Free Surface Flow Model Research Articles

Modelling of high-velocity free-surface flows: a revised interfacial area approach

The specific interface area transport equation is derived from first principles, and a closure model for turbulent flow is proposed. The flow is modelled by a two-phase method providing velocities of both phases at the bubbly, spray and the intermediate large scale interface (LSI) regimes. The interface area is transported by the interface velocity – a linear combination of corresponding phase velocities and a turbulent drift velocity. The resulting equation includes the source terms due to the bubble (droplet) breakup and coalescence, and turbulent splashes at the interface. A model is proposed for the source term in the LSI regime. The model is implemented in the developer's version of the Simcenter STAR-CCM+ ® CFD code. The new model is applied to self-aerating water flows down stepped chutes; the results are in reasonable agreement with the experimental data.

Mixed material point method formulation, stabilization, and validation for a unified analysis of free-surface and seepage flow

This paper presents a novel stabilized mixed material point method (MPM) designed for the unified modeling of free-surface and seepage flow. The unified formulation integrates the Navier-Stokes equation with the Darcy-Brinkman-Forchheimer equation, effectively capturing flows in both non-porous and porous domains. In contrast to the conventional Eulerian computational fluid dynamics (CFD) solver, which solves the velocity and pressure fields as unknown variables, the proposed method employs a monolithic displacement-pressure formulation adopted from the mixed-form updated-Lagrangian finite element method (FEM). To satisfy the discrete inf-sup stability condition, a stabilization strategy based on the variational multiscale method (VMS) is derived and integrated into the proposed formulation. Another distinctive feature is the implementation of blurred interfaces, which facilitate a seamless and stable transition of flows between free and porous domains, as well as across two distinct porous media. The efficacy of the proposed formulation is verified and validated through several benchmark cases in 1D, 2D, and 3D scenarios. Conducted numerical examples demonstrate enhanced accuracy and stability compared to analytical, experimental, and other numerical solutions.

Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel–Bulkley fluids

This paper presents mathematical modelling and simulation of thin free-surface flows of viscoplastic fluids with a Herschel–Bulkley rheology over complex topographies with basal perturbations. Using the asymptotic expansion method, depth-averaged models (lubrication and shallow water type models) are derived for 3D (three-dimensional) multi-regime flows on non-flat inclined topographies with varying basal slipperiness. Starting from the Navier–Stokes equations, two flow regimes corresponding to different balances between shear and pressure forces are presented. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. In the second regime case, a pressure term is non-negligible. Mathematically, it leads to a corrective term to the classical regime equations. Flow solutions of the two regimes are compared; the difference appears in particular in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement. Furthermore, numerical examples are shown to illustrate the robustness of the present shallow water models to simulate viscoplastic flows in 3D and over an inclined topography with local perturbations in basal elevation and basal slipperiness. The derived models are adequate for direct (engineering and geophysical) applications to real-world flow problems presenting Herschel–Bulkley rheology like lava and mud flows.

Multi-layer non-hydrostatic free-surface flow model with kinematic seafloor for seismic tsunami generation

Earthquake induced tsunamis pose devastating threat to coastal communities worldwide. Accurate description of tsunami generation by kinematic fault rupture is of importance to investigate earthquake events and evaluate tsunami impacts. The paper develops a multi-layer non-hydrostatic model with a kinematic bottom boundary condition and conducts comprehensive validation to examine the model’s capability in resolving seismic tsunami generation. The two-dimensional governing equations of the multi-layer non-hydrostatic free-surface flow system equipped with kinematic seafloor displacement is first introduced in Cartesian coordinates. A combined finite difference and finite volume scheme is utilized to discretize the governing equations with flow variables arranged on a staggered Arakawa C-grid. Application of pressure correction technique to the discretized formulations yields Poisson-type equations from which the non-hydrostatic pressure is solved for the next time step to complete the temporal integration. Based on existing analytical solutions, four groups of tsunami generation cases considering broad ranges of source parameters, including horizontal scale, rise time, and rupture velocity, are designed to demonstrate performance of the proposed non-hydrostatic model with one-, two-, and three-layers as well as the hydrostatic one. Comparison of 59 generation cases including extreme scenarios indicates the non-hydrostatic model performs better than the hydrostatic model in reproducing the entire waveform and predicting the maximum wave amplitude. High modeling accuracy can be achieved through incorporation of more layers. The proposed multi-layer non-hydrostatic model is a powerful tool for investigating earthquake source mechanisms and evaluating coastal tsunami hazards.

