Stable, entropy-consistent, and localized artificial-diffusivity method for capturing discontinuities

In this work, a localized artificial-viscosity/diffusivity method is proposed for accurately capturing discontinuities in compressible flows. There have been numerous efforts to improve the artificial diffusivity formulation in the last two decades, through appropriate localization of the artificial bulk viscosity for capturing shocks. However, for capturing contact discontinuities, either a density or internal energy variable is used as a detector. An issue with this sensor is that it not only detects contact discontinuities but also falsely detects the regions of shocks and vortical motions. Using this detector to add artificial mass/thermal diffusivity for capturing contact discontinuities is hence unnecessarily dissipative. To overcome this issue, we propose a sensor similar to the Ducros sensor (for shocks) to detect contact discontinuities and further localize artificial mass/thermal diffusivity for capturing contact discontinuities. The proposed method contains coefficients that are less sensitive to the choice of the flow problem. This is achieved by improved localization of the artificial diffusivity in the present method. A discretely consistent dissipative flux formulation is presented and is coupled with a robust low-dissipative scheme, which eliminates the need for filtering the solution variables. The proposed method also does not require filtering for the discontinuity detector/sensor functions, which is typically done to smear out the artificial fluid properties and obtain stable solutions. Hence, the challenges associated with extending the filtering procedure for unstructured grids are eliminated, thereby making the proposed method easily applicable for unstructured grids. Finally, a straightforward extension of the proposed method to two-phase flows is also presented.

Figure 1

Modified version of the Sod shock-tube test case, simulated with the sensor in Eq. (17) (proposed method) and without the sensor (without $f_D$), showing (a) density, $\rho$, (b) artificial mass diffusivity, $D^{*}$, and (c) artificial bulk viscosity, ${\beta }^{*}$.

Figure 2

Modified version of the Sod shock-tube test case, showing density, for (a) $C_{\beta }=10$, 100 (proposed method), 200, and 1000, and (b) $C_D=0.01$, 0.5 (proposed method), 1, and 100.

Figure 3

Simulation of decaying HIT with shocklets, with both the $f_D$ and $f_{\beta }$ sensors (proposed method), without the $f_D$ sensor (similar to the Kawai et al. [31] formulation), without $f_{\beta }$ sensor, and without both the $f_D$ and $f_{\beta }$ sensors (similar to the Mani et al. [30] formulation), and its comparison with the DNS and filtered DNS. The plotted nondimensional quantities: (a) mean square velocity, (b) density variance, (c) dilatational variance, and (d) vorticity variance.

Figure 4

Two-dimensional slices from the simulations of decaying HIT with shocklets, showing (a)–(d) dilatation, ABV, density, and AMD with the $f_D$ sensor, and (e)–(h) dilatation, ABV, density, and AMD without the $f_D$ sensor Note that the resolution of these images is low due to the chosen grid resolution in LES calculations. Raw data have directly been plotted in these figures without any interpolation.

Figure 5

Simulation of decaying HIT with shocklets, with both the $f_D$ and $f_{\beta }$ sensors, and without DSM (proposed method) and with DSM (proposed method $+$ DSM), showing the nondimensional quantities: (a) mean square velocity, (b) density variance, (c) dilatational variance, and (d) vorticity variance.

Figure 6

Simulation of decaying HIT with shocklets, with both the $f_D$ and $f_{\beta }$ sensors, and without DSM (proposed method) and with DSM (proposed method $+$ DSM), showing the spectrum of (a) kinetic energy and (b) density variance.

Figure 7

Simulation of decaying HIT with shocklets with $f_D, f_{\beta }$, and DSM active, for $a=10$, 100 (proposed method $+$ DSM), 200, and 1000, showing the nondimensional quantities: (a) mean square velocity, (b) density variance, (c) dilatational variance, and (d) vorticity variance.

Figure 8

Schematic of the problem setup for the shock-vortex interaction simulation, showing the moving vortex and the deformed stationary shock after the vortex has passed through the shock.

Figure 9

Simulation of a shock-vortex interaction, showing (a) the sound pressure field, $\mathrm{\Delta }p=(p-p_{\infty })/p_{\infty }$ and (b) the artificial bulk viscosity, ${\beta }^{*}$, at $t=6$.

Figure 10

Simulation of a shock-vortex interaction, showing the sound pressure field, $\mathrm{\Delta }p=(p-p_{\infty })/p_{\infty }$ along the radial direction $r$ from the center of the vortex for three grid resolutions, along with the reference solution from Inoue and Hattori [84], and at (a) $t=6$ (b) $t=8$.

Figure 11

Simulation of a one-dimensional drop advection, showing the volume fraction and pressure fields at time $t=2$, with (a) ATD, (b) no ATD or AMD, and (c) AMD.

Figure 12

Initial setup for the Riemann problem, showing four quadrants with constant states.

Figure 13

Simulation of a 2D Riemann problem, showing (a) density, $\rho$, and (b) artificial bulk viscosity, ${\beta }^{*}$, at $t=0.2$.