Steady Free Surface Flow Research Articles

An Exact Solution to the Inverse Problem of Steady Free-Surface Flow over Topography

A simple exact solution is presented to the inverse problem in steady, two-dimensional idealised flow over topography that seeks the bottom profile given knowledge of the free-surface data. Attention is focused on the case when a uniform stream flows over a localised obstacle, although the solution is not restricted to this case. The inverse problem is formulated as a Stieltjes integral equation which is solved exactly using a Fourier transform. The solution requires the analytic continuation of two real functions representing the surface speed and the angle between the surface velocity vector and the horizontal. Some example surface profiles and their corresponding bottom topographies are discussed. Although the solution requires the prescription of the surface as a function of the velocity potential, it is shown to closely resemble the corresponding profile in physical space, even for quite large surface displacements, while significant discrepancy occurs at the bottom. Inference of the bottom profile from discrete surface data is accomplished by way of polynomial interpolation and rational approximation in the complex plane for the sample case of a hydraulic fall.

Steady free-surface flow in vegetation debris pad clogging trash racks

ABSTRACT We consider a steady water flow in a channel where a vertical grid is clogged by a rectangular patch of aquatic vegetation. Laboratory experiments are conducted to observe the decrease in the water table within the patch, as well as the subsequent head loss, as a function of the patch length, the flowrate and the upstream water height. Various models of porous media are used to produce theoretical formulae describing the water surface profile within the patch, among which the Barree–Conway model proves to perform reasonably well. However, for large enough Froude numbers a seepage face or water chute appears past the vegetation patch while non-hydrostatic effects become important. Under the latter condition, the accuracy of our analytic solution is less satisfactory, while remaining accurate enough for practical purposes. As confirmed by numerical simulations, with the specific aquatic plants used in our experiments the porous flow is not Darcian, so that the seepage and head loss could not be explained by the exact Polubarinova–Kochina theory.

Obstructed free-surface viscoplastic flow on an inclined plane

The interaction of steady free-surface flows of viscoplastic material with a surface-piercing obstruction of square cross-section on an inclined plane is investigated theoretically. The flow thickness increases upstream of the obstruction and decreases in its lee. The flow depends on two dimensionless parameters: an aspect ratio that relates the flow thickness, the obstruction width and the plane inclination; and a Bingham number that quantifies the magnitude of the yield stress relative to the gravitationally induced stresses. Flows with a non-vanishing yield stress always form a static ‘dead’ zone in a neighbourhood of the upstream and downstream stagnation points. For relatively wide obstructions, a deep ‘ponded’ region develops upstream with a small dead zone, while the deflected flow reconnects over relatively long distances downstream. The depth of the upstream pond increases with both the dimensionless yield stress and width of the obstruction, while the unyielded dead zone varies primarily with the yield stress. Both are predicted asymptotically by balancing the volume flux of fluid into and out of the ponded region. When the obstruction is narrow, the perturbation to the depth of the oncoming flow is reduced. It exhibits fore–aft antisymmetry, while the dead zone is symmetric to leading order. Increasing the yield stress leads to larger dead zones that eventually encompass all of the upstream- and downstream-facing boundaries of the obstruction and fully divert the flow. Results for obstructions with circular and rhomboidal cross-sections are also presented and illustrate the effects of boundary shape on the properties of the steady flow.

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.

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.

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

New exotic capillary free-surface flows

Abstract

Forces on buildings with openings and orientation in a steady post-tsunami free-surface flow

Steady free-surface flows around buildings occurring during flood or tsunami events can produce major damages and a quantification of the post-peak forces is essential for safety and resilience of coastal structures. The loading process is highly affected by the flow Froude number and the drag coefficient, commonly defined for highly subcritical flows, while field observations of tsunami flows reported Froude numbers close to one, visually appearing as a choked regime. The present experimental study addresses the hydrodynamic forces on emerging buildings in a subcritical choked regime, focusing on the effect of openings and orientation. Laboratory experiments indicated a substantial difference in flow depths between the up- and downstream side of the building for increasing Froude numbers. The presence of openings induced a flow through the building lowering the difference in flow depth, limiting the effect of the hydrostatic component of the loading process. The formulation of an empirical resistance coefficient CR allowed a combination of both form drag and hydrostatic forces. Whilst for impervious buildings CR was consistent with reference studies, for buildings with openings CR was directly dependent upon porosity. Contrarily to the unsteady flow conditions, results showed that the sidewalls also played a role in the loading process. Buildings with a rotated orientation resulted in slightly larger surfaces exposed to the flow and larger horizontal forces. Nevertheless, these were applied at lower cantilever arms, thus reducing the tilting moment. Altogether, this study provides experimentally-derived parameters that will support hydraulic engineers in the design of coastal structures.

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.

