A robust, colocated, implicit algorithm for direct numerical simulation of compressible, turbulent flows

High-order gas-kinetic scheme with parallel computation for direct numerical simulation of turbulent flows

The direct numerical simulation (DNS) offers useful information about the understanding and modeling of turbulent flow. Few direct numerical simulations of wall-bounded compressible turbulent flows have been performed however. The objective of our study is to construct the DNS algorithm which can simulate the compressible turbulent flow between adiabatic and isothermal walls. The flow is selected since it is the simplest turbulent flow with an adiabatic wall, and therefore it is ideal for the modeling of compressible turbulent flow near the adiabatic wall. From the applicability and efficiency, we construct the new DNS algorithm based on the B-spline collocation method in the wall-normal direction. The validity and efficiency of the DNS algorithm are confirmed by comparing our results with those of an existing DNS in the compressible turbulent flow between isothermal walls. Then the DNS of compressible turbulent flow between the adiabatic and isothermal walls exhibits the applicability and usefulness of the algorithm.

Data-parallel DNS of tubrulent flow

This paper presents a dynamic one-equation eddy viscosity model for large-eddy simulation (LES) of compressible flows. The transport equation for subgrid-scale (SGS) kinetic energy is introduced to predict SGS kinetic energy. The exact SGS kinetic energy transport equation for compressible flows is derived formally. Each of the unclosed terms in the SGS kinetic energy equation is modelled separately and dynamically closed, instead of being grouped into production and dissipation terms, as in the Reynolds averaged Navier–Stokes equations. All of the SGS terms in the filtered total energy equation are found to reappear in the SGS kinetic energy equation. Therefore, these terms can be included in the total energy equation without adding extra computational cost.A prioritests using direct numerical simulation (DNS) of decaying isotropic turbulence show that, for a Smagorinsky-type eddy viscosity model, the correlation between the SGS stress and the model is comparable to that from the original model. Also, the suggested model for the pressure dilatation term in the SGS kinetic energy equation is found to have a high correlation with its actual value. Ina posterioritests, the proposed dynamic$k$-equation model is applied to decaying isotropic turbulence and normal shock–isotropic turbulence interaction, and yields good agreement with available experimental and DNS data. Compared with the results of the dynamic Smagorinsky model (DSM), the$k$-equation model predicts better energy spectra at high wavenumbers, similar kinetic energy decay and fluctuations of thermodynamic quantities for decaying isotropic turbulence. For shock–turbulence interaction, the$k$-equation model and the DSM predict similar evolutions of turbulent intensities across shocks, owing to the dominant effect of linear interaction. The proposed$k$-equation model is more robust in that local averaging over neighbouring control volumes is sufficient to regularize the dynamic procedure. The behaviour of pressure dilatation and dilatational dissipation is discussed through the budgets of the SGS kinetic energy equation, and the importance of the dilatational dissipation term is addressed.

Time irreversibility of compressible homogeneous isotropic turbulence (HIT) is investigated from a Lagrangian point of view and single-particle statistics. For this purpose, direct numerical simulation (DNS) is implemented for compressible HIT at Taylor-mircoscale Reynolds number Reλ∼100 and turbulent Mach number Mt up to 1.01, in which tracers and inertial particles in a wide Stokes number (St) range are instantaneously tracked with time. The statistics of instantaneous power of particles corroborates that the violation of detailed balance of turbulence in compressible HIT is much stronger than in incompressible HIT. It turns out that the third moment of dimensionless instantaneous power (Ir) of tracers scales as Mt4. A possible explanation based on Kolmogorov-like argument proves to be plausible due to the lacks of direct verification and generality for other moments of the power. A further analysis from an Eulerian point of view suggests that the underlying mechanism for time irreversibility of highly compressible turbulence is quite different from that for weakly compressible or incompressible turbulence. For inertial particles, the moments of instantaneous power are suggested to scale as St−2n/3 at relatively large St numbers, and be dependent only on Mach number or Reynolds number at the small-St number end, which are manifested by the present numerical data. It is further shown that the empirical Mt4 scaling of Ir also approximately applies to various inertial particles, but only at high Mt numbers due to the diminishing effect of compressibility to the low-Mach number end. The time irreversibility announced by the Lagrangian statistical properties of particles of different inertias is shown to be highly associated with their responses to the vortex and shocklet structures in compressible turbulence.

