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We have developed a new type of simple lattice Boltzmann (LB) model for the compressible Euler equations based on the collisionless kinetic-equation approach proposed by Sone. The model uses the collisionless kinetic equation in the streaming process, and modifies the distribution function to its Chapman-Enskog type at each time step. Compared with the current LB models which solve the kinetic equation of the BGK type, the proposed model is superior in the following two points: (i) Inviscid flows can be computed stably while there is no such model in the current LBM; (ii) The velocity distribution function does not need to be memorized in computation. We calculate various inviscid compressible flows described by the compressible Euler equations using our new one-, two-, and three-dimensional models. Numerical results show the capability of our scheme to simulate high-speed supersonic flows with shock waves.

We have developed a fundamentally new type of simple lattice Boltzmann (LB) model for the compressible Navier‐Stokes equations based on the kinetic system proposed by Sone. The model uses the kinetic equation of free‐molecular type in the streaming process and modifies the distribution function to its Chapman‐Enskog type at each time step. Compared with the current LB models, the proposed model is superior in the following two points: (i) there are no inherent errors associated with the Knudsen number; (ii) any flow parameters, including three transport coefficients, can be chosen freely according to our convenience. Numerical tests and error estimates confirm the above statements.

The mean flow induced by a three-dimensional propagating internal gravity wave beam in a uniformly stratified fluid is studied experimentally and theoretically. Previous related work concentrated on the early stage of mean-flow generation, dominated by the phenomenon of streaming – a horizontal mean flow that grows linearly in time – due to resonant production of mean potential vorticity in the vicinity of the beam. The focus here, by contrast, is on the long-time mean-flow evolution. Experimental observations in a stratified fluid tank for times up to is the beam period, reveal that the induced mean flow undergoes three distinct stages: (i) resonant growth of streaming in the beam vicinity; (ii) saturation of streaming and onset of horizontal advection; and (iii) establishment of a quasi-steady state where the mean flow is highly elongated and stretches in the along-tank horizontal direction. To capture (i)–(iii), the theoretical model of Fan et al. (J. Fluid Mech., vol. 838, 2018, R1) is extended by accounting for the effects of horizontal advection and viscous diffusion of mean potential vorticity. The predictions of the proposed model, over the entire mean-flow evolution, are in excellent agreement with the experimental observations as well as numerical simulations based on the full Navier–Stokes equations.

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

The coupled nonlinear interaction of three-dimensional gravity–inertia internal wavepackets, in the form of beams with nearly monochromatic profile, with their induced mean flow is discussed. Unlike general three-dimensional wavepackets, such modulated nearly monochromatic beams are not susceptible to modulation instability from their inviscid, purely modulation-induced mean flow. However, streaming – the induced mean flow associated with the production of mean potential vorticity via the combined action of dissipation and nonlinearity – can cause cross-beam bending, transverse broadening and increased along-beam decay of the beam profile, in qualitative agreement with earlier laboratory experiments. For wavepackets with general three-dimensional modulations, by contrast, streaming does arise, but plays a less prominent role in the interaction dynamics.

The three-dimensional propagation of internal gravity wave beams in a uniformly stratified Boussinesq fluid is discussed, assuming that variations in the along-beam and transverse directions are of long length scale relative to the beam width. This situation applies, for instance, to the far-field behaviour of a wave beam generated by a horizontal line source with weak transverse dependence. In contrast to the two-dimensional case of purely along-beam variations, where nonlinear effects are minor even for beams of finite amplitude, three-dimensional nonlinear interactions trigger the transfer of energy to a circulating horizontal time-mean flow. This resonant beam–mean-flow coupling is analysed, and a system of two evolution equations is derived for the propagation of a small-amplitude beam along with the induced mean flow. This model explains the salient features of the experimental observations of Bordes et al. (Phys. Fluids, vol. 24, 2012, 086602).

