Guillaume Blanquart

High order conservative finite difference scheme for variable density low Mach number turbulent flows

The high order conservative finite difference scheme of Morinishi et al. Y. Morinishi, O.V. Vasilyev, T. Ogi, Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations, J. Comput. Phys. 197 (2004) 686] is extended to simulate variable density flows in complex geometries with cylindrical or cartesian non-uniform meshes. The formulation discretely conserves mass, momentum, and kinetic energy in a periodic domain. In the presence of walls, boundary conditions that ensure primary conservation have been derived, while secondary conservation is shown to remain satisfactory. In the case of cylindrical coordinates, it is desirable to increase the order of accuracy of the convective term in the radial direction, where most gradients are often found. A straightforward centerline treatment is employed, leading to good accuracy as well as satisfactory robustness. A similar strategy is introduced to increase the order of accuracy of the viscous terms. The overall numerical scheme obtained is highly suitable for the simulation of reactive turbulent flows in realistic geometries, for it combines arbitrarily high order of accuracy, discrete conservation of mass, momentum, and energy with consistent boundary conditions. This numerical methodology is used to simulate a series of canonical turbulent flows ranging from isotropic turbulence to a variable density round jet. Both direct numerical simulation (DNS) and large eddy simulation (LES) results are presented. It is observed that higher order spatial accuracy can improve significantly the quality of the results. The error to cost ratio is analyzed in details for a few cases. The results suggest that high order schemes can be more computationally efficient than low order schemes.

Flux corrected finite volume scheme for preserving scalar boundedness in reacting large-eddy simulations

Preserving scalar boundedness is an important prerequisite to performing large-eddy simulations of turbulent reacting flows. A number of popular combustion models use a conserved-scalar, mixture-fraction to parameterize reactions that, by definition, is bound between zero and one. To avoid unphysical clipping, the numerical scheme solving the conserved-scalar transport equation must preserve these bounds, while minimizing the amount of numerical diffusivity. To this end, a flux correction method is presented and applied to the quadratic-upwind biased interpolative convective scheme that ensures preservation of the scalars physical bounds while retaining the low numerical diffusivity of the original quadratic-upwind biased interpolative convective scheme. It is demonstrated that this bounded quadratic-upwind biased interpolative convective scheme outperforms the third-order weighted essentially nonoscillatory scheme in maintaining spatial accuracy and reducing numerical dissipation errors both in generic test cases as well as direct numerical simulation of canonical flows.

A proposed modification to Lundgren's physical space velocity forcing method for isotropic turbulence

As an alternative to spectral space velocity field forcing techniques commonly used in simulation studies of isotropic turbulence,Lundgren [Linearly forced isotropic turbulence,” in Annual Research Briefs (Center for Turbulence Research, Stanford, 2003), pp. 461–473] proposed and Rosales and Meneveau [“Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties,” Phys. Fluids17, 095106 (2005)] validated a physical space forcing method termed “linear forcing.” Linear forcing has the advantages of being less memory intensive, less computationally expensive, and more easily extended to variable density simulations. However, this forcing method generates turbulent statistics that are highly oscillatory, requiring extended simulation run times to attain time-invariant properties. A slight modification of the forcing term is proposed, and it is shown to reduce this oscillatory nature without altering the turbulent physics.

A computationally-efficient, semi-implicit, iterative method for the time-integration of reacting flows with stiff chemistry

