free surface flow

Now, we will summaries the three main results of this Chapter. First, the linear free surface flow problem has mixed boundary conditions, and is therefore much more difficult to solve than the well-studied problems of flow over a bottom topography, or flow due to a pressure distribution, for example. Even taking into account the normal issues arising from applying the Wiener–Hopf technique, a challenging feature of the present problem is that for plate shapes that have slopes with compact support (dη/dx = 0 for x < − L with L > 0), we are unable to close the contour of integration in the more natural upper half-plane, and instead have to deal with the significant algebra that accompanies closing the contour in the lower half-plane (note that this aspect was not present in the flat plate problem considered by McCue and Stump [85], since the function J (k) was identically zero in that case). Furthermore, an additional subtlety associated with the present problem is that the term J + (k) provides the leading order behaviour in (2.42) for

Sco = γ /a ρ g . which is the reciprocal of Bo, or 1/Bo, but it has a different physic connotation during the free surface flow flowing in magnetic field. When Sco is about 1, the free surface flow will be in the full-cover free surface flow. The Ha 2 Ca/Fr is defined as rivulet flow index to the superficial layer of full-cover free surface flow, Rin, or 3 2

Peterson, R.C., Jimack, P.K. and Kelmanson, M.A. (2001) On the stability of viscous free-surface flow supported by a rotating cylinder. Proceedings of the Royal Society Series A: Mathematical, Physical and Engineering Sciences, 457 (2010). pp. 1427-1445. ISSN 1471-2946

In this paper we have introduced an adaptive algorithm for the solution of three-dimensional moving-boundary problems based primarily upon the use of mesh movement but also exploit- ing discrete remeshing when necessary. Previous work in this area has been typically based upon linear tetrahedral or hexahedral finite element approximations, however here the gov- erning equations are discretised with an a priori stable tetrahedral Taylor-Hood finite element method. This method requires no additional artificial stabilisation and includes an isopara- metric quadratic approximation of the problem geometry allowing a more accurate model of the evolving free surface and of the surface tension forces acting on it. Application of this algorithm to two fully three-dimensional problems has been described in order to demonstrate the significant potential of the approach. At this stage, however, further development is still required in order to ensure that this potentially powerful approach can be made sufficiently robust to be applied to the widest possible range of free-surface flow problems.

In recent years there has been a significant interest in the computational study of such flows using an arbitrary Lagrangian–Eulerian ( ale ) finite el- ement methods and it is this approach that is pursued here. Cairncross et al. [3] used a linear hexahedral finite element method to solve the incom- pressible Navier–Stokes equations and also introduced a dynamic-contact- angle model [1] to describe the evolution of a coating flow. B¨ansch [2] de- veloped and analysed a tetrahedral Taylor–Hood finite element method for the Navier–Stokes equations. This model included a static contact angle, allowing fluid slip along solid boundaries, but did not account for dynamic- contact-angle effects. Zhou and Derby [15] describe a linear tetrahedral fi- nite element model for the Stokes equations and apply this to the sintering of two spherical particles. Here, as with to B¨ansch [2], a three-dimensional incompressible free-surface flow solver based upon the use of implicitly stable elements (the so-called Taylor–Hood element) is developed. These isopara- metric elements represent the three-dimensional free surface using piecewise quadratics which is of particular significance when the curvature-dependent surface-tension effects are important.

ANSYS ICEM is used for the high quality hexa mesh generation. To generate the structured grid with hexahedral cells commercially available grid generation software ANSYS ICEM CFD V 12.1 is used. Free surface computations require fine mesh in the region around the free surface so as to capture the free surface flow, which can be observed in figures. Mesh on the ship hull is shown in fig 1.Total number of cells in the fluid region: 1427564. Minimum mesh quality – 0.27

