2D brine sewage after impinging on a shallow sea free surface

For a strength analysis of under-water (hydrofoil) and above-water wings, the magnitude of the overloads during impact on the free fluid surface must be known. Incidence of bodies of different shapes on a fluid surface has been investigated in [1–4]. Theoretical and experimental results on determining the impact overloads during flat plate incidence on a free fluid surface are contained in this paper and the possibility of approximately modeling this phenomenon is indicated.

Numerical solutions are derived for a viscous, buoyant drop spreading below a free fluid surface. The drop has zero interfacial tension, and we consider viscosity contrasts 0.1 < λ < 10 with the surrounding fluid half-space. The density contrast between the drop and outer fluid is assumed to be small compared with the density contrast at the fluid surface. The numerical solutions for the approach and initial spread of the drop below the fluid surface are obtained using the boundary integral method. To facilitate an investigation over a larger range of viscosity contrasts and for longer time periods, we solve for the motion of gravity currents at the fluid surface. For this geometry we also solve the boundary integral equations for the cases λ = 0 and 1/λ = 0.For extensive drop spreading, the motion is described by asymptotic solutions. Three asymptotic solutions are derived, which apply for different values of the viscosity contrast relative to the aspect ratio ((radial extent R)/(drop thickness a)). For very low-viscosity drops (λ [Lt ] a/R[ln(R/a)]-1), the greatest resistance to spreading occurs at the drop rim, and the asymptotic solution is found using slender body theory. Drops with intermediate viscosity contrast (a/R [Lt ] λ [Lt ] R/a) are slowed primarily by shear stresses at the lower drop surface, and a lubrication solution (Lister & Kerr 1989) applies. The greatest resistance to the spread of very viscous drops (λ [Gt ] R/a) comes from the radial stresses within the drop, and the asymptotic solution is independent of the outer fluid. All drops having 0 [Lt ] λ [Lt ] ∞ will eventually spread according to lubrication theory, when their aspect ratio becomes sufficiently large relative to viscosity contrast.Theoretical results are compared with numerical and experimental results for drops and gravity currents spreading at a fluid surface. The solutions can be applied to aspects of planetary mantle flow where temperature variations cause significant viscosity contrasts. The low-viscosity solution has been applied to study the encounter of a hot, low-viscosity upwelling plume with a planet surface (Koch 1994). Here we apply the high-viscosity asymptotic solution to study how cold downwelling slabs spread at a depth of neutral buoyancy in the Earth's mantle.

We investigated theoretically and experimentally internal solitary waves (ISWs) in a two-layer fluid system with a top free surface. Laboratory experiments are performed by lock-release, under Boussinesq and non-Boussinesq conditions. Experimental results are compared with those obtained by the analytical solution of the Korteweg–de Vries (KdV) weakly nonlinear equation and by the strongly nonlinear Miyata-Choi-Camassa (MCC) model. We analyze the initial conditions which allow to find soliton solutions for both rigid-lid (-RL) and free-surface (-FS) boundary conditions. For the MCC-FS model, we employ a new mathematical procedure to derive the ISW-induced free surface displacement. The density structure strongly affects the elevation of the free surface predicted by the MCC-FS model. The free surface maximum displacement increases mostly with the density difference, assuming non-negligible values also for smaller interfacial amplitudes. Larger displacements occur for thinner upper layer thickness. The MCC-FS model gives the best prediction in terms of both internal waves geometric/kinematic features and surface displacements. The existence of a free surface allows the ISW to transfer part of its energy to the free surface: the wave celerity assumes lower values with respect to ISW speed resulting from the MCC-RL model. For ISWs with a very large amplitude, this behavior tends to fade, and the MCC-RL and the MCC-FS model predict approximately the same celerity and interfacial geometric features. For small-amplitude waves also, the predictions of the KdV-RL equation are consistent with experimental results. Thus, ISWs with an intermediate amplitude should be modeled taking into account a free top surface as the boundary condition.

