Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

A lattice Boltzmann flux solver (LBFS) is presented in this work for simulation of incompressible viscous and inviscid flows. The new solver is based on Chapman-Enskog expansion analysis, which is the bridge to link Navier-Stokes (N-S) equations and lattice Boltzmann equation (LBE). The macroscopic differential equations are discretized by the finite volume method, where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh, tie-up of mesh spacing and time interval, limitation to viscous flows. LBFS is validated by its application to simulate the viscous decaying vortex flow, the driven cavity flow, the viscous flow past a circular cylinder, and the inviscid flow past a circular cylinder. The obtained numerical results compare very well with available data in the literature, which show that LBFS has the second order of accuracy in space, and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.     

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter ν = 10^?3, the two-dimensional flow past a cylinder at Reynolds number Re = 200, and the three-dimensional flow past a cylinder at Reynolds number Re = 1000.     

Rectangular Lattice Boltzmann Equation for Gaseous Microscale Flow

The lattice Boltzmann equation (LBE) is considered as a promising approach for simulating flows of liquid and gas. Most of LBE studies have been devoted to regular square LBE and few works have focused on the rectangular LBE in the simulation of gaseous microscale flows. In fact, the rectangular LBE, as an alternative and efficient method, has some advantages over the square LBE in simulating flows with certain computational domains of large aspect ratio (e.g., long micro channels). Therefore, in this paper we expand the application scopes of the rectangular LBE to gaseous microscale flow. The kinetic boundary conditions for the rectangular LBE with a multiple-relaxation-time (MRT) collision operator, i.e., the combined bounce-back/specular-reflection (CBBSR) boundary condition and the discrete Maxwell's diffuse-reflection (DMDR) boundary condition, are studied in detail. We observe some discrete effects in both the CBBSR and DMDR boundary conditions for the rectangular LBE and present a reasonable approach to overcome these discrete effects in the two boundary conditions. It is found that the DMDR boundary condition for the square MRT-LBE can not realize the real fully diffusive boundary condition, while the DMDR boundary condition for the rectangular MRT-LBE with the grid aspect ratio a≠1 can do it well. Some numerical tests are implemented to validate the presented theoretical analysis. In addition, the computational efficiency and relative difference between the rectangular LBE and the square LBE are analyzed in detail. The rectangular LBE is found to be an efficient method for simulating the gaseous microscale flows in domains with large aspect ratios.     

Lattice Boltzmann modeling of microchannel flow in slip flow regime

We present the lattice Boltzmann equation (LBE) with multiple relaxation times (MRT) to simulate pressure-driven gaseous flow in a long microchannel. We obtain analytic solutions of the MRT-LBE with various boundary conditions for the incompressible Poiseuille flow with its walls aligned with a lattice axis. The analytical solutions are used to realize the Dirichlet boundary conditions in the LBE. We use the first-order slip boundary conditions at the walls and consistent pressure boundary conditions at both ends of the long microchannel. We validate the LBE results using the compressible Navier–Stokes (NS) equations with a first-order slip velocity, the information-preservation direct simulation Monte Carlo (IP-DSMC) and DSMC methods. As expected, the LBE results agree very well with IP-DSMC and DSMC results in the slip velocity regime, but deviate significantly from IP-DSMC and DSMC results in the transition-flow regime in part due to the inadequacy of the slip velocity model, while still agreeing very well with the slip NS results. Possible extensions of the LBE for transition flows are discussed.     

A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows

We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss–Lobatto–Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge–Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re = 400–5000 and flows around an impulsively started cylinder for Re = 550–9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.     

A deconvolution enhancement of the Navier–Stokes-αβ model

A deconvolution enhancement of the Navier–Stokes-αβ model for turbulent flow is introduced. The energy and energy-dissipation rate for the enhanced model are derived. It is also shown that the consistency error, relative to the Navier–Stokes equations, and the microscale of the enhanced model are less than those of the Navier–Stokes-αβ model. The proposed model is used to simulate the Taylor–Green vortex problem and results show a qualitatively improved representation of the mean-square vorticity when compared to the Navier–Stokes-αβ model. Numerical studies of the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor for three-dimensional turbulent flows with Re = 200 are used to explore the utility of the model. A benchmark problem of a two-dimensional channel flow over a step for Re = 600 also indicates that this model can be applied to more general flows than those involving periodic boundary conditions.     