The sketch of the geometrical arrangement of the channel with an inclined backward- facing step is given in Fig. 1. The distance of the edge of the inclined step from the channel entry was 0.9 m. The height of the inclined step is H = 100 mm. The whole length of the investigated section was 2.1 m. Boundary conditions for the numerical simulation were examined in detail in the cross-sections x = -0.9 and -0.6 m using the PIV and LDA methods. At the chosen mean bulk velocity, the turbulence level in the stream core was about 2.5 %. The PIV measurements were taken at vertical and horizontal planes parallel with the channel axis with the aim to determine the general view of the flow with separation and secondary flows behind the step as well. In the mean vertical plane, profiles of mean and fluctuation longitudinal velocities were determined by the LDA method at selected sections x = const. The measurement of the subcritical free-surface flow was carried out for the mean bulk velocity U m = 0.38 m/s

The use of boundary-conforming finite-element methods is considered for the solution of surface-tension- dominated free-surface flow problems in three dimensions. This class of method is based upon the use of a moving mesh whose velocity is driven by the motion of the free surface, which is in turn determined via a kinematic boundary condition for the normal velocity. The significance of the method used to compute the normal direction at the finite-element node points for a C 0 piecewise-polynomial free surface is investigated. In particular, it is demonstrated that the concept of mass-consistent normals on an isoparametric quadratic tetrahedral mesh is flawed. In this case an alternative, purely geometric, normal is shown to lead to a far more robust numerical algorithm. Copyright c ­ 2003 John Wiley & Sons, Ltd.

Slug ow is the most complicated pattern in a two- phase ow and includes extreme conditions. Mc- Corquodale and Hamam [6] simulated the transition from free-surface to pressurized ow by assuming a hypothetical, stationary air pocket inside the pipe. Using their assumption on the fully developed slug ow model, the ow is then divided into three rigid water columns with uniform velocities (Figure 3). Each water column is assumed to be enclosed by a xed control volume. Continuity and momentum equations are then derived for each water column, the interface between columns and the headwater (at the upper reservoir).

To vary the mesh density along the domain volumetric controls were built which work in conjunction with volume shapes. These volume shapes encompass more computationally demanding and computationally important spaces of the mesh, e.g., the space around the bow and around the stern, as well as the space around the free surface up to the height of the generated waves. Through the volumetric controls, those spaces, covered by the shapes, were specified to have a more refined mesh.

Complementary indications on modelling strategies are pro- vided by dimensional analysis, to delineate the domains of validity of the selected flow models (NS, RANS, SV, or ASV) across their multiple spatiotemporal scales of appli- cation but in a powerful scale-independent analysis. Jus- tifications for the use of dimensionless numbers may be sought in the developments of similitude laws (Fourier, 1822; Rayleigh, 1877; Bertrand, 1878; Vaschy, 1892; Riabouchin- sky, 1911), later extended to dimensional analysis, provid- ing guidance for the sizing of experimental facilities used in reduced-scale modelling as well as more general arguments for the choice of adequate sets of dimensionless quantities (Buckingham’s (1914) π-theorem; Bridgman, 1922, 1963; Langhaar, 1951; Barenblatt, 1987). Throughout history, the establishment of dimensionless numbers has led to the recog- nition of contextually dominant terms in the flow equations, rendering them prone to dedicated simplifications, provided these would not be used outside their conditions of validity, following successive hypotheses made during their deriva- tion. On the one hand the dimensionless numbers arise in the non-dimensionalization of the systems of governing equa- tions, being an inherent feature of the model. On the other hand only the selected dimensionless numbers appear in the non-dimensional formulation of the equations, from appro- priate arrangements of their terms, and this choice indicates which are the physical processes of interest for the modeller. Finally, not all dimensionless numbers can be made explicit

The research examines an influence of a platform shape on free surface waves generated behind a semi-infinite two-dimensional platform moving with a constant speed on a water surface of a finite depth h. The fluid is assumed to be inviscid, incompressible and irrotational; the surface tension effect is neglected. The aim of the research is to find analytically and numerically such platform shape which minimizes generated waves and reduces wave drug exerting on a moving platform when the Froude number is relatively small, F < 1. It is shown that for certain platform shapes, generated waves can be minimised or even eliminated, at least, within the framework of a linearized theory.

