Spectral element-Fourier methods for incompressible turbulent flows

A mixed spectral element-(Fourier) spectral method is proposed for solution of the incompressible Navier-Stokes equations in general, curvilinear domains. The formulation is appropriate for simulations of turbulent flows in complex geometries with only one homogeneous flow direction. The governing equations are written in a form suitable for both direct (DNS) and large-eddy (LES) simulations allowing a unified implementation. The method is based on skew-symmetric convective operators that induce minimal aliasing errors and fast Helmholtz solvers that employ efficient iterative algorithms (e.g. multigrid). Direct numerical simulations of channel flow verified that the proposed method can sustain turbulent fluctuations even at ‘marginal’ Reynolds numbers. The flexibility of the method to efficiently simulate complex-geometry flows is demonstrated through an example of transitional flow in a grooved channel and an example of transitional-turbulent flow over rough wall surfaces.     

多重网格技术对SIMPLER算法的加速性能研究

将多重网格技术引入SIMPLER算法以加快其收敛速度,从而节约计算时间。通过计算不同雷诺数下的二维方腔顶盖驱动流,研究了多重网格方法中的V循环、W循环对SIMPLER算法的加速效果,并讨论了网格层数对加速性能的影响。研究结果表明,在不同雷诺数下,多重网格方法均可以起到良好的加速效果;在相同雷诺数和精度要求下,W循环方式的外迭代次数少于V循环方式的外迭代次数,而且网格层数对多重网格加速性能的影响并不显著。     

Effect of internal mass in the simulation of a moving body by the immersed boundary method

We investigate the effect of internal mass in the simulation of a moving body by the immersed boundary method. In general, the force and the torque acting on the body are influenced by the internal mass, if they are obtained by the negative of the sum of body forces which are applied near the boundary in order to enforce the no-slip condition on the boundary. In this study, the following schemes for approximating the internal mass effect are introduced; no internal mass effect, rigid body approximation, and Lagrangian points approximation. By comparing these schemes through the simulations of a moving body, we examine the internal mass effect. The simulations of the flow around an oscillating circular cylinder and of the sedimentations of an elliptical cylinder and a sphere are performed by using an immersed boundary–lattice Boltzmann method, and it is found that the internal mass effect is significant to unsteady body motions for the Reynolds numbers over 10 and grows as the Reynolds number increases. We also find that for the angular motions of the body, the rigid body approximation causes errors for the rotational Reynolds numbers over 10.     

A zonal RANS-LES method to determine the flow over a high-lift configuration

High Reynolds number flows are particularly challenging problems for large-eddy simulations (LES) since small-scale structures in thin and often transitional boundary layers are to be resolved. The range of the turbulent scales is enormous, especially when high-lift configuration flows are considered. For this reason, the prediction of high Reynolds number flow over the entire airfoil using LES requires huge computer resources. To remedy this problem a zonal RANS-LES method for the flow over an airfoil in high-lift configuration at Re_c=1.0×10⁶ is presented. In a first step, a 2D RANS solution is sought, from which boundary conditions are formulated for an embedded LES domain, which comprises the flap and a sub-part of the main airfoil. The turbulent fluctuations in the boundary layers at the inflow region of the LES domain are generated by controlled forcing terms, which use the turbulent shear stress profiles obtained from the RANS solution. The comparison with an LES solution for the full domain and with experimental data shows likewise results for the velocity profiles and wall pressure distributions. The zonal RANS-LES method reduces the computational effort of a full domain LES by approx. 50%.     

Adherence and bouncing of liquid droplets impacting on dry surfaces

The paper explores liquid drop dynamics over a solid surface, focusing on adherence and bouncing phenomena. The study relies on detailed interface tracking simulations using the Level Set approach incorporated within a Navier–Stokes solver. The investigation deals with moderate Reynolds number droplet flows, for which two-dimensional axisymmetric simulations can be performed. The modelling approach has been validated against experiments for axisymmetric and full three-dimensional impact upon dry surfaces. A drop-impact regime map is generated for axisymmetric conditions, in which the impact dynamics is characterized as a function of Weber number and equilibrium contact angle, based on about 60 simulations. The detailed simulations also helped validate a new mechanistic model based on energy-balance analysis, delimiting the boundary between adherence and bouncing zones at low Weber numbers. The mechanistic model is only valid for moderate droplet Reynolds numbers and it complements existing models for higher Reynolds numbers.     