The linear stability of uniform, plane internal wave beams with locally confined spatial profile, in a stratified fluid of constant buoyancy frequency, is discussed. The associated eigenvalue problem is solved asymptotically, assuming perturbations of long wavelength relative to the beam width. In this limit, instability is found only for oblique disturbances which vary in the along-beam and the horizontal transverse directions. The mechanism of instability is a first-harmonic–mean resonant interaction between the underlying wave beam and three-dimensional perturbations that comprise a time-harmonic component, with the beam frequency, and a mean flow. Progressive beams which transport energy in one direction, in particular, are unstable if the beam steepness exceeds a certain threshold value, whereas purely standing beams are unstable even at infinitesimal steepness. A distinguishing feature of this three-dimensional modulational instability is the generation of circulating horizontal mean flows at large distances from the vicinity of the beam.

The instability due to resonant interactions of finite-amplitude water waves is examined in the long-wave limit. In contrast to the well-known case of a small-amplitude limit in which the resonance is considered for a flat surface, here we consider a periodic approximate of the finite-amplitude solitary wave which is the long-wave limit of the periodic wave. The resonance conditions for the corresponding perturbations yield a new family of resonance curves that are totally different from those of the small-amplitude limit obtained by Phillips and Mclean. Under these resonance conditions, we conduct a systematic asymptotic analysis for small wavenumbers to obtain the growth rates of the perturbations explicitly and clarify whether each resonance curve is associated with instability. These results are verified numerically by showing that the instability bands for finite-amplitude periodic waves in shallow water are located along these unstable resonance curves.

In a previous work, Kataoka & Tsutahara (J. Fluid Mech., vol. 512, 2004a, p. 211) proved the existence of longitudinally stable but transversely unstable surface solitary waves by asymptotic analysis for disturbances of small transverse wavenumber. In the present paper, the same transverse instability is examined numerically for the whole range of solitary-wave amplitudes and transverse wavenumbers of disturbances. Numerical results show that eigenvalues and eigenfunctions of growing disturbance modes agree well with those obtained by the asymptotic analysis if the transverse wavenumber of the disturbance is small. As the transverse wavenumber increases, however, the growth rate of the disturbance, which is an increasing function for small wavenumbers, reaches a maximum and finally falls to zero at some finite wavenumber. Thus, there is a high-wavenumber cutoff to the transverse instability. For higher amplitude, solitary waves become longitudinally unstable, and the dependence of the eigenvalues on the transverse wavenumber exhibits various complicated patterns. We found that such eigenvalues versus transverse wavenumber can be simply grouped into three basic classes.

系全体が剛体回転している状況下における密度成層流体の点吸込み流れを静止状態から始まる初期値問題として取り扱い，その流れの特徴を数値的に調べた．吸込みを開始すると同時に慣性内部波が生成され，円筒状に水平方向に広がって伝播する．この波の振舞いを詳細に調べ，選択取水流れが形成されるまでの過程を明らかにした．さらに時間が経過すると，剛体回転の影響により吸込点に流れ込む流体層の厚さ（吸込層厚さ）が徐々に増加していく．この吸込層厚さの増加する速度は粘性の影響を受けると遅くなる．本研究ではこの増加速度が，回転角速度により無次元化した時間を基準に取れば，レイノルズ数のみの関数として表わされ，その他のパラメータには依存しないことを見出した．

The linear stability of finite-amplitude interfacial solitary waves in a two-layer fluid of finite depth is examined analytically on the basis of the Euler equations. An asymptotic analysis is performed, which provides an explicit criterion of instability in the case of long-wavelength transverse disturbances. This result leads to the general statement that, when the amplitude of the solitary wave is increased, the solution becomes transversely unstable before an exchange of longitudinal stability occurs.

An unsteady two-dimensional inviscid blade-vortex interaction has been calculated using the finite difference lattice Boltzmann method of the compressible Euler model. The perturbed discrete Boltzmann equation based on a prescribed vortex approach has been proposed to prevent a vortex from diffusing by numerical dissipation. The discretization of the governing equation is based on a second-order-accurate explicit Runge-Kutta time integration and a fifth-order-accurate upwind scheme that includes additional terms to capture the shock waves clearly. Subsonic and transonic flows around an airfoil were simulated to validate the perturbed discrete Boltzmann equation system. The numerical results were compared with other numerical data, and good agreement has been obtained. In the simulation of the transonic blade-vortex interaction, an instantaneous pressure coefficient, a time history of a lift coefficient, and patterns of acoustic waves were compared with other numerical results, and were found to agree with them very well. We have also investigated a generation mechanism of acoustic wave caused by the blade-vortex interaction, the effect of the flow Mach number, and the influence of the vortex miss distance.