Thick interface coupling technique for weakly dispersive models of waves

The primary focus of this work is the coupling of dispersive free-surface flow models through the utilization of a thick interface coupling technique. The initial step involves introducing a comprehensive framework applicable to various dispersive models, demonstrating that classical weakly dispersive models are encompassed within this framework. Next, a thick interface coupling technique, well-established in hyperbolic framework, is applied. This technique enables the formulation of unified models across different subdomains, each corresponding to a specific dispersive model. The unified model preserves the conservation of mechanical energy, provided it holds for each initial dispersive model. We propose a numerical scheme that preserve the projection structure at the discrete level and as a consequence is entropy-satisfying when the continuous model conserve the mechanical energy. We perform a deep numerical analysis of the waves reflected by the interface. Finally, we illustrate the usefulness of the method with two applications known to pose problems for dispersive models, namely the imposition of a time signal as a boundary condition or the imposition of a transparent boundary condition, and wave propagation over a discontinuous bathymetry.

An immersed boundary method coupled non-hydrostatic model for free surface flow

An efficient numerical model has been developed for the study of three-dimensional free surface flow, which combines a non-hydrostatic model with the immersed boundary method (IBM). The model utilizes the transformed σ coordinate in the vertical direction, which has been found to be highly effective in accurately capturing the free surface and uneven bottom. In addition, IBM is implemented by introducing a virtual boundary force into the momentum equations to emulate the natural boundary conditions. This implementation enhances the ability of the model in simulating the interactions of wave and solid structures. A robust boundary recognition method is used to identify the location of IB cells neighboring the virtual boundary. Furthermore, a water elevation correction method has been proposed to handle the integral discontinuity in the vertical direction caused by the submerged solid bodies. By adding the virtual boundary forces to the IB cells, the no-slip velocity condition at the solid boundary is effectively enforced. To validate the performance of the model, the numerical results have been compared with experimental data. The benchmark tests focused on the free surface evolution, velocity distributions and the dynamic pressures verified the performance of the numerical model and the robust strategy of the IBM.

Numerical analysis of 3D hydrodynamics and performance of an array of oscillating water column wave energy converters integrated into a vertical breakwater

Performance and hydrodynamics of an array of Oscillating Water Column (OWC) Wave Energy Converter (WEC) integrated into a vertical breakwater is studied. The FLUENT® software, in which the numerical model is based on the Reynolds-Averaged Navier-Stokes equations and the Volume of Fluid method for free surface flow modeling, is used in a 3D numerical wave tank. Three vertical breakwater configurations subject to the action of incident regular waves with periods from 6 to 12 s are studied: normal breakwater, with vertical walls parallel to the direction along the breakwater length; and two novel breakwater geometries, partially and fully convergent breakwaters, whose converging vertical walls are inclined θ in relation to this direction. Different spacing S from 0 to 20 m between the array of OWC devices and two converging wall angles θ, 30 and 45°, are investigated. Firstly, analysis of the influence of S for the normal breakwater shows that the vertical wall concentrates naturally a higher quantity of the incident wave energy inside OWC chamber devices and, consequently, increases their efficiencies. This effect is intensified as the spacing S increases. Secondly, analyses of the partially and fully convergent breakwaters allow concluding that these novel geometries, which direct an amount of incident wave energy into the OWC chamber, increase significantly the efficiency of the array of the OWC devices at the range of the wave periods. The highest performance of OWC device is obtained by the fully convergent breakwater with S = 20 m and θ = 45°, once 10 OWC devices inserted in a breakwater 300 m long have the same efficiency of 20 OWC devices inserted into the normal breakwater.

