Free Surface Flow Equations Research Articles

Assessment of simplified momentum equations for free surface flows through rigid porous media

In many applications, free surface flow through rigid porous media has to be modeled. Examples refer to coastal engineering applications as well as geotechnical or biomedical applications. Albeit the frequent applications, slight inconsistencies in the formulation of the governing equations can be found in the literature. The main goal of this paper is to identify these differences and provide a quantitative assessment of different approaches. Following a review of the different formulations, simulation results obtained from three alternative formulations are compared with experimental and numerical data. Results obtained by 2D and 3D test cases indicate that the predictive differences returned by the different formulations remain small for most applications, in particular for small porous Reynolds number ReP < 5000. Thus it seems justified to select a simplified formulation that supports an efficient algorithm and coding structure in a computational fluid dynamics environment. An estimated accuracy depending on the porous Reynolds number or the mean grain diameter is given for the simplified formulation.

Upwind finite element-PML approximation of a novel linear potential model for free surface flows produced by a floating rigid body

A novel linear potential model is presented to compute free surface flows of incompressible fluids produced by the motion of a floating rigid body in the presence of an underlying non-uniform flow. In particular, the proposed model enables the accurate numerical simulation of the Kelvin wake pattern in a computational domain of reduced size. The governing equations are obtained by using an Arbitrary Lagrangian Eulerian (ALE) formulation which involves the underlying velocity of the fluid around the floating body assuming flat free surface, and a non-dimensional analysis to derive the novel linear system of equations for free surface flows. The discretization of the proposed model is made by a standard Galerkin finite element method, where a SUPG-inspired upwinding strategy has been used in combination with a Perfectly Matched Layer technique which allows truncating the original unbounded fluid domain without introducing spurious reflections in the Kelvin wake pattern. The numerical simulations computed with the proposed approach are compared with the results obtained by the classical linear potential model with uniform underlying flow and also with those from the full incompressible Navier–Stokes equations equipped with the k−ω SST turbulent model. This numerical comparison is discussed in terms of a classical hydrodynamic floating body benchmark involving the Wigley hull.

Wave–current interaction on a free surface

AbstractThe classical water wave equations (CWWEs) comprise two boundary conditions for the two‐dimensional flow on the free surface of a bulk three‐dimensional (3D) incompressible potential flow in the volume bounded by the free surface, which itself moves under the restoring force of gravity. One of these two boundary conditions provides the kinematic definition of the vertical velocity of the surface elevation. The other boundary condition is the dynamic Bernoulli law that governs the evaluation of the bulk velocity potential on the free surface. The present paper applies these two boundary conditions as constraints in the action integral for Hamilton's variational principle, along with a non‐hydrostatic pressure constraint that imposes incompressible flow on the free surface. The stationary variations in Hamilton's principle then yield closed dynamical equations of free surface flow whose divergence‐free velocity admits nonzero vorticity and whose nonhydrostatic pressure matches the pressure of the 3D bulk flow when evaluated on the free surface. A minimal coupling approach is proposed to model the mutual interactions of the waves and currents. The dynamical effects of horizontal buoyancy gradients are also considered in this context. For any combination of these model variables, the resulting system of variational equations admits a Lie–Poisson Hamiltonian formulation. Finally, stochastic versions of these model equations are derived by assuming that the material loop for their Kelvin circulation theorem in each case follows stochastic Lagrangian histories in a Stratonovich sense.

A comparative study on wave-deck interactions of T-type and box girder decks under regular waves

Extreme wave forces have damaged massive coastal bridges in the past decades. To the best knowledge of the authors, a direct comparison of dynamic wave-deck interactions between T-type and Box girder decks under the same wave conditions has not been studied. This paper establishes a numerical model solving the Navier-Stokes equations for viscous and incompressible free surface flows, which is verified by existing experimental data. A total of 54 numerical experiments based on this model are then carried out to investigate the dynamic responses of two bridge decks under various wave conditions, and the influences of hydrodynamic parameters on the wave-deck interaction are studied. Two peak values are found in the curve of lateral displacement versus dimensionless submersion depth for the T-type girder deck, while one peak value is found in the curve for the Box girder deck. The relations between lateral displacement and period for two decks are both uncertain, which highly depends on the submersion depth and girder type. The lateral displacement is positively correlated with the wave height for two bridge decks, which is only valid when the horizontal wave force in the specified wave height is large enough to overcome the frictional force. This study provides researchers an insight into the difference of instantaneous lateral displacement between two most commonly used cross-section shapes of bridge deck, and helps to further understand the fragility of T-type and Box girder decks under extreme waves.

