Lagrangian motion of fluid particles in gravity–capillary standing waves

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period.     

Perturbation analysis of short-crested waves in Lagrangian coordinates

A new second-order asymptotic solution that describes short-crested waves is derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition and there being zero pressure at the free surface, in contrast with the Eulerian solution, in which there is residual pressure at the free surface. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level. The mass transport velocity and Lagrangian mean level associated with particle displacement can also be obtained directly. In particular, the mean level of the particle motion in a Lagrangian form differs that of the Eulerian form. The new formulation reduces to second-order standing or progressive wave solutions in Lagrangian coordinates at the limiting angles of approach. Expressions for kinematic quantities are also presented.     

On frequency–amplitude dependences for surface and internal standing waves

The analytic approach proposed by Sekerzh-Zenkovich [On the theory of standing waves of finite amplitude, Dokl. Akad. Nauk USSR 58 (1947) 551–554] is developed in the present study of standing waves. Generalizing the solution method, a set of standing wave problems are solved, namely, the infinite- and finite-depth surface standing waves and the infinite- and finite-depth internal standing waves. Two-dimensional wave motion of an irrotational incompressible fluid in a rectangular domain is considered to study weakly nonlinear surface and internal standing waves. The Lagrangian formulation of the problems is used and the fifth-order perturbation solutions are determined. Since most of the approximate analytic solutions to these problems were obtained using the Eulerian formulation, the comparison of the results, as an example the analytic frequency–amplitude dependences, obtained in Lagrangian variables with the corresponding ones known in Eulerian variables has been carried out in the paper. The analytic frequency–amplitude dependences are in complete agreement with previous results known in the literature. Computer algebra procedures were written for the construction of asymptotic solutions. The application of the model constructed in Lagrangian formulation to a set of different problems shows the ability to correctly reproduce and predict a wide range of situations with different characteristics and some advantages of Lagrangian particle models (for example, the bigger radius of convergence of an expansion parameter than in Eulerian variables, simplification of the boundary conditions, parametrization of a free boundary).     

Higher-order asymptotic solutions in fluids of various configurations

Applying perturbation methods, symbolic computation, and generalizing the solution method, higher-order asymptotic solutions are constructed in Lagrangian variables for several models describing 2D standing wave motions in fluids of various configurations. Three main parameters of the fluid configuration, depth, capillarity, and stratification layer, are considered. The frequency-amplitude dependences are obtained and compared with those known in the literature in Eulerian and Lagrangian variables. The comparison shows that the analytical frequency-amplitude dependences are in complete agreement with previous results known in the literature and with the results obtained for other models. A generalization allows us to investigate critical phenomena for standing waves in fluids of various configurations. Namely, special attention is focused on critical values of one parameter, the fluid depth. The frequency-amplitude dependences are analyzed from the point of view of critical values: critical points and critical curves are determined for several models describing standing waves in fluids of various configurations.     

Third order approximation to capillary gravity short crested waves with uniform currents

A third-order analytical solution for the capillary gravity short crested waves with uniform current (the main current direction is along the vertical wall) in front of a vertical wall is derived through a perturbation expanding technique. The validity and advantage of the new solution were proved by comparing the results of wave profiles and wave pressures with those of Huang and Jia [H. Huang, F. Jia, The patterns of surface capillary gravity short-crested waves with uniform current fields in coastal waters, Acta Mech. Sinica 22 (2006) 433–441] and Hsu [J.R.C. Hsu, Y. Tsuchiya, R. Silvester, Third-order approximation to short-crested waves, J. Fluid Mech 90 (1979) 179–196]. The important influences of currents on the wave profiles, wave frequency, the ratio of maximum crest height to the total wave height, and wave pressure are investigated for both small-scale (for example, the incident wave wavelength is 9.35 cm) and larger-scale (for example, the incident wave wavelength is 5 m) short crested wave. By numerical computation, we find wave frequency of short crested wave system is greatly affected by incident wave amplitude, incident angle, water depth, surface tension coefficient and the strength of the currents for small-scale incident wave. Furthermore, for the larger-scale short crested wave system, the higher-order solution with uniform current is particularly important for the prediction of wave profile and wave pressure for different water depth and incident angle. Computational results show that with the increase of the current speed, the crests of wave profile and wave pressure become more and more steep. In some cases, the crest value of wave pressure with strong current would be larger about six times than that of no current. Therefore, ocean engineers should consider the short crested wave-current load on marine constructs carefully.     

