Numerical prediction of isothermally reacting mixing layer using vortex‐in‐cell and filtered density function

A large eddy simulation based on the filtered vorticity transport equation and the filtered density function (FDF) transport equation developed in an earlier study is extended to predict a chemically reacting flow with no heat release. The filtered vorticity transport equation is solved using the vortex‐in‐cell scheme in conjunction with the dynamic eddy viscosity subgrid‐scale models. The transport equation for FDF is solved using the Lagrangian Monte‐Carlo method. The methodology is tested on a chemically reacting spatially growing mixing layer with no heat release. The effects of Damköhler number (Da) on the concentration structure of the reacting mixing layer, the mean reactant and product concentrations and on the reactant FDF are investigated. It is shown that mixing has a greater effect on scalar field within the vortex structure as compared with the braid regions. Also for high Da, the reaction zones are mainly limited to the thin reacting interfacial zones, i.e. the contact zone between the reactants, whereas for low Da, the reacting zones are spread as reacting pockets within the vortex structure. The effects of Da on mean reactant and product concentrations, root‐mean‐square concentration fluctuations and probability density are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Large eddy simulation (2D) using diffusion–velocity method and vortex‐in‐cell

A large eddy Simulation based on the diffusion‐velocity method and the discrete vortex method is presented. The vorticity‐based and eddy viscosity type subgrid scale model simulating the enstrophy transfer between the large and small scale appears as a convective term in the diffusion‐velocity formulation. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method and the Smagorinsky subgrid scale model. The effects on the vorticity contours, momemtum thickness, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations and negative cross‐stream correlation are discussed. Comparison is made with experiment and numerical work where diffusion is simulated using random walk. Copyright © 2004 John Wiley & Sons, Ltd.     

Large Eddy Simulation (2D) of a Reacting Plane Mixing Layer Using Filtered Density Function Closure

The filtered density function (FDF) is implemented for a two-dimensional, large eddy simulation (LES) of a gas phase, spatially developing, reacting and non-reacting, constant-density, plane mixing layer in a flow regime prior to the mixing transition where the flow is mainly two-dimensional. The unresolved scalar fluctuations are taken into account by considering the probability density function (PDF) of subgrid scale (SGS) scalar quantities following the FDF approach. In the derived FDF transport equation, the effect of chemical reactions appears in a closed form. The Lagrangian Monte Carlo scheme is used to solve the FDF transport equation. The applicability and performance of the FDF for LES of a reacting plane mixing layer are assessed by comparisons with experimental measurements. In non-reacting flow, the calculated mean streamwise velocity profiles and mean mixture fraction profiles relax to self-similarity, which is in satisfactory agreement with the measurements. In reacting flow, the FDF calculation provided a satisfactory accuracy in comparison with measurements of mean reactant and product concentration. The increase in the total amount of product formation in the flip case demonstrates the asymmetric characteristics of the entrainment and mixing characteristics in the mixing layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Vortex-In-Cell and Probability Density Function Approach for a Passive Scalar Field in a Mixing Layer

A Reynolds-averaged simulation based on the vortex-in-cell (VIC) and the transport equation for the probability density function (PDF) of a scalar has been developed to predict the passive scalar field in a two-dimensional spatially growing mixing layer. The VIC computes the instantaneous velocity and vorticity fields. Then the mean-flow properties, i.e. the mean velocity, the root-mean-square (rms) longitudinal and lateral velocity fluctuations, the Reynolds shear stress, and the rms vorticity fluctuations are computed and used as input to the PDF equation. The PDF transport equation is solved using the Monte Carlo technique. The convection term uses the mean velocities from the VIC. The turbulent diffusion term is modeled using the gradient transport model, in which the eddy diffusivity, computed via the Boussinesq's postulate, uses the Reynolds shear stress and gradients of mean velocities from the VIC. The molecular mixing term is closed by the modified Curl model.

The computational results were compared with two-dimensional experimental results for passive scalar. The predicted turbulent flow characteristics, i.e. mean velocity and rms longitudinal fluctuations in the self-preserving region, show good agreement with the experimental measurements. The profiles of the mean scalar and the rms scalar fluctuations are also in reasonable agreement with the experimental measurements. Comparison between the mean scalar and the mean velocity profiles shows that the scalar mixing region extends further into the free stream than does the momentum mixing region, indicating enhanced transport of scalar over momentum. The rms scalar profiles exhibit an asymmetry relative to the concentration centerline, and indicate that the fluid on the high-speed side mixes at a faster rate than the fluid on the low-speed side. The asymmetry is due to the asymmetry in the mixing frequency cross-stream profiles. Also, the PDFs have peaks biased toward the high-speed side.     