A high order Discontinuous Galerkin – Fourier incompressible 3D Navier–Stokes solver with rotating sliding meshes

We present the development of a sliding mesh capability for an unsteady high order (order ? 3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier–Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular–quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian–Eulerian form of the incompressible Navier–Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x–y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier–Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier–Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.     

A generalized wall boundary condition for smoothed particle hydrodynamics

In this paper we present a new formulation of the boundary condition at static and moving solid walls in SPH simulations. Our general approach is both applicable to two and three dimensions and is very simple compared to previous wall boundary formulations. Based on a local force balance between wall and fluid particles we apply a pressure boundary condition on the solid particles to prevent wall penetration. This method can handle sharp corners and complex geometries as is demonstrated with several examples. A validation shows that we recover hydrostatic equilibrium conditions in a static tank, and a comparison of the classical dam break simulation with state-of-the-art results in literature shows good agreement. We simulate various problems such as the flow around a cylinder and the backward facing step at Re = 100 to demonstrate the general applicability of this new method.     

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.     

On the application of the two-level large-eddy simulation method to turbulent free-shear and wake flows

We present application of the hybrid two-level large-eddy simulation (TLS-LES) method, a multi-scale simulation model, to turbulent free-shear and wake flows at moderately high Reynolds number. The TLS-LES method combines the scale-separation-based two-level simulation (TLS) model with the spatial-filtering-based conventional large-eddy simulation (LES) model in an additive manner using a normalised blending function. The additive blending can be performed in a static or a dynamic manner. We demonstrate that the method, which has been originally developed for wall-bounded flows, can be used to simulate flows in complex configurations without requiring any further adjustments to the model. In this study, three canonical flows are simulated, which are representative of free-shear and wake flows. These cases include a temporally evolving mixing layer, flow past a circular cylinder in a uniform flow and flow past a finite-span airfoil placed in a uniform flow at three different angle of attacks. We analyse the role of static and dynamic blending functions, large-scale grid resolution and the effect of small scales on the instantaneous flow features and turbulence statistics. The results obtained from these cases demonstrate robustness, accuracy and consistency of the multi-scale TLS-LES method and show that the method is suitable for investigation of turbulent flows that encompass features such as massive separation, reattachment, transition to turbulence and unsteady wake, which are challenging to model numerically.     

A computational study of the HyShot II combustor performance

Experimental and flight data for hypersonic air-breathing engines are both difficult and extremely expensive to obtain, motivating the use of computational models to enhance the understanding of the complex physics involved. Here, a comprehensive numerical study has been carried out for the HyShot II scramjet combustor. This study makes use of Reynolds Average Navier Stokes (RANS) based models and Large Eddy Simulation (LES) based models with semi-detailed reaction kinetics. In this investigation we focus on the underlying flow-mixing-combustion physics at different operating conditions tested in the High Enthalpy Shock Tunnel Göttingen (HEG). To account for the complex flow in the HEG facility a zonal approach is employed in which RANS is used to simulate the flow in the HEG nozzle and test-section, providing the necessary inflow boundary conditions for the combustor RANS and LES, being the focus of this analysis. Specifically, we here combine results from RANS and LES computations with data from the HEG experiments and the target HyShot II flight-tests at two different flight-altitudes (28 and 33 km). The LES model is observed to capture the experimental wall-pressure and heat-flux data very well for both the 33 and 28 km altitude cases, whereas the RANS model is only able to predict the wall-pressure and heat flux data for the 28 km altitude case. Based on the LES results, the flow at both altitudes is found to be unsteady, but with unsteady transitional flow features dominating the 33 km case. Moreover, these results show that the equivalence ratio is of key importance to the resulting flow, mixing and combustion physics, with richer mixtures being prone to transitional flow features. The LES results are also used to describe the flow physics in detail for both altitudes, and the key processes responsible for the transition between the two combustion modes observed.     