A numerical study on statistical temporal scales in inertia particle dispersion

The statistical temporal scales involved in inertia particle dispersion are analyzed numerically. The numerical method of large eddy simulation, solving a filtered Navier-Stokes equation, is utilized to calculate fully developed turbulent channel flows with Reynolds numbers of 180 and 640, and the particle Lagrangian trajectory method is employed to track inertia particles released into the flow fields. The Lagrangian and Eulerian temporal scales are obtained statistically for fluid tracer particles and three different inertia particles with Stokes numbers of 1, 10 and 100. The Eulerian temporal scales, decreasing with the velocity of advection from the wall to the channel central plane, are smaller than the Lagrangian ones. The Lagrangian temporal scales of inertia particles increase with the particle Stokes number. The Lagrangian temporal scales of the fluid phase ‘seen’ by inertia particles are separate from those of the fluid phase, where inertia particles travel in turbulent vortices, due to the particle inertia and particle trajectory crossing effects. The effects of the Reynolds number on the integral temporal scales are also discussed. The results are worthy of use in examining and developing engineering prediction models of particle dispersion.     

Exact geophysical waves in stratified fluids

When studying water waves travelling over an inviscid fluid at the Earth's surface there are additional Coriolis and centrifugal forces which influence the motion of the fluid particles. In particular, for waves propagating near the Equator the geophysical wave problem can be modelled by the so-called f-plane approximation. In this paper, we provide an explicit exact solution to the edge wave problem for stratified geophysical flows in the equatorial f-plane approximation.     

A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does not exist Eulerian stable flow which is Lagrangian exponential unstable.We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.     

Material stability analysis of particle methods

Material instabilities are precursors to phenomena such as shear bands and fracture. Therefore, numerical methods that are intended for failure simulation need to reproduce the onset of material instabilities with reasonable fidelity. Here the effectiveness of particle discretizations in reproducing of the onset of material instabilities is analyzed in two dimensions. For this purpose, a simplified hyperelastic law and a Blatz–Ko material are used. It is shown that the Eulerian kernels used in smooth particle hydrodynamics severely distort the domain of material stability, so that material instabilities can occur in stress states that should be stable. In particular, for the uniaxial case, material instabilities occur at much lower stresses, which is often called the tensile instability. On the other hand, for Lagrangian kernels, the domain of material stability is reproduced very well. We also show that particle methods without stress points exhibit instabilities due to rank deficiency of the discrete equations. AMS subject classification 74S30     

Simulation of free turbulent particle-laden jet using Reynolds-stress gas turbulence model

Free two-phase flows occur in many practical applications, such as sprays or particle drying and combustion. This paper deals with mathematical modelling of a free turbulent two-phase jet. A steady, axisymmetric, dilute, monodisperse, particle-laden, turbulent jet injected into a still environment, has been considered. The model treats the gas-phase from an Eulerian standpoint and the motion of particles from a Lagrangian one. Closure of the system of time averaged transport equations has been accomplished by using a Reynolds-stress turbulence model. The particles–fluid interaction has been considered by the PSI-Cell concept. Both the effect of interphase slip and the effect of particle dispersion have been taken into account.     