A fully Lagrangian DEM-MPS mesh-free model for ice-wave dynamics

This paper develops and evaluates a novel three-dimensional fully-Lagrangian (particle-based) numerical model, based on the hybrid discrete element method (DEM) and moving particle semi-implicit (MPS) mesh-free techniques, for modeling the highly-dynamic ice-wave interactions. Both MPS and DEM belong to mesh-free Lagrangian (particle) techniques. The model considers ice-wave dynamics as a multiphase continuum-discrete system. While the MPS solves the continuum equations of free-surface flow in a Lagrangian particle-based domain, the DEM uses a multi-sphere Hertzian contact dynamic model to simulate the ice floes motion and interaction. The hybrid model predicts the motion and collision of ice floes as well as their interaction with water, boundaries, and any obstacle in their way. Considering the mesh-free Lagrangian nature of both DEM and MPS, the developed model has an inherent ability to predict the free-drift (absence of internal stress) movements of the ice floes, e.g., sliding, rolling, colliding, and piling-up, in violent free-surface flow. A small-scale and challenging experiment based on dam-break flow over dry and wet beds with floating block floes, which mimics the characteristics of an idealized jam release, has been conducted to provide useful and comprehensive quality data for the validation of the proposed model, as well as other numerical models. Experimental and numerical results of the free-surface profile and the position of the blocks are compared. The results show the ability of the model to numerically reproduce and predict the complex three-dimensional dynamic behavior of wave-ice floes interaction. Overall, this study is a first effort toward developing an ice-wave dynamic within a fully Lagrangian framework (i.e. both flow hydrodynamics and ice dynamics in the Lagrangian particle-based system), and its results can be extended to bring an in-depth understanding of the physics of the real-scale ice-wave or river ice dynamic problems in the future.

Numerical modeling of the interaction between submerged floating tunnel and surface waves

This paper presents a numerical model for analyzing the nonlinear interaction between the moored Submerged Floating Tunnel (SFT) and surface waves. The mechanics model of the moored floating body driven by wave forces is built, and an efficient mesh update method is employed to dynamically configure the computational meshes solving the Navier-Stokes equations for viscous and incompressible free surface flows with the volume of fluid (VOF) method. Two laboratory experiments are used for validating the numerical model in terms of surface elevations, motion responses and mooring forces of the SFT, indicating the proposed model is capable of simulating the dynamics of the moored floating body under the wave action. This hydrodynamic model is then utilized to simulate the wave-structure interaction of the prototype SFT designed for Funka Bay, Hokkaido located in Japan. A total of 49 cases are designed for the numerical simulation to investigate the characteristics of the wave-tunnel interaction for different hydrodynamic parameters, including wave height, wave period, immersion depth and buoyancy-weight ratio (BWR). The numerical experiments not only shed light on the mooring forces, as well as pitch, sway and heave responses of the SFT in different wave conditions, but also provide guidance for the choice of BWR in engineering design. A medium value of BWR is suggested to be suitable, which is useful for avoiding the happening of snap forces in mooring chains and preventing SFT from experiencing large movement under external forces in severe wave conditions. As the correlation between the motion responses and BWR is not merely linear or quadratic but parabolic with a peak value, the design of BWR should avoid the case where peak motion responses of SFT happen.

Research on Flood Propagation for Different Dam Failure Modes: A Case Study in Shenzhen, China

Dam-break flood simulation can evaluate the impact of a dam-break, and is significant in conducting risk analyses, creating emergency plans, and mitigating calamities. This study investigates the dam-break flood propagation in the downstream urban areas of the Minzhi Reservoir in Shenzhen, China, based on a two-dimension shallow-water model (SWM-2D). The SWM-2D solves two-dimensional shallow water equations for free surface flow using the finite-volume method. A refined grid with an unstructured mesh of triangular cells, constrained by building walls, is constructed to represent urban structures in this model. The flood hydrographs of dam-break and the flood propagation under different failure modes are calculated by adopting two breach mechanisms, namely instantaneous dam-break and gradual dam-break. The results indicate that the peak flow of the instantaneous dam-break is relatively large at the beginning of the dam-break. The gradual dam-break peak flow is relatively small when seepage failure deformation develops towards the upper part of the dam. The inundation information during the evolution of the dam-break flood, including the regularity of the flood propagation, distribution of maximum water depth, and variation of depth and velocity with time are also analysed.

