Finite Difference Lattice Boltzmann Method Research Articles

Hybrid multiple-relaxation-time lattice-Boltzmann finite-difference method for axisymmetric multiphase flows

We propose a hybrid lattice-Boltzmann finite-difference method to simulate axisymmetric multiphase flows. The hydrodynamics is simulated by the lattice-Boltzmann equations with the multiple-relaxation-time (MRT) collision model and suitable forcing terms that account for the interfacial tension and axisymmetric effects. The interface dynamics is captured by the finite-difference solution of the convective Cahn–Hilliard equation. This method is applied to simulate a quiescent drop, an oscillating drop, a drop spreading on a dry surface and a drop accelerated by a constant body force. It is validated through comparisons of the computed results for these problems with analytical solutions or numerical solutions by other different methods. It is shown that the MRT-based method is able to handle more challenging cases than that with the single-relaxation-time collision model for axisymmetric multiphase flows due to its improved stability.

G1001 送風機配管系の自励振動により圧縮した空気を駆動源とする船舶用推進装置 (管内に配置した弁の効果)(GS10 超音速・音・振動)

The purpose of this study is to investigate the characteristic of the thrust for the propulsive nozzle driven by the high-pressure air compressed by the self-excited vibration in the blower piping system. In order to increase the performance, two valves were arranged in the piping system. The pressure and the thrust were measured for various parameters such as the number of revolution for the blower, the period of the opening and closing action of the two valves, and the water depth. The following conclusions were obtained. The peak of thrust for the nozzle with valve moved to the side of the deep depth. In the range of the present experiment the value of the thrust increased in comparison with that of original nozzle without valve. The finite difference lattice Boltzmann method is helpful to investigate the characteristics of the flow in the nozzle.

On the Stability of the Finite Difference based Lattice Boltzmann Method

This paper is devoted to determining the stability conditions for the finite difference based lattice Boltzmann method (FDLBM). In the current scheme, the 9-bit two-dimensional (D2Q9) model is used and the collision term of the Bhatnagar- Gross-Krook (BGK) is treated implicitly. The implicitness of the numerical scheme is removed by introducing a new distribution function different from that being used. Therefore, a new explicit finite-difference lattice Boltzmann method is obtained. Stability analysis of the resulted explicit scheme is done using Fourier expansion. Then, stability conditions in terms of time and spatial steps, relaxation time and explicitly-implicitly parameter are determined by calculating the eigenvalues of the given difference system. The determined conditions give the ranges of the parameters that have stable solutions.

A NOVEL LESS DISSIPATION FINITE-DIFFERENCE LATTICE BOLTZMANN SCHEME FOR COMPRESSIBLE FLOWS

In this paper, a new smoothness indicator is proposed to improve the finite-difference lattice Boltzmann method (FDLBM). The necessary and sufficient conditions for convergence are derived. A detailed analysis reveals that the convergence order is higher than that of the previous finite-difference scheme. The coupled double distribution function (DDF) model is used to describe discontinuity flows and verify the improvement. Numerical simulations of compressible flows with shock wave show that the improved finite-difference lattice Boltzmann scheme is accurate and has less dissipation. The numerical results are found to be in good agreement with the analytical results and better than those of the previous scheme.

Development of numerical analysis method of flow-acoustic resonance in stub pipes of safety relief valves

The boiling water reactors (BWRs) have steam dryer in the upper part of the pressure vessel to remove moisture from the steam. The steam dryer in the Quad Cities Unit 2 nuclear power plant was damaged by high-cycle fatigue due to acoustic-induced vibration during extended power uprate operation. The principal source of the acoustic-induced vibration was flow-acoustic resonance at the stub pipes of the safety relief valves (SRVs) in the main steam lines (MSLs). The acoustic wave generated at the SRV stub pipes propagates throughout the MSLs and eventually reaches and damages the steam dryer. Therefore, for power uprate operation of the BWRs, it has been required to predict the flow-acoustic resonance at the SRV stub pipes. The purpose of this article was to propose a numerical analysis method for evaluating the flow-acoustic resonance in the SRV stub pipes. The proposed method is based on the finite difference lattice Boltzmann method (FDLBM). So far, the FDLBM has been applied to flow-acoustic simulations of laminar flows around simple geometries at low Reynolds number. In order to apply the FDLBM to the flow-acoustic resonance simulations of turbulent flows around complicated geometries at the high Reynolds number, we developed computationally efficient model by introducing new function into the governing equation. The proposed method was compared with the conventional FDLBM in the cavity-driven flow simulation. The proposed method was validated by comparisons with the experimental data in the 1/10-scale test of BWR-5 under atmosphere condition. The following three results were obtained; the first is that the proposed method can reduce the computing time by 30% compared with the conventional FDLBM; the second is that the proposed method successfully simulated the flow-acoustic resonance in the SRV stub pipes of the BWR-5, and the pressure fluctuations of the simulation results agreed well with those of the experimental data; and the third is the mechanism of the flow-acoustic resonance in the SRV stub pipes. Acoustic waves causing the flow-acoustic resonance in the SRV stub pipes are generated by the unsteady vortices in the SRV stub pipes.

