Steady Free Surface Research Articles

Методология влияния свободной порозности и водоотдачи на процессы, происходящие в почвогрунтах оросительных каналов с земляным руслом

Purpose of the research: to consider the influence of free porosity and water loss on the filtration of water from channels with an earthen bed. Explore existing calculation methods taking into account the parameters of free porosity and fluid loss. Collect calculation material that can form the basis of a program for calculating the filtration of channels with an earthen bed, taking into account the factors influencing the first and second stages of filtration in the channels. Materials and research methods: The work used theoretical methods for calculating filtration in channels with an earthen bed, presented the calculation of the loss of flow rate Qf for filtration per 1 km of length with a steady free surface for different types of channel cross-sections, considered the basic formulas proving that water supply changes with changes in transverse cross-section of the canal, the change in the stability of the canal slopes is shown, a test calculation of the canal for filtration strength according to the maximum pressure gradients is presented, the need to compare the excess of the critical filtration pressure gradient with the permissible pressure gradient is considered. Results and its discussion. The relationship between the total, capillary and minimum moisture capacity and water yield in the thickness of soils is presented, the dependence of free porosity on the groundwater level, and the change in water permeability of soils over time are considered. As a result of the research, material was collected on the influence of free porosity and water loss on the movement of water in soils and the influence on the increase in groundwater levels in the zone of influence of canals, a methodology was presented for the influence of free porosity and water loss on the processes occurring in the soils of irrigation canals in an earthen channel, which will allow its use for modeling these processes on a computer.

A Simple Line-Element Model for Three-Dimensional Analysis of Steady Free Surface Flow through Porous Media

Considering the fact that only pores can transport water, pores in the homogeneous control volume are conceptualized as a three-dimensional orthogonal network of line elements, which is in contrast to the continuum hypothesis in traditional numerical approaches. The related flow velocity, hydraulic conductivity and continuity equation equivalent to the continuum model are formulated based on the principle of flow balance. Subsequently, the unified form for flow velocity and continuity equation is established based on the local coordinate system, and a finite line-element method is developed, in which three-dimensional steady free surface flow is reduced to one-dimensional form, and the numerical difficulty is greatly decreased. The proposed line-element model is validated by the good agreements of free surface locations with other methods through steady flow in a rectangular dam and a right trapezoidal dam, respectively. It is found that the proposed line-element model is not heavily dependent on the mesh size and penalty parameter. Steady free surface flow on the left bank abutment slope of the Kajiwa Dam in Southwestern China is further evaluated, and a parabolic variational inequality algorithm based on the continuum model is also employed for comparison. The consistent results indicate that the proposed line-element model can capture the steady free surface flow behavior as well as the continuum-based method. Moreover, the proposed line-element model can rapidly achieve accurate solutions whether for simple examples or for complicated engineering applications.

Laminar planar hydraulic jump in thin film flow of power-law liquids—Experimental, analytical and numerical study

This study explores the physics of laminar planar hydraulic jump of power-law liquids in a horizontal channel through shallow water analysis. The theory is supplemented and validated by numerical simulation performed by the phase-field method in COMSOL Multiphysics. The analysis is applicable for shear thinning as well as thickening liquids and reduces to Newtonian equations when the flow behavior index (n) is unity. The analytical and numerical results are further validated with experimental data of the present study. The steady state free surface profile in conjunction with a modified Bélanger type equation predicts jump location and strength (the ratio of film height just after and before the jump). The analysis provides a modified scaled relationship between the jump location and the liquid flow rate compared to that proposed for inviscid and viscous Newtonian liquids. An order of magnitude analysis expresses the scaled jump location and strength as a function of inlet Reynolds number (Rein), inlet Froude number (Frin), scaled channel length, and n. We find that an increase in the flow consistency index and/or n favors jump formation and the jump shifts toward the channel exit with increasing n for the same channel geometry, Rein and Frin. The influence of scaled channel length is more pronounced for n < 1 as compared to n > 1 while shear thickening liquids are more significantly influenced by Frin at fixed Rein and scaled channel length. The coupled effect of flow parameters on jump characteristics is depicted in the Rein–Frin plane, which denote the range of the existence of different jump types as the function of n.