Modeling the Flow and Geomorphic Heterogeneity Induced by Salt Marsh Vegetation Patches Based on Convolutional Neural Network UNet‐Flow

AbstractThe two‐way interactions between biological and physical processes, bio‐geomorphic feedback, play a vital role in landscape formation and evolution in salt marshes. Patchy vegetation represents a typical form of scale‐dependent feedback in salt marshes and is primarily responsible for the formation of efficient drainage networks. The intuitive manifestation of scale‐dependent feedback is the heterogeneity of flow and landscape. Process‐based modeling is an essential tool for exploring flow heterogeneity, but calculations for small spatial scales and over long time frames can be prohibitively costly. In this study, we proposed a deep learning model architecture, UNet‐Flow, based on convolutional neural networks (CNNs), which is used to build a surrogate model to simulate a flow field induced by salt marsh patchy vegetation. After optimizing and evaluating the model, we discovered that UNet‐Flow exhibits a speed improvement of over four orders of magnitude compared to single‐process simulations using the free surface flow model TELEMAC‐2D, with acceptable levels of error. Furthermore, we proposed an approach that combines the process‐based model SISYPHE with the deep learning method to model geomorphic heterogeneity. After numerous simulations of flow heterogeneity modeling using UNet‐Flow, we obtained a significant logarithmic relationship between scale‐dependent feedback strength and vegetation stem density, and an ascending‐descending trend in feedback strength was observed as the number or surface area of vegetation patches increased. Finally, we investigated the relationship between geomorphic heterogeneity and vegetation‐related variables. This study represents a noteworthy effort to study bio‐geomorphology using deep learning methods.

Analysis of Convergence Behavior for the Overset Mesh Based Numerical Wave Tank in openfoam

Abstract This paper presents a solution verification and validation study for an overset mesh based numerical wave tank in openfoam, which considers the coupling between a free-surface hydrodynamic flow model, a rigid body motion model, and an overset mesh. The coupling between the rigid body motion solver and the free-surface flow solver was achieved in a segregated manner. Free decay of roll motion of a barge was modeled using the numerical wave tank, and the damping coefficient was selected as the target quantity for solution verification. The least-square based solution verification procedure was adopted, where one of the four types of error estimators was fit to the data in the least-square sense. Both structured and unstructured mesh were tested, and their effects on the convergence order, numerical uncertainty, and error were carefully investigated. From the numerical tests, it is found that the numerical wave tank exhibits a very good convergence property for the floating body problems with structured mesh, i.e., nearly second order in space and first order in time. However, when switching the body-fitted mesh to unstructured mesh, the grid convergence is reduced to first order. Unstructured mesh does not significantly affect the convergence order in time domain, but results in a larger uncertainty due to data scattering.

Assessment of interfacial turbulence treatment models for free surface flows

The modelling of complex free surface flows is challenging due to the mobility and deformability of the interface and air entrainment characteristics, which are highly affected by turbulence. With the framework of Reynolds averaged Navier–Stokes (RANS) models and the volume of fluid (VOF) method, turbulence quantities at the air–water interface tend to be over-estimated. In this study, interfacial turbulence treatment methods including the buoyancy modification model based on the simple gradient diffusion hypothesis (SGDH) and Egorov’s turbulence damping model are investigated. Furthermore, due to the unconditionally unstable characteristics of the standard k-ε turbulence model, the stabilized k-ε turbulence model is applied as a comparison. The turbulence attenuation performance using different interfacial turbulence treatment methods in the vicinity of the interface is compared and discussed for stratified flows and free overflow weirs for aerated and non-aerated nappe scenarios. The turbulence quantities and free surface profile under different flow conditions are validated against experimental data and an analytical model. The results show that for free surface waves, both the SGDH model and the turbulence damping model give strong improvements in turbulence production compared with the standard k-ε model. The SGDH model augments the turbulence kinetic energy (TKE) in the unstable stratification, leading to unphysical behaviour for the partially dispersed and separated flow.

Development of a single-phase free-surface flow model with the improved lattice kinetic scheme

Development of a single-phase free-surface flow model with the improved lattice kinetic scheme

Free-surface flow simulations with floating objects using lattice Boltzmann method

ABSTRACTIn tsunami inundations or slope disasters of heavy rain, a lot of floating debris or driftwood logs are included in the flows. The damage to structures from solid body impacts is more severe than the damage from the water pressure. In order to study free-surface flows that include floating debris, developing a high-accurate simulation code of free-surface flows with high performance for large-scale computations is desired. We propose the single-phase free-surface flow model based on the cumulant lattice Boltzmann method coupled with a particle-based rigid body simulation. The discrete element method calculates the contact interaction between solids. An octree-based AMR (Adaptive Mesh Refinement) method is introduced to improve computational accuracy and time-to-solution. High-resolution grids are assigned near the free surfaces and solid boundaries. We conducted two kinds of tsunami flow experiments in the 15 and 70 m water tanks at Hachinohe Institute of Technology and Kobe University to validate the accuracy of the proposed model. The simulation results have shown good agreement with the experiments for the drifting speed, the number of trapped wood pieces, and the stacked angles.

