Large eddy simulation (2D) of spatially developing mixing layer using vortex‐in‐cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion‐velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte‐Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations profiles and negative cross‐stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root‐mean‐square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

The effect of air-water interface on the vortex shedding from a vertical circular cylinder

The flow past an interface piercing circular cylinder at the Reynolds number Re=2.7×10⁴ and the Froude numbers Fr=0.2 and 0.8 is investigated using large-eddy simulation. A Lagrangian dynamic subgrid-scale model and a level set based sharp interface method are used for the spatially filtered turbulence closure and the air-water interface treatment, respectively. The mean interface elevation and the rms of interface fluctuations from the simulation are in excellent agreement with the available experimental data. The organized periodic vortex shedding observed in the deep flow is attenuated and replaced by small-scale vortices at the interface. The streamwise vorticity and the outward transverse velocity generated near the edge of the separated region, which enforces the separated shear layers to deviate from each other and restrains their interaction, are primarily responsible for the devitalization of the periodic vortex shedding at the interface. The lateral gradient of the difference between the vertical and transverse Reynolds normal stresses, increasing with the Froude number, is the main source of the streamwise vorticity and the outward transverse velocity at the interface.     

Low Reynolds number k–ε model for near‐wall flow

A wall‐distance free k–ε turbulence model is developed that accounts for the near‐wall and low Reynolds number effects emanating from the physical requirements. The model coefficients/functions depend non‐linearly on both the strain rate and vorticity invariants. Included diffusion terms and modified C_ε(1,2) coefficients amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct Numerical Simulation of the Puffing Phenomenon of an Axisymmetric Thermal Plume

A spatial direct numerical simulation of an axisymmetric buoyant thermal plume is presented. The governing flow field equations at the centerline are put into a special form to circumvent the axis singularity associated with the cylindrical coordinates and the high order accuracy of the numerical scheme is preserved at the centerline. Boundary conditions associated with the spatial DNS of open-boundary buoyant flows and compatible with the modern nondissipative high-order finite difference schemes have been developed. The fluid exhibits a periodic oscillatory motion known as the puffing phenomenon, which is the formation and convection of vortex at the near field of the plume. Budgets of the vorticity transport are determined to examine the mechanisms leading to the puffing phenomenon. The analysis on vorticity transport shows that vorticity is created mainly by the gravitational term which is due to the interaction between the radial density gradients and gravity at the initial stage of the establishment of the puffing structure, while the baroclinic torque dominates the vorticity transport when the flow is established. Density stratification in the radial direction close to the plume base is found to be essential to the development of the buoyant flow instability. Simulations with different initial temperature ratios reveal that entrainment close to the plume base is enhanced at a higher temperature ratio despite the fact that the puffing structures and the plume pulsation frequency only vary very weakly with the initial temperature ratio. The predicted puffing frequencies are in agreement with the values from experimental correlations for fire and isothermal helium/air plumes. Received 12 May 1999 and accepted 9 December 1999     

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.     

Transonic viscous-inviscid interaction by a finite element method

A method is outlined for solving two-dimensional transonic viscous flow problems, in which the velocity vector is split into the gradient of a potential and a rotational component. The approach takes advantage of the fact that for high-Reynolds-number flows the viscous terms of the Navier-Stokes equations are important only in a thin shear layer and therefore solution of the full equations may not be needed everywhere. Most of the flow can be considered inviscid and, neglecting the entropy and vorticity effects, a potential model is a good approximation in the flow core. The rotational part of the flow can then be calculated by solution of the potential, streamfunction and vorticity transport equations. Implementation of the no-slip and no-penetration boundary conditions at the walls provides a simple mechanism for the interaction between the viscous and inviscid solutions and no extra coupling procedures are needed. Results are presented for turbulent transonic internal choked flows.     