Non-linear wave equations for free surface flow over a bump

ABSTRACT This study aims to develop a new wave equation model by modifying the Fully-nonlinear and strongly-Dispersive Surface wave (FDS) equations. The modification was performed by applying a new expansion in a series of the vertical coordinate, zμ , to the velocity potential while a simple expansion in a series of z was applied to the FDS equations. Verification of the model was conducted by comparing with the theoretical solutions of surface solitary waves. We applied the modified FDS equations to wave fields over a bump under conditions with and without currents, which agreed very well with the time series of wave heights and velocity obtained from laboratory experiments. The dispersion relationship computed using the normalized modified FDS equations also agreed very well with the theoretical solution when we gave the number of expansion terms as 3 with μ = 2.5. Additionally, the profile of surface waves computed with the modified FDS equation was shown to have a larger width ridge, a bulbous-type wave, by comparing with a Trochoidal wave under the condition of waves against a current.

On the hydrodynamic performance of a vertical pile-restrained WEC-type floating breakwater

This paper presents a numerical study on the hydrodynamic performance of a vertical pile-restrained wave energy converter type floating breakwater. The aims are to further understand the characteristics of such integrated system in terms of both wave energy extraction and wave attenuation, and to provide guidance for optimising the shape of the floating breakwater for more energy absorption and less wave transmission at the same time. The numerical model solves the incompressible Navier-Stokes equations for free-surface flows using the particle-in-cell method and incorporates a Cartesian cut cell based strong coupling algorithm for fluid-structure interaction. The numerical model is first validated against an existing experiment, consisting of a rectangular box as the floating breakwater and a power take-off system installed above the breakwater, for the computation of the capture width ratio and wave transmission coefficients. Following that, an optimisation study based on the numerical model is conducted focusing on modifying the shape of the floating breakwater used in the experiment. The results indicate that by changing only the seaward side straight corner of the rectangular box to a small curve corner, the integrated system achieves significantly more wave energy extraction at the cost of only a slight increase in wave transmission.

Collocated discrete subdomain meshless method for dam-break and dam-breaching modelling

This paper introduces a ‘collocated discrete subdomain meshless method’ to solve shallow water equations for free-surface flows with conservation of sediment mass. The computational domain in the method is discretised by distributed nodal points. Shape functions are made using the moving least-squares method for efficiency and simplicity. A number of collocation points, independent of nodal points, are used to form the specified residual function. The sum of residuals at the collocation points for each subdomain is zero in the problem domain and boundaries. In addition, boundary conditions are readily enforced. The model was verified by comparing results of simulated and physically modelled dam-break flows. There was good agreement between the analytical solution and experimental measured data, and a high convergence rate and efficiency were achieved. The method appears to have significant potential for investigating a variety of problems concerning coupled flow with sediment transport.

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

In geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model. In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to model the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.

A 3D parallel Particle-In-Cell solver for wave interaction with vertical cylinders

In this paper, the Particle-In-Cell (PIC) based PICIN solver is extended to three spatial dimensions and parallelised using the Message Passing Interface (MPI) approach. The PICIN solver employs both Eulerian grid and Lagrangian particles to solve the incompressible Navier-Stokes equations for free-surface flows. The particles are employed to carry all the fluid properties, solve the non-linear advection term and track the free-surface, while the grid is used solely for computational efficiency in solving the non-advection terms. Validation of the new 3D model concentrates on test cases involving multiple wave types (including regular waves, focused waves and solitary waves) interacting with vertical cylinders in several spatial configurations. The results are compared with laboratory data and numerical results from state-of-the-art Volume of Fluid (VOF) based Eulerian solvers such as those from the OpenFOAM® suite. It is shown that the 3D parallel PICIN model is able to well simulate highly non-linear water waves, and the interaction of such waves with vertical cylinders, with a CPU efficiency similar to Eulerian solvers. Moreover, the innovative use of particles in PICIN, akin to meshless Lagrangian solvers, gives the model a particular flexibility in handling complex, full 3D, water-wave scenarios involving large free-surface deformations.

