Vortices, turbulence, and unsteady non-laminar flows are likely both prominent and dynamically important features of astrophysical disks. Such strongly nonlinear phenomena are often difficult, however, to simulate accurately, and are generally amenable to analytic treatment only in idealized form. In this project (Seligman & Laughlin, 2017), we explore the evolution of compressible two-dimensional flows using an implicit dual-time hydrodynamical scheme that strictly conserves vorticity (if applied to simulate inviscid flows for which Kelvin’s Circulation Theorem is applicable). The algorithm is based on the work of Lerat, Falissard & Sidé (2007), who proposed it in the context of terrestrial applications such as the blade-vortex interactions generated by helicopter rotors. We present several tests of Lerat et al.’s vorticity-preserving approach, which we have implemented to second-order accuracy, providing side-by-side comparisons with other algorithms that are frequently used in protostellar disk simulations. The comparison codes include one based on explicit, second-order van-Leer advection and another that implements a higher-order Godunov solver. Our results suggest that Lerat et al’s algorithm may be useful for simulations of astrophysical environments in which vortices play a dynamical role, and where strong shocks are not expected.

Here is a link to the Paper accepted to the Astrophysical Journal

Left Panels: x-directional and y-directional cross sections of the tangential velocity of the seeded Yee et al. (1999) vortex following significant hydrodynamical evolution. The seeded, analytic velocity profile is shown in the black solid line, and is unchanging in time. The red squares, yellow diamonds and orange circles show the vortex’s velocity profile after 100 orbits as evolved with the RBV2, PLUTO (WENO+RK3) and ZEUS (van Leer) codes respectively. Right Panels: color-coded maps of the vorticity after 100 orbits of the entire vortex in RBV2 (upper right), PLUTO (middle right) and ZEUS (lower right). The maximum vorticity in the initial and RBV2, PLUTO and ZEUS evolved vortex is 2.62, 2.62,1.05 and 1.09 respectively. All three simulations were run with 256x256 zones and PLUTO was run with RK3 time stepping and the WENO3 (weighted essentially non-oscillatory) reconstruction scheme (Liu et al. 1994).

Vorticity in shearing box simulations of two dynamically interacting vortices evolved with RBV2 (top) and PLUTO (bottom). The simulations were run on 512x512 zones with an extent of ∆R = .04 AU centered at r = 1 AU with a sound speed, cs = 0.05. Not only is the substructure that develops significantly different, but in RBV2 the vortices complete fourteen orbits before merger, while in PLUTO, they merge after seven orbits.

Vorticity evolution in two-dimensional fluid simulations seeded with random, subsonic velocity fields smoothed with a low pass filter. Results are shown, in the columns running from left to right, for RBV2, PLUTO, ZEUS and Stable Fluids respectively, all on 1024x1024 zone grids. The initial vorticity conditions are displayed in the top row, and the middle and bottom rows show the results after 20 sound crossing times of the simulation box. In the top and middle row, the vorticity color scale is held constant between ±.01 for all eight simulations. In the bottom row, the color scale is adjusted in PLUTO, ZEUS and Stable Fluids to ±− 0.001, ±− 0.001 and ±− 0.002 respectively, in order to show the evolution of the fluid vorticity structure. A magnification of a smaller portion of each simulation is shown in the upper left corner of each panel, corresponding to the region encapsulated by the smaller squares shown in the lower right of each panel. The simulations took 40 hours, 29 hours, 14 hours and 22 minutes for RBV2, PLUTO, ZEUS and Stable Fluids respectively.