Assessment of dual plane PIV measurements in wall turbulence using DNS data

Experimental dual plane particle image velocimetry (PIV) data are assessed using direct numerical simulation (DNS) data of a similar flow with the aim of studying the effect of averaging within the interrogation window. The primary reason for the use of dual plane PIV is that the entire velocity gradient tensor and hence the full vorticity vector can be obtained. One limitation of PIV is the limit on dynamic range, while DNS is typically limited by the Reynolds number of the flow. In this study, the DNS data are resolved more finely than the PIV data, and an averaging scheme is implemented on the DNS data of similar Reynolds number to compare the effects of averaging inherent to the present PIV technique. The effects of averaging on the RMS values of the velocity and vorticity are analyzed in order to estimate the percentage of turbulence intensity and enstrophy captured for a given PIV resolution in turbulent boundary layers. The focus is also to identify vortex core angle distributions, for which the two-dimensional and three-dimensional swirl strengths are used. The studies are performed in the logarithmic region of a turbulent boundary layer at z ⁺ = 110 from the wall. The dual plane PIV data are measured in a zero pressure gradient flow over a flat plate at Re _τ = 1,160, while the DNS data are extracted from a channel flow at Re _τ = 934. Representative plots at various wall-normal locations for the RMS values of velocity and vorticity indicate the attenuation of the variance with increasing filter size. Further, the effect of averaging on the vortex core angle statistics is negligible when compared with the raw DNS data. These results indicate that the present PIV technique is an accurate and reliable method for the purposes of statistical analysis and identification of vortex structures.     

基于深度神经网络的横流转捩预测

基于深度神经网络DNN构建了从层流流场无量纲速度梯度、流向涡强度等物理量到横流转捩模态下间歇因子间的映射关系,获得一种新的数据驱动转捩模型.通过将数据驱动转捩模型与SST k-ω湍流模型耦合,有效简化了转捩模型输运方程求解,实现高效、准确的亚音速三维边界层横流转捩流场计算. DNN训练数据来自变雷诺数的NLF(2)-0415无限展长后掠翼计算结果,并以两种工况进行测试,数据驱动转捩模型预测精度与γ-Reθ转捩模型近似.将数据驱动转捩模型用于其他典型横流转捩算例的计算,以验证其泛化能力.对于变后掠角的NLF(2)-0415后掠翼,数据驱动转捩模型与γ-Reθt-CF模型预测的转捩位置几乎一致,并且能够预测出后掠角从45°增长到65°的过程中,转捩位置先向前再向后移动的现象;对于标准椭球体,使用低分辨率网格进行计算,数据驱动转捩模型依然能够实现转捩位置预测,对椭球体表面Cf的计算结果与多个平台的横流转捩模型、实验结果基本一致.研究表明,以横流转捩相关物理量作为输入对DNN进行训练,并将获得的数据驱动转捩模型与SST k-ω湍流模型耦合,可以实现对横流转捩的有效预测,且具有较强的泛化能力.数...     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

Large-eddy simulation of nonlinear evolution and breakdown to turbulence in high-speed boundary layers

The nonlinear evolution and laminar-turbulent breakdown of a boundary-layer flow along a cylinder at Mach 4.5 is investigated with large-eddy temporal simulation. The results are validated using the direct numerical simulation data of Pruett and Zang. The structure of the flow during the transition process is studied in terms of the vorticity field. The subgrid scales are modeled dynamically, where the model coefficients are determined as part of the solution from the local resolved field. In the numerical simulation the dynamic-model coefficients are obtained by using both the strain-rate contraction of Germano et al. and the least-squares contraction of Lilly; they produced some differences in the details of the vorticity structure inside the transition region. A new dynamic model that utilizes the second-order velocity structure function is used to parametrize the small-scale field. The evolution to turbulence is successfully simulated with dynamic subgrid-scale modeling at least in terms of average quantities as well as vorticity fields. This is achieved with one-sixth of the grid resolution used in direct numerical simulation.This work was sponsored by the Theoretical Flow Physics Branch of the Fluid Mechanics Division of NASA Langley Research Center under Contract NAS1-19320.     

