Subgrid-scale characterization and asymptotic behavior of multidimensional upwind schemes for the vorticity transport equations

We study the subgrid-scale characteristics of a vorticity-transport-based approach for large-eddy simulations. In particular, we consider a multidimensional upwind scheme for the vorticity transport equations and establish its properties in the under-resolved regime. The asymptotic behavior of key turbulence statistics of velocity gradients, vorticity, and invariants is studied in detail. Modified equation analysis indicates that dissipation can be controlled locally via nonlinear limiting of the gradient employed for the vorticity reconstruction on the cell face such that low numerical diffusion is obtained in well-resolved regimes and high numerical diffusion is realized in under-resolved regions. The enstrophy budget highlights the remarkable ability of the truncation terms to mimic the true subgrid-scale dissipation and diffusion. The modified equation also reveals diffusive terms that are similar to several commonly employed subgride-scale models including tensor-gradient and hyperviscosity models. Investigations on several canonical turbulence flow cases show that large-scale features are adequately represented and remain consistent in terms of spectral energy over a range of grid resolutions. Numerical dissipation in under-resolved simulations is consistent and can be characterized by diffusion terms discovered in the modified equation analysis. A minimum state of scale separation necessary to obtain asymptotic behavior is characterized using metrics such as effective Reynolds number and effective grid spacing. Temporally -evolving jet simulations, characterized by large-scale vortical structures, demonstrate that high Reynolds number vortex-dominated flows are captured when criteria is met and necessitate diffusive nonlinear limiting of vorticity reconstruction be employed to realize accuracy in under-resolved simulations.

Figure 1

Sketch of multidimensional scheme using both normal ${\mathit{F}}_1$ and transverse ${\mathit{G}}_2$ fluxes.

Figure 2

Energy spectra $E\left(k\right)$ of simulations at several under-resolved grid resolutions at $\mathrm{Re}=1600$ at (a) $t{u}_0{k}_0=5$ and (b) $t{u}_0{k}_0=10$. Red: no-limiter, black: minmod limiter. The magenta dotted line is the energy spectra from DNS results.

Figure 3

Energy spectra $E\left(k\right)$ of simulations at several under-resolved grid resolutions $N=128$ (solid line) and $N=64$ (dashed line) at $\mathrm{Re}=1600,2000,3000,5000$, and $\infty$ at times (a) $t{u}_0{k}_0=5$ and (b) $t{u}_0{k}_0=10$.

Figure 4

Dissipation spectra $\mathcal{E}\left(k\right)$ from the resolved vorticity of simulations at several under-resolved grid resolutions at $Re=1600$ at (a) $t{u}_0{k}_0=5$ and (b) $t{u}_0{k}_0=10$. Red: no-limiter, black: minmod limiter. The magenta dotted line is the dissipation spectra from DNS results.

Figure 5

The filtered dissipation ${\tilde{\epsilon }}^{*}$ as a function of time for (a) $N=256$, (b) $N=128$, and (c) $N=64$. Red dashed line: no-limiter, black solid line: minmod limiter. The magenta line is the dissipation from DNS results spatially filtered with a sharp-cutoff spectral filter with several filter widths $\mathrm{\Delta }=2{\mathrm{\Delta }}_x$ (solid line), $4{\mathrm{\Delta }}_x$ (dashed line), and $8{\mathrm{\Delta }}_x$ (dotted line).

Figure 6

The filtered dissipation ${\tilde{\epsilon }}^{*}$ as a function of time for (a) $N=128$ and (b) $N=64$ at $\mathrm{Re}=1600,2000,3000,5000$, and $\infty$. The magenta line is the dissipation from DNS results spatially filtered with a sharp-cutoff spectral filter with several filter widths $\mathrm{\Delta }=2{\mathrm{\Delta }}_x$ (solid line), $4{\mathrm{\Delta }}_x$ (dashed line), and $8{\mathrm{\Delta }}_x$ (dotted line).