Nested Cartesian grid method in incompressible viscous fluid flow

In this work, the local grid refinement procedure is focused by using a nested Cartesian grid formulation. The method is developed for simulating unsteady viscous incompressible flows with complex immersed boundaries. A finite-volume formulation based on globally second-order accurate central-difference schemes is adopted here in conjunction with a two-step fractional-step procedure. The key aspects that needed to be considered in developing such a nested grid solver are proper imposition of interface conditions on the nested-block boundaries, and accurate discretization of the governing equations in cells that are with block-interface as a control-surface. The interpolation procedure adopted in the study allows systematic development of a discretization scheme that preserves global second-order spatial accuracy of the underlying solver, and as a result high efficiency/accuracy nested grid discretization method is developed. Herein the proposed nested grid method has been widely tested through effective simulation of four different classes of unsteady incompressible viscous flows, thereby demonstrating its performance in the solution of various complex flow–structure interactions. The numerical examples include a lid-driven cavity flow and Pearson vortex problems, flow past a circular cylinder symmetrically installed in a channel, flow past an elliptic cylinder at an angle of attack, and flow past two tandem circular cylinders of unequal diameters. For the numerical simulations of flows past bluff bodies an immersed boundary (IB) method has been implemented in which the solid object is represented by a distributed body force in the Navier–Stokes equations. The main advantages of the implemented immersed boundary method are that the simulations could be performed on a regular Cartesian grid and applied to multiple nested-block (Cartesian) structured grids without any difficulty. Through the numerical experiments the strength of the solver in effectively/accurately simulating various complex flows past different forms of immersed boundaries is extensively demonstrated, in which the nested Cartesian grid method was suitably combined together with the fractional-step algorithm to speed up the solution procedure.     

A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows

An immersed boundary method (IBM) with second-order spatial accuracy is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method is based on the computationally efficient direct-forcing method of Uhlmann [M. Uhlmann, An immersed boundary method with direct forcing for simulation of particulate flows, J. Comput. Phys. 209 (2005) 448–476] that is embedded in a finite-volume/pressure-correction method. The IBM consists of two grids: a fixed uniform Eulerian grid for the fluid phase and a uniform Lagrangian grid attached to and moving with the particles. A regularized delta function is used to communicate between the two grids and proved to be effective in suppressing grid locking. Without significant loss of efficiency, the original method is improved by: (1) a better approximation of the no-slip/no-penetration (ns/np) condition on the surface of the particles by a multidirect forcing scheme, (2) a correction for the excess in the effective particle diameter by a slight retraction of the Lagrangian grid from the surface towards the interior of the particles with a fraction of the Eulerian grid spacing, and (3) an enhancement of the numerical stability for particle–fluid mass density ratios near unity by a direct account of the inertia of the fluid contained within the particles. The new IBM contains two new parameters: the number of iterations N_s of the multidirect forcing scheme and the retraction distance r_d. The effect of N_s and r_d on the accuracy is investigated for five different flows. The results show that r_d has a strong influence on the effective particle diameter and little influence on the error in the ns/np condition, while exactly the opposite holds for N_s. A novel finding of this study is the demonstration that r_d has a strong influence on the order of grid convergence. It is found that for spheres the choice of r_d = 0.3Δx yields second-order accuracy compared to first-order accuracy of the original method that corresponds to r_d = 0. Finally, N_s = 2 appears optimal for reducing the error in the ns/np condition and maintaining the computational efficiency of the method.     

A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows

A momentum exchange-based immersed boundary-lattice Boltzmann method is presented in this Letter for simulating incompressible viscous flows. This method combines the good features of the lattice Boltzmann method (LBM) and the immersed boundary method (IBM) by using two unrelated computational meshes, an Eulerian mesh for the flow domain and a Lagrangian mesh for the solid boundaries in the flow. In this method, the non-slip boundary condition is enforced by introducing a forcing term into the lattice Boltzmann equation (LBE). Unlike the conventional IBM using the penalty method with a user-defined parameter or the direct forcing scheme based on the Navier–Stokes (NS) equations, the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. Numerical examples show that the present method can provide very accurate numerical results.     

An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows

The recently proposed boundary condition-enforced immersed boundary-lattice Boltzmann method (IB-LBM) [14] is improved in this work to simulate three-dimensional incompressible viscous flows. In the conventional IB-LBM, the restoring force is pre-calculated, and the non-slip boundary condition is not enforced as compared to body-fitted solvers. As a result, there is a flow penetration to the solid boundary. This drawback was removed by the new version of IB-LBM [14], in which the restoring force is considered as unknown and is determined in such a way that the non-slip boundary condition is enforced. Since Eulerian points are also defined inside the solid boundary, the computational domain is usually regular and the Cartesian mesh is used. On the other hand, to well capture the boundary layer and in the meantime, to save the computational effort, we often use non-uniform mesh in IB-LBM applications. In our previous two-dimensional simulations [14], the Taylor series expansion and least squares-based lattice Boltzmann method (TLLBM) was used on the non-uniform Cartesian mesh to get the flow field. The final expression of TLLBM is an algebraic formulation with some weighting coefficients. These coefficients could be computed in advance and stored for the following computations. However, this way may become impractical for 3D cases as the memory requirement often exceeds the machine capacity. The other way is to calculate the coefficients at every time step. As a result, extra time is consumed significantly. To overcome this drawback, in this study, we propose a more efficient approach to solve lattice Boltzmann equation on the non-uniform Cartesian mesh. As compared to TLLBM, the proposed approach needs much less computational time and virtual storage. Its good accuracy and efficiency are well demonstrated by its application to simulate the 3D lid-driven cubic cavity flow. To valid the combination of proposed approach with the new version of IBM [14] for 3D flows with curved boundaries, the flows over a sphere and torus are simulated. The obtained numerical results compare very well with available data in the literature.     