Numerical Investigation on Temporal Evolution Behavior for Triad Resonant Interaction Induced by Steady Free-Surface Flow over Rippled Bottoms

Investigating the wave hydrodynamics of free-surface flow over rippled bottoms is a continuing concern due to the existence of submarine multiple sandbars and ambient flow in coastal and estuarial areas. More attention to free-surface wave stimulation has been received from the perspective of resonant wave-wave interaction, which is an intensive way for wave energy transfer and a potential way for wave component generation. However, the basic behavior of the triad resonant interaction of this problem is still limited and unclear. In this study, the triad resonant interaction induced by steady free-surface flow over rippled bottoms is numerically investigated by means of the High-Order Spectral (HOS) method. By considering the interactions among free-surface waves, ambient current, and rippled bottoms, the numerical model is applied for this situation based on Zakharov equation with ambient flow term. The temporal evolution of the triad resonant wave amplitude has been numerically investigated and compared well with multiple-scale expansion perturbation theory. Specifically, the temporal evolution behaviors of all six triad resonant wave components are confirmed by both numerical simulation and nonlinear perturbation analysis.

A Vortex Lattice Method for the Hydrodynamic Solution of Lifting Bodies Traveling Close and Across a Free Surface

The hydrodynamics performance of submerged and surface-piercing lifting bodies is analyzed by a potential flow model based on a Vortex Lattice Method (VLM). Such a numerical scheme, widely applied in aerodynamics, is particularly suitable to model the lifting effects thanks to the vortex distribution used to discretize the boundaries of the lifting bodies. The method has been developed with specific boundary conditions to account for the development of steady free surface wave patterns. Both submerged bodies, such as flat plates and hydrofoils, as well as planing hulls can be studied. The method is validated by comparison against available experimental data and other Computational Fluid Dynamic (CFD) results from Reynolds Averaged Navier Stokes (RANS) approaches. In all the analyzed cases, namely 2D and 3D flat plates, a NACA hydrofoil, planning flat plates and prismatic planing hulls, results have been found to be consistent with those taken as reference. The obtained hydrodynamic predictionsare discussed highlighting the advantages and the possible improvements of the developed approach.

An efficient quasi‐Newton method for three‐dimensional steady free surface flow

AbstractIn this article, we introduce the Free surface Quasi‐Newton method with Least‐Squares and Surrogate (FreQ‐LeSS) to solve steady free surface flows, as encountered in, for example, marine applications. The method has two important advantages over other methods. First of all it is fast thanks to the efficient iterative scheme based on quasi‐Newton iterations that make use of an advanced surrogate model. Second, it has good compatibility because it poses no special requirements on the flow solver that is called in the algorithm. The excellent convergence speed of the FreQ‐LeSS method is demonstrated for a shallowly submerged submarine: good quality wave patterns are obtained in few iterations. These results match with ISIS‐CFD, validating the FreQ‐LeSS method. The total cost of solving this steady free surface problem is only a few times that of solving a steady single‐phase flow with fixed free surface position. We conclude that FreQ‐LeSS promises to be much faster than the two‐phase time‐stepping methods typically used to solve these problems, and provides a sharper interface at the same time thanks to the fitting approach. In the present work, we restrict ourselves to problems without surface‐penetrating objects, to avoid the associated complications related to the contact‐line motion.

Pressure-balanced Saint–Venant equations for improved asymptotic modelling of pipe flow

In urban drainage networks, steady free surface flows in circular-shaped pipes are commonly observed given low inflow discharges. Therefore, accurate computation of such steady states is of great practical significance, especially considering bent pipes. This requires numerical schemes to be capable of achieving a delicate balance between slope source terms and fluxes in the discrete framework. To this end, in the current study, a new method using pressure balanced Saint–Venant equations and positivity preserving Godunov-type finite volume scheme, is developed for 1D flow simulation in bent circular pipes. Using pressure balancing, a set of pre-balanced Saint–Venant equations is derived. The numerical discretization method uses an approximate Riemann solver and is capable of dealing with wetting–drying process stably in frictional pipes. As verified by several test cases, the proposed method shows better performances than existing methods in modelling the asymptotic flow behavior and preserving steady states in bent circular pipes.