Identification of local extinction topology in axisymmetric bluff‐body diffusion flames with a reactedness‐mixture fraction presumed probability density function model

The effects of finite‐rate chemistry, such as partial extinctions and re‐ignitions, are investigated in turbulent non‐pre‐mixed reacting flows stabilized in the wake of an axisymmetric bluff‐body burner. A two‐dimensional large‐eddy simulation procedure is employed that uses a partial equilibrium/two‐scalar reactedness mixture fraction probability density function (PDF) combustion sub‐model, which is applied at the sub‐grid scale (SGS) level. An anisotropic sub‐grid eddy–viscosity and two equations for the SGS turbulence kinetic and scalar energies complete the SGS closure model. The scalar covariances required in the joint PDF formulation are obtained from an extended scale‐similarity assumption between the resolved and the sub‐grid fluctuations. Extinction due to strong turbulence/chemistry interactions is recognized with the help of a ‘critical’, locally variable, turbulent Damkohler number criterion, while transient localized extinctions and re‐ignitions are treated with a Lagrangian transport equation for a reactedness progress variable. Comparisons with available experimental data suggested that the formulated approach was capable of identifying the effects of large‐scale vortex structure activity, which were inherent in the reacting wake and dominant in the counterpart isothermal flows that otherwise would have been obscured if a standard time‐averaged procedure had been used. Additionally, the post‐extinction and re‐ignition behaviour and its time‐varying interaction with the large‐scale structure dynamics were more appropriately addressed within the context of the present time‐dependent method. Copyright © 2001 John Wiley & Sons, Ltd.     

A Non-Adiabatic Flamelet Progress�CVariable Approach for LES of Turbulent Premixed Flames

A progress variable/flame surface density/probability density function method has been employed for a Large Eddy Simulation of a CH₄/Air turbulent premixed bluff body flame. In particular, both mean and variance of the progress variable are transported and subgrid spatially filtered gradient contributes to model the flame surface density (that introduces the effect of the subgrid flame reaction zone) and to presume a probability density function (that introduces the effect of subgrid fluctuations on chemistry). Chemistry is preliminarly tabulated in terms of laminar premixed flames and enthalpy is included as a new coordinate in their tabulation to take into account heat losses in the flowfield. Then, the PDF is used to build a turbulent flamelet library. The filtered mass, momentum, enthalpy and scalar equations mentioned above are integrated by an explicit scheme using finite differences, 2nd?Corder accurate in space and third order in time, over a cylindrical non-uniform grid using a staggered mesh. The bluff-body geometry is modelled by using the Immersed Boundary Method. The numerical predictions are compared with the available experimental data.     

Assessment of stretched vortex subgrid‐scale models for LES of incompressible inhomogeneous turbulent flow

The physical space version of the stretched vortex subgrid scale model is tested in LES of the turbulent lid‐driven cubic cavity flow. LES is carried out by using a higher order finite‐difference method. The effects of different vortex orientation models and subgrid turbulence spectrums are assessed through comparisons of the LES predictions against DNS. Three Reynolds numbers 12000, 18000, and 22000 are studied. Good agreement with the DNS data for the mean and fluctuating quantities is observed. Copyright © 2013 John Wiley & Sons, Ltd.     

Residual‐based variational multiscale methods for laminar,transitional and turbulent variable‐density flow at low Mach number

In the present study, residual‐based variational multiscale methods are developed for and applied to variable‐density flow at low Mach number. In particular, two different formulations are considered in this study: a standard stabilized formulation featuring SUPG/PSG/grad‐div terms and a complete residual‐based variational multiscale formulation additionally containing cross‐ and Reynolds‐stress terms as well as subgrid‐scale velocity terms in the energy‐conservation equation. The proposed methods are tested for various laminar flow test cases as well as a test case at laminar via transitional to turbulent flow stages. Stable and accurate results are obtained for all numerical examples. Substantial differences in the results between the two approaches do not become notable until a high temperature gradient is applied and the flow reaches a turbulent flow stage. The more pronounced influence of adding subgrid‐scale velocity terms to the energy‐conservation equation on the results than adding analogous terms to the momentum‐conservation equation in this situation appears particularly noteworthy. Copyright © 2009 John Wiley & Sons, Ltd.     