[2001] that if the water surface pattern is not associated with gravity waves then the spatial correlation pattern is related to the underlying turbulence. It is now useful to examine the spatial correlation parameters listed in Table 3 as a function of the corresponding hydraulic parameters listed in Table 1 for the 16 experimental flow conditions. A simple examina- tion of the data in Table 1 shows that the root mean square of the water surface elevation depends almost linearly with the depth-averaged velocity and flow depth. This simple rela- tionship also indicates that there is likely to be a physical connection between the bulk flow parameters and the water surface pattern. From Table 3 it can be seen that the values of the characteristic period and the correlation radius generally increase with depth although this pattern is less clear. Given the range of slope, velocity, and depth combinations, it was possible to determine generalized nondimensional relation- ships between the surface wave pattern and the underlying flow. Figure 12 shows that the nondimensional characteristic period demonstrates a strong consistent nonlinear relation- ship with the ratio of depth-averaged flow velocity to shear velocity. This figure indicates that the spatial characteristic length generally carries information on the shape of the ver- tical velocity profile and the underlying bed roughness for a range of hydraulic conditions. Figure 13 shows that the correlation radius, which reflects the dissipation of the sur- face pattern, can be nondimensionalized with the equivalent hydraulic roughness and can be shown to be a linear function of the flow Reynolds number and so carries a clear physi- cal sense that the spatial organization of the water surface waves is related to the dissipation of energy by the turbulent flow. Although the main aim of this paper was to study and quantify the spatial correlation patterns of free surface waves in shallow rough flows, the data collected and the analyti- cal function used to describe the spatial correlation provide clear evidence of a link between the surface pattern and the underlying turbulence and character of the shearing flow. By comparing the length scale of the water surface waves

Abstract. Most of the existing models of speleogenesis are limited to situations where flow in all conduits is pressurized. The feedback between the distribution of hydraulic head and growth of new solution conduits determines the geometry of the resulting conduit network. We present a novel modeling approach that allows a transition from pressurized (pipe) flow to a free-surface (open-channel) flow in evolving discrete conduit networks. It calculates flow, solute transport and dis- solution enlargement within each time step and steps through time until a stable flow pattern is established. The flow in each time step is calculated by calling the US Environmen- tal Protection Agency Storm Water Management Model (US Environmental Protection Agency, 2014), which efficiently solves the 1-D Saint-Venant equations in a network of con- duits. Two basic scenarios are modeled, a low-dip scenario and a high-dip scenario. In the low-dip scenario a slightly in- clined plane is populated with a rectangular grid of solution conduits. The recharge is distributed to randomly selected junctions. The results for the pressurized flow regime resem- ble those of the existing models. When the network becomes vadose, a stable flow pathway develops along a system of conduits that occupy the lowest positions at their inlet junc- tions. This depends on the initial diameter and inlet position of a conduit, its total incision in a pressurized regime and its alignment relative to the dip of the plane, which plays im- portant role during the vadose entrenchment. In the high-dip scenario a sub-vertical network with recharge on the top and outflow on the side is modeled. It is used to demonstrate the vertical development of karst due to drawdown of the water table, development of invasion vadose caves during vadose

to hold stability. If the real fluid surface prediction falls below this tolerance, the cell is perceived to be dry and no water is within the cell. Bradford and Sanders (2002b) justifies this because the model accurately predicts wave propagation. Whereas with other models (Anastasiaduo-Partheniou, Banti & Aissis 2010; Zhang and Cundy 1989; Playan et al. 1994) which also assume a thin layer of water result in the incorrect propagation of waves. For example, bores that reach the shoreline will collapse and intrude on a dry beach as a depression wave but where there is a thin layer of water on the shore, a model will wrongly predict that the bore will travel up the beach Bradford and Sanders (2002b). In addition, such models will have instability when grid cells become dry during a simulation Bradford and Sanders (2002b).

has been analysed both in the steady state (e.g. Coyle, Macosko & Scriven 1986; Gaskell, Savage & Thompson 1998b) and to predict the onset of interfacial instability (e.g. Coyle, Macosko & Scriven 1990). Figure 1 shows the experimental realization of the flow and the associated computational domain. The rolls are half-submerged in a bath of liquid and counter-rotate to produce a narrow ‘bead’ of liquid in the small gap between them. A thin film is drawn out of the bead by each roll, and under normal operating conditions a substrate would be wound around one of the rolls so that it is continuously coated. Generally, a recirculation zone is present in the ‘film-split’ region, as indicated in figure 1. Regions of recirculation are undesirable in coating flows (Gaskell et al. 1999; Gutoff 1993), however, since fluid trapped inside them can degrade and form solid deposits whose later ejection is detrimental to the quality of the finished coating. In some cases it may be possible to redesign the geometry of the coater to reduce the occurrence of such features (Noakes et al. 2002), but where this is not possible it is of interest to investigate possibilities for inducing fluid exchange between the recirculation region and the surrounding flow. The method of flow modulation is one potential means of achieving this.