A self-gain random distributed feedback fiber laser based on stimulated Rayleigh scattering

We reported a narrow linewidth (~ 4 kHz) fiber laser with a mirror-less open cavity based on stimulated Rayleigh scattering (STRS) in a non-uniform fiber. Because of its variable core size and dispersion along the fiber, the threshold of the stimulated Brillouin scattering was increased by ~ 7 dB compared with that of conventional single mode fiber, which allows higher order Rayleigh scattering. The self-gain is initiated by the spontaneous Rayleigh scattering and amplified via STRS, and the distributed feedback mechanism is formed by different orders of Rayleigh scattering counter-propagating as the “random mirror reflection” in the non-uniform fiber.     

A case study of localization and identification of noise sources from a pitch and a stall regulated wind turbine

We have experimentally identified the noise-generation mechanisms of large modern upwind wind turbines (WTs). First, the sound measurement procedures of IEC 61400-11 were used in the field test, and noise emissions from two WTs were evaluated: a stall-controlled WT with powers of 1.5 MW and a pitch-regulated WT with powers of 660 kW. One-third octave band levels were normalized using the scale law for the velocity dependence of the inflow broadband noise and airfoil self-noise. The results showed that for the 1.5 MW WT, inflow turbulence noise was dominant over the whole frequency range. For the 660 kW WT, the inflow broadband noise did not contribute across the whole audible frequency range. The distribution of noise sources in the rotor plane was visualized using a beam-forming measurement system (B&K 7768, 7752, and WA0890) consisting of 48 microphones. The array results for the 660 kW WT indicated that all noise was produced during the downward movement of the blades. This finding was in good agreement with theoretical results obtained using an empirical formula that includes the effects of the convective amplification, directivity, and flow-speed dependence of the turbulence boundary-layer trailing edge noise. This agreement implies that this trailing edge noise is dominant over the whole frequency range in the case of the 660 kW WT.     

A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers

We present a validation study for the hybrid particle-mesh vortex method against a pseudo-spectral method for the Taylor–Green vortex at Re_Γ = 1600 as well as in the collision of two antiparallel vortex tubes at Re_Γ = 10,000. In this study we present diagnostics such as energy spectra and enstrophy as computed by both methods as well as point-wise comparisons of the vorticity field. Using a fourth order accurate kernel for interpolation between the particles and the mesh, the results of the hybrid vortex method and of the pseudo-spectral method agree well in both flow cases. For the Taylor–Green vortex, the vorticity contours computed by both methods around the time of the energy dissipation peak overlap. The energy spectrum shows that only the smallest length scales in the flow are not captured by the vortex method.In the second flow case, where we compute the collision of two anti-parallel vortex tubes at Reynolds number 10,000, the vortex method results and the pseudo-spectral method results are in very good agreement up to and including the first reconnection of the tubes. The maximum error in the effective viscosity is about 2.5% for the vortex method and about 1% for the pseudo-spectral method. At later times the flows computed with the different methods show the same qualitative features, but the quantitative agreement on vortical structures is lost.     

Strain engineering tunable nanotemplates on the Cu(110)–(2 × 1)–O surface

We present an STM study on the domain pattern formation of the Cu(110)–O surface. We found that the separation of the oxygen adsorbate phase into the domain pattern is consistent with a phenomenological model of size-dependent elastic relaxation of the strained surface. We developed a thermally assisted oxygen adsorption procedure aiming to control the size of the two-dimensional (2 × 1)–O islands nucleated at the surface under oxygen adsorption serving as precursors for domain pattern formation. We engineered wide range tuneable (2 × 1)–O domain patterns by controlling the nuclei size and the oxygen coverage at the pattern formation stage.