A semi-implicit preconditioned iterative method is proposed for the time-integration of the stiff chemistry in simulations of unsteady reacting flows, such as turbulent flames, using detailed chemical kinetic mechanisms. Emphasis is placed on the simultaneous treatment of convection, diffusion, and chemistry, without using operator splitting techniques. The preconditioner corresponds to an approximation of the diagonal of the chemical Jacobian. Upon convergence of the sub-iterations, the fully-implicit, second-order time-accurate, Crank-Nicolson formulation is recovered. Performance of the proposed method is tested theoretically and numerically on one-dimensional laminar and three-dimensional high Karlovitz turbulent premixed n-heptane/air flames. The species lifetimes contained in the diagonal preconditioner are found to capture all critical small chemical timescales, such that the largest stable time step size for the simulation of the turbulent flame with the proposed method is limited by the convective CFL, rather than chemistry. The theoretical and numerical stability limits are in good agreement and are independent of the number of sub-iterations. The results indicate that the overall procedure is second-order accurate in time, free of lagging errors, and the cost per iteration is similar to that of an explicit time integration. The theoretical analysis is extended to a wide range of flames (premixed and non-premixed), unburnt conditions, fuels, and chemical mechanisms. In all cases, the proposed method is found (theoretically) to be stable and to provide good convergence rate for the sub-iterations up to a time step size larger than 1 µs. This makes the proposed method ideal for the simulation of turbulent flames.

Vorticity transformation in high Karlovitz number premixed flames

To better understand the two-way coupling between turbulence and chemistry, the changes in turbulence characteristics through a premixed flame are investigated. Specifically, this study focuses on vorticity, ω, which is characteristic of the smallest length and time scales of turbulence, analyzing its behavior within and across high Karlovitz number (Ka) premixed flames. This is accomplished through a series of direct numerical simulations (DNS) of premixed n-heptane/air flames, modeled with a 35-species finite-rate chemical mechanism, whose conditions span a wide range of unburnt Karlovitz numbers and flame density ratios. The behavior of the terms in the enstrophy, ω^2 = ω ⋅ ω, transport equation is analyzed, and a scaling is proposed for each term. The resulting normalized enstrophy transport equation involves only a small set of parameters. Specifically, the theoretical analysis and DNS results support that, at high Karlovitz number, enstrophy transport obtains a balance of the viscous dissipation and production/vortex stretching terms. It is shown that, as a result, vorticity scales in the same manner as in homogeneous, isotropic turbulence within and across the flame, namely, scaling with the inverse of the Kolmogorov time scale, τ_η. As τ_η is a function only of the viscosity and dissipation rate, this work supports the validity of Kolmogorov’s first similarity hypothesis in premixed turbulentflames for sufficiently high Ka numbers. Results are unaffected by the transport model, chemical model, turbulent Reynolds number, and finally the physical configuration.

Assessment of the constant non-unity Lewis number assumption in chemically-reacting flows

Accurate computation of molecular diffusion coefficients in chemically reacting flows can be an expensive procedure, and the use of constant non-unity Lewis numbers has been adopted often as a cheaper alternative. The goal of the current work is to explore the validity and the limitations of the constant non-unity Lewis number approach in the description of molecular mixing in laminar and turbulent flames. To carry out this analysis, three test cases have been selected, including a lean, highly unstable, premixed hydrogen/air flame, a lean turbulent premixed n-heptane/air flame, and a laminar ethylene/air coflow diffusion flame. For the hydrogen flame, both a laminar and a turbulent configuration have been considered. The three flames are characterised by Lewis numbers which are less than unity, greater than unity, and close to unity, respectively. For each flame, mixture-averaged transport simulations are carried out and used as reference data. The current analysis suggests that, for numerous combustion configurations, the constant non-unity Lewis number approximation leads to small errors when the set of Lewis numbers is chosen properly. For the selected test cases and our numerical framework, the reduction of computational cost is found to be minimal.

The effect of velocity field forcing techniques on the Karman–Howarth equation

Many velocity field forcing methods exist to simulate isotropic turbulence, but no in-depth analysis of the effects that these methods have on the generated turbulence has been performed. This work contains such a detailed study. It focuses on Lundgren’s linear and Alvelius’ spectral velocity forcing methods. Based on the constraints imposed on their associated forcing terms, these two are representative of the numerous forcing methods available in the literature. This study is conducted in the context of the Karman–Howarth equation, which, as a consequence of velocity forcing, has a source term appended to it. The expressions for the forcing method-specific Karman–Howarth source terms are derived, and their effect on key turbulent metrics, e.g. structure functions and spectra, is investigated. The behaviour of these source terms determines the state to which all turbulent metrics evolve, allowing for the differences noted between linearly and spectrally forced turbulent fields to be explained.