Figure 7

(a) The temporal evolution of terms in the enstrophy transport equation for the $N=64$ test case with minmod limiter. (b) The temporal evolution of the calculated SGS dissipation and diffusion and the temporal evolution of the SGS dissipation and diffusion obtained from the modified equation for the $N=64$ test case with minmod limiter.

Figure 8

(a) The temporal evolution of terms in the enstrophy transport equation for the $N=128$ test case with minmod limiter. (b) The temporal evolution of the calculated SGS dissipation and diffusion and the temporal evolution of the SGS dissipation and diffusion obtained from the modified equation for the $N=128$ test case with minmod limiter.

Figure 9

The temporal evolution enstrophy transport of components of the SGS dissipation and diffusion obtained from modified equation in Eqs. (32), (34), and (35) for (a) $N=64$ and (b) $N=128$ test cases with minmod limiter.

Figure 10

The temporal evolution of terms in the enstrophy transport equation for (a) $N=64$ and (b) $N=128$ test cases with no-limiter.

Figure 11

The SGS dissipation ${\epsilon }_{\text{SGS}}$ for the Taylor-Green vortex for (a) $N=256$, (b) $N=128$, and (c) $N=64$. The SGS dissipation obtained from the modified equation analysis is shown for the $N=128$ and $N=64$.

Figure 12

The filtered dissipation ${\tilde{\epsilon }}^{*}$ as a function of time for (a) $N=64$ and (b) $N=128$ comparing test case with minmod limiter, two explicit LES cases with $C_s=0.1$ and 0.3, and DNS results from a spectral method. The temporal evolution of the calculated SGS dissipation and diffusion obtained from the present method, the modified equation, and Smagorinsky model for explicit LES with grid discretization of (c) $N=64$ and (d) $N=128$.

Figure 13

The effective Re, ${\text{Re}}_f$, as a function of twice the enstrophy throughout the runtime of the simulations for several grid resolutions and viscosities. ${\text{Re}}_f$ at (b) $t=5$ and (c) maximum dissipation.

Figure 14

The ratio of the effective grid spacing and Kolmorgorov lengthscale (from DNS) $\mathrm{\Delta }{x}_f/\eta$ for the $\text{Re}=1600$ cases. Red: no-limiter, black: minmod limiter.

Figure 15

Longitudinal (left) and transverse (right) velocity gradient probability density functions normalized by the rms vorticity ${\omega }^{\prime }$ with several grid resolutions. Markers indicate PDFs captured from DNS results from Ref. [89] are at ${\text{Re}}_{\lambda }=35,61,94$, and 168 and experimental measurements from Ref. [90] are at ${\text{Re}}_{\lambda }=852$. Each PDF is labeled.

Figure 16

(a) Velocity magnitude, (b) vorticity magnitude, (c) vortex stretching, and (d) strain rate magnitude probability density function normalized by the rms vorticity ${\omega }^{\prime }$ of forcing case for several grid resolutions. The case $N=256$ with no-limiter is shown as a solid cyan line. Marked indicate DNS results from Ref. [91] are at ${\text{Re}}_{\lambda }=37,62,95,142$, and 168. Each PDF is labeled.

Figure 17

Turbulence statistics of the forced turbulence simulations for grid resolutions of $N=256,128,64$, and 32. Contour lines for each plot are shown at $\{{10}^0,{10}^{-1},{10}^{-2},{10}^{-3}\}$.

Figure 18

Joint PDF of velocity gradient invariants $Q$ and $R$ normalized by the resolved enstrophy $\tilde{\mathcal{E}}$ of the forced turbulence simulations for grid resolutions of (a) $N=64$ and (b) $N=128$. Contour lines for each plot are shown in log space between 0 and ${10}^{-5}$.

Figure 19

Longitudinal (left) and transverse (right) velocity gradient probability density functions normalized by the rms vorticity ${\omega }^{\prime }$ for grid resolutions of $N=64$ at several initial Re numbers. Markers indicate DNS results from Ref. [89] are at ${\text{Re}}_{\lambda }=35,61,94$, and 168. Experimental measurements from Ref. [90] are at ${\text{Re}}_{\lambda }=852$. Each PDF is labeled.