Inertial thin film flow on planar surfaces featuring topography

A range of problems is investigated, involving the gravity-driven inertial flow of a thin viscous liquid film over an inclined planar surface containing topographical features, modelled via a depth-averaged form of the governing unsteady Navier-Stokes equations. The discrete analogue of the resulting coupled equation set, employing a staggered mesh arrangement for the dependent variables, is solved accurately using an efficient full approximation storage (FAS) algorithm and a full multigrid (FMG) technique; together with error-controlled automatic adaptive time-stepping and proper treatment of the associated nonlinear convective terms. An extensive set of results is presented for flow over both one- and two-dimensional topographical features, and errors quantified via detailed comparisons drawn with complementary experimental data and predictions from finite element analyses where they exist. In the case of one-dimensional (spanwise) topography, moderate Reynolds numbers and shallow/short topographical features, the results obtained are in close agreement with corresponding finite element solutions of the full free-surface problem. For the case of flow over two-dimensional (localised) topography, it is shown that the free-surface disturbance is influenced significantly by the presence of inertia leading, as in the case of spanwise topography, to an increase in the magnitude and severity of the resulting capillary ridge and trough formations: the effect of inclination angle and topography aspect ratio are similarly explored.     

Efficient unsteady high Reynolds number flow computations on unstructured grids

Despite the advances in computer power and numerical algorithms over the last decades, solutions to unsteady flow problems remain computing time intensive. Especially for high Reynolds number flows, nonlinear multigrid, which is commonly used to solve the nonlinear systems of equations, converges slowly. The stiffness induced by the high-aspect ratio cells and turbulence is not tackled well by this solution method.In this paper, it is investigated if a Jacobian-free Newton-Krylov (jfnk) solution method can speed up unsteady flow computations at high Reynolds numbers. Preconditioning of the linear systems that arise after Newton linearization is commonly performed with matrix-free preconditioners or approximate factorizations based on crude approximations of the Jacobian. Approximate factorizations based on a Jacobian that matches the target residual operator are unpopular because these preconditioners consume a large amount of memory and can suffer from robustness issues. However, these preconditioners remain appealing because they closely resemble A^-1.In this paper, it is shown that a jfnk solution method with an approximate factorization preconditioner based on a Jacobian that approximately matches the target residual operator enables a speed up of a factor 2.5-12 over nonlinear multigrid for two-dimensional high Reynolds number flows. The solution method performs equally well as nonlinear multigrid for three-dimensional laminar problems. A modest memory consumption is achieved with partly lumping the Jacobian before constructing the approximate factorization preconditioner, whereas robustness is ensured with enhanced diagonal dominance.     

Evaluation of multigrid acceleration for preconditioned time-accurate Navier-Stokes algorithms

The development of a two-dimensional time-accurate dual time step Navier-Stokes flow solver with time-derivative preconditioning and multigrid acceleration is described. The governing equations are integrated in time with both an explicit Runge-Kutta scheme and an implicit lower-upper symmetric-Gauss-Seidel scheme in a finite volume framework, yielding second-order accuracy in space and time. Issues concerning the implementation of multigrid for preconditioned, dual time step algorithms are discussed. Steady and unsteady computations were made of lid driven cavity flow, thermally driven cavity flow and pulsatile channel flow for a variety of conditions to validate the schemes and evaluate the effectiveness of multigrid for time-accurate simulations. Significant speedups were observed for steady and unsteady simulations. The speedups for unsteady simulations were problem dependent, a function of how rapidly the flow varied in time and the size of the allowable time step.     

A Fourth Order Scheme for Incompressible Boussinesq Equations

A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.     