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

A Non-Hydrostatic Depth-Averaged Model for Hydraulically Steep Free-Surface Flows

This study describes the results of a numerical investigation aimed at developing and validating a non-hydrostatic depth-averaged model for flow problems where the horizontal length scales close to flow depth. For such types of problems, the steep-slope shallow-water equations are inadequate to describe the two-dimensional structure of the curvilinear flow field. In the derivation of these equations, the restrictive assumptions of negligible bed-normal acceleration and bed curvature were employed, thus limiting their applicability to shallow flow situations. Herein, a Boussinesq-type model is deduced from the depth-averaged energy equation by relaxing the weakly-curved flow approximation to deal with the non-hydrostatic steep flow problems. The proposed model is solved with an implicit finite difference scheme and then applied to simulate steady free-surface flow problems with strong curvilinear effects. The numerical results are compared to experimental data, resulting in a reasonable overall agreement. Further, it is shown that the discharge characteristics of free flow over a round-crested weir are accurately described by using a Boussinesq-type approximation, and the drawbacks arising from a standard hydrostatic approach are overcome. The suggested numerical method to determine the discharge coefficient can be extended and adopted for other types of short-crested weirs.

A general relation for standing normal jumps in both hydraulic and dry granular flows

Steady free-surface flows can produce sudden changes in height and velocity, namely standing jumps, which demarcate supercritical from subcritical flows. Standing jumps have traditionally been observed and studied experimentally with water in order to mimic various hydraulic configurations, for instance in the vicinity of energy dissipators. More recently, some studies have emerged that investigate standing jumps formed in flows of dry granular materials, which are relevant to the design of protection dams against avalanches. In the present paper, we present a new explicit relation for the prediction of the height of standing jumps. We demonstrate the robustness of the new relation proposed by revisiting and cross-comparing a great number of data sets on standing jumps formed in water flows on horizontal and inclined smooth beds, in water flows on horizontal rough beds, and in flows of dry granular materials down smooth inclines. Our study reveals the limits of the traditional one-to-one relation between the sequent depth ratio of the jump and the Froude number of the incoming supercritical flow, namely the Bélanger equation. The latter is a Rankine–Hugoniot relation which does not take into account the gravitational and frictional forces acting within the jump volume, over the jump length, as well as the possible density change across the jump when the incoming fluid is compressible. The newly proposed relation, which is exact for grains and a reasonable approximation for water, can solve all of these issues. However, this relation can predict the height of the standing jump only if another length scale, namely the length of the jump, is known. We conclude our study by discussing empirical but simple closure relations to get a reasonable estimate of the jump length for water flows and dry granular flows. These closure relations can be used to feed the general jump relation and then predict with accuracy the heights of the jumps in a number of situations, provided that well-calibrated friction laws – described in the present study – are considered.

Steady free-surface flow over spatially periodic topography

Two-dimensional free-surface flow over a spatially periodic channel bed topography is examined using a steady periodically forced Korteweg–de Vries equation. The existence of new forced solitary-type waves with periodic tails is demonstrated using recently developed non-autonomous dynamical-systems theory. Bound states with two or more co-existing solitary waves are also identified. The solution space for varying amplitude of forcing is explored using a numerical method. A rich bifurcation structure is uncovered and shown to be consistent with an asymptotic theory based on small forcing amplitude.

On steady non-breaking downstream waves and the wave resistance

In this work we have obtained exact analytical formulae expressing the wave resistance of a two-dimensional body by the parameters of the downstream non-breaking waves. The body moves horizontally at a constant speed $c$in a channel of finite depth $h$. We have analysed the relationships between the parameters of the upstream flow and the downstream waves. Making use of some results by Keady & Norbury (J. Fluid Mech., vol. 70, 1975, pp. 663–671) and Benjamin (J. Fluid Mech., vol. 295, 1995, pp. 337–356), we have rigorously proved that realistic steady free-surface flows with a positive wave resistance exist only if the upstream flow is subcritical, i.e. the Froude number$\mathit{Fr}=c/\sqrt{gh}<1$. For all solutions with downstream waves obtained by a perturbation of a supercritical upstream uniform flow the wave resistance is negative. Applying a numerical technique, we have calculated accurate values of the wave resistance depending on the wavelength, amplitude and mean depth.

Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton–Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

Non-uniqueness of steady free-surface flow at critical Froude number

Free-surface flow past a disturbance at critical Froude number is commonly found to be unsteady with complex wave patterns both upstream and downstream of the disturbance. Such flows can be undesirable as the waves that are generated can have a negative impact in applications including the erosion of waterway banks and energy loss through wave drag on a ship. This motivates us to develop a new approach to obtain steady solutions at critical Froude number that are wave free in the far field. Under the assumption of two-dimensional, irrotational, incompressible fluid flow, we show that both weakly and fully nonlinear solutions to the problem are non-unique. A range of qualitatively different types of numerical solutions and analytical approximations are discovered, for example for flow over a corrugated channel bottom.

Dynamic Response of a Sphere Immersed in a Shallow Water Flow

This work analyzes the dynamic response of a sphere located close to the floor of a hydraulic channel within steady free-surface current flows. The sphere is free to move in transverse (y) and streamwise (x) directions, and it is characterized by a mass ratio m* equal to 1.34. The oscillation amplitudes and the frequencies of the sphere have been measured by means of the image analysis of a charge coupled device (CCD) camera. The experimental data show a significant influence of the free surface on the sphere movement and highlight a different behavior of the dynamic response to the increasing of the water level on the upper part of the body.