Zuoli Xiao

Effective eddy viscosities in implicit large eddy simulations of turbulent flows

We propose a method for computing effective numerical eddy viscosity acting in dissipative numerical schemes used in monotonically integrated large eddy simulations of turbulence. The method is evaluated on an example of a specific nonoscillatory finite volume scheme MPDATA developed for simulations of geophysical flows.

A hybrid numerical simulation of isotropic compressible turbulence

A novel hybrid numerical scheme with built-in hyperviscosity has been developed to address the accuracy and numerical instability in numerical simulation of isotropic compressible turbulence in a periodic domain at high turbulent Mach number. The hybrid scheme utilizes a 7th-order WENO (Weighted Essentially Non-Oscillatory) scheme for highly compressive regions (i.e., shocklet regions) and an 8th-order compact central finite difference scheme for smooth regions outside shocklets. A flux-based conservative and formally consistent formulation is developed to optimize the connection between the two schemes at the interface and to achieve a higher computational efficiency. In addition, a novel numerical hyperviscosity formulation is proposed within the context of compact finite difference scheme for the smooth regions to improve numerical stability of the hybrid method. A thorough and insightful analysis of the hyperviscosity formulation in both Fourier space and physical space is presented to show the effectiveness of the formulation in improving numerical stability, without compromising the accuracy of the hybrid method. A conservative implementation of the hyperviscosity formulation is also developed. Combining the analysis and test simulations, we have also developed a criterion to guide the specification of a numerical hyperviscosity coefficient (the only adjustable coefficient in the formulation). A series of test simulations are used to demonstrate the accuracy and numerical stability of the scheme for both decaying and forced compressible turbulence. Preliminary results for a high-resolution simulation at turbulent Mach number of 1.08 are shown. The sensitivity of the simulated flow to the detail of thermal forcing method is also briefly discussed.

Physical mechanism of the inverse energy cascade of two-dimensional turbulence: a numerical investigation

We report an investigation of inverse energy cascade in steady-state two-dimensional turbulence by direct numerical simulation (DNS) of the two-dimensional Navier–Stokes equation, with small-scale forcing and large-scale damping. We employed several types of damping and dissipation mechanisms in simulations up to 2048 2 resolution. For all these simulations we obtained a wavenumber range for which the mean spectral energy flux is a negative constant and the energy spectrum scales as k −5/3 , consistent with the predictions of Kraichnan ( Phys. Fluids , vol. 439, 1967, p. 1417). To gain further insight, we investigated the energy cascade in physical space, employing a local energy flux defined by smooth filtering. We found that the inverse energy cascade is scale local, but that the strongly local contribution vanishes identically, as argued by Kraichnan ( J. Fluid Mech ., vol. 47, 1971, p. 525). The mean flux across a length scale l was shown to be due mainly to interactions with modes two to eight times smaller. A major part of our investigation was devoted to identifying the physical mechanism of the two-dimensional inverse energy cascade. One popular idea is that inverse energy cascade proceeds via merger of like-sign vortices. We made a quantitative study employing a precise topological criterion of merger events. Our statistical analysis showed that vortex mergers play a negligible direct role in producing mean inverse energy flux in our simulations. Instead, we obtained with the help of other works considerable evidence in favour of a ‘vortex thinning’ mechanism, according to which the large-scale strains do negative work against turbulent stress as they stretch out the isolines of small-scale vorticity. In particular, we studied a multi-scale gradient (MSG) expansion developed by Eyink ( J. Fluid Mech ., vol. 549, 2006 a , p. 159) for the turbulent stress, whose contributions to inverse cascade can all be explained by ‘thinning’. The MSG expansion up to second order in space gradients was found to predict well the magnitude, spatial structure and scale distribution of the local energy flux. The majority of mean flux was found to be due to the relative rotation of strain matrices at different length scales, a first-order effect of ‘thinning’. The remainder arose from two second-order effects, differential strain rotation and vorticity gradient stretching. Our findings give strong support to vortex thinning as the fundamental mechanism of two-dimensional inverse energy cascade.

Constrained subgrid-scale stress model for large eddy simulation

In this letter, we propose to impose physical constraints in the dynamic procedure of the dynamic subgrid-scale (SGS) stress model in large eddy simulation, and to calculate the SGS model coefficients using a constrained variation. Numerical simulations of forced and decaying isotropic turbulence demonstrate that the constrained dynamic mixed model predicts the energy evolution and the SGS energy dissipation well. The constrained SGS model also shows a strong correlation with the real stress and is able to capture the energy backscatter, manifesting a desirable feature of combining the advantages of dynamics Smagorinsky and mixed models.

Effect of shocklets on the velocity gradients in highly compressible isotropic turbulence

The effect of randomly generated shocklets on velocity gradients in a three-dimensional compressible isotropic turbulence was systematically studied. The forced flows obtained from high-resolution simulations had a turbulent Mach number of 1.0 and a Taylor microscale Reynolds number around 180. The shock detection algorithm developed by Samtaney et al. [“Direct numerical simulation of decaying compressible turbulence and shocklet statistics,” Phys. Fluids 13, 1415 (2001)] was applied to extract the shocklets. Using reference frames moving with the detected shocks, we obtained statistical properties of velocity and its gradients both upstream and downstream of the shocks. It was shown that the shocks induced flow modulation at a wide range of length scales, including the inertial subrange scales. The shocks intensified enstrophy in the shock regions and this enhanced enstrophy production was partially redistributed over various scales and dissipated by straining and viscous effects outside the shock region...