Application of single-level,pointwise algebraic,and smoothed aggregation multigrid methods to direct numerical simulations of incompressible turbulent flows

Single- and multi-level iterative methods for sparse linear systems are applied to unsteady flow simulations via implementation into a direct numerical simulation solver for incompressible turbulent flows on unstructured meshes. The performance of these solution methods, implemented in the well-established SAMG and ML packages, are quantified in terms of computational speed and memory consumption, with a direct sparse LU solver (SuperLU) used as a reference. The classical test case of unsteady flow over a circular cylinder at low Reynolds numbers is considered, employing a series of increasingly fine anisotropic meshes. As expected, the memory consumption increases dramatically with the considered problem size for the direct solver. Surprisingly, however, the computation times remain reasonable. The speed and memory usage of pointwise algebraic and smoothed aggregation multigrid solvers are found to exhibit near-linear scaling. As an alternative to multi-level solvers, a single-level ILUT-preconditioned GMRES solver with low drop tolerance is also considered. This solver is found to perform sufficiently well only on small meshes. Even then, it is outperformed by pointwise algebraic multigrid on all counts. Finally, the effectiveness of pointwise algebraic multigrid is illustrated by considering a large three-dimensional direct numerical simulation case using a novel parallelization approach on a large distributed memory computing cluster.     

A Robust Multigrid Algorithm for the Simulation of a Yawed Flat Plate

This paper presents a full multigrid solver for the simulation of a flow over a yawed flat plate. The two problems associated with this simulation; boundary layers and entering flows with non-aligned characteristics, have been successfully overcome through the combination of a plane-implicit solver and semicoarsening. In fact, this multigrid algorithm exhibits a textbook multigrid convergence rate, i.e., the solution of the discrete system of equations is obtained in a fixed amount of computational work, independently of the grid size, grid stretching factor and non-alignment parameter. Also, a parallel variant of the smoother based on a four-color ordering of planes is investigated.     

Thin film flow on surfaces containing arbitrary occlusions

The flow of a continuous liquid film, only several microns in thickness, on a plane surface containing occlusions, is modelled using lubrication theory. The resultant time-dependent non-linear equation set is discretised using both finite difference and finite element analogues; the former is solved using a very efficient full approximation storage (FAS) multigrid algorithm employing both automatic spatial and temporal adaptivity, the latter exploits the COMSOL suite of software to solve a weak form of the governing equations. The efficiencies and accuracies of both approaches are compared and a variety of problems, of increasing complexity, explored. Although the adaptive multigrid approach is clearly the most efficient, with the two sets of solutions indistinguishable, COMSOL offers an attractive alternative to the non-specialist user. The results, the first of their kind, further demonstrate: (i) the power of using adaptive multigriding, in time as well as space, for the simultaneous control of both spatial and temporal errors; (ii) the significant reduction in memory requirements and CPU time that can be achieved. It is revealed that occlusions lead to many of the features inherent in the flow of thin liquid films over fully submerged micro-scale topographic features; namely, the presence of capillary ridges linked to “bow-wave” plus “comet-tail” free-surface disturbances.     

Numerical simulation of gas flow in rough microchannels: hybrid kinetic–continuum approach versus Navier–Stokes

The flow field in a rough microchannel is numerically analyzed using a hybrid solver, dynamically coupling kinetic and Navier–Stokes solutions computed in rarefied and continuum subareas of the flow field, respectively, and a full Navier–Stokes solver. The rough surface is configured with triangular roughness elements, with a maximum relative roughness of 5 % of the channel height. The effects of Mach number, Knudsen number (or Reynolds number) and roughness height are investigated and discussed in terms of Poiseuille number and mass flow rate. Discrepancies between full Navier–Stokes and hybrid solutions are analyzed, assessing the range of validity of Navier–Stokes equations provided with first-order slip boundary conditions for modeling gas flow along a rough surface. Results indicate that the roughness increases Poiseuille number and decreases mass flux in comparison with those for the smooth microchannel. Increasing rarefaction results in further enhancement of roughness effect. At the same time, the compressibility effect is more noticeable than the roughness one, although the compressibility effect is alleviated by increase in the rarefaction. It was found that, although the Navier–Stokes solution of the flow in a smooth channel is accurate up to Kn = 0.1, when relative roughness height is higher than 1.25 % significant errors already appear at Kn = 0.02.     

