Numerical study of the properties of the central moment lattice Boltzmann method

Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the lattice kinetic schemes for computational fluid dynamics. A key element in this approach is the use of central moments to specify the collision process and forcing, and thereby naturally maintaining Galilean invariance, an important characteristic of fluid flows. When the different central moments are relaxed at different rates like in a standard multiple relaxation time (MRT) formulation based on raw moments, it is endowed with a number of desirable physical and numerical features. Because the collision operator exhibits a cascaded structure, this approach is also known as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a systematic study of its numerical properties, such as the accuracy, grid convergence, and stability for well‐defined canonical problems is lacking, and the present work is intended to fulfill this need. We perform a quantitative study of the performance of the cascaded LBM for a set of benchmark problems of differing complexity, viz., Poiseuille flow, decaying Taylor–Green vortex flow, and lid‐driven cavity flow. We first establish its grid convergence and demonstrate second‐order accuracy under diffusive scaling for both the velocity field and its derivatives, that is, the components of the strain rate tensor, as well. The method is shown to quantitatively reproduce steady/unsteady analytical solutions or other numerical results with excellent accuracy. The cascaded MRT LBM based on the central moments is found to be of similar accuracy when compared with the standard MRT LBM based on the raw moments, when a detailed comparison of the flow fields are made, with both reproducing even the small scale vortical features well. Numerical experiments further demonstrate that the central moment MRT LBM results in significant stability improvements when compared with certain existing collision models at moderate additional computational cost. Copyright © 2015 John Wiley & Sons, Ltd.     

Discretization of incompressible vorticity–velocity equations on triangular meshes

This paper describes a new approach to discretizing first- and second-order partial differential equations. It combines the advantages of finite elements and finite differences in having both unstructured (triangular/tetrahedral) meshes and low-order physically intuitive schemes. In this ‘co-volume’ framework, the discretized gradient, divergence, curl, (scalar) Laplacian, and vector Laplacian operators satisfy relationships found in standard vector field theory, such as a Helmholtz decomposition. This article focuses on the vorticity–velocity formulation for planar incompressible flows. The algorithm is described and some supporting numerical evidence is provided.     

A novel non-primitive Boundary Integral Equation Method for three-dimensional and axisymmetric Stokes flows

A new Boundary Integral Equation (BIE) formulation for Stokes flow is presented for three-dimensional and axisymmetrical problems using non-primitive variables, assuming velocity field is prescribed on the boundary. The formulation involves the vector potential, instead of the classical stream function, and all three components of the vorticity are implied. Furthermore, following the Helmholtz decomposition, a scalar potential is added to represent the solenoidal velocity field. Firstly, the BIEs for three-dimensional flows are formulated for the vector potential and the vorticity by employing the fundamental solutions in free space of vector Laplace and biharmonic equations. The equations for axisymmetric flows are then derived from the three-dimensional formulation in a second step. The outcome is a domain integral free BIE formulation for both three-dimensional and axisymmetric Stokes flows with prescribed velocity boundary condition. Numerical results are included to validate and show the efficiency of the proposed axisymmetric formulation.     

The Fast Exact Closure for Jeffery’s equation with diffusion

Jeffery’s equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold’s numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a ‘closure’ that approximates the distribution function’s fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery’s equation. Numerical examples for dense fiber suspensions are provided with both a Folgar–Tucker (1984) [3] diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009) [9]. Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.     

Solutions of the moment hierarchy in the kinetic theory of Maxwell models

In the Maxwell interaction model the collision rate is independent of the relative velocity of the colliding pair and, as a consequence, the collisional moments are bilinear combinations of velocity moments of the same or lower order. In general, however, the drift term of the Boltzmann equation couples moments of a given order to moments of a higher order, thus preventing the solvability of the moment hierarchy, unless approximate closures are introduced. On the other hand, there exist a number of states where the moment hierarchy can be recursively solved, the solution generally exposing non-Newtonian properties. The aim of this paper is to present an overview of results pertaining to some of those states, namely the planar Fourier flow (without and with a constant gravity field), the planar Couette flow, the force-driven Poiseuille flow, and the uniform shear flow.     