An efficient quasi‐Newton method for two‐dimensional steady free surface flow

An efficient quasi‐Newton method for two‐dimensional steady free surface flow

An efficient quasi‐Newton method for two‐dimensional steady free surface flow

SummarySteady free surface flows are of interest in the fields of marine and hydraulic engineering. Fitting methods are generally used to represent the free surface position with a deforming grid. Existing fitting methods tend to use time‐stepping schemes, which is inefficient for steady flows. There also exists a steady iterative method, but that one needs to be implemented with a dedicated solver. Therefore a new method is proposed to efficiently simulate two‐dimensional (2D) steady free surface flows, suitable for use in conjunction with black‐box flow solvers. The free surface position is calculated with a quasi‐Newton method, where the approximate Jacobian is constructed in a novel way by combining data from past iterations with an analytical model based on a perturbation analysis of a potential flow. The method is tested on two 2D cases: the flow over a bottom topography and the flow over a hydrofoil. For all simulations the new method converges exponentially and in few iterations. Furthermore, convergence is independent of the free surface mesh size for all tests.

Steady free surface potential flow of an ideal fluid due to a singular sink on the flat bottom

A two-dimensional steady problem of a potential free-surface flow of an ideal incompressible fluid caused by a singular sink is considered. The sink is placed at the horizontal bottom of the fluid layer. With the help of the Levi-Civita technique, the problem is rewritten as an operator equation in a Hilbert space. It is proved that there exists a unique solution of the problem provided that the Froude number is greater than some particular value. The free boundary corresponding to this solution is investigated. It has a cusp over the sink and decreases monotonically when going from infinity to the sink point. The free boundary is an analytic curve everywhere except for the cusp point. It is established that the inclination angle of the free boundary is less than π∕2 everywhere except for the cusp point, where this angle is equal to π∕2. The asymptotics of the free boundary near the cusp point is investigated.

Breaking of waves on thin films over topographies

In the present study, we performed experiments on the evolution of gravity-driven nonlinear traveling free surface waves over strongly undulated inclines. We focused on the impact of the excitation frequency and amplitude and the substrates’ shape and periodicity on the traveling wave. Thereby, we revealed phenomena concerning the amplitude evolution of convectively unstable waves. We can classify the wave evolution in three categories: (1) A normal exponential growth of the wave until it reaches a saturation amplitude. (2) An exponential growth of the wave and an abrupt collapse of the wave leading to a wave with a saturated amplitude, which is sensibly smaller than the maximal amplitude. (3) An alternating exponential growth and breaking of the wave. By using Fourier analysis, we investigated the waves in more detail. Furthermore, we report (a) a selection of excitation frequencies for the wave breaking, (b) a major impact of the steady state free surface, especially its mean or global curvature, (c) a bubble formation occurring at the wave breaking, (d) an overlap of the wave front, and (e) a formation of a jet during wave breaking.

Three-dimensional free-surface flow over arbitrary bottom topography

We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton–Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.

Stability analysis of a partitioned iterative method for steady free surface flow

Stability analysis of a partitioned iterative method for steady free surface flow

Dip-coating with prestructured substrates: transfer of simple liquids and Langmuir–Blodgett monolayers

When a plate is withdrawn from a liquid bath, either a static meniscus forms in the transition region between the bath and the substrate or a liquid film of finite thickness (a Landau–Levich film) is transferred onto the moving substrate. If the substrate is inhomogeneous, e.g. has a prestructure consisting of stripes of different wettabilities, the meniscus can be deformed or show a complex dynamic behavior. Here we study the free surface shape and dynamics of a dragged meniscus occurring for striped prestructures with two orientations, parallel and perpendicular to the transfer direction. A thin film model is employed that accounts for capillarity through a Laplace pressure and for the spatially varying wettability through a Derjaguin (or disjoining) pressure. Numerical continuation is used to obtain steady free surface profiles and corresponding bifurcation diagrams in the case of substrates with different homogeneous wettabilities. Direct numerical simulations are employed in the case of the various striped prestructures.The final part illustrates the importance of our findings for particular applications that involve complex liquids by modeling a Langmuir–Blodgett transfer experiment. There, one transfers a monolayer of an insoluble surfactant that covers the surface of the bath onto the moving substrate. The resulting pattern formation phenomena can be crucially influenced by the hydrodynamics of the liquid meniscus that itself depends on the prestructure on the substrate. In particular, we show how prestructure stripes parallel to the transfer direction lead to the formation of bent stripes in the surfactant coverage after transfer and present similar experimental results.

