Stable free surface flows with the lattice Boltzmann method on adaptively coarsened grids

In this paper we will present an algorithm to perform free surface flow simulations with the lattice Boltzmann method on adaptive grids. This reduces the required computational time by more than a factor of three for simulations with large volumes of fluid. To achieve this, the simulation of large fluid regions is performed with coarser grid resolutions. We have developed a set of rules to dynamically adapt the coarse regions to the movement of the free surface, while ensuring the consistency of all grids. Furthermore, the free surface treatment is combined with a Smagorinsky turbulence model and a technique for adaptive time steps to ensure stable simulations. The method is validated by comparing the position of the free surface with an uncoarsened simulation. It yields speedup factors of up to 3.85 for a simulation with a resolution of 480³ cells and three coarser grid levels, and thus enables efficient and stable simulations of free surface flows, e.g. for highly detailed physically based animations of fluids.     

A finite volume method parallelization for the simulation of free surface shallow water flows

We construct a parallel algorithm, suitable for distributed memory architectures, of an explicit shock-capturing finite volume method for solving the two-dimensional shallow water equations. The finite volume method is based on the very popular approximate Riemann solver of Roe and is extended to second order spatial accuracy by an appropriate TVD technique. The parallel code is applied to distributed memory architectures using domain decomposition techniques and we investigate its performance on a grid computer and on a Distributed Shared Memory supercomputer. The effectiveness of the parallel algorithm is considered for specific benchmark test cases. The performance of the realization measured in terms of execution time and speedup factors reveals the efficiency of the implementation.     

The efficient prediction of shallow water flows Part 1: Theory

Finite difference schemes are presented for the solution of the one-dimensional shallow water equations in open channels, and the two-dimensional shallow water equations. The numerical results are subsequently compared for a range of problems.     

Generalized vortex methods for free surface flow problems. II: Radiating waves

Generalized vortex methods are applied to the study of free surface motion in incompressible, irrotational, inviscid, layered flows in which waves are generated by external means, such as the motion of submerged bodies. Radiation conditions are developed that allow outward-traveling waves to escape the computational domain without significant reflection. Numerical sponge layers are used to absorb outward-traveling waves. Applications are given to wave generation by surface pressure distributions and by translating submerged bodies.     

Accounting for surface–groundwater interactions and their uncertainty in river and groundwater models: A case study in the Namoi River,Australia

Surface–groundwater (SW–GW) interactions constitute a critical proportion of the surface and groundwater balance especially during dry conditions. Conjunctive management of surface and groundwater requires an explicit account of the exchange flux between surface and groundwater when modelling the two systems. This paper presents a case study in the predominantly gaining Boggabri–Narrabri reach of the Namoi River located in eastern Australia. The first component of the study uses the Upper Namoi numerical groundwater model to demonstrate the importance of incorporating SW–GW interactions into river management models. The second component demonstrates the advantages of incorporating groundwater processes in the Namoi River model.Results of the numerical groundwater modelling component highlighted the contrasting groundwater dynamics close to, and away from the Namoi River where lower declines were noted in a near-field well due to water replenishment sourced from river depletion. The contribution of pumping activities to river depletion was highlighted in the results of the uncertainty analysis, which showed that the SW–GW exchange flux is the most sensitive to pumping rate during dry conditions. The uncertainty analysis also showed that after a drought period, the 95% prediction interval becomes larger than the simulated flux, which implies an increasing probability of losing river conditions. The future prospect of a gaining Boggabri–Narrabri reach turning into losing was confirmed with a hypothetical extended drought scenario during which persistent expansion of groundwater pumping was assumed. The river modelling component showed that accounting for SW–GW interactions improved the predictions of low flows, and resulted in a more realistic calibration of the Namoi River model.Incorporating SW–GW interactions into river models allows explicit representation of groundwater processes that provides a mechanism to account for the impacts of additional aquifer stresses that may be introduced beyond the calibration period of the river model. Conventional river models that neglect the effects of such future stresses suffer from the phenomenon of non-stationarity and hence have inferior low flow predictions past the calibration period of the river model. The collective knowledge acquired from the two modelling exercises conducted in this study leads to a better understanding of SW–GW interactions in the Namoi River thus leading to improved water management especially during low flow conditions.     

Radial basis functions: Developments and applications to planetary scale flows

Radial basis functions (RBFs) can be seen as a major generalization of pseudospectral (PS) methods, abandoning the orthogonality of the basis functions and in return obtaining much improved simplicity and geometric flexibility. Spectral accuracy becomes now easily available also when using completely unstructured node layouts, permitting local node refinements in critical areas. The first major PDE applications for which RBFs have been shown to compete successfully against the best currently available numerical approaches can be found in the geosciences. Examples that are discussed here include translating vortex roll-ups (cyclogenesis), nonlinear flows on the sphere modeled by the shallow water equations, and 3D convection in the earth’s mantle.     

Hybrid LES/RANS modelling of free surface flow through vegetation

Vegetated channels are environmentally friendly and frequently used to convey water for drainage and recreational purposes. The design and assessment of these channels often requires the use of numerical models which are based on the Reynolds Averaged Navier-Stokes (RANS) approach or Large Eddy Simulations (LES). It is well accepted that both approaches have their advantages and disadvantages. To overcome these disadvantages a hybrid model combining the RANS and LES methodologies is proposed in this work. The major task for the model development is to couple the RANS and the LES models effectively. Various methods have been investigated and the results are as follows. At the inflow boundary of the computational domain, a semi-analytical velocity profile for submerged vegetation is used as the RANS inflow condition to shorten the unrealistic flow transition region. At the interface of the upstream RANS region and the downstream LES region, turbulence fluctuations are artificially generated using a spectral line processor, with the mean velocity determined by using the frozen cloud assumption. At the interface of the upstream LES region and the downstream RANS region, a virtual momentum sink is imposed to dissipate the sub-grid scale fluctuations and to shorten the transition region. The final model has been verified against experiments of flow through submerged and emergent vegetation, as well as a partly vegetated channel.     

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

In the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten, this paper presents an extension of a previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier–Stokes equations. Second, it reports numerical simulation results for 1D shock tube problem, 2D impinging jet and 2D/3D broken dam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems.     

Study of a Jacobian-free approach in the simulation of compressible fluid flows in porous media using a derivative-free spectral method

The development of Jacobian-free software for solving problems formulated by nonlinear partial differential equations is of increasing interest to simulate practical engineering processes