Figure 20

Joint PDF of velocity gradient invariants $Q$ and $R$ normalized by the resolved enstrophy $\tilde{\mathcal{E}}$ of the forced turbulence simulations for grid resolutions of $N=64$ at several Re numbers. Contour lines for each plot are shown in log space between 0 and ${10}^{-5}$.

Figure 21

(a) The temporal evolution of terms in the enstrophy transport equation for the $N=64$ test case with minmod limiter. (b) The temporal evolution of the calculated SGS dissipation and diffusion and the temporal evolution of the SGS transfer and diffusion obtained from the modified equation for the $N=128$ test case with minmod limiter.

Figure 22

(a) The temporal evolution of terms in the enstrophy transport equation for the $N=128$ test case with minmod limiter. (b) The temporal evolution of the calculated SGS dissipation and diffusion and the temporal evolution of the SGS transfer and diffusion obtained from the modified equation for the $N=128$ test case with minmod limiter.

Figure 23

Energy spectra of forced turbulence simulations. The thin dotted line is at a $-5/3$ slope. The case $N=256$ with no-limiter is shown as a solid cyan line.

Figure 24

Energy spectra of forced turbulence simulations at $\mathrm{Re}=6000$ and $\mathrm{Re}=\infty$. The solid line is $N=128$ and the dashed line is $N=64$. The diamond, square, and triangle markers are forced turbulence cases 1, 2, and 3, respectively, from Ref. [86]. Circle and plus markers denote $N=64$ and $N=128$, respectively, and correspond to explicit LES of case 2 from Ref. [93].

Figure 25

Velocity ${\omega }_3$ normalized by the jet half-width ${\delta }_{1/2}$ and centerline velocity for the temporally evolving jet contours for the temporally evolving jet at $t/T_{\text{ref}}=15$. (a) $N=64\left(\mathrm{ENO}\right)$, (b) $N=128\left(\mathrm{ENO}\right)$, (c) $N=256\left(\mathrm{ENO}\right)$, (d) $N=64\left(\mathrm{no}\text{−}\mathrm{limiter}\right)$, (e) $N=128\left(\mathrm{no}\text{−}\mathrm{limiter}\right)$, and (f) $N=256\left(\mathrm{no}\text{−}\mathrm{limiter}\right)$.

Figure 26

Profiles of the (a) mean streamwise velocity $\langle u_1\rangle$ normalized by the centerline velocity $U_c$ compared with experimental results from Refs. [95] (circles) and [96] (squares) and (b) mean vorticity $\langle {\omega }_3\rangle$ normalized by the jet half-width ${\delta }_{1/2}$ and centerline velocity for the temporally evolving jet. Blue—no-limiter, black—ENO limiter.

Figure 27

Profiles of the (a) rms (root-mean-square) streamwise velocity ${\langle u_1^2\rangle }^{1/2}$ and (b) rms normal velocity ${\langle u_2^2\rangle }^{1/2}$ normalized by the centerline velocity $U_c$ compared with experimental results from Refs. [95] (circles) and [96] (squares). Blue—no-limiter, black—ENO limiter.

Figure 28

Temporal evolution of the vorticity profiles $\langle {\omega }_3\rangle$ for (a) $N=256$, (b) $N=128$ and (c) $N=64$. Each case employs the ENO limiter.

Figure 29

The temporal evolution of the (a) jet half-width, (b) effective viscosity, and (c) effective Re. Blue—no-limiter, black—ENO limiter.

Figure 30

Energy spectra $E\left(k\right)$ of simulations temporal simulations: (a) $\mathrm{Re}=3200$ at several discretizations, blue—no-limiter, black—ENO limiter, (b) $N=128$ case at different $\mathrm{Re}$'s, and (c) $N=64$ case at different $\mathrm{Re}$'s.