The finite-difference lattice Boltzmann method and its application in computational aero-acoustics

The application of the finite-difference lattice Boltzmann method in computational aero-acoustics is reviewed, mainly on the basis of the work of the author and his colleagues. Some models of thermal and isothermal fluids are described and the constraints for recovering the Euler equations and the Navier–Stokes equations are described. The arbitrary Lagrangian Eulerian technique is used for high Mach number flows and for simulations of moving bodies. A model of gas–liquid two-phase fluid is introduced in which the density difference is 800 times and the sound velocity difference is 4 times. Some applications of aero-acoustic problems are briefly described and the simultaneous simulation of underwater sound and sound propagating in air is also presented. The difference between the thermal model and the isothermal model is shown in the aero-acoustic problems.

Rotational Slip Flow in Coaxial Cylinders by the Finite-Difference Lattice Boltzmann Methods

AbstractRecent studies on applications of the lattice Boltzmann method (LBM) and the finite-difference lattice Boltzmann method (FDLBM) to velocity slip simulations are mostly on one-dimensional (1D) problems such as a shear flow between parallel plates. Applications to a 2D problem may raise new issues. The author performed numerical simulations of rotational slip flow in coaxial cylinders as an example of 2D problem. Two types of 2D models were used. The first were multi-speed FDLBM models proposed by the author. The second was a standard LBM, the D2Q9 model. The simulations were performed applying a finite difference scheme to both the models. The study had two objectives. The first was to investigate the accuracies of LBM and FDLBM on applications to rotational slip flow. The second was to obtain an experience on application of the cylindrical coordinate system. The FDLBM model with 8 directions and the D2Q9 model showed an anisotropic flow pattern when the relaxation time constant or the Knudsen number was large. The FDLBM model with 24 directions showed accurate results even at large Knudsen numbers.

Unsteady Potential Flow Analysis of Tunnel Compression Wave (Comparison with Computational Results by the Lattice Boltzmann Method)

The generation of compression wave produced when a high-speed train enters a tunnel is studied by solving the two dimensional incompressible irrotational flow over the train nose. The Schwarz-Christoffel conformal mapping is applied to model the train and tunnel portal configuration. The complex velocity potential of the flow field is defined by a distribution of sources on the nose. The contribution from the exit-flow shear layer vortex is estimated by modeling a pair of point vortices placed at short distances from the corner of the tunnel portal and imposing the Kutta condition that the velocity should be finite at the corner. The unsteady potential equation is solved to obtain the pressure field at each step of train translation. The results are compared with numerical calculations of the pressure rise obtained by the finite difference lattice Boltzmann method (FDLBM), which recovers the compressible Navier-Stokes equation of fluids. It is shown that the contribution from the exit-flow vortex increases up to approximately 7 % during the time in which the train nose enters the tunnel. The prediction with point vortices shows close agreement with the FDLBM.

Immersed Boundary-Finite Difference Lattice Boltzmann Method for Liquid-Solid Two-Phase Flows

An immersed boundary method based on a finite difference lattice Boltzmann method (IB-FDLBM) is presented. The FDLBM solves the discrete Boltzmann equation including an additional collision term by using finite difference schemes. The additional term works as a negative viscosity in the macroscopic level and allows us to alter the fluid viscosity while keeping the other relevant parameters of the simulation fixed. The immersed boundary method employs a direct-forcing method, which utilizes external forces at Lagrangian points embedded on immersed boundaries to impose the no-slip boundary condition. Several benchmark simulations are carried out to validate the developed method, i.e., flows past a circular cylinder, a falling particle, and interaction between two falling particles. Couette flows between a stationary and a rotating cylinder are also simulated at various values of the relaxation time for collision. The main conclusions obtained are as follows: (1) steady flows past a stationary circular cylinder are well predicted, (2) the motions of particles falling through liquids predicted using IB-FDLBM quantitatively agree well with those obtained using immersed boundary methods based on the lattice Boltzmann equation (IB-LBM), (3) the developed method well predicts the interaction between two particles falling through a liquid, e.g., the drafting-kissing-tumbling motion, and (4) distortion of velocity fields in circular Couette flows at high relaxation times is removed by the additional collision term.