Flow over submerged obstacles is one of the classical problems in fluid mechanics. This problem has many related physical applications ranging from the flow of water over rocks to atmospheric, and oceanic stratified flows encountering topographic obstacles, or even a moving pressure distribution over a free surface. Free surface flows over an obstacle have been investigated for different bottom topography by many researchers. Forbes and Schwartz [3] used the boundary integral method to find fully nonlinear solutions of subcritical and supercritical flows over a semi-circular obstacle. Their results confirmed and extended Lamb’s solutions. In 2002, Dias and Vanden-Broeck [2] found a new solution called the ”generalized hydraulic fall”. Such solutions are characterized by downstream supercritical flow and a train of waves on the upstream side. This type of solution can be obtained by removing the radiation condition on the far upstream of the obstacle. Forbes [3] calculated numerical solutions of gravity-capillary flows over a semicircular obstruction. The fluid was subject to the combined effects of gravity and surface tension. Three different branches of solution were presented and compared between linear and fully nonlinear problems. In this work we compute accurate numerical solutions for the fully nonlinear problem. The problem is first formulated as an integral equation for the unknown shapes of the free surface. This equation is then discretized and the resulting algebraic equations are solved by Newton’s method. Later on, we found numerical solutions of

A hydrodynamic channel, of length 2 m, width 20 cm and depth 15 cm, contains a free surface flow generated by an adjustable flow supply. The channel is partitioned by vertical partitions with a thickness of 10 mm and a height of 7 cm separated from each other by 25 cm. A honeycomb conditioner is placed at the inlet of the channel to obtain a weak turbulent flow. A cylinder with a mass of 50 grams, a length of 15 cm, a diameter of 4 cm is placed between two compartments and is capable of oscillating in a pendulum movement. The horizontal polystyrene hollow cylinder is connected at its two ends to an axis of rotation by two rods of 30 cm long. The axis is itself rigidly connected to the side walls of the channel by an adjustable link in vertical and horizontal translation. This set is placed in a display tank that continuously feeds the hydrodynamic channel. The objective of the work is first to experimentally characterise the velocity field around the cylinder and its dynamics by measuring its displacement as well as the applied forces. In a second step, we propose a numerical model simulating the fluid-structure interaction resulting from the wake oscillator model. The originality of the work lies in the fact that the cylinder is placed in a confined flow limited by the bottom, the two vertical walls and the free surface and that its initial position is regulated by a quantity of water inside.

small positive quantity predicted for uncontaminated films ) and for these surface elasticity fulfils a destabilising role. Thia component reinforces the tangential stress component due to the air flow and together they cause liquid to be dragged towards the crests and away from the troughs of a snail wave-like disturbance ( see figure 2.1). The approximate condition for instability is then a simple kinematic one: namely, that there should be a net horizontal volume flux towards the crests and away from the troughs of this disturbance. This destabilising role of surface elasticity

Recently,,V.Ravikumar.et.al(2012) investigated the heat and mass transfer effects on MHD flow of viscous incompressible and electrically conducting fluid through a non homogeneous porous medium in the presence of heat source, oscillatory suction velocity. A uniform transverse magnetic field is applied in the direction of the flow perpendicular to the plates. S. K. Ghosh(2012) study the boundary layer flow of a steady incompressible and visco- elastic fluid with short memory (obeying Walters’ B fluid model) passing over a hot vertical porous plate in the presence of transverse magnetic field. In addition Tiwari and Kapoor(2011) has done some work in this direction but this work is basically the extension of Rawat and Kapoor(2012) in which they considered the viscous model of the above problem they also shown the effect of radiation in there study. Here we have taken the porous model without the taken the external effect.