The linear stability of finite-amplitude interfacial gravity solitary waves propagating in a two-layer fluid is investigated analytically focusing on the occurrence of an exchange of stability. We make an asymptotic analysis for small growth rates of infinitesimal disturbances, and explicitly obtain their growth rates near an exchange of stability. The result indicates that an exchange of stability occurs at every stationary value of the total energy of the solitary waves. It also gives us information whether the number of growing modes increases or decreases after experiencing the exchange of stability. We apply these analytical results to specific interfacial solitary waves, and find various features on their stability that are not seen in the case of surface solitary waves.

The linear stability of two-dimensional surface gravity waves on fluid of finite depth is investigated for superharmonic disturbances. For this problem, Zufiria & Saffman suggested that an exchange of stability occurs when the total wave energy becomes stationary as a function of wave speed for fixed 'Bernoulli constant'. In defining the potential energy of the above total wave energy, the surface displacement was measured from the quiescent surface with the same 'Bernoulli constant'. We have re-examined this problem both analytically and numerically

The linear stability of finite-amplitude surface solitary waves with respect to long-wavelength transverse perturbations is examined by asymptotic analysis for small wavenumbers of perturbations. The sufficient condition for the transverse instability is explicitly derived

Selective withdrawal through a line sink of both non-rotating and rotating stratified fluid in a reservoir of finite depth is studied under an assumption that viscosity and diffusivity of the fluid are negligible. This flow is characterized by the two parameters: the Froude number F; and the ratio of the inertial frequency f to the buoyancy frequency N. Following the initiation of discharge from the sink, internal (or inertial) gravity wave modes propagate upstream. We first consider the case of F→0 to get linearized governing equations and seek a linear asymptotic solution for large t ∗ (where t ∗ is time after starting the discharge) which is uniformly valid in space. The obtained solution shows the propagation of the individual modes clearly and it is found that the amplitude of the modal front is kept constant with time t ∗ when f=0, whereas it decreases as t ∗−1/3 when f>0. Next, numerical calculation is conducted to study the case of F>0. Specifically, the validity of both our linear solution and the theory suggested by Clarke and Imberger in describing the mode propagation for F>0 is explored. Investigations are also made of the flow patterns constructed by the passage of these modes. It is then found that the withdrawal-layer thickness shows strong time dependence whose period is about 6/f.

The linear stability of planar solitary waves with respect to long-wavelength transverse perturbations is studied in the framework of the generalized Kadomtsev-Petviashvili equation. It is newly discovered that for some nonlinearities in this family, the solitary waves could be transversely unstable even in a medium with negative dispersion. In the case of positive dispersion, they are found to be always unstable.

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.

The time development of two-dimensional fluid motion induced by a line sink of a rotating linearly stratified Boussinesq fluid in a finite-depth reservoir is numerically studied. The flow is characterized by a Froude number, Fr and the ratio of the inertial frequency, f, to the buoyancy frequency, N. Following the initiation of the sink, the inertial gravity wave modes are generated to propagate upstream, which results in the narrowing of a withdrawal-layer thickness. Attention is focused on investigating the evolution of the mode propagation and the resultant flow patterns for various Fr and f/N. The numerical results clarify that the nature of the mode propagation has little dependence on the parameter Fr if the distance from the sink is larger than a critical one, Nd/f where d is the reservoir's depth. The withdrawal layer thickness for Fr=0 is linearly proportional to the distance from the sink and the nonlinear effect (Fr〓0) can be easily incorporated only by adding the term representing the thickness under no rigid rotation. Furthermore, profiles of the withdrawal layer thickness show strong time-dependence whose period is about 2π/f.