Four free surface algorithms for the 3D Navier–Stokes equations

Four algorithms are described for computing a steady free water surface with the solution of the three-dimensional (3D) Navier–Stokes equations. The numerical methods are used in hydraulic engineering cases, typically spillways and river modelling. The algorithms were tested against a laboratory experiment of a v-shaped broad-crested weir. The complex geometry of the weir introduced three-dimensional effects, which the numerical methods handled with varying degrees of success. One of the methods tested was the classical volume of fluid (VOF) approach, implemented in the OpenFOAM software with a fixed grid. The other three algorithms used an adaptive grid that followed the free water surface. These methods were coded in the SSIIM 2 program and were based on water continuity, pressure differences and an implicit solution of the diffusive wave equation. The VOF method gave the best results compared with the experiments. However, this method requires a very short time step. Two of the investigated methods compute the water surface location implicitly and can therefore use a much longer time step. The method based on the diffusive wave equation has the disadvantage that the results depend on a calibrated friction factor. All four methods predicted the water depth over the weir with an average accuracy below 14%.

Sessile drop deformations under an impinging jet

The problem of steady axisymmetric deformations of a liquid sessile drop on a flat solid surface under an impinging gas jet is of interest for understanding the fundamental behavior of free surface flows as well as for establishing the theoretical basis in process design for the Aerosol \({{\rm Jet}^{\circledR}}\) direct-write technology. It is studied here numerically using a Galerkin finite-element method, by computing solutions of Navier–Stokes equations. For effective material deposition in Aerosol \({{\rm Jet}^{\circledR}}\) printing, the desired value of Reynolds number for the laminar gas jet is found to be greater than ~500. The sessile drop can be severely deformed by an impinging gas jet when the capillary number is approaching a critical value beyond which no steady axisymmetric free surface deformation can exist. Solution branches in a parameter space show turning points at the critical values of capillary number, which typically indicate the onset of free surface shape instability. By tracking solution branches around turning points with an arc-length continuation algorithm, critical values of capillary number can be accurately determined. Near turning points, all the free surface profiles in various parameter settings take a common shape with a dimple at the center and bulge near the contact line. An empirical formula for the critical capillary number for sessile drops with \({45^{\circ}}\) contact angle is derived for typical ranges of jet Reynolds number and relative drop sizes especially pertinent to Aerosol \({{\rm Jet}^{\circledR}}\) printing.

Depth-averaged model for undular hydraulic jump

ABSTRACTAn undular hydraulic jump corresponds to the weak transition from super- to subcritical-flow in the form of steady free surface undulations. Previous models on undular hydraulic jumps employed the potential flow theory, i.e., the solitary and cnoidal wave theories. Experimental observations indicate the inadequacy of this theory, which motivated the development of more advanced approximations. Basic flow features including friction effects on the velocity profile, modelling of the bed-shear stress, and Reynolds stresses are considered. However, none of the models currently available include all these aspects. In this study, a general depth-averaged model is developed based on the k-ϵ turbulence closure. The general depth-averaged equations are applied to the undular jump problem, introducing a suitable time-averaged velocity distribution based on a composite power-law model, in which both streamline curvature and vorticity are accounted for. The bed-shear stress closure is included by a boundary layer method. Predictions of the depth-averaged Reynolds averaged Navier-Stokes (RANS) model are shown to be close to the 2D RANS solution and to experimental observations.

The second variation of the stream function energy of water waves with vorticity

We compute the second variation of the stream function energy of two-dimensional steady free surface gravity water waves with vorticity in the stream function formulation. We prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with Stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point. The particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy.

Development and evaluation of a numerical model for steady state interface and/or free surface groundwater flow

A 3D numerical procedure is developed to solve free surface and/or freshwater–saltwater interface groundwater flow problems under steady state conditions, using the Galerkin finite element technique and the Picard linearization method to solve nonlinear systems. The presented approach is an improvement of the Larabi and De Smedt (1997) scheme, and is showed to be robust, more efficient, more stable and more accurate than the latter. Benchmark tests involving different features are presented and discussed to validate the present approach.

Free surface flow past topography: A beyond-all-orders approach

The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves.J. Fluid Mech.567, 299–326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.