A new phase-screen method for electromagnetic wave propagation in turbulent flows using large-eddy simulation

Use of large-eddy simulation (LES) data in electromagnetic wave propagation modeling is not very common because of the high computational cost involved. A new phase-screen method is proposed to model radio wave propagation, in the atmospheric turbulence, using the resolved scale refractivity obtained from LES. The proposed method offers the same level of accuracy, as the one already existing in the literature, at much cheaper cost.     

Suitability of artificial bulk viscosity for large-eddy simulation of turbulent flows with shocks

The artificial bulk viscosity method to numerically capture shocks is investigated for large-eddy simulation (LES). Different variations of this method are tested on a turbulent flow over a cylinder at Reynolds number of 10,000 and free-stream Mach number of 0.85. The artificial bulk viscosity model by Cook and Cabot, which is parameterized by the strain rate magnitude, is found to provide unnecessary bulk viscosity in turbulent regions away from shocks. While developed turbulent structures are found unaffected, this extra bulk viscosity is shown to significantly damp the sound field. An alternative formulation of the model which is parameterized by the rate of dilatation is proposed. This formulation is shown to avoid the unnecessary bulk viscosity and enhance the sound-prediction capability of the model. It was found that standard LES combined with artificial bulk viscosity is a promising approach for simulation of turbulent flows with shocks. The formulation of the model on curvilinear coordinates is presented in the appendix.     

Large-eddy simulation of turbulent channel flows with conservative IDO scheme

The resolution of a numerical scheme in both physical and Fourier spaces is one of the most important requirements to calculate turbulent flows. A conservative form of the interpolated differential operator (IDO-CF) scheme is a multi-moment Eulerian scheme in which point values and integrated average values are separately defined in one cell. Since the IDO-CF scheme using high-order interpolation functions is constructed with compact stencils, the boundary conditions are able to be treated as easy as the 2nd-order finite difference method (FDM). It is unique that the first-order spatial derivative of the point value is derived from the interpolation function with 4th-order accuracy and the volume averaged value is based on the exact finite volume formulation, so that the IDO-CF scheme has higher spectral resolution than conventional FDMs with 4th-order accuracy. The computational cost to calculate the first-order spatial derivative with non-uniform grid spacing is one-third of the 4th-order FDM. For a large-eddy simulation (LES), we use the coherent structure model (CSM) in which the model coefficient is locally obtained from a turbulent structure extracted from a second invariant of the velocity gradient tensor, and the model coefficient correctly satisfies asymptotic behaviors to walls.     

A-priori evaluations of subgrid-scale terms for large-eddy simulation of compressible turbulent flows

The subgrid-scale terms for different formulations of the energy equation are evaluated from a-priori tests using the direct numerical simulation (DNS) data of a compressible mixing layer at a moderate Mach number of M = 0.65. To extend the generality of the results, the simulations were performed with three different initial conditions for the velocity fields. To examine the impact of strong temperature variations on the subgrid scales, a non-isothermal mixing layer with lower to upper free-stream temperature ratio of 3 is also considered. For cold simulations, with equal free-stream temperatures, the total energy equation is shown to be the best choice in view of the accuracy and the subgrid-scale modelling requirements. For hot simulations, with the free-stream temperature ratio equal to 3, the total enthalpy equation is found to be the best formulation for the energy equation. Furthermore, it is shown that the subgrid-scale pressure dilatation term, which has been largely neglected so far, is of the same order of the subgrid-scale heat flux. Based on the present results, the contribution of the subgrid-scale pressure dilatation can be up to 46% of the total sugbrid-scale activity. Moreover, the time evolutions of the volume-average mean kinetic energy, turbulent kinetic energy, production, dissipation, and pressure dilatation terms are considered. Unlike the subgrid-scale pressure dilatation term, the volume-average pressure dilatation terms are negligible, and compressibility does not affect the large-scale evolutions of the mean and turbulent kinetic energies.     