涡识别技术及其在船用螺旋桨伴流场中的应用

涡识别是很重要的流体问题, 为了在船用螺旋桨伴流场中寻找一种合理的涡识别方法, 本文结合实践, 研究了六种涡识别技术理论, 其中使用Burgers涡流和Lamb-Oseen涡流作了必要的解释, 讨论了各种识别方法的优缺点. 局部低压标准比较直观, 但深究其黏性和非定常影响后, 明显不足; 迹线或流线显然不能满足伽利略不变性, 会使辨别变得混乱; 涡度大小需要规定其阈值, 具有一定不确定性, 且也会识别不是涡的涡片; 速度梯度张量的复特征值也会有识别不出的区域; 速度梯度张量的第二不变量标准和对称张量的第二特征值标准能更好地识别涡核, 这两种标准有时等效. 螺旋桨伴流场的数值模拟是在开源软件OpenFOAM平台上实现的, 湍流大涡模型由一种局部动态方程建模, 此模型优于动态Smagorinsky模型. 最终的结果显示: 对于船用螺旋桨伴流场中的涡, 采用速度梯度张量的第二不变量和对称张量的第二特征值的结果基本一致, 而最小压力标准、流线或迹线标准、涡度值标准和速度张量的复特征值标准都存在一定的缺陷, 不适用于船用螺旋桨伴流场中的涡识别.      

Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

A velocity–vorticity formulation of the Navier–Stokes equations is presented as an alternative to the primitive variables approach. The velocity components and the vorticity are solved for in a fully coupled manner using a Newton method. No artificial viscosity is required in this formulation. The pressure is updated by a method allowing natural imposition of boundary conditions. Incompressible and subsonic results are presented for two-dimensional laminar internal flows up to high Reynolds numbers.     

Accuracy of tomographic particle image velocimetry data on a turbulent round jet

Tomographic particle image velocimetry (Tomo-PIV) was applied on a turbulent round air jet to quantitatively assess the accuracy of velocity gradients obtained in the self-similar turbulent region. The jet Reynolds number based on the nozzle diameter (d) was Re_d = 3000. Mean velocity, turbulent intensities, and Reynolds shear stress at the center plane of the jet were measured. In addition, statistical results of Tomo-PIV along the axial direction were assessed by performing a separate set of two-dimensional two-component PIV experiments on a “side view” plane along the jet axis. Moreover, the probability distribution functions of four components of the measured velocity gradients in the axial and radial directions were validated by these “side view” planar PIV data. The root mean square of the velocity divergence values relative to the norm of the velocity gradient tensor was 0.36. Furthermore, the on- and off-diagonal components of the velocity gradients satisfied the axisymmetric isotropy conditions. The divergence error in the data affected only areas with low gradient magnitude. Therefore, turbulent structures in the regions with intense vorticity and dissipation can be closely monitored. On this basis, the joint pdfs of the invariants of the velocity gradient and strain and rotation tensor rates were produced and compared well with those in isotropic turbulence studies.     

On the role of unsteady forcing of tracer gradient in local stirring

Local stirring properties in two basic mixing flows – namely, the blinking vortex and the sine flow – are studied through the tracer gradient approach. The velocity gradient tensor and related quantities such as the strain persistence parameter are derived from the analytical velocity fields. Numerical Lagrangian tracking of the gradient of a tracer shows how local stirring is affected by forcing experienced through strain persistence. In both flows Lagrangian variations of strain persistence occuring on a time scale shorter than the response time scale of the tracer gradient lead the latter to align close to the direction determined by the mean strain persistence. It is the special alternating behaviour of strain persistence resulting from flow operation that makes this direction coincide with the local compressional strain direction for both the sine flow and the clockwise/counterclockwise blinking vortex. The rise of the tracer gradient and thus local stirring are in turn promoted by this statistical alignment.     

Interplay of surface geometry and vorticity dynamics in incompressible flows on curved surfaces

Incompressible viscous flows on curved surfaces are considered with respect to the interplay of surface geometry, curvature, and vorticity dynamics. Free flows and cylindrical wakes over a Gaussian bump are numerically solved using a surface vorticitystream function formulation. Numerical simulations show that the Gaussian curvature can generate vorticity, and non-uniformity of the Gaussian curvature is the main cause.In the cylindrical wake, the bump dominated by the positive Gaussian curvature can significantly affect the vortex street by forming velocity depression and changing vorticity transport. The results may provide possibilities for manipulating surface flows through local change in the surface geometry.     