Large eddy simulation using a dynamic mixing length subgrid‐scale model

A novel dynamic mixing length (DML) subgrid‐scale model for large eddy simulations is proposed in this work to improve the cutoff length of the Smagorinsky model. The characteristic mixing length (or the characteristic wave number) is dynamically estimated for the subgrid‐scale fluctuation of turbulence by the cutoff wave‐number, k_c, and the dissipation wave‐number, k_d. The dissipation wave number is derived from the kinetic energy spectrum equation and the dissipation spectrum equation. To prove the promise of the DML model, this model is used to simulate the lid‐driven cubical cavity with max‐velocity‐based Reynolds numbers 8850 and 12,000, the channel flows with friction‐velocity‐based Reynolds numbers 180, 395, 590, and 950, and the turbulent flow past a square cylinder at the higher Reynolds number 21,400, respectively, compared with the Smagorinsky model and Germano et al.'s dynamic Smagorinsky model. Different numerical experiments with different Reynolds numbers show that the DML model can be used in simulations of flows with a wide range of Reynolds numbers without the occurrence of singular values. The DML model can alleviate the dissipation of the Smagorinsky model without the loss of its robustness. The DML model shows some advantages over Germano et al.'s dynamic Smagorinsky model in its high stability and simplicity of calculation because the coefficient of the DML model always stays positive. The characteristic mixing length in the DML model reflects the subgrid‐scale fluctuation of turbulence in nature and thus the characteristic mixing length has a spatial and temporal distribution in turbulent flow. Copyright © 2011 John Wiley & Sons, Ltd.     

An improved near‐wall modeling for large‐eddy simulation using immersed boundary methods

An improved near‐wall modeling for large‐eddy simulation using the immersed boundary method is proposed. It is shown in this study that the existing near‐wall modeling for the immersed boundary (IB) methods that imposes the velocity boundary condition at the IB node is not sufficient to enforce a correct wall shear stress at the IB node. A new method that imposes a shear stress condition through the modification of the subgrid scale‐eddy viscosity at the IB node is proposed. In this method, the subgrid eddy viscosity at the IB node is modified such that the viscous flux at the face adjacent to the IB node correctly approximates the total shear stress. The method is applied to simulate the fully developed turbulent flows in a plane channel and a circular pipe. It is demonstrated that the new method improves the prediction of the mean velocity and turbulence stresses in comparison with the existing wall modeling based solely on the velocity boundary condition. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐order filtering for control volume flow simulation

A general methodology is presented in order to obtain a hierarchy of high‐order filter functions, starting from the standard top‐hat filter, naturally linked to control volumes flow simulations. The goal is to have a new filtered variable better represented in its high resolved wavenumber components by using a suitable deconvolution. The proposed formulation is applied to the integral momentum equation, that is the evolution equation for the top‐hat filtered variable, by performing a spatial reconstruction based on the approximate inversion of the averaging operator. A theoretical analysis for the Burgers' model equation is presented, demonstrating that the local de‐averaging is an effective tool to obtain a higher‐order accuracy. It is also shown that the subgrid‐scale term, to be modeled in the deconvolved balance equation, has a smaller absolute importance in the resolved wavenumber range for increasing deconvolution order. A numerical analysis of the procedure is presented, based on high‐order upwind and central fluxes reconstruction, leading to congruent control volume schemes. Finally, the features of the present high‐order conservative formulation are tested in the numerical simulation of a sample turbulent flow: the flow behind a backward‐facing step. Copyright © 2001 John Wiley & Sons, Ltd.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

Evaluation of Smagorinsky‐based subgrid‐scale models in a finite‐volume computation

Smagorinsky‐based models are assessed in a turbulent channel flow simulation at Re_b=2800 and Re_b=12500. The Navier–Stokes equations are solved with three different grid resolutions by using a co‐located finite‐volume method. Computations are repeated with Smagorinsky‐based subgrid‐scale models. A traditional Smagorinsky model is implemented with a van Driest damping function. A dynamic model assumes a similarity of the subgrid and the subtest Reynolds stresses and an explicit filtering operation is required. A top‐hat test filter is implemented with a trapezoidal and a Simpson rule. At the low Reynolds number computation none of the tested models improves the results at any grid level compared to the calculations with no model. The effect of the subgrid‐scale model is reduced as the grid is refined. The numerical implementation of the test filter influences on the result. At the higher Reynolds number the subgrid‐scale models stabilize the computation. An analysis of an accurately resolved flow field reveals that the discretization error overwhelms the subgrid term at Re_b=2800 in the most part of the computational domain. Copyright © 2002 John Wiley & Sons, Ltd.     