High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition

The direct numerical simulation of receptivity, instability and transition of hypersonic boundary layers requires high-order accurate schemes because lower-order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields. The main limiting factor in the application of high-order schemes to practical boundary-layer flow problems is the numerical instability of high-order boundary closure schemes on the wall. This paper presents a family of high-order non-uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary-layer transition. By using an appropriate grid stretching, and clustering grid points near the boundary, high-order schemes with stable boundary closures can be obtained. The order of the schemes ranges from first-order at the lowest, to the global spectral collocation method at the highest. The accuracy and stability of the new high-order numerical schemes is tested by numerical simulations of the linear wave equation and two-dimensional incompressible flat plate boundary layer flows. The high-order non-uniform-grid schemes (up to the 11th-order) are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge. The steady and unsteady results show that the new high-order schemes are stable and are able to produce high accuracy for computations of the nonlinear two-dimensional Navier–Stokes equations for the wall bounded supersonic flow.     

A wavelet-based multiresolution approach to large-eddy simulation of turbulence

The wavelet-based multiresolution analysis (MRA) technique is used to develop a modelling approach to large-eddy simulation (LES) and its associated subgrid closure problem. The LES equations are derived by projecting the Navier–Stokes (N–S) equations onto a hierarchy of wavelet spaces. A numerical framework is then developed for the solution of the large and the small-scale equations. This is done in one dimension, for the Burgers equation, and in three dimensions, for the N–S problem. The proposed methodology is assessed in a priori tests on an atmospheric turbulent time series and on data from direct numerical simulation. A posteriori (dynamic) tests are also carried out for decaying and force-driven Burgers turbulence.     

On the application of the two-level large-eddy simulation method to turbulent free-shear and wake flows

We present application of the hybrid two-level large-eddy simulation (TLS-LES) method, a multi-scale simulation model, to turbulent free-shear and wake flows at moderately high Reynolds number. The TLS-LES method combines the scale-separation-based two-level simulation (TLS) model with the spatial-filtering-based conventional large-eddy simulation (LES) model in an additive manner using a normalised blending function. The additive blending can be performed in a static or a dynamic manner. We demonstrate that the method, which has been originally developed for wall-bounded flows, can be used to simulate flows in complex configurations without requiring any further adjustments to the model. In this study, three canonical flows are simulated, which are representative of free-shear and wake flows. These cases include a temporally evolving mixing layer, flow past a circular cylinder in a uniform flow and flow past a finite-span airfoil placed in a uniform flow at three different angle of attacks. We analyse the role of static and dynamic blending functions, large-scale grid resolution and the effect of small scales on the instantaneous flow features and turbulence statistics. The results obtained from these cases demonstrate robustness, accuracy and consistency of the multi-scale TLS-LES method and show that the method is suitable for investigation of turbulent flows that encompass features such as massive separation, reattachment, transition to turbulence and unsteady wake, which are challenging to model numerically.     

Effects of buoyancy on turbulent premixed V-flames by large-eddy simulation

Buoyancy effects on turbulent premixed V-flames are investigated under normal gravity (+g) and reversed gravity (–g). Numerical simulations employ large eddy simulation (LES) with a dynamic model for sub-grid scale stress. With the assumption of fast chemistry combustion, a progress variable c-equation is applied to describe the flame front propagation. The equations are solved using a projection-based fractional step method in two dimensions for low-Mach number flows. Computed LES results of buoyancy effects on flame angle and flame brush thickness are consistent with those obtained from experiments. In both +g and –g conditions, the effects of buoyancy become important with increase in Richardson number (Ri). Buoyancy force tends to close up the flame under +g, but has the opposite effect under –g. Buoyancy force also suppresses flame wrinkling in +g and enhances wrinkling in –g. While there is a lack of experimental data available, computed axial velocity is shown to be significantly affected by buoyancy downstream from the flame holder under moderate Reynolds number.     

Construction of explicit and implicit dynamic finite difference schemes and application to the large-eddy simulation of the Taylor–Green vortex

A general class of explicit and implicit dynamic finite difference schemes for large-eddy simulation is constructed, by combining Taylor series expansions on two different grid resolutions. After calibration for Re→∞, the dynamic finite difference schemes allow to minimize the dispersion errors during the calculation through the real-time adaption of a dynamic coefficient. In case of DNS resolution, these dynamic schemes reduce to Taylor-based finite difference schemes with formal asymptotic order of accuracy, whereas for LES resolution, the schemes adapt to Dispersion-Relation Preserving schemes. Both the explicit and implicit dynamic finite difference schemes are tested for the large-eddy simulation of the Taylor–Green vortex flow and numerical errors are investigated as well as their interaction with the dynamic Smagorinsky model and the multiscale Smagorinsky model. Very good results are obtained.     

