0;
    v[BX1] =  Bx*cos(x2);
    v[BX2] = -Bx*sin(x2);
    v[PRS] = 1.0;
   
    if (r <= r0) {
      v[RHO] = 10.0;
      v[VX2] = omega*r/r0;
    }
 
    v[AX1] = v[AX2] = 0.0;
    v[AX3] = - r*sin(x2)*Bx;
 
   #endif
 
 }

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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 135 of file init.c.

 {
   int   i, j, k, nv;
   double  *r, slp;
 
   if (side == X1_BEG){
     r = grid[IDIR].x;
     if (box->vpos == CENTER){
       BOX_LOOP(box,k,j,i){
         slp = r[i]/r[IBEG];
         d->Vc[RHO][k][j][i] = d->Vc[RHO][k][j][IBEG];
         d->Vc[VX1][k][j][i] = slp*d->Vc[VX1][k][j][IBEG];
         d->Vc[VX2][k][j][i] = slp*d->Vc[VX2][k][j][IBEG]; 
         d->Vc[PRS][k][j][i] = d->Vc[PRS][k][j][IBEG];
         d->Vc[BX1][k][j][i] = d->Vc[BX1][k][j][IBEG];
         d->Vc[BX2][k][j][i] = d->Vc[BX2][k][j][IBEG];
       }
     }else if (box->vpos == X2FACE){
       #ifdef STAGGERED_MHD
        BOX_LOOP(box,k,j,i) d->Vs[BX2s][k][j][i] = d->Vs[BX2s][k][j][IBEG];
       #endif
     }
   }
 }

Functions

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