A CNN-supported Lagrangian ISPH model for free surface flow

As a popular method for modeling violent free surface flow, the incompressible Smoothed Particle Hydrodynamics (ISPH) based on the Lagrangian formulation has attracted a great attention worldwide. The Lagrangian ISPH solves the unsteady Navier-Stokes and continuity equations using the projection method, in which the pressure is obtained by solving the pressure Poisson's equation (PPE) that is the most time-consuming part in the ISPH procedure. In this paper, the Convolutional Neural Network (CNN) is combined with ISPH and used to predict the fluid pressure instead of solving the PPE directly. Although limited attempts of using CNN for solving the PPE in Eulerian formulation (referred to as the Eulerian CNN framework) in mesh-based methods are found in the public domain, the present model is the first ISPH model supported by CNN in a Lagrangian formulation. The proposed model overcome several challenges associated with combining CNN with ISPH, including selecting the input parameters, formulating the objective functions, producing the training dataset and dealing with boundary conditions. Two classic free surface problems, i.e. the dam breaking and the wave propagation, are simulated to evaluate the performance of the present model. Quantitative assessments of the numerical error in terms of both the free surface profile and the pressure field are carried out. The assessments show that the new model does not only give results with satisfactory accuracy, but also requires much less computation time for estimating pressure if the number of particles is large, e.g., 100 thousands particles that is usually required in the practical ISPH simulation for free surface flow.

A study on the performance of circular and rectangular submerged breakwaters using nun-uniform FGVT method

ABSTRACT Submerged circular breakwaters are laid to a point at the bottom, simplifying their installation compared to the rigid rectangular ones. In the present study, numerical simulations were performed to evaluate the circular and rectangular submerged breakwaters transmission coefficient, changing different hydraulic/geometric effective parameters. A sensitivity analysis was performed to evaluate the effectiveness of various parameters on the transmission coefficient for different wave heights, water depths, and bed slopes. Numerical simulations were performed using a two-phase free surface flow model. Interface tracking was also performed using the modified fine grid volume tracking-volume of fluid (FGVT-VOF) method. Experimental measurements were employed to verify the numerical results, suggesting that the numerical model accurately predicts the transmission coefficient; thereby, the numerical results are useful for designing submerged breakwaters. An equation was finally derived to determine the transmission coefficient of the circular submerged breakwaters.

Modeling and stability investigation on the governor-turbine-hydraulic system with a ceiling-sloping tail tunnel

For a hydropower system with a relatively short tail tunnel and wide variation of tail water level, the ceiling-sloping tail tunnel (CSTT) is often recommended. Considering various flow patterns and traveling free-surface-pressurized interface in the CSTT, stability modeling and analysis of the governor-turbine-hydraulic system shows more challenging. Based on state equations analysis, an improved nonlinear model considering transition part, and a new linear model for downstream free-surface flow are derived. Further comparative analysis indicates that, by introducing these models, water flow characteristics in transition part have positive effect on operation stability with an around 4.0% increase of attenuation rate, while that of downstream free-surface flow has a negative effect with a 21.8% decrease rate on the absolute value of negative attenuation factor. For stability optimization of the CSTT, to properly increase its bottom width is preferable, herein with a 6.2% increase of attenuation rate for a 2.0 m increase of bottom width, while a reasonable ceiling slope should be optimized considering the counteraction effect of upstream pressurized flow and downstream free-surface flow. The new developed models and further discussion for the hydropower systems with a CSTT will greatly improve system's stability evaluation and advance the development of renewable energy.

Numerical simulating wave propagation over solid obstructions using a non-hydrostatic free surface flow model combined immersed boundary method

The vertical transformed σ-coordinate adopted by a family of non-hydrostatic free surface water flow models is highly efficient in capturing the free surface and uneven bottom. Nevertheless, it is difficult to simulate flows around complex solid obstructions. In this paper, the immersed boundary method (IBM) is developed and implemented in a non-hydrostatic wave model to promote the ability in simulating wave interactions with solid structures. In the IBM spirit, a virtual force term is introduced into the momentum equations to mimic the natural physical boundary conditions. The virtual force terms are added on the named ghost cells neighboring the solid boundary to enforce the no-slip velocity condition on the solid wall. An efficient and robust method is promoted to identify the virtual solid boundary and the ghost cells. A set of test cases are implemented to verify and validate the proposed model in modeling water wave interaction with solid obstructions. The free surface flow model coupled with IBM is well verified, accurate and efficient in modeling wave-structure interaction and is expected to promote the model in engineering applications.