Méthode particulaire anisotropeAnisotropic particle method

A particle method has been used to simulate the vorticity transport in a two-dimensional flow of an incompressible inviscid fluid. In this method, not only the location and the circulation of the particle are used but also the moments of the internal vorticity. The transport equation for these moments has been derived from the vorticity transport equation. The method has been compared to the usual particle method as well as to Teng's elliptic vortex model. The test case is that of the evolution of two circular patches of vorticity already used by Teng. To cite this article: A. Beaudoin et al., C. R. Mecanique 330 (2002) 51–56     

An explicit algebraic Reynolds stress model in turbulence

A new algebraic Reynold stress model is constructed with recourse to the realizability constraints. Model coefficients are made a function of strain and vorticity invariants through calibration by reference to homogeneous shear flow data. The anisotropic production in near‐wall regions is accounted for substantially by modifying the model constants C_{ε(1, 2)} and adding a secondary source term in the ε equation. Hence, it reduces the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. To facilitate the evaluation of the turbulence model, some extensively used benchmark cases in the turbulence modelling are convoked. The comparisons demonstrate that the new model maintains qualitatively good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

Curved shock theory

Curved shock theory (CST) is introduced, developed and applied to relate pressure gradients, streamline curvatures, vorticity and shock curvatures in flows with planar or axial symmetry. Explicit expressions are given, in an influence coefficient format, that relate post-shock pressure gradient, streamline curvature and vorticity to pre-shock gradients and shock curvature in steady flow. The effect of pre-shock flow divergence/convergence, on vorticity generation, is related to the transverse shock curvature. A novel derivation for the post-shock vorticity is presented that includes the effects of pre-shock flow non-uniformities. CST applicability to unsteady flows is discussed.     

Laminar mixing of homogeneous currents in the presence of a pressure gradient

The problem of the mixture of a current with a quiescent gas was solved by Chapman [1]. In this study, the results of certain calculations on the laminar mixing zone of homogeneous gas currents with a pressure gradient will be presented. On the basis of the data calculated and evaluations, it is shown that the concept of similarity, formulated by Less [2], is applicable to the problem of mixture with a pressure gradient. In variable similarities, the velocity profiles for gradient flow practically coincide with the profiles of nongradient flow for the same parameter values at the interior and exterior boundaries of the mixture zone. Moreover, it proves to be the case that the excess velocity profile depends weakly on the specific parameters of the problem.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 67–71, March–April, 1972.     

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.     

Local vorticity computation approach in double distribution functions based lattice Boltzmann methods for flow and scalar transport

Computation of vorticity, or the skew-symmetric velocity gradient tensor, in conjunction with the strain rate tensor, plays an important role in the flow classification, in vortical structure identification and in the modeling of various complex fluids and flows. For the simulation of flows accompanied by the advection-diffusion transport of a scalar field (e.g., temperature), double distribution functions (DDF) based lattice Boltzmann (LB) methods, involving a pair of LB schemes are commonly used. We present a new local vorticity computation approach by introducing an intensional anisotropy of the scalar flux in the third order, off-diagonal moment equilibria of the LB scheme for the scalar field, and then combining the second order non-equilibrium components of both the LB methods. As such, any pair of lattice sets in the DDF formulation that can independently support the third order off-diagonal moments would enable local determination of the complete flow kinematics, with the LB methods for the fluid motion and the transport of the passive scalar respectively providing the necessary moment relationships to determine the symmetric and skew-symmetric components of the velocity gradient tensor. Since the resulting formulation is completely local and do not rely on any finite difference approximations for velocity derivatives, it is by design naturally suitable for parallel computation. As an illustration of our approach, we formulate a DDF-LB scheme for local vorticity computation using a pair of multiple relaxation times (MRT) based collision approaches on two-dimensional, nine velocity (D2Q9) lattices, where the necessary moment relationships to determine the velocity gradient tensor and the vorticity are established via a Chapman-Enskog analysis. Simulations of various benchmark flows demonstrate good accuracy of the predicted vorticity fields using our approach against available solutions, including numerical results, with a second order convergence. Furthermore, extensions of our formulation for a variety of collision models, including those based on cascaded and non-cascaded central moments, to enable local vorticity computation are presented.     

