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Snapshot at time ∼ π : The animation movie shows the temperature for the normalized time 0 to 10, simulated with 256x256 grids covering 2π-by-2π space. The colors in the movie runs from 0.2(white) to 1.2(dark blue) in dimensionless value, divided by 0.6 (the initial value of temperature). |
Left plot and the background image of this page show the snapshot of temperature at t ∼ 1 and 1.5, respectively. |
Temperature at t ~ 0.055. |
HD case mpeg file (679kB) |
Vertical mpeg file (191kB) |
Horizontal mpeg file (117kB) |
Oblique mpeg file (151kB) |
From left shown are the snapshots at t ∼ 12.75 of hydrodynamics case (without magnetic field), and three magnetohydrodynamics cases starting with vertical, horizontal and oblique magnetic field. The initial magnetic field in all MHD cases is uniform and, its strength is set so that the beta ratio will be 250 at the boundary between the high- and low-density regimes. All four cases start with the identical initial plasma distribution with tiny "single"-mode symmetric fluctuation of bulk flow. The movies in the left show the temporal evolutions from t = 0 to 15. Here again, (tension of) magnetic field works to suppress instability with different ways depending on the direction of field. |
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When the initial magnetic field has only the component perpendicular to the plotted plane, the magnetic field will behave as if it is a plasma with specific heat ratio of 2. In this case, no magnetic tension will arise, and thus instability will not be suppressed. |
HD case mpeg file (995kB) |
MHD case mpeg file (429kB) |
The evolutions of RT instability of HD (left) and MHD (right) cases starting with multiple-wavelength small-amplitude fluctuations of plasma : The MHD case starts with uniform, horizontal, and very weak magnetic field (beta-ratio is about 1000). |
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The initial values at left and right side, in a order [Rho, Vx, Vy, Vz, Bx, By, Bz, Pg], are [1, 0, 0, 0, 1.3, 1.0, 0, 1.0] and [0.4, 0, 0, 0, 1.3, -1, 0, 0.4], respectively. The specific heat ratio (γ) is set to be 5/3. The non-reflection boundary treatment based on the method of characteristics is applied at both ends. |
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A plane pressure perturbation is given to the left side of plano-convex/concave magnetic lens with the magnetic field vector perpendicular to the plane. Being set the background plasma temperature and total pressure constant (except the pressure-perturbation site), the wave driven by the pressure perturbation propagates rightward then undergoes refractions at the contact with the magnetic column. Because the propagation speeds inside the lens are faster than the one outside, the refractions are in opposite sense than the optic ones. The left plot and a movie show the temporal evolution of the gas pressure (propagating from left to right). |
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The two plots show the snapshot of the gas pressure; the MHD (left) and HD(right) waves propagate, responding to the impulse given at the center. The non-reflection boundary treatment based on the "projected normal characteristics method" is implemented on the four "free" boundaries which limit the domain of computation. Mainly because this method uses only the normal component of the characteristic curves for simplicity, reflection slightly remains. However, most of wave energy goes outside the computation domain, and such small reflection waves vanish after reflecting twice or so. |
The snapshots of Orszag-Tang vortex problem at time ∼ π.
The left plot is a snapshot of the simulated temperature with the 5th-order weighted essentially non-oscillatory (WENO) method and the Roe's method. The right one is obtained with 3rd-order TVD-MUSCL (here, with "minmod" function) and thus identical to the plot appearing in the topmost part of this page. |
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An asymmetric heat source is given on the bottom boundary to induce asymmetric motions.
Particle motions are traced with Runge-Kutta method.
The Lax-Wendroff method is numerically diffusive, though, has been long used because it is simple and much less computationally expensive. Computations with this method are generally 4 ∼ 6 times faster than those with the TVD method of Godunov type approximate Riemann solver. This is an advantage for various simulation problems. |
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A tiny density enhancement is given near the injection site (on the left boundary), so that symmetry will be broken. With a very-simplified Riemann solver, the Lax-Friedrichs method is a quick but sufficiently accurate scheme. |
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Just for fun.
Collision of two objects is simulated with compressive fluid code. With the initial pressure being uniform everywhere in the simulation box, the objects are featured by different density (thus, different temperature) values. The ratio of initial densities at the core (inner part) of the left object, the outer part of the left object, the right object, and the surrounding space, is 20 : 10 : 15 : 1. The right object is moving leftward to collide onto the left one. Periodic boundary is applied on all four boundaries. No gravity is taken into account. |