Numerical studies of establishment of selective withdrawal from a point sink of a linearly stratified Boussinesq fluid in a reservour of uniform depth have been performed. The flow, which is axisymmetric with respect to the vertical axis passing through the point sink, is mainly characterized by the Froude number (Fr). When the Froude number is small, the columnar disturbance modes propagate upstream in the same manner as in the line sink flow (two dimensional) case and that the selective withdrawal is established. We have found, however, the major difference in the large Froude number flows, where in the two dimensional case the columnar disturbances can not propagate upstream, while in the axisymmetric case these can propagate upstream for any finite Froude number except infinity. Amplitude of these columnar disturbances relative to a basic current velocity becomes smaller as the Froude number increases so that the selective withdrawal ceases to occur at Fr=1.5.

Thermal free convection is simulated by the Lattice Boltzmann Method using two types of particles, that is red particles and blue ones. The gravitational force is introduced into the collision stage, at which the impulsive force is added by making the population of blue particles moving downward increase. The buoyancy force is taken into consideration in the translating stage of the particles. The velocity field and the density or the temperature field are obatained by averaging these calculated two stage fields. The results agree well with those obtained by the finite difference method.

Numerical investigations are made of establishment of two-dimensional selective withdrawal through a line sink of a linearly stratified Boussinesq fluid in a finite-depth duct. The flow is mainly characterized by a Froude number. When the Froude number is small, the internal wave modes of horizontal wavenumber and frequency 0 which are also called columnar disturbances propagate horizontally upstream against the induced uniform velocity and the selective withdrawal is established. We introduce the new method to specify the propagation of internal wave mode from numerically obtained velocity distribution. Using this method, we investigate its behavior especially focusing our attention on its vertical structure, amplitude and horizontal group velocity. The numerical results show that the amplitude becomes smaller as the Froude number or the mode number increases, and the group velocity of higher modes is inversely proportional to the mode number.

Acoustic waves in the phenomenon of aeolian tone is simulated by the finite difference lattice Boltzmann method (FDLBM) using a model for thermal compressible fluids and a new scheme to stabilize the calculation. Two-dimensional flow past a circular cylinder is calculated and the sound waves whose pressure is 10-3 times smaller than the pressure fluctuation of the flow field are successfully detected. Accuracy of the scheme is also studied in two-dimensional Couette flow.

Rotating fluids are simulated numerically by the lattice Boltzmann method. The 3D 15V incompressible lattice Boltzmann model is applied to three-dimensional fluid flows. In a rotating coordinate system, the effect of Coriolis force is incorporated into the local equilibrium distribution function. The averaging method is used in the calculation. The Taylor column is detected by the simulation. The Ekman spiral and the spin-down are simulated and results agree well with theoretical ones.

The present paper is a more detailed article of the previously published Letter [Phys. Rev. Lett. 84, 3065 (2000)]], which discovered the transversely unstable solitary wave solutions of the generalized Kadomtsev-Petviashvili (GKP) equation with negative dispersion. In addition to detailed explanation of stability analysis to long-wavelength transverse perturbations, numerical calculation of the GKP equation is carried out here to study the transverse stability to perturbations of finite wavelength. The numerical results show that there is a short-wavelength cutoff to the transverse instability. Moreover, we reveal the existence of transversely unstable solitons.

The evolution of selective withdrawal through a point sink of horizontally unbounded, linearly stratified fluid of finite depth is studied as an initial-value problem. Following the initiation of discharge from the sink, internal gravity wave modes propagate radially upstream to change the flow pattern. These modes are called cylindrical modes. We first consider the case of F→0 (where F is the Froude number) to get linearized governing equations, and seek a linear asymptotic solution for large times t* after starting the discharge, of the cylindrical modes in a stratified fluid where viscous and diffusive effects are negligible. The obtained solution shows that the strength of the modal front grows like t*1/3, unlike the case of two-dimensional modes whose strength at the front is kept constant. Numerical calculations are also performed to study the case of F>0. The results then indicate that all the cylindrical modes can propagate upstream for any F except infinity. The steady-state withdrawal-layer thic...