A Quasi-Realizable Cubic Low-Reynolds Eddy-Viscosity Turbulence Model with a New Dissipation Rate Equation

A non-linear relationship of the Reynolds stresses in function of the strain rate and vorticity tensors, with terms up to third order, is developed. Anisotropies in the normal stresses, influence from streamline curvature or rotation of the reference frame, and swirl effects are accounted for. The relationship is linked to ak–ε model with a modified transport equation for the dissipation rate. A new low-Reynolds source term is introduced and a model parameter is written in terms of dimensionless rate-of-strain and vorticity. The model is checked on different realizability constraints. It is shown that practically all constraints are fulfilled. The model is numerically tested on a fully developed channel and pipe flow, both stationary and rotating. The plane jet–round jet anomaly is addressed. Finally, the model is applied to the flow over a backward-facing step. Results are compared with a linear low-Reynolds k–ε model and the shear stress transport model. This revised version was published online in July 2006 with corrections to the Cover Date.     

Simulating two-dimensional turbulent flow by using the κ–ϵ model and the vorticity–streamfunction formulation

A numerical procedure to solve turbulent flow which makes use of the κ–? model has been developed. The method is based on a control volume finite element method and an unstructured triangular domain discretization. The velocity-pressure coupling is addressed via the vorticity-streamfunction and special attention is given to the boundary conditions for the vorticity. Wall effects are taken into account via wail functions or a low-Reynolds-number model. The latter was found to perform better in recirculation regions. Source terms of the κ and ε transport equations have been linearized in a particular way to avoid non-realistic solutions. The vorticity and streamfunction discretized equations are solved in a coupled way to produce a faster and more stable computational procedure. Comparison between the numerical predictions and experimental data shows that the physics of the flow is correctly simulated.     

Vorticity–velocity formulation of the 3D Navier–Stokes equations in cylindrical co‐ordinates

A finite difference method is presented for solving the 3D Navier–Stokes equations in vorticity–velocity form. The method involves solving the vorticity transport equations in ‘curl‐form’ along with a set of Cauchy–Riemann type equations for the velocity. The equations are formulated in cylindrical co‐ordinates and discretized using a staggered grid arrangement. The discretized Cauchy–Riemann type equations are overdetermined and their solution is accomplished by employing a conjugate gradient method on the normal equations. The vorticity transport equations are solved in time using a semi‐implicit Crank–Nicolson/Adams–Bashforth scheme combined with a second‐order accurate spatial discretization scheme. Special emphasis is put on the treatment of the polar singularity. Numerical results of axisymmetric as well as non‐axisymmetric flows in a pipe and in a closed cylinder are presented. Comparison with measurements are carried out for the axisymmetric flow cases. Copyright © 2003 John Wiley & Sons, Ltd.     

Mixing in a stably stratified medium by horizontal shear near vertical walls

Stratified environmental flows near boundaries can have a horizontal mean shear component, orthogonal to the vertical mean density gradient. Vertical transport, against the stabilizing force of gravity, is possible in such situations if three-dimensional turbulence is sustained by the mean shear. A model problem, water with a constant mean density gradient flowing in a channel between parallel vertical walls, is examined here using the technique of large eddy simulation (LES). It is found that, although the mean shear is horizontal, the fluctuating velocity field has significant vertical shear and horizontal vorticity, thereby causing small-scale vertical mixing of the density field. The vertical stirring is especially effective near the boundaries where the mean shear is large and, consequently, the gradient Richardson number is small. The mean stratification is systematically increased between cases in our study and, as expected, the buoyancy flux correspondingly decreases. Even so, horizontal mean shear is found to be more effective than the well-studied case of mean vertical shear in inducing vertical buoyancy transport as indicated by generally larger values of vertical eddy diffusivity and mixing efficiency.