Relationship between accuracy and number of velocity particles of the finite-difference lattice Boltzmann method in velocity slip simulations

Relationship between accuracy and number of velocity particles in velocity slip phenomena was investigated by numerical simulations and theoretical considerations. Two types of 2D models were used: the octagon family and the D2Q9 model. Models have to possess the following four prerequisites to accurately simulate the velocity slip phenomena: (a) equivalency to the Navier–Stokes equations in the N-S flow area, (b) conservation of momentum flow Pxy in the whole area, (c) appropriate relaxation process in the Knudsen layer, and (d) capability to properly express the mass and momentum flows on the wall. Both the octagon family and the D2Q9 model satisfy conditions (a) and (b). However, models with fewer velocity particles do not sufficiently satisfy conditions (c) and (d). The D2Q9 model fails to represent a relaxation process in the Knudsen layer and shows a considerable fluctuation in the velocity slip due to the model’s angle to the wall. To perform an accurate velocity slip simulation, models with sufficient velocity particles, such as the triple octagon model with moving particles of 24 directions, are desirable.

Finite Difference Lattice Boltzmann Method for Compressible Thermal Fluids

a splitting method to solve the modeled lattice Boltzmann equation. The splitting technique permits the boundary conditions for the lattice Boltzmann equation to be set as conveniently as those required for the finite difference solution of the Navier–Stokes equations. It is shown that the compressible Navier–Stokes equation can be recovered fully from this approach; however, the formulation requires the solution of a Poisson equation governing a secondorder tensor. Thus constructed, the method has no arbitrary constants. The proposed method is used to simulate thermalCouette flow,aeroacoustics,andshockstructureswithanextendedthermodynamicsmodel.Thesimulations are carried out using a high-order finite difference scheme with a two-dimensional, nine-velocity lattice. All simulations are performed using a single relaxation time and a set of constants deduced from the derivation. It is found that the finite difference lattice Boltzmann method is able to correctly replicate viscous effects in thermal Couette flows,aeroacoustics,andshockstructures.Thesolutionsobtainedareidenticaleithertoanalyticalresults,or obtained by solving the compressible Navier–Stokes equations using a direct numerical simulation technique.

Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows

With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.

Finite Difference Lattice Boltzmann Method Applied to Acoustic-Scattering Problems

This paper reports on an attempt to simulate acoustic waves scattering using a finite-difference lattice Boltzmann method based on an alternative lattice equilibrium particle distribution function constructed for compressible thermal fluids. The studies focus on acoustics scattering by a zero-circulation vortex and by an isolated thermal source with no heat gain/loss. Two limiting cases of each type of scattering are examined; one is the case of an incoming acoustic wave with a short wavelength, and the other has a relatively long wavelength compared with the characteristic dimension of the obstacle. These scattering problems have been treated previously using a conventional lattice Boltzmann method and a gas-kinetic scheme. The results showed that these methods were only able to simulate the short wavelength limit case with fair accuracy for the two types of acoustics scattering considered. Because the present approach is able to recover the compressible Navier―Stokes equations with correct fluid properties, the finite-difference solution of the proposed alternative modeled lattice Boltzmann equation allows the limiting cases of the acoustics scattering problems to be calculated without numerical instability. The results thus obtained are in agreement either with analysis or with results obtained from direct aeroacoustics simulations employing the compressible Navier―Stokes equations.

Direct simulation of Aeolian tones emitted from a circular cylinder in transonic flows using the finite difference lattice Boltzmann method

We present an application of the finite difference lattice Boltzmann method to direct simulations of aerodynamic sound in transonic flows. The arbitrary Lagrangian Eulerian formulation is introduced and the Aeolian tones emitted from a moving circular cylinder are successfully simulated. It is shown that the sound sources change depending on the flow Mach numbers as follows: the oscillating flow just behind the cylinder is the sound source for low Mach number flows. For flow of Mach numbers about 0.7, in which the supersonic region appears without shock waves, an additional source, interaction between the vortices, becomes significant. For Mach numbers about 0.9, a shock wave appears and interaction between the vortex and the weak shock wave is shown to be the sound source.