An Improved Bounded Semi-Lagrangian Scheme for the Turbulent Transport of Passive Scalars

An improved bounded semi-Lagrangian scalar transport scheme based on cubic Hermite polynomial reconstruction is proposed in this paper. Boundedness of the scalar being transported is ensured by applying derivative limiting techniques. Single sub-cell extrema are allowed to exist as they are often physical, and help minimize numerical dissipation. This treatment is distinct from enforcing strict monotonicity as done by D.L. Williamson and P.J. Rasch [5], and allows better preservation of small scale structures in turbulent simulations. The proposed bounding algorithm, although a seemingly subtle difference from strict monotonicity enforcement, is shown to result in significant performance gain in laminar cases, and in three-dimensional turbulent mixing layers. The scheme satisfies several important properties, including boundedness, low numerical diffusion, and high accuracy. Performance gain in the turbulent case is assessed by comparing scalar energy and dissipation spectra produced by several bounded and unbounded schemes. The results indicate that the proposed scheme is capable of furnishing extremely accurate results, with less severe resolution requirements than all the other bounded schemes tested. Additional simulations in homogeneous isotropic turbulence, with scalar timestep size unconstrained by the CFL number, show good agreement with spectral scheme results available in the literature. Detailed analytical examination of gain and phase error characteristics of the original cubic Hermite polynomial is also included, and points to dissipation and dispersion characteristics comparable to, or better than, those of a fifth order upwind Eulerian scheme.

Effect of a splitter plate on the dynamics of a vortex pair

An experimental and numerical study was performed to investigate and compare the behavior of a counter-rotating vortex pair and a single vortex in the vicinity of a wall. This analysis is motivated by the theoretical equivalence, in the inviscid limit, between these two configurations. A wind tunnel with two NACA0012 profiles mounted vertically with an optional splitter plate in the center and a stereoscopic particle image velocimetry system was used to experimentally study these interactions. Many significant differences were found between the two configurations, including the growth of the Crow instability in the two vortex configuration, but not in the one vortex/wall configuration. The numerical results re-enforced the experimental results, and emphasized the fundamental physical differences between the two configurations. While modeling a vortex wall system with an image vortex may give correct integral results for loads experienced by blades, this model does not accurately describe the downstream dynamics of the vortex system.

A novel forcing technique to simulate turbulent mixing in a decaying scalar field

To realize the full potential of Direct Numerical Simulation in turbulent mixing studies, it is necessary to develop numerical schemes capable of sustaining the flow physics of turbulent scalar quantities. In this work, a new scalar field forcing technique, termed “linear scalar forcing,” is presented and evaluated for passive scalars. It is compared to both the well-known mean scalar gradient forcing technique and a low waveshell spectral forcing technique. The proposed forcing is designed to capture the physics of one-time scalar variance injection and the subsequent self-similar turbulent scalar field decay, whereas the mean scalar gradient forcing and low waveshell forcing techniques are representative of continuous scalar variance injection. The linear scalar forcing technique is examined over a range of Schmidt numbers, and the behavior of the proposed scalar forcing is analyzed using single and two-point statistics. The proposed scalar forcing technique is found to be perfectly isotropic, preserving accepted scalar field statistics (fluxes) and distributions (scalar quantity, dissipation rate). Additionally, it is found that the spectra resulting from the three scalar forcing techniques are comparable for unity Schmidt number conditions, but differences manifest at high Schmidt numbers. These disparities are reminiscent of those reported between scaling arguments suggested by theoretical predictions and experimental results for the viscous-convective subrange.