A hybrid vortex method for the simulation of three‐dimensional flows

This paper presents an integral vorticity method for solving three‐dimensional Navier–Stokes equations. A finite volume scheme is implemented to solve the vorticity transport equation, which is discretized on a structured hexahedral mesh. A vortex sheet algorithm is used to enforce the no‐slip boundary condition through a vorticity flux at the boundary. The Biot–Savart integral is evaluated to compute the velocity field, in conjunction with a fast algorithm based on multipole expansion. This method is applied to the simulation of uniform flow past a sphere. Copyright © 2007 John Wiley & Sons, Ltd.     

Use of a Monte Carlo PDF method in a study of the influence of turbulent fluctuations on selectivity in a jet-stirred reactor

A model for single-phase turbulent reacting flow is presented and a solution algorithm is described. The model combines the standardk - model for the velocity field with a transport equation for the probability density function (PDF) of the thermochemical variables. In this equation terms describing spatial transport by velocity fluctuations and mixing on the smallest scales are modelled. The essential advantage of this approach is that the effect of nonlinear kinetics appears in closed form and that the influence of turbulent fluctuations on mean reaction rates is included. A stochastic algorithm for the solution of the PDF transport equation, essentially due to Pope, is described. Cylindrical symmetry is assumed. The PDF is represented by ensembles ofN representative values of the thermochemical variables in each cell of a nonuniform finite-difference grid and operations on these elements representing convection, diffusion, mixing and reaction are derived. A simplified model and solution algorithm which neglects the influence of turbulent fluctuations on mean reaction rates is also described. Both algorithms are applied to a selectivity problem in a real reactor studied earlier by Liu and Barkelew. Spatial profiles of mean species mole fractions and of relative selectivity to the target product are obtained. The profiles are clearly different in both models but at the end of the reactor the same selectivity is predicted.Presented at the Shell Conference on Computational Fluid Dynamics for Petrochemical Process Equipment, Hoenderloo, December 10–12, 1989.     

超声速来流横向狭缝喷流标量输运的混合RANS/LES模拟

采用基于时均湍流模型的混合RANS/LES模拟方法对超声速来流狭缝横向喷流标量输运过程进行大涡模拟,控制方程对流项采用五阶精度的WENO格式求解,过滤后的组分方程中的亚格子组分对流通量采用梯度扩散模化.模拟得到了超声速来流狭缝横向喷流大尺度涡和组分的演化过程,研究表明喷流具有明显的非定常周期性特点,并且组分随时间的空间分布特征决定于流动的大尺度结构.模拟得到的压力的统计时均结果和实验结果定性一致,但喷流前后的分离区较实验值过大,这表明混合RANS/LES需要进一步改进.     

模型燃烧室内瞬态紊流的大涡模拟

本文采用三种不同亚网格尺度模型对带有V型稳定器的模型燃烧室二维瞬态紊流流动进行了大涡模拟。并在交错网格系下用SIMPLE算法和混合差分格式求解离散方程。数值研究拟不同型式入口速度分布和不同亚网格尺度模型下模型燃烧室二维瞬态紊流流场。计算结果表明不同入口速度分布和不同亚网格尺度模型对瞬态流场和出口速度分布有一定的影响。本文通过数值模拟,揭示了V型稳定器后旋涡的产生和脱落过程。通过计算结果及实验数据的比较可知,本文采用的亚网格尺度模型可以用来模拟模型燃烧室紊流流场及稳定器后面回流区的流动情况。     

A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

A new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.     

Shear‐improved Smagorinsky modeling of turbulent channel flow using generalized Lattice Boltzmann equation

Generalized Lattice Boltzmann equation (GLBE) was used for computation of turbulent channel flow for which large eddy simulation (LES) was employed as a turbulence model. The subgrid‐scale turbulence effects were simulated through a shear‐improved Smagorinsky model (SISM), which is capable of predicting turbulent near wall region accurately without any wall function. Computations were done for a relatively coarse grid with shear Reynolds number of 180 in a parallelized code. Good numerical stability was observed for this computational framework. The results of mean velocity distribution across the channel showed good correspondence with direct numerical simulation (DNS) data. Negligible discrepancies were observed between the present computations and those reported from DNS for the computed turbulent statistics. Three‐dimensional instantaneous vorticity contours showed complex vortical structures that appeared in such flow geometries. It was concluded that such a framework is capable of predicting accurate results for turbulent channel flow without adding significant complications and the computational cost to the standard Smagorinsky model. As this modeling was entirely local in space it was therefore adapted for parallelization. Copyright © 2010 John Wiley & Sons, Ltd.