The present paper addresses some topical issues in modelling compressible turbulent shear flows. The work is based on direct numerical simulation (DNS) of two supersonic fully developed channel flows between very cold isothermal walls. Detailed decomposition and analysis of terms appearing in the mean momentum and energy equations are presented. The simulation results are used to provide insights into differences between conventional Reynolds and Favre averaging of the mean-flow and turbulent quantities. Study of the turbulence energy budget for the two cases shows that compressibility effects due to turbulent density and pressure fluctuations are insignificant. In particular, the dilatational dissipation and the mean product of the pressure and dilatation fluctuations are very small, contrary to the results of simulations for sheared homogeneous compressible turbulence and to recent proposals for models for general compressible turbulent flows. This provides a possible explanation of why the Van Driest density-weighted transformation (which ignores any true turbulent compressibility effects) is so successful in correlating compressible boundary-layer data. Finally, it is found that the DNS data do not support the strong Reynolds analogy. A more general representation of the analogy is analysed and shown to match the DNS data very well.

For the unsteady simulation of compressible subsonic flows (Large Eddy Simulation or Direct Numerical Simulation), the proper handling of the inlet boundary is a challenging task. Indeed, inflow generation through imposition of the velocity may lead to a non-physical reflection of the upstream acoustic waves. In the present contribution, a method that allows both filtering of these waves and proper imposition of the required variables is proposed. This method is based on identification of the roles of the temporal rate of change of wave amplitudes at the inlet in the low Mach number regime. The formulation obtained is tested numerically on unsteady one-dimensional flows at low Mach number for which the unsteady inlet velocity signal is purely harmonic or harmonic with the superimposition of synthetic turbulence.

Two-dimensional compressible turbulent flow simulations are performed for the open cavities (L/D=0.787 and 2.36) with an EASM (Explicit Algebraic Stress Model) second-moment turbulence closure model. For direct numerical simulations of the flow-sound wave interactions, the full compressible Navier-Stokes equations are solved with a sixth-order compact scheme and a fourthorder Runge-Kutta method. The present study is focused on prediction of the cavity tones at low subsonic Mach numbers (0.1 ≤ M ≤ 0.4), for which a mode shifting from an acoustic feedback to a depth mode occurs. In this study, the present URANS-based DaNS(Direct Acoustic Numerical Simulation) results are validated with the experimental data and theories by comparing the Strouhal numbers of the cavity tones, and the vortical structures of the turbulent shear layer are studied for each mode with considering the effect of cavity length to depth ratio.

Computational simulation for analysis and synthesis of impact resilient structure

A database of transitional direct numerical simulation (DNS) realizations of a supercritical mixing layer is analysed for understanding small-scale behaviour and examining subgrid-scale (SGS) models duplicating that behaviour. Initially, the mixing layer contains a single chemical species in each of the two streams, and a perturbation promotes roll-up and a double pairing of the four spanwise vortices initially present. The database encompasses three combinations of chemical species, several perturbation wavelengths and amplitudes, and several initial Reynolds numbers specifically chosen for the sole purpose of achieving transition. The DNS equations are the Navier-Stokes, total energy and species equations coupled to a real-gas equation of state; the fluxes of species and heat include the Soret and Dufour effects. The large-eddy simulation (LES) equations are derived from the DNS ones through filtering. Compared to the DNS equations, two types of additional terms are identified in the LES equations: SGS fluxes and other terms for which either assumptions or models are necessary. The magnitude of all terms in the LES conservation equations is analysed on the DNS database, with special attention to terms that could possibly be neglected. It is shown that in contrast to atmospheric-pressure gaseous flows, there are two new terms that must be modelled: one in each of the momentum and the energy equations. These new terms can be thought to result from the filtering of the nonlinear equation of state, and are associated with regions of high density-gradient magnitude both found in DNS and observed experimentally in fully turbulent high-pressure flows. A model is derived for the momentum-equation additional term that performs well at small filter size but deteriorates as the filter size increases, highlighting the necessity of ensuring appropriate grid resolution in LES. Modelling approaches for the energy-equation additional term are proposed, all of which may be too computationally intensive in LES. Several SGS flux models are tested on an a priori basis. The Smagorinsky (SM) model has a poor correlation with the data, while the gradient (GR) and scale-similarity (SS) models have high correlations. Calibrated model coefficients for the GR and SS models yield good agreement with the SGS fluxes, although statistically, the coefficients are not valid over all realizations. The GR model is also tested for the variances entering the calculation of the new terms in the momentum and energy equations; high correlations are obtained, although the calibrated coefficients are not statistically significant over the entire database at fixed filter size. As a manifestation of the small-scale supercritical mixing peculiarities, both scalar-dissipation visualizations and the scalar-dissipation probability density functions (PDF) are examined. The PDF is shown to exhibit minor peaks, with particular significance for those at larger scalar dissipation values than the mean, thus significantly departing from the Gaussian behaviour.