We propose an approach for temporally coherent patch‐based texture synthesis on the free surface of fluids. Our approach is applied as a post‐process, using the surface and velocity field from any fluid simulator. We apply the texture from the exemplar through multiple local mesh patches fitted to the surface and mapped to the exemplar. Our patches are constructed from the fluid free surface by taking a subsection of the free surface mesh. As such, they are initially very well adapted to the fluid's surface, and can later deform according to the free surface velocity field, allowing a greater ability to represent surface motion than rigid or 2D grid‐based patches. From one frame to the next, the patch centers and surrounding patch vertices are advected according to the velocity field. We seek to maintain a Poisson disk distribution of patches, and following advection, the Poisson disk criterion determines where to add new patches and which patches should e flagged for removal. The removal considers the local number of patches: in regions containing too many patches, we accelerate the temporal removal. This reduces the number of patches while still meeting the Poisson disk criterion. Reducing areas with too many patches speeds up the computation and avoids patch‐blending artifacts. The final step of our approach creates the overall texture in an atlas where each texel is computed from the patches using a contrast‐preserving blending function. Our tests show that the approach works well on free surfaces undergoing significant deformation and topological changes. Furthermore, we show that our approach provides good results for many fluid simulation scenarios, and with many texture exemplars. We also confirm that the optical flow from the resulting texture matches the fluid velocity field. Overall, our approach compares favorably against recent work in this area.

We consider a thin fluid film flowing down an inclined substrate subjected to localized external sources of momentum and heat flux that induce deformations of the fluid's free surface. This scenario is encountered in several industrial processes and of particular interest is the case where these deformations are undesirable. When the substrate is thin and the temperature along its underside is freely imposed by an active cooling mechanism, temperature gradients are generated at the fluid surface which drive a thermocapillary flow and influence the deformations. This naturally leads us to pose the optimal control problem of choosing the temperature profile that minimizes the unwanted free-surface deformations. Numerical computations reveal that the external forces generate deflections in a region near their peak beyond which all deflections are suppressed by the optimal control. Where nonzero deflections occur, the control is of bang-bang type (taking either its upper or lower bound), while the control is obtained in closed form for regions where the deflections are suppressed. Strikingly, in switching between these regions the optimal control chatters, that is, it switches infinitely many times over a finite interval. By appealing to Pontryagin's maximum principle and leveraging a symmetry embedded in the adjoint problem we uncover the underlying fractal structure of the chattering. Finally, we present practical approaches to avoid the infinite switching while retaining significantly reduced free-surface deformations.

A transformation of gravitational waves in fluid of constant depth with a crushed ice layer floating on the free fluid surface is considered. The propagating waves undergo a slight damping along their path of propagation. The main goal of the study is to construct an approximate descriptive model of this phenomenon.With regard to small displacements of the free surface, a viscous type model of damping is considered, which corresponds to a continuous distribution of dash-pots at the free surface of the fluid. A constant parameter of the dampers is assumed in advance as known parameter of damping. This parameter may be obtained by means of experiments in a laboratory flume.

Droplet impact on solid surface is a common natural phenomenon and industrial process which is also a complex polyphase conditions coupling process with gas, liquid and solid. And during the droplet’s deformation and movement, it is often affected by the wettability of solid wall. In this paper, a lagrange SPH numerical method, with the attraction of the van der Waals state equation added to simulate the surface tension of the droplet, is applicated to simulate the process of single droplet impact on wettable surface and investigate the influence mechanism of surface wettability to the droplet’s deformation and movement. Several improvements for traditional SPH methd are presented such as, a Lagrange wettable solid wall boundary condition of which solid wall pariticles’ hydrophilia and capillary action are unified supposed to be a adsorption force to the liquid particles of support domains. The force is deemed relevant to fluid pressure, saturation and solid wall’s hydrophilia. Then a stress correction method is proposed for the stress instability caused by the SPH kernel function. The method uses two different shapes of kernel functions to calculate the tensile and compressive stresses, respectively. After that, a SPH model compiled in Fortran language and based on the above two improved methods is established to simulate static and dynamic droplets’ deformation processes on different wettable wall. The following studies were carried out based on the simulation results. Firstly, the effect of stress correction method is investigated by the comparison between simulating results with the stress correction before and after used. Secondly, the validity of the wettable solid wall boundary condition is studied based on the simulation results of static droplet deformation. Finally, the influence of the wall wettability on the droplet movement process after the collision is analyzed. Researches show that, the stress correction method improves the stress instability problem in the traditional SPH method, even in the large deformation movement can still maintain a uniform particle distribution. In addition, the stress correction method can better simulate the surface tension of free surface flow, and can get a more smooth free surface. According to the change characteristics of droplet’s static contact angle, the mentioned wettable solid wall boundary condition can clearly reflect the wall’s wettability. Simulation of doplet impact on wettable wall shows good agreement with the experiment result. During rebound stage, a large enough wettability will induce droplet deforming into liquid column. The wettability have a little influence to droplet’s spreading stage, the wall friction force plays a major role. While during retraction and rebound stages, the influence of wettability is obvious. This study improves the theory of the interaction between free surface fluid and solid wall, and can provide a reference for further study of the effects between free surface fluids and wettable media or discrete particles.