A 3D second‐order accurate projection‐based Finite Volume code on non‐staggered,non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows

It is well known that exact projection methods (EPM) on non‐staggered grids suffer for the presence of non‐solenoidal spurious modes. Hence, a formulation for simulating time‐dependent incompressible flows while allowing the discrete continuity equation to be satisfied up to machine‐accuracy, by using a Finite Volume‐based second‐order accurate projection method on non‐staggered and non‐uniform 3D grids, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by first solving an elliptic equation on a compact stencil that is by performing a standard approximate projection method (APM). In such a way, three sets of divergence‐free normal‐to‐face velocities can be computed. Then, a second elliptic equation for a scalar field is derived by prescribing that its additional discrete gradient ensures the continuity constraint based on the adopted linear interpolation of the velocity. Characteristics of the double projection method (DPM) are illustrated in details and stability and accuracy of the method are addressed. The resulting numerical scheme is then applied to laminar buoyancy‐driven flows and is proved to be stable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

Topological visualisation of focal structures in free shear flows

This paper describes a method for identifying and visualising the three-dimensional geometry of focal (vortex) structures in complex flows. The method is based primarily on the classification of the local topology as it is identified from the values of the velocity gradient tensor invariants. The identification of the local topology is reference frame invariant. Therefore, focal (vortex) structures can be unambiguously identified in these flows. A novel flow visualisation method is introduced whereby focal structures are rendered using a solid model view of the local topology. This new approach is applied to the identification of focal structures in three-dimensional plane mixing layer and plane wake flows.     

Geometrical Properties of the Resolved-Scale Velocity and Temperature Fields Predicted using Large-Eddy Simulation

In this paper, local geometrical properties of the velocity and temperature fields of combined forced and natural convection in a vertical slot are studied using large-eddy simulation based on both numerical and analytical approaches. Previous studies on turbulence geometrical statistics appearing in the literature have primarily focused on either isothermal or isotropic turbulent flows; whereas in this work, we extend the scope of research to investigation of a wall-bounded thermal flow. In particular, we focus on studying the resolved helicity, enstrophy generation, local vortex stretching, and a variety of characteristic geometrical alignment patterns between the resolved velocity, vorticity, temperature gradient, subgrid-scale heat flux and the eigenvectors of the resolved strain rate tensor. In order to quantify the effect of buoyancy on the geometrical properties of the thermal flow field, a systematic comparative analysis has been conducted based on three different flow regimes (i.e., viscous sublayer, buffer layer and logarithmic layer) in both the hot and cold wall regions. The near-wall restriction on the geometrical property of the thermal flow field has been analyzed and some interesting wall-limiting geometrical alignment patterns in the form of Dirac delta functions are also reported.     

Simulation of Dilute Acetone Spray Flames with LES-CMC Using Two Conditional Moments

Large-eddy simulations (LES) have been coupled with a conditional moment closure (CMC) method for the computation of a series of turbulent spray flames. An earlier study by Ukai et al. (Proc. Combust. Inst. 34(1),1643–1650, 2013) gave reasonable results for the prediction of temperature and velocity profiles, but some limitations of the method became apparent. These limitations are primarily related to the upper limit in mixture fraction space. In order to enhance the applicability of the LES-CMC model, this paper proposes a two-conditional moment approach to account for the existence of pre-evaporated fuel by introducing two sets of conditional moments based on different mixture fractions. The two-conditional moment approach is first tested for a non-reacting test case. The results indicate that the spray evaporation induces relatively large conditional fluctuations within a CMC cell, and one set of conditional moments might not be sufficient. The upper limit of the mixture fraction space is dynamically selected for the solution of the second set of conditional moments, and the corresponding CMC solution in a CFD cell is estimated by interpolation between the two conditional moments weighted by the amount of vapour emitted within the domain. The cell-filtered value is given by integration of the conditional moment across mixture fraction space using a bounded β-FDF for the distribution of the scalar. As a result, the fuel concentration profiles given by LES and the two-conditional moment approach are shown to agree well. Then, the two-conditional moment approach is applied to four different flame configurations. The comparison of LES cell quantities and conditionally averaged moments indicates that the two sets of conditional moments are necessary for accurate predictions in zones where gas phase mixture fraction is significantly increased by droplet evaporation within the computational domain. The unconditional temperature profiles clearly show that the new approach improves the predictions of mean temperature especially along the centerline. Also, the better predictions of the temperature field improve the accuracy of the predicted mean axial droplet velocities. Overall, good agreement with the experimental results is found for all four cases, and the methodology is shown to be applicable to flames with a relatively wide range of fuel vapour concentrations.     