High-resolution finite difference schemes using curvilinear coordinate grids for DNS of compressible turbulent flow over wavy walls

Direct Numerical Simulation (DNS) and linear analysis of a shock interacting with incompressible and compressible isotropic turbulence is conducted. A dependence of amplification ratios on the degree of compressibility of the incoming flow is found. It can be shown that the enhancement of r.m.s.-values of turbulent quantities across the shock varies according to the ratio of compressible to incompressible kinetic energy χ. Inflow conditions with high values of χ display reduced amplification ratios of TKE and thermodynamic quantities while vorticity fluctuations are enhanced more strongly. The different behaviour of the TKE is due to the reduced pressure diffusion term in the TKE-equation. Experiments show qualitatively a similar behaviour as the simulation with incompressible inflow conditions, but they could so far not confirm our findings of reduced amplification rates in the compressible case, one of the reasons being the lack of knowledge of all flow parameters upstream of the shock front and the inability to generate isotropic turbulence in real life experiments. For the DNS we use a third order in space shock-capturing scheme based on the ENO algorithm of Harten [10] together with an approximate Riemann solver. This non-TVD scheme turned out to have many advantages over other common Godunov-type high resolution schemes for the specific problem of a shock interacting with turbulent fields.

Features of the simulation of low-velocity inviscid and viscous compressible gas flows are considered, and a finite-volume discretization of gas dynamics equations at low Mach numbers on unstructured meshes is discussed. Preconditioning based on the use of physical variables is used to speed up the convergence of time marching to a steady state and to improve the accuracy of the steady-state solution. The structure of the preconditioning matrix and the diagonalization of the Jacobian of the preconditioned system of equations are discussed. The capabilities of this approach are demonstrated using model gasdynamic simulations ing a wide range of Mach numbers.

The dynamic subgrid-scale (SGS) model of Germano et al. [Phys. Fluids A 3, 1760 (1991)] is generalized for the large eddy simulation (LES) of compressible flows and transport of a scalar. The model was applied to the LES of decaying isotropic turbulence, and the results are in excellent agreement with experimental data and direct numerical simulations. The expression for the SGS turbulent Prandtl number was evaluated using direct numerical simulation (DNS) data in isotropic turbulence, homogeneous shear flow, and turbulent channel flow. The qualitative behavior of the model for turbulent Prandtl number and its dependence on molecular Prandtl number, direction of scalar gradient, and distance from the wall are in accordance with the total turbulent Prandtl number from the DNS data.

Many electrically and heat conducting fluid flows cannot be described within the framework of incompressible medium or by compressible magnetohydrodynamic equations on the assumption of polytropic (or adiabatic) process. Therefore, we consider a heat conducting compressible fluid with the use of an energy equation. Application of large eddy simulation approach to heat conducting compressible magnetohydrodynamics is considered. The system of the filtered magnetohydrodynamic equations with the total energy equation using the mass-weighted filtering procedure has been obtained. It is shown that novel subgrid-scale terms arise in the Favre-filtered equations due to the presence of a magnetic field in the total energy equation. Parametrizations of these extra terms are developed. In order to derive these subgrid-scale terms, we use an approach based on generalized central moments. Computations at various Mach numbers are made for decaying compressible magnetohydrodynamic turbulence. The obtained numerical large eddy simulation results are analyzed on the basis of comparison with results of numerical experiments performed by direct numerical simulation. Validity of large eddy simulation method is thus demonstrated. It is shown that consideration of the subgrid-scale terms in the total energy equation scarcely affects the kinetic and the magnetic energy for low and even high Mach number, while for the temperature the presence of subgrid-scale models in the energy equation is an important condition for improvement in calculation accuracy of thermodynamic quantities. The technique with the mass-weighted filtering and with the use of various types of subgrid-scale models provides good calculation accuracy for different problems for compressible fluid in the absence of discontinuities, associated with the appearance of shocks, in other words, when the value of the Mach number is low or moderate (that is, the flow is subsonic). For supersonic magnetohydrodynamic flows, it is necessary to use special numerical methods.