When two cylinders are counter-rotated at low Reynolds number about parallel horizontal axes below the free surface of a viscous fluid, the rotation being such as to induce convergence of the flow on the free surface, then above a certain critical angular velocity Ωc, the free surface dips downwards and a cusp forms. This paper provides an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques. Surface tension effects are included, but gravity is neglected. The solution is analytic for finite capillary number [Cscr ], but the radius of curvature on the line of symmetry on the free surface is proportional to exp (−32π[Cscr ]) and is extremely small for [Cscr ] [gsim ] 0.25, implying (in a real fluid) the formation of a cusp. The equation of the free surface is cubic in (x, y) with coefficients depending on [Cscr ], and with a cusp singularity when [Cscr ] = ∞.The influence of gravity is considered through a stability analysis of the free surface subjected to converging uniform strain, and a necessary condition for the development of a finite-amplitude disturbance of the free surface is obtained.An experiment was carried out using the counter-rotating cylinders as described above, over a range of capillary numbers from zero to 60; the resulting photographs of a cross-section of the free surface are shown in figure 13. For Ω < Ωc, a rounded crest forms in the neighbourhood of the central line of symmetry; for Ω > Ωc, the downward-pointing cusp forms, and its structure shows good agreement with the foregoing theory.

The stability of capillary-gravity wave motion on horizontal free surface of viscous noncompressible fluid in the presence of magnetic surfactant in an external magnetic field was studied. It is shown that for normal as well as for tangential external magnetic field the horizontal free liquid surface is unstable for field strength exceeding some critical value that does not depend on the elastic constant of the surfactant film. However, for oblique external magnetic field the stability of the free surface depends not only on the field value but also on the surfactant elastic constant.

Novel numerical method to simulate hydrodynamic characteristic of moving body through free surface and stratified-fluid interface

The profiles of standing gravity waves of maximum height, parametrically excited on the free surface of a deep fluid in a vertically oscillating rectangular vessel (Faraday waves), are investigated experimentally. For a small modulation index of the excitation parameter, waves of three types are distinguished: regular, temporally periodic and symmetric about the vertical line passing through their crest; irregular but retaining the connectivity of the liquid volume; and breaking waves with drops separating from the free surface of the fluid. It is established that the profile of the maximum-height regular waves is smooth with a steepness of 0.255 and a limiting angle at the crest of less than 80°. Certain realizations of irregular and breaking waves, with profiles similar to those of regular waves but with much smaller steepnesses, 0288 and 0.429, respectively, are detected.