A 3-D numerical study of solitary wave interaction with vertical cylinders using a parallelised particle-in-cell solver

This paper aims to provide a better understanding of the interaction between solitary waves and vertical circular cylinders. This is achieved via process based numerical modelling using the parallel particle-in-cell based incompressible flow solver PICIN. The numerical model solves the Navier-Stokes equations for free-surface flows and incorporates a Cartesian cut cell method for fluid-structure interaction. Solitary waves are generated using a piston-type wave paddle. The PICIN model is first validated using a test case that involves solitary wave scattering by a single vertical cylinder. Comparisons between the present results and experimental data show good agreement for the free surface elevations around the cylinder and the horizontal wave force on the cylinder. The model is then employed to investigate solitary wave interaction with a group of eleven vertical cylinders. The wave run-up and wave forces on the cylinders are discussed.

A well-balanced numerical scheme for debris flow run-out prediction in Xiaojia Gully considering different hydrological designs

To simulate debris flow run-out, the governing equations for free-surface shallow flow are corrected by setting the basal flow resistance coefficients with the quadratic rheological friction model. A well-balanced numerical scheme is developed for its run-out simulation over irregular topography. A linear reconstruction is adopted for improving the spatial accuracy of the numerical scheme. Considering the complex friction terms of governing equations of debris flow, they are estimated with a full implicit scheme for ensuring stability of the numerical scheme. The validity check of run-out simulation is implemented based on general knowledge of fluid, and a well-studied case occurred in the Xiezi Gully in Yingxiu Town, Sichuan Province of China. For practical purpose, the present numerical scheme is used for run-out prediction of debris flow in Xiaojia Gully of Panzhihua City, Sichuan Province of China. Our work aims to present a well-balanced numerical scheme for debris flow run-out simulation prediction, which can be applied quite conveniently to solve other kinds of debris flow models and helpful to promote the development in debris flow numerical calculation.

A numerical method for the solution of the nonlinear turbulent one-dimensional free surface flow equations

Free surface flow of an incompressible fluid over a shallow plane/undulating horizontal bed is characteristically turbulent due to disturbances generated by the bed resistance and other causes. The governing equations of such flows in one dimension, for finite amplitude of surface elevation over the bed, are the Continuity Equation and a highly nonlinear Momentum Equation of order three. The method developed in this paper introduces the “discharge” variable q = η U, where η = elevation of the free surface above the bed level, and U = average stream-wise forward velocity. By this substitution, the continuity equation becomes a linear first-order PDE and the momentum equation is transformed after introduction of a small approximation in the fifth term. Next, it is shown by an invertibility argument that q can be a function of η: q = F(η), rendering the momentum equation as a first order, second degree ODE for F(η), that can be be integrated by the Runge-Kutta method. The continuity equation then takes the form of a first order evolutionary PDE that can be integrated by a Lax-Wendroff type of scheme for the temporal evolution of the surface elevation η. The method is implemented for two particular cases: when the initial elevation is triangular with vertical angle of 120 ∘ and when it has a sinusoidal form. The computations exhibit the physically interesting feature that the frontal portion of the propagating wave undergoes a sharp jump followed by tumbling over as a breaker. Compared to other discretization methods, the application of the Runge-Kutta and an extended version of the Lax-Wendroff scheme is much easier.

Simulación numérica con RiverFlow2D de posibles soluciones de mitigación de avenidas en el tramo medio del río Ebro

&lt;p class="Resumen"&gt;En este trabajo se presenta un estudio de medidas orientadas a la mitigación de avenidas en el tramo medio del río Ebro: limpieza de vegetación del cauce, uso de zonas de inundación controlada y construcción o re-adaptación de motas. Para ello se utiliza el software RiverFlow2D que resuelve las ecuaciones conservativas del flujo de superficie libre con un método de volúmenes finitos realizando los cálculos sobre GPU. Se comparan los resultados con medidas en estaciones de aforo e información extraída de ortofotos. La medida más efectiva, de las analizadas, ha resultado ser la eliminación de la vegetación en el cauce. Se demuestra que no sólo el área máxima inundada es menor en todo el tramo sino que también reduce la altura de agua hasta en 1 m. El resto de medidas tienen consecuencias locales y de poca entidad cuando los caudales pico son altos, aunque podrían resultar de utilidad para avenidas con caudales más bajos.&lt;/p&gt;

A modified hydrodynamic model for routing unsteady flow in a river having piedmont zone