Analysis of small-scale scalar mixing processes in highly under-expanded jets

In the present study, we numerically investigate turbulent scalar mixing taking place downstream of highly under-expanded jets. The focus is placed on two inter-related issues: (i) the closure of the mean scalar dissipation rate (SDR) and (ii) the turbulence–scalar interaction (TSI) term. It is indeed commonly admitted that the former, i.e., the SDR, which is defined as the product of the scalar diffusivity with the squared scalar gradient, provides a good measure of the mixing efficiency. In turbulent flows, the mean (turbulent) SDR requires a specific closure to be settled. It is generally obtained within the approximation of a linear relaxation of scalar fluctuations or linear relaxation model. We will first evaluate herein the performance of this widely used closure. The analysis is further developed by means of the consideration of the mean SDR transport equation which shows that, in gaseous conditions, the SDR is mainly driven by two terms: (i) a dissipation contribution and (ii) the third-order correlation between the velocity gradient tensor and small-scale scalar anisotropy tensor. The scalar mixing efficiency thus appears to be controlled by the latter quantity, which is often denoted as the TSI term. It can be shown that only the symmetric part (rate of strain) of the velocity gradient tensor contributes to this term; the anti-symmetric part modifies, indeed, the orientation of the scalar gradient but not its magnitude. The classical approach is to analyze this contribution in the eigenframe of the rate of strain tensor. Such analyses show that, in homogeneous isotropic turbulence, the scalar gradient tends to align with the most compressive direction, thus leading to SDR production. However, the present conditions, which are far from homogeneity and involve strong density variations, may modify this classical picture. The present study analyzes this possible influence.     

A vorticity formulation for computational fluid dynamic and aeroelastic analyses of viscous flows

The paper deals with a novel computational formulation for the analysis of viscous flows past a solid body. The formulation is based upon a convenient decomposition of the flow field into potential and rotational velocity contributions, which has the distinguishing feature that the rotational velocity vanishes in much of, if not all, the region in which the vorticity is negligible. Contrary to related formulations implemented by the authors in the past, in the proposed approach, discontinuities of the potential and rotational velocity fields across a prescribed surface emanating from the trailing edge (such as the wake mid-surface) are eliminated, thereby facilitating numerical implementations. However, the main novelty is related to the application of the boundary condition: first, the expression for the velocity used for the condition on the body boundary is consistent with that for the velocity in the field; also—contrary to related formulations used by the authors in the past—in the proposed approach, the condition on the body boundary does not require the evaluation of the total vorticity (inside and outside the computational domain). The proposed algorithm, valid for three-dimensional compressible flows, is validated—as a first step—for the case of two-dimensional incompressible flows. Specifically, numerical results are presented for the aerodynamic analysis of two-dimensional incompressible viscous flows past a circular cylinder and past a Joukowski airfoil. In order to verify the desirable absence of artificial damping, we present also results pertaining to the flutter (i.e., dynamic aeroelastic) analysis of a spring-mounted circular cylinder in a viscous flow, free to move in a direction orthogonal to the unperturbed flow. In both cases (aerodynamics and aeroelasticity), the results are in good agreement with existing literature data.     

Vortex identification methods based on temporal signal-processing of time-resolved PIV data

The lack of a universally accepted mathematical definition of a vortex structure has led to a considerable number of Eulerian criteria to identify coherent structures. Most are derived from the instantaneous local velocity gradient tensor and its derivatives and require appropriate thresholds to extract the boundaries of the structures. Notwithstanding their great potential for studying coherent structures, most criteria are not frame-independent and they lack a clear physical meaning. The Lyapunov exponent, a popular tool in dynamical system theory, appears as a promising alternative. This Lagrangian criterion does not suffer from the drawbacks of the Eulerian criteria and is constructed on a simple physical interpretation that includes information on the history of the flow. However, since the computation of the Lyapunov exponent involves the knowledge of fluid particle trajectories, experimental applications are currently restricted to laminar flows and two-dimensional turbulence, provided that velocity fields are time-resolved. In this work, we explore temporal post-treatment methods to extract vortical structures developing in a flow through a smooth axisymmetric constriction. Data from planar time-resolved Particle image velocimetry, measuring two or three components of the velocity vectors, are transformed via the Taylor hypothesis to quasi-instantaneous three-dimensional velocity field and are interpreted in terms of the discrete wavelet decomposition, the finite-time Lyapunov exponent, and the linear stochastic estimation. It appears that these methods can concurrently provide very rich and complementary scalar fields representing the effects of the vortical structures and their interactions in the flow.     

Vorticity of the velocity field for flow in a porous medium with random inhomogeneities

The vorticity field of the flow velocity in a porous medium with random inhomogeneities is considered in the correlation approximation of perturbation theory. The correlation tensor of the vorticity, the correlation between the vorticity and the permeability field, and the circulation of the velocity are calculated for three- and two-dimensional flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 157–160, July–August, 1982.