Stabilization mechanism of turbulent premixed flames in strongly swirled flows

The stabilization of turbulent premixed flames in strongly swirled flows undergoing vortex breakdown is studied in the case of the ALSTOM En-Vironmental (EV) double cone burner using a simple one-dimensional boundary layer type model and computational fluid dynamics, mainly at the level of large-eddy simulation. The analysis shows that, due to flame curvature effects, the flame speed on the combustor axis is 2 D _t/R _F lower than the turbulent burning rate, where D _t is a characteristic turbulent diffusion coefficient and R _F the flame radius of curvature. Flame propagation with negative speed observed in the experiments, i.e. the flame completely embedded in the central recirculation zone on the symmetry axis, is explained with the one-dimensional model as caused by the factor 2 D _t/R _F being larger than the characteristic turbulent burning rate. A peculiar sudden displacement of the flame anchoring location deep into the burner, which takes place experimentally at a critical value of the equivalence ratio, cannot however be explained with the present one-dimensional approach due to the modelling assumptions. The mathematical analysis is supported in this case with large-eddy simulation which can accurately reproduce the flame behaviour across the full operating range. It is finally shown that steady RANS methods cannot cope with the problem due to their inability to correctly predict the velocity flowfield in this burner.     

Estimation of the Kolmogorov constant by large-eddy simulation in the stable PBL

In this paper is evaluated the inertial subrange Kolmogorov constant C₀ in a stable boundary-layer. The importance of the constant C₀ is well known as predictions of turbulent dispersion by means of Lagrangian stochastic models depend upon its value. Different values of the C₀ constant has been proposed along the years, most of them determined at low Reynolds number and/or under different techniques. Here we estimate the constant C₀ by tracking an ensemble of Lagrangian particles injected in a stable planetary boundary-layer simulated with a large-eddy simulation model and analyzing the ensemble-averaged Lagrangian velocity structure function in the inertial subrange. Our estimative of C_0,w is 3.7, which is consistent with values found in literature. The evaluation of C_0,u and C_0,v cannot be easily accomplished since it is difficult to identify an inertial subrange for the wind field horizontal components.     

The role of a split injection strategy in the mixture formation and combustion of diesel spray: A large-eddy simulation

The role of a split injection in the mixture formation and combustion characteristics of a diesel spray in an engine-like condition is investigated. We use large-eddy simulations with finite rate chemistry in order to identify the main controlling mechanism that can potentially improve the mixture quality and reduces the combustion emissions. It is shown that the primary effect of the split injection is the reduction of the mass of the fuel-rich region where soot precursors can form.Furthermore, we investigate the interaction between different injections and explain the effects of the first injection on the mixing and combustion of the second injection. Results show that the penetration of the second injection is faster than that of the first injection. More importantly, it is shown that the ignition delay time of the second injection is much shorter than that of the first injection. This is due to the residual effects of the ignition of the first injection which increases the local temperature and maintains a certain level of combustion some intermediates or radical which in turn boosts the ignition of the second injection.     

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.     

Large eddy simulation of orientation and rotation of ellipsoidal particles in isotropic turbulent flows

The rotational motion and orientational distribution of ellipsoidal particles in turbulent flows are of significance in environmental and engineering applications. Whereas the translational motion of an ellipsoidal particle is controlled by the turbulent motions at large scales, its rotational motion is determined by the fluid velocity gradient tensor at small scales, which raises a challenge when predicting the rotational dispersion of ellipsoidal particles using large eddy simulation (LES) method due to the lack of subgrid scale (SGS) fluid motions. We report the effects of the SGS fluid motions on the orientational and rotational statistics, such as the alignment between the long axis of ellipsoidal particles and the vorticity, the mean rotational energy at various aspect ratios against those obtained with direct numerical simulation (DNS) and filtered DNS. The performances of a stochastic differential equation (SDE) model for the SGS velocity gradient seen by the particles and the approximate deconvolution method (ADM) for LES are investigated. It is found that the missing SGS fluid motions in LES flow fields have significant effects on the rotational statistics of ellipsoidal particles. Alignment between the particles and the vorticity is weakened; and the rotational energy of the particles is reduced in LES. The SGS-SDE model leads to a large error in predicting the alignment between the particles and the vorticity and over-predicts the rotational energy of rod-like particles. The ADM significantly improves the rotational energy prediction of particles in LES.     