A hybrid level-set/moving-mesh interface tracking method

An approach for combining Arbitrary–Lagrangian–Eulerian (ALE) moving-mesh and level-set interface tracking methods is presented that allows the two methods to be used in different spatial regions and coupled across the region boundaries. The coupling allows interface shapes to be convected from the ALE method to the level-set method and vice-versa across the ALE/level-set boundary. The motivation for this is to allow high-order ALE methods to represent interface motion in regions where there is no topology change, and the level-set function to be used in regions where topology change occurs. The coupling method is based on the characteristic directions of information propagation and can be implemented in any geometrical configuration. In addition, an iterative method for the hybrid formulation has been developed that can be combined with pre-existing solution methods. Tests of a propagating interface in a uniform flow show that the hybrid approach provides accuracy equivalent to what one is able to obtain with either of the methods individually.     

On peculiarities of the application of Lagrangian variables in problems of time-dependent supersonic flow past bodies

The paper considers particularities in applying Lagrangian variables in problems of hypersonic flow past bodies. It is pointed out that, in problems with intense shock waves, it is advisable to introduce Lagrangian variables as the values of parameters that characterize a particle not on surface t = t ₀ (t ₀ = const) but on surface t = σ, where σ is the time instant at which the particle meets the surface of discontinuity. Considering examples of two-dimensional flow past two-dimensional and axisymmetric bodies moving with high time-dependent velocity, we show that the passage to Lagrangian variables enables us to obtain a system of equations describing the gas flow behind the front of an intense shock wave, which is suitable for the application of the thin-shock-layer flow method. A solution is constructed in the form of series in powers of a small parameter which characterizes the ratio of the gas densities at the front of the leading shock wave. We remark that all nonlinear effects of the problem are concentrated in the equation to determine the law of motion of a particle in zero-order approximation. The cases in which this equation can be integrated are pointed out. For the remaining unknowns, the solution is taken in quadratures. We study the rearrangement of the gas flow in the shock layer when the motion of the body is changed. The flow rearrangement zone is distinguished. Also, a condition to determine the life time of that region (the time of establishment of the new flow regime) is obtained. In a specific case of passing from a steady motion of a wedge to a uniformly accelerated motion, the time of establishment of the uniformly accelerated motion is determined from a quadratic equation.     

Solitary wave solutions to the Isobe-Kakinuma model for water waves

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of a flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

An Eulerian‐Lagrangian control volume scheme for two‐dimensional unsteady advection‐diffusion problems

We develop a mass conservative Eulerian‐Lagrangian control volume scheme (ELCVS) for the solution of the transient advection‐diffusion equations in two space dimensions. This method uses finite volume test functions over the space‐time domain defined by the characteristics within the framework of the class of Eulerian‐Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight‐node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well‐received classes of numerical methods for the equations considered. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

An Eulerian–Lagrangian Weighted Essentially Nonoscillatory scheme for nonlinear conservation laws

We develop a formally high order Eulerian–Lagrangian Weighted Essentially Nonoscillatory (EL‐WENO) finite volume scheme for nonlinear scalar conservation laws that combines ideas of Lagrangian traceline methods with WENO reconstructions. The particles within a grid element are transported in the manner of a standard Eulerian–Lagrangian (or semi‐Lagrangian) scheme using a fixed velocity v. A flux correction computation accounts for particles that cross the v‐traceline during the time step. If v = 0, the scheme reduces to an almost standard WENO5 scheme. The CFL condition is relaxed when v is chosen to approximate either the characteristic or particle velocity. Excellent numerical results are obtained using relatively long time steps. The v‐traceback points can fall arbitrarily within the computational grid, and linear WENO weights may not exist for the point. A general WENO technique is described to reconstruct to any order the integral of a smooth function using averages defined over a general, nonuniform computational grid. Moreover, to high accuracy, local averages can also be reconstructed. By re‐averaging the function to a uniform reconstruction grid that includes a point of interest, one can apply a standard WENO reconstruction to obtain a high order point value of the function. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 651–680, 2017     

Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time

We consider the approximation of the unsteady Stokes equations in a time dependent domain when the motion of the domain is given. More precisely, we apply the finite element method to an Arbitrary Lagrangian Eulerian (ALE) formulation of the system. Our main results state the convergence of the solutions of the semi-discretized (with respect to the space variable) and of the fully-discrete problems towards the solutions of the Stokes system.