差分格子ボルツマン法による円柱周りの放射音解析(<小特集>マルチフィジックスCFD/EFDの最前線)

In this paper, we investigated an advantage of the finite difference lattice Boltzmann method (FDLBM) to direct numerical simulations of sound propagation including scattering by a body. In order to incorporate, monopole sound source into the flow, a new FDLB formulation was proposed using the Chapman-Enskog expansion. We calculated the acoustic scattering problem around a circular cylinder using the FDLBM discretized by non-compact finite difference scheme which is faster than compact finite difference scheme. The results agreed with both the exact solution and the accurate calculation by the macroscopic equations based on the discretizations of compact finite difference scheme. As a result, the FDLBM was proved to have a merit to a calculation of sound propagation. Also, we estimated error of the FDLBM in the acoustic pulse problem.

1017 差分格子ボルツマン法による気液界面と音との相互作用に関する研究(OS-5 可変形性境界を含む流れ(1))

1017 差分格子ボルツマン法による気液界面と音との相互作用に関する研究(OS-5 可変形性境界を含む流れ(1))

差分格子ボルツマン法によるトンネル圧縮波の発生と伝播計算(流体工学,流体機械)

Direct numerical simulation of compression wave produced when a high-speed train enters a tunnel, distortion of the compression wave front as it travels in the tunnel, radiation of micropressure wave from the tunnel exit are performed using the finite difference lattice Boltzmann method. The discrete Boltzmann equation for the 3D39Q thermal BGK model is solved in three-dimensional space using a second-order Runge-Kutta scheme in time and a third-order-upwind finite difference scheme in space. The arbitrary Lagrangian-Eulerian formulation (ALE) is applied to model the interaction of the moving train nose and the tunnel portal. Detailed numerical calculations were carried out for axisymmetric trains with the blockage of 0.2 and various nose profiles entering a long circular cylindrical tunnel with straight and stepwise flared portals. The predicted compression wave profiles are found to be in good agreement with linear theory predictions obtained from the analytic expression derived by Howe. It is shown that the distortion of the compression wave front is consistent with the time-domain computation of one-dimensional Burgers equation. Longer nose profiles and tunnel entrance with flared portals are confirmed not only to decrease the initial steepness of the compression wave front but also to counteract the effect of nonlinear steepening.

High velocity flow simulation using lattice Boltzmann method with no-free-parameter dissipation scheme

A finite difference lattice Boltzmann method of second-order accuracy in time is developed based on non-oscillatory scheme with no-free-parameter dissipation (NND) difference scheme in this paper. The NND lattice Boltzmann method is used to simulate high-speed flows by constructing a new equilibrium distribution function of the lattice Boltzmann method. Compared with a variation of lattice Boltzmann method developed by Qu, et al., the present method can capture shock waves and handle oscillations of high velocity flows accurately in larger time steps and in shorter computing time. Numerical results indicate the correctness and capability of simulating shock wave interactions of the NND lattice Boltzmann method.

Numerical simulation of sound generation in a mixing layer by the finite difference lattice Boltzmann method

We simulated aerodynamic sound in a two-dimensional mixing layer using the finite difference lattice Boltzmann method (FDLBM). We introduced a finite difference scheme, called the dispersion relation preserving (DRP) scheme, into the FDLBM to carry out an accurate simulation of aerodynamic problems. The scheme is designed such that the dispersion relation of the finite difference scheme is the same as that of the original partial differential equations and is useful for acoustic simulations. A turbulent flow field was simulated by large-eddy simulation (LES), using the Smagorinsky model, and the results were compared with those from a direct simulation based on the Navier–Stokes equations to confirm the usefulness of this method. The combination of the FDLBM and the DRP scheme was shown to be very effective for direct simulations of aerodynamic sound.

Direct simulation of sound and underwater sound generated by a water drop hitting a water surface using the finite difference lattice Boltzmann method

The sound and underwater sound emitted from a water drop colliding with a water surface are simulated by a new model of the finite difference lattice Boltzmann method. The two-particle immiscible fluid model is modified to simulate sound in the gas phase and underwater simultaneously. In the very early stage after the collision, sounds propagating into the gas and liquid phases are successively detected, and the effects of drop shape and gas bubbles are also observed.