Abstract Existence of piedmont zone in a river bed is a critical parameter from among numerous variations of topographical, geological and geographical conditions that can significantly influence the river flow scenario. Downstream flow situation assessed by routing of upstream hydrograph may yield higher flow depth if existence of such high infiltration zone is ignored and therefore it is a matter of concern for water resources planning and flood management. This work proposes a novel modified hydrodynamic model that has the potential to accurately determine the flow scenario in presence of piedmont zone. The model has been developed using unsteady free surface flow equations, coupled with Green-Ampt infiltration equation as governing equation. For solution of the governing equations Beam and Warming implicit finite difference scheme has been used. The proposed model was first validated from the field data of Trout Creek River showing excellent agreement. The validated model was then applied to a hypothetical river reach commensurate with the size of major tributaries of Brahmaputra Basin of India. Results indicated a 10% and 14% difference in the maximum value of discharge and depth hydrograph in presence and absence of piedmont zone respectively. Overall this model was successfully used to accurately predict the effect of piedmont zone on the unsteady flow in a river.

Numerical simulations of 2-D floating body driven by regular waves

An improved meshing method based on Fluent is used to update the computational meshes in solving the Navier-Stokes (N-S) equations for viscous and incompressible free surface flows with the volume of fluid (VOF) method. To maintain the mesh quality when updating meshes for a moving structure, the computational domain is separated into several parts and each part corresponds to a specific type of body motion. The numerical results of the interaction between the floating body and the regular waves agree well with the experimental data. A total of eight typical motion types are simulated separately to understand the correlation between the motion types and the wave transmission as well as the forces acting on the floating body. Numerical experiments show that the wave transmission increases in the case of sway and heave motions and decreases in the case of pitch motion as compared with the stationary case. It is also found that the sway motion reduces the horizontal wave force acting on the floating body, while the heave motion enhances the vertical wave force.

Determinants of modelling choices for 1-D free-surface flow and morphodynamics in hydrology and hydraulics: a review

Abstract. This review paper investigates the determinants of modelling choices, for numerous applications of 1-D free-surface flow and morphodynamic equations in hydrology and hydraulics, across multiple spatiotemporal scales. We aim to characterize each case study by its signature composed of model refinement (Navier–Stokes: NS; Reynolds-averaged Navier–Stokes: RANS; Saint-Venant: SV; or approximations to Saint-Venant: ASV), spatiotemporal scales and subscales (domain length: L from 1 cm to 1000 km; temporal scale: T from 1 s to 1 year; flow depth: H from 1 mm to 10 m; spatial step for modelling: δL; temporal step: δT), flow typology (Overland: O; High gradient: Hg; Bedforms: B; Fluvial: F), and dimensionless numbers (dimensionless time period T*, Reynolds number Re, Froude number Fr, slope S, inundation ratio Λz, Shields number θ). The determinants of modelling choices are therefore sought in the interplay between flow characteristics and cross-scale and scale-independent views. The influence of spatiotemporal scales on modelling choices is first quantified through the expected correlation between increasing scales and decreasing model refinements (though modelling objectives also show through the chosen spatial and temporal subscales). Then flow typology appears a secondary but important determinant in the choice of model refinement. This finding is confirmed by the discriminating values of several dimensionless numbers, which prove preferential associations between model refinements and flow typologies. This review is intended to help modellers in positioning their choices with respect to the most frequent practices, within a generic, normative procedure possibly enriched by the community for a larger, comprehensive and updated image of modelling strategies.

Numerical Modeling of Wave Forces on Movable Bridge Decks

This paper presents a numerical analysis of the nonlinear interaction between water waves and the movable superstructure of a coastal bridge. An improved mesh update method is used to dynamically configure the computational meshes solving the Navier-Stokes equations for viscous and incompressible free surface flows with the volume of fluid (VOF) method. To maintain the mesh quality as the bridge deck is displaced, the computational domain is separated into several parts corresponding to a specific type of body motion. The numerical model results have been compared with the laboratory experiments conducted at Oregon State University with good agreement. This hydrodynamic model is then used to simulate the failure process of the prototype U.S. Highway 90 (US 90) Bridge during Hurricane Katrina (2005). It is found that the bridge deck is most vulnerable to be displaced by the waves when the bridge is partially submerged because of the combined effects of the vertical and horizontal wave forces. Numerical experiments of the US 90 Bridge shed light on the sway and heave responses of the bridge deck to different wave height conditions. It is shown that a significant wave height of 2.6 m would result in a failure of the bridge deck. A critical wave height (2 m) based on the incipient motion of the bridge deck is recommended for screening the vulnerability of other bridges similar to the US 90 Bridge in hurricane-prone areas. Moreover, the computed maximum heave response can be used as a reference for engineers to design the shear key height to prevent a bridge deck from horizontal displacement.