In this study, the total velocity potential, for the swaying and rolling cylinder expressed as a sum of a series of linear multi-pole potentials and a dipole potential situated at the origin for incom- pressible, inviscid and irrotational flow. The non-dimensional expression of the dipole potential and its conjugated stream functions are re-obtained by using the mathematical expressions with complete alge- braic manipulations. The total velocity potential UT ðx; y; tÞ, for the swaying and rolling cylinder which is a periodic harmonic function of time t, may be expressed as a sum of a series of linear multi-pole potentials and a dipole potential situated at the origin (1),(2). The x-axis is horizontal which coincides with the free surface of the fluid and is perpendicular to the axis of the cylinder and the y-axis is vertical, positive downward and going through the mean position of the axis of the cylinder. De Jong (1) formulated the problem as follows: In a fluid of infinite depth a cylinder is considered which is oscil- lating one-dimensionally and harmonically with a frequency of x while the mean posi- tion of its axis is assumed to lie in the free surface of the undisturbed fluid. De Jong assumed the amplitude of the oscillation to be small with respect to the diam- eter of the cylinder and the length of the waves generated by the oscillation, so that in the linearized approximation, the values of all physical quantities could be referred to the mean position of the cylinders. Taking the cylinder very long with respect to the breadth or enclosing the cylinder at both ends between two infinitely long walls per- pendicular to the axis of the cylinder, De Jong neglected the velocity components par- allel to the axis of the cylinder and consequently the motion is two-dimensional. The determination of the motions of the fluid under influence of the harmonic oscillation of the cylinder could be reduced to the solution of a boundary-value prob- lem from the linear potential theory. So, the velocity potential UT ðx; y; tÞ is also a periodic harmonic function of time. Therefore, the potential expressed is given in the following form, using complex notation, UTðx; y; t Þ¼� i/ T ðx; yÞe ixt : ð1Þ

1. Many investigations have been made to determine the wave resistance acting on a body moving horizontally and uniformly in a heavy, perfect fluid. Lamb obtained a first approximation for the wave resistance on a long circular cylinder, and this was later confirmed to be quite sufficient over a large range. In 1926 and 1928, Havelock (4, 5) obtained a second approximation for the wave resistance and a first approximation for the vertical force or lift. Later, in 1936(6), he gave a complete analytical solution to this problem, in which the forces were expressed in the form of infinite series in powers of the ratio of the radius of the cylinder to the depth of the centre below the free surface of the fluid. General expressions for the wave resistance and lift of a cylinder of arbitrary cross-section were found by Kotchin (7) using integral equations, and the special case of a flat plate was evaluated. He continued with a discussion of the motion of a three-dimensional body. More recently, Haskind (3) has examined the same problem when the stream has a finite depth.

The formation of capillary ridges is the typical features of thin viscous or viscoelastic fluids over a locally heated plate. This ridge leads to the nonuniformity in the thin film coating. In this work, the formation of capillary ridges on the free surface of thin second-grade non-Newtonian fluid flowing over an inclined heated plate is discussed. The flow is modelled by two-dimensional laws of conservation of mass, momentum, and energy with corresponding boundary conditions at the plate and the free surface. An evolution equation for the description of the liquid thin film height is derived from the two-dimensional balance equations using the long-wave approximation. The resulting nonlinear dynamic equation is discretised implicitly on a uniform grid using the finite volume method. The obtained results on the capillary ridge in the free surface are discussed for the different flow parameters. It is noted that the capillary ridge height is higher for the second-grade viscoelastic fluid in comparison to the Newtonian one. This study can be a starting point to investigate the influence of second-grade viscoelastic parameter on the free surface instability and other phenomena of interest.

The effects of a sphere and a cylinder sinking beneath the surface of a viscous fluid are examined, and two results are found: (1) the horizontal divergence of the velocity at the surface of the fluid is related to the measured free-air anomaly if the viscosity of the fluid is uniform, and (2) the vertical load supported at the surface is essentially independent of any vertical variation of the viscosity of the fluid. The vertical load supported at the surface, Fzz, or the magnitude of the depression if the surface is a free surface, is found by an image solution. The solution formally has a pressure term and a velocity term. If the viscosity is constant, we find through an identity of integrals that the contribution to the surface load represented by the pressure term exactly cancels the gravitational attraction of the density inhomogeneities beneath the surface for any arbitrary density distribution. The net attracting mass, which produces the net free-air anomaly, is then related only to the velocity divergence. This theorem is limited in its application because it is sensitive to viscosity variations with depth. It is found that Fzz is independent of the viscosity of the fluid and also of the boundary conditions of the top surface—a free surface or a rigid plate give the same Fzz. A two-layer model in which a fluid of one viscosity lies over a fluid of another viscosity is examined, and it is found that Fzz is relatively independent of the thickness of the top layer or the ratio of the two viscosities, especially if the top layer has the greater viscosity. Whereas the horizontal velocity at the surface is very dependent upon the assumed viscosity pattern, the vertical load Fzz is not, and this enables us to find the approximate mass of a sinking body and its depth beneath the surface independently of the viscosity pattern of the earth.