Jianchun Wang

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.

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...

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.

Study of the instability of the Poiseuille flow using a thermodynamic formalism

Significance The stability of Poiseuille flow is among the most classical problems in fluid mechanics going back to the pioneering work of Reynolds. Despite a huge amount of effort, particularly recent advances in experimental studies, we still lack a suitable theoretical framework. In this paper, we develop a thermodynamic formalism for studying this problem. We show that this formalism allows us to identify a unique critical Reynolds number at which the laminar flow loses stability. We also study the free energy, action, and other thermodynamic relations for the macroscopic quantities associated with the flow. The formalism developed here should be applicable to a large class of problems exhibiting subcritical instabilities. The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier–Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec≈2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

Analysis of Reynolds number scaling for viscous vortex reconnection

A theoretical analysis of viscous vortex reconnection is developed based on scale separation, and the Reynolds number, Re (= circulation/viscosity), scaling for the reconnection time Trec is derived. The scaling varies continuously as Re increases from Trec∼ Re −1 to Trec∼ Re −1/2. This theoretical prediction agrees well with direct numerical simulations by Garten et al. [J. Fluid Mech. 426, 1 (2001)]10.1017/S0022112000002251 and Hussain and Duraisamy [Phys. Fluids 23, 021701 (2011)]10.1063/1.3532039. Moreover, our analysis yields two Re’s, namely, a characteristic Re Re 0.75∈O102,O103 for the Trec∼ Re −0.75 scaling given by Hussain and Duraisamy and the critical Re Re c∼O104 for the transition after which the first reconnection is completed. For Re > Re c, a quiescent state follows, and then, a second reconnection may occur.

Constrained Large Eddy Simulation of Wall-Bounded Turbulent Flows

We present a novel simulation tool-constrained large eddy simulation (CLES), for numerical experiments on the wall-bounded turbulent flows. Different from the traditional large eddy simulation(LES) and the available hybrid RANS/LES approaches, the CLES method computes the whole flow domain by solving the LES equations with a Reynolds-stress-constrained (RSC) subgrid-scale (SGS) stress model in the near-wall region and a traditional SGS stress model in the rest.The CLES approach is validated by simulating the turbulent channel flow and flow around a circular cylinder. With the same grid resolutions, CLES can successfully simulate all these flow regimes as well as DES and other available methods. For the case of attached flows, CLES is able to eliminate the non-physical Log-Layer Mismatch problem in traditional hybrid RANS/LES methods successfully, and to predict mean velocity profile, turbulent stresses and skin friction coefficient more accurately compared with the DES. For the case of detached flows, the performance of CLES is comparable to DES.

Spectra and Mach number scaling in compressible homogeneous shear turbulence

The effects of Mach number on the spectra and statistics of stationary compressible homogeneous shear turbulence (HST) are studied using numerical simulations in a rectangular domain (Lx = 4π, Ly = Lz = 2π) at turbulent Mach numbers from 0.05 to 0.66 and Taylor Reynolds numbers from 40 to 100. Long-term simulation results show that a statistically stationary state is obtained and the flow meets the strong acoustic equilibrium assumption at Mt ≈ 0.4. The analysis of spectral properties indicates that velocity and pressure tend toward a Mach number scaling of Mt2 suggested by acoustic dynamics at Mt ≳ 0.3. As for one-point statistics, it is found that a Mt4 scaling predicted by pseudo-sound theory holds for normalized compressible kinetic energy, Kc/Ks, at the small turbulent Mach number Mt ≲ 0.1. Kc/Ks approaches a Mt2 scaling at relatively higher turbulent Mach numbers, which is consistent with the spectral results. The normalized compressible dissipation rate, ϵc/ϵs, is nearly independent of Taylor Reynolds number and exhibits the same Mt4 scaling at small turbulent Mach numbers. The root mean square values of pressure, density, and temperature of compressible HST show good agreement with the Mt2 scaling, with the coefficient approximately doubled as compared with the compressible isotropic turbulence.The effects of Mach number on the spectra and statistics of stationary compressible homogeneous shear turbulence (HST) are studied using numerical simulations in a rectangular domain (Lx = 4π, Ly = Lz = 2π) at turbulent Mach numbers from 0.05 to 0.66 and Taylor Reynolds numbers from 40 to 100. Long-term simulation results show that a statistically stationary state is obtained and the flow meets the strong acoustic equilibrium assumption at Mt ≈ 0.4. The analysis of spectral properties indicates that velocity and pressure tend toward a Mach number scaling of Mt2 suggested by acoustic dynamics at Mt ≳ 0.3. As for one-point statistics, it is found that a Mt4 scaling predicted by pseudo-sound theory holds for normalized compressible kinetic energy, Kc/Ks, at the small turbulent Mach number Mt ≲ 0.1. Kc/Ks approaches a Mt2 scaling at relatively higher turbulent Mach numbers, which is consistent with the spectral results. The normalized compressible dissipation rate, ϵc/ϵs, is nearly independent of Taylor Reyno...

A modified optimal LES model for highly compressible isotropic turbulence

An energy budget analysis and a posteriori tests of subgrid-scale (SGS) models for large eddy simulation (LES) of stationary highly compressible homogeneous isotropic turbulence are carried out at the turbulent Mach number Mt ranging from 0.4 to 1.0 and the Taylor Reynolds number Reλ ranging from 180 to 250. An energy budget analysis shows that the SGS stress τij and the SGS heat flux Qj are dominant terms in the current Mt and Reλ ranges, while other terms are significantly smaller than the divergence of the SGS heat flux Qj and can be neglected in LES. We perform LES of compressible isotropic turbulence by using several SGS models including a dynamic Smagorinsky model, a dynamic mixed model, and an optimal model. In addition, a modified optimal model is constructed based on the magnitude of the filtered strain-rate tensor |S|, inspired by the physical insight that the region of the large magnitude of the filtered strain-rate tensor plays a significant role in kinetic energy transfer. Spectra, statistics, and scaling of velocity and thermodynamic variables from LES are tested. The modified optimal model performs better than other models, especially for the spectrum of the compressible velocity component at relatively low turbulent Mach numbers and high Taylor Reynolds numbers. The probability density function and the structure functions of velocity and thermodynamic variables are further studied, demonstrating that the statistical properties of the simulated flows are improved by the modified optimal model.An energy budget analysis and a posteriori tests of subgrid-scale (SGS) models for large eddy simulation (LES) of stationary highly compressible homogeneous isotropic turbulence are carried out at the turbulent Mach number Mt ranging from 0.4 to 1.0 and the Taylor Reynolds number Reλ ranging from 180 to 250. An energy budget analysis shows that the SGS stress τij and the SGS heat flux Qj are dominant terms in the current Mt and Reλ ranges, while other terms are significantly smaller than the divergence of the SGS heat flux Qj and can be neglected in LES. We perform LES of compressible isotropic turbulence by using several SGS models including a dynamic Smagorinsky model, a dynamic mixed model, and an optimal model. In addition, a modified optimal model is constructed based on the magnitude of the filtered strain-rate tensor |S|, inspired by the physical insight that the region of the large magnitude of the filtered strain-rate tensor plays a significant role in kinetic energy transfer. Spectra, statistic...