Constrained large-eddy simulation of separated flow in a channel with streamwise-periodic constrictions

Constrained large-eddy simulation (CLES) method has been recently developed by Chen and his colleagues for simulating attached and detached wall-bounded turbulent flows. In CLES, the whole domain is simulated using large-eddy simulation (LES) while a Reynolds stress constraint is enforced on the subgrid-scale (SGS) stress model for near wall regions. In this paper, CLES is used to simulate the separated flow in a channel with streamwise-periodic constrictions at Re = 10,595. The results of CLES are compared with those of Reynolds-averaged Navier-Stokes (RANS) method, LES, detached eddy simulation (DES) and previous LES results by Breuer et al. and Ziefle et al. Although a coarse grid is used, our results from the present LES, DES and CLES do not show large deviations from the reference results using much finer grid resolution. The comparison also shows that CLES performs the best among different turbulence models tested, demonstrating that the CLES provides an excellent alternative model for separated flows. Furthermore, the cross-comparisons among different CLES implementations have been carried out. Our simulation results are in favor of using the constraint from algebraic RANS model or solving the RANS model equations in the whole domain with a length scale modification according to the idea from DES.

Dissipation-energy flux correlations as evidence for the Lagrangian energy cascade in turbulence

We study the spatial and temporal evolution of kinetic energy flux at different scales using direct numerical simulations of isotropic turbulence. The correlation coefficients at different times, between the molecular energy dissipation and local energy fluxes across inertial-range scales, are computed in both Eulerian and Lagrangian frames. The Eulerian correlation coefficients are found to decay monotonically backward in time. However, the Lagrangian correlation coefficients peak after a certain time delay. The peak time delay is found to be proportional to the local eddy turnover time (it scales with wave number k according to k−2/3), consistent with Kolmogorov’s theory. Conditional sampling is used to isolate effects of strong rotation. The results presented provide strong evidence of the Lagrangian nature of turbulent energy cascade.We study the spatial and temporal evolution of kinetic energy flux at different scales using direct numerical simulations of isotropic turbulence. The correlation coefficients at different times, between the molecular energy dissipation and local energy fluxes across inertial-range scales, are computed in both Eulerian and Lagrangian frames. The Eulerian correlation coefficients are found to decay monotonically backward in time. However, the Lagrangian correlation coefficients peak after a certain time delay. The peak time delay is found to be proportional to the local eddy turnover time (it scales with wave number k according to k−2/3), consistent with Kolmogorov’s theory. Conditional sampling is used to isolate effects of strong rotation. The results presented provide strong evidence of the Lagrangian nature of turbulent energy cascade.

Constrained large-eddy simulation of wall-bounded compressible turbulent flows

A constrained large-eddy simulation (CLES) approach is developed for wall-bounded compressible turbulent flows based on its incompressible analogue [Chen et al., “Reynolds-stress-constrained large-eddy simulation of wall-bounded turbulent flows,” J. Fluid Mech. 703, 1–28 (2012)]. In the new CLES approach, both the subgrid-scale (SGS) stress and the SGS heat flux are decomposed into an averaged part and a fluctuating part in the near-wall region with the mean SGS stress and heat flux constrained by prescribed Reynolds stress model and turbulent heat flux model, respectively. The Smagorinsky SGS models are employed to approximate the SGS stress and heat flux in the remaining region of the flow domain. The present CLES method is validated by simulating the compressible turbulent channel flows at various Reynolds numbers and Mach numbers. The mean velocity profiles, mean temperature profiles, and other statistical quantities and turbulent structures are obtained and well compared among the present approach, d...

Interactions between inertial particles and shocklets in compressible turbulent flow

Numerical simulations are conducted to investigate the dynamics of inertial particles being passively convected in a compressible homogeneous turbulence. Heavy and light particles exhibit very different types of non-uniform distributions due to their different behaviors near shocklets. Because of the relaxation nature of the Stokes drag, the heavy particles are decelerated mainly at downstream adjacent to the shocklets and form high-number-density clouds. The light particles are strongly decelerated by the added-mass effect and stay in the compression region for a relatively long time period. They cluster into thin filament structures near shocklets.

Statistics and structures of pressure and density in compressible isotropic turbulence

We study statistics and structures of pressure and density in the presence of large-scale shock waves in a forced compressible isotropic turbulence using high-resolution numerical simulation. The spectra for pressure and density exhibit a −2 scaling over an operational definition of the inertial range. Both the numerical simulation and a heuristic PDF model reveal that the PDFs of pressure increment exhibit a −2 power law region for the separation in the operational definition of inertial range, quantitatively similar to the PDF of pressure gradient, which also displays a −2 power law region. Moreover, the statistical relation between density increment and pressure increment has been investigated through a shock-relation model. There is a positive correlation between the vorticity magnitude and pressure, which is different from the case of incompressible turbulence. We argue that this difference is due to large-scale shock waves, another type of intermittent structures in addition to vortex structures in incompressible turbulence.