Semicoarsening smoothers for the simulation of a flat plate at yaw

This paper presents a full multigrid solver for the simulation of flow over a yawed flat plate. The two problems associated with this simulation; boundary layers and entering flows with non-aligned characteristics, have been successfully overcome through the combination of a plane-implicit solver and semicoarsening. In fact, this multigrid algorithm exhibits a textbook multigrid convergence rate, i.e., the solution of the discrete system of equations is obtained in a fixed amount of computational work, independently of the grid size, grid stretching factor and non-alignment parameter. Also, a parallel variant of the smoother based on a four-color ordering of planes is investigated.     

Multigrid solution of the steady-state lattice Boltzmann equation

Efficient solution strategies for the steady-state lattice Boltzmann equation are investigated. Stable iterative methods for the linearized lattice Boltzmann equation are formulated based on the linearization of the lattice Boltzmann time-stepping procedure. These are applied as relaxation methods within a linear multigrid scheme, which itself is used to drive a Newton solver for the full non-linear problem. Although the linear multigrid strategy provides rapid convergence, the cost of a linear residual evaluation is found to be substantially higher than the cost of evaluating the non-linear residual directly. Therefore, a non-linear multigrid approach is adopted, which makes use of the non-linear LBE time-stepping scheme on each grid level. Rapid convergence to steady-state is achieved by the non-linear algorithm, resulting in one or more orders of magnitude increase in solution efficiency over the LBE time-integration approach. Grid-independent convergence rates are demonstrated, although degradation with increasing Reynolds number is observed, as in the case of the original LBE time-stepping scheme. The multigrid solver is implemented in a modular fashion by calling an existing LBE time-stepping routine, and delivers the identical steady-state solution as the original LBE time-stepping approach.     

Implicit multigrid computation of unsteady flows past cylinders of square cross-section

An implicit, multigrid scheme has been extended to treat unsteady flows using the concept of temporal subiteration (or dual-time stepping, as it sometimes is called). The efficiency and accuracy of the subiterated, multigrid approach has been discussed in a previous paper. Here, the scheme is applied to compute the unsteady flow past fixed cylinders of square cross-section at moderate Reynolds numbers. The observed pattern of periodic vortex shedding is computed and the dimensionless frequency of this phenomenon (the Strouhal number) is compared with experimentally determined values. Results of coupled aeroelastic computations also are presented that illustrate a hysteresis phenomenon as the structural frequency is varied.     

Hybrid vortex method for high Reynolds number flows around three-dimensional complex boundary

The hybrid vortex method, in which the vortex particle method was combined with the vortex sheet method, was extended to flows around a three-dimensional complex geometry boundary at high Reynolds number of O(10⁶). The computing domain was decomposed into an interior domain of vortex blobs and a thin numerical boundary layer of vortex sheets. The core radii of shedding blobs were related to the size of the vortex sheets. As the result of numerical experiments on the flow over a ship propeller, the hybrid vortex method was found to be acceptable for simulations of unsteady separated flows around a solid body at high Reynolds number, since the computed results showed reasonable agreement with the measured data.     

Optimal relaxation collisions for lattice Boltzmann methods

The optimal relaxation time of about 0.8090 has been proposed to balance the efficiency, stability, and accuracy at a given lattice size of numerical simulations with lattice Boltzmann methods. The optimal lattice size for a desired Reynolds number can be refined by reducing the Mach number for incompressible flows. The functioned polylogarithm polynomials are defined and used to express the lattice Boltzmann equations at different time scales and to analyze the impact of relaxation times and lattice sizes on truncation errors. Smaller truncation errors can be achieved when relaxation times are greater than 0.5 and less than 1.0. The steady-state lid-driven cavity flow was chosen to validate the code of lattice Boltzmann procedures. The applications of the optimal relaxation parameters numerically balance the stability, efficiency, and accuracy through Hartmann flow. The optimal relaxation time can also be used to select the initial lattice size for the channel flow over a square cylinder with a given Mach number.