An implicit high-order spectral difference approach for large eddy simulation

The filtered fluid dynamic equations are discretized in space by a high-order spectral difference (SD) method coupled with large eddy simulation (LES) approach. The subgrid-scale stress tensor is modelled by the wall-adapting local eddy-viscosity model (WALE). We solve the unsteady equations by advancing in time using a second-order backward difference formulae (BDF2) scheme. The nonlinear algebraic system arising from the time discretization is solved with the nonlinear lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. In order to study the sensitivity of the method, first, the implicit solver is used to compute the two-dimensional (2D) laminar flow around a NACA0012 airfoil at Re = 5 × 10⁵ with zero angle of attack. Afterwards, the accuracy and the reliability of the solver are tested by solving the 2D “turbulent” flow around a square cylinder at Re = 10⁴ and Re =  2.2 × 10⁴. The results show a good agreement with the experimental data and the reference solutions.     

Vorticity,backscatter and counter-gradient transport predictions using two-level simulation of turbulent flows

The two-level simulation (TLS) method evolves both the large-and the small-scale fields in a two-scale approach and has shown good predictive capabilities in both isotropic and wall-bounded high Reynolds number (Re) turbulent flows in the past. Sensitivity and ability of this modelling approach to predict fundamental features (such as backscatter, counter-gradient turbulent transport, small-scale vorticity, etc.) seen in high Re turbulent flows is assessed here by using two direct numerical simulation (DNS) datasets corresponding to a forced isotropic turbulence at Taylor’s microscale-based Reynolds number Re_λ ≈ 433 and a fully developed turbulent flow in a periodic channel at friction Reynolds number Re_τ ≈ 1000. It is shown that TLS captures the dynamics of local co-/counter-gradient transport and backscatter at the requisite scales of interest. These observations are further confirmed through a posteriori investigation of the flow in a periodic channel at Re_τ = 2000. The results reveal that the TLS method can capture both the large- and the small-scale flow physics in a consistent manner, and at a reduced overall cost when compared to the estimated DNS or wall-resolved LES cost.     

Modelling alkali metal emissions in large-eddy simulation of a preheated pulverised-coal turbulent jet flame using tabulated chemistry

The numerical modelling of alkali metal reacting dynamics in turbulent pulverised-coal combustion is discussed using tabulated sodium chemistry in large eddy simulation (LES). A lookup table is constructed from a detailed sodium chemistry mechanism including five sodium species, i.e. Na, NaO, NaO₂, NaOH and Na₂O₂H₂, and 24 elementary reactions. This sodium chemistry table contains four coordinates, i.e. the equivalence ratio, the mass fraction of the sodium element, the gas-phase temperature, and a progress variable. The table is first validated against the detailed sodium chemistry mechanism by zero-dimensional simulations. Then, LES of a turbulent pulverised-coal jet flame is performed and major coal-flame parameters compared against experiments. The chemical percolation devolatilisation (CPD) model and the partially stirred reactor (PaSR) model are employed to predict coal pyrolysis and gas-phase combustion, respectively. The response of the five sodium species in the pulverised-coal jet flame is subsequently examined. Finally, a systematic global sensitivity analysis of the sodium lookup table is performed and the accuracy of the proposed tabulated sodium chemistry approach has been calibrated.     

Conditional statistics of nonreacting and reacting sprays in turbulent flows by direct numerical simulation

Conditional statistics concerning evaporation and combustion of a spray are investigated in homogeneous, isotropic, and decaying two-dimensional (2D) turbulence. Randomly distributed, polydisperse droplets of n-heptane go through single-step combustion chemistry. Attention is focused on parametric effects of initial Sauter mean radius (SMR), turbulence level and droplet velocity in both reacting and nonreacting cases. A simple linear model for the conditional evaporation rate is proposed and validated against DNS data. A conventional β-probability density function (pdf) is shown to be valid with no peak occurring on the fuel side. The amplitude mapping closure (AMC) model works well for the conditional scalar dissipation rate with evaporating and reacting sprays. Parametric study shows that initial SMR and droplet velocity are major factors affecting conditional flame structures, whereas the effect of reaction is not significant except during autoignition.