A model for computing thermally-driven shallow flows

In many natural disasters such as overland oil spills or lava flows, physical fluid properties as density change when considering non-homogeneous spatial and time variable distributions of the temperature. This effect is even more remarkable when these flows show a non-Newtonian behaviour due to the sensitivity of their rheological properties as viscosity or yield stress to temperature. In these cases, temperature becomes a significant variable that drives the fluid behaviour, which must be solved using an energy equation coupled with the free surface flow system. Special attention is devoted to thermal source terms which must include all the heat fluid exchanges, and their modelling sometimes can govern the complete flow behaviour. Fluid density, viscosity and yield stress, also affected by temperature, must be recomputed every time step. Summarizing, this work presents a 2D free surface flow model considering density and temperature variations, which could even modify viscosity and yield stress, with heat transfer mechanisms. The model is applied to oil spill overland simulations and heating/cooling test cases are carried out to ensure the system energy balance. As conclusions, it can be said that the numerical results demonstrate the importance of the heat exchange effects and those of the density, viscosity and yield stress variations.

A coupled SPH-DEM approach for modeling of free-surface debris flows

A Lagrangian mesh-less model is proposed to simulate fluid–solid flows with multiple-sized solids, i.e., millimeter-sized particle and larger-sized debris. Considering the difference in the size of solid phases, a hybrid resolved and unresolved model is established based on the coupling method of smoothed particle hydrodynamics (SPH) and discrete element method (DEM). SPH is used to model fluid, and the locally averaged Navier–Stokes equations are adopted as governing equations. DEM is used to model the particle–particle interactions, and the unresolved description of hydrodynamic forces including drag and buoyancy is established. The large-sized debris is modeled as the rigid body, which is discretized by particle elements having both SPH and DEM characteristics, where SPH particle elements are involved in the closure of the SPH fluids, and DEM particle elements interact with the solid particles following the contact law. The numerical model is validated and verified by several examples, including single-particle sedimentation, collapse of cylinder columns, and debris dam break. Results show that the present model reproduces general features of the complex fluid–solid flow with free surfaces. The advantage of the hybrid model is that it can deal with the fluid–solid flow problem with both small particles and large objects at a suitable resolution, and it is especially good at dealing with the free surface flow problem. A discretization for the modeling of debris flows is proposed based on the coupled SPH-DEM method. The novelty of the work is a coupled resolved–unresolved scheme for the free surface flow with multi-sized solids. The present scheme allows using a uniform resolution by bridging the size difference between small-scale solid particles and large-scale debris. The unresolved model of fluid-particle flow is efficient because the fluid resolution can be configured comparably to the particle size. The unified nature of the model allows the combination of resolved and unresolved simulations in the same computational domain.

A 3D Fully Non-Hydrostatic Model for Free-Surface Flows with Complex Immersed Boundaries

A fully non-hydrostatic hydrodynamic model is developed to simulate a three-dimensional, incompressible, and viscous free-surface flow passing downstream rigid rectangular and circular cylinders. A direct numerical simulation (DNS) based on the volume of fluid (VOF) and immersed boundary (IB) method is presented for solving the Navier–Stokes equations. The numerical scheme provides accurate solutions with high efficiency using the novel computational procedure to model severe surface deformations. A staggered finite difference method with a Cartesian mesh coordinate system is used to discretize the governing equations with the complexity of the deformed free-surface flow, for which the numerical schemes include a free-surface tracking technique based on the VOF and a VOS-based IB method to simulate 3D dam-break flows passing the slender objects. Additionally, the case studies demonstrate the accuracy and flexibility of the proposed model to predict the impact forces of the surface flow against the different configurations of structures. The results reveal that the temporal variation of the impact force acted on the rectangular obstacle is dominated by the aspect ratio. The force increases with the increase in the shape parameter. The resistance caused by a thin obstacle is considerably less than the blunt shape.

Two-dimensional modelling of free-surface flows in presence of a spherical object using the Modified Volume of Fluid technique

Two-dimensional modelling of free-surface flows in presence of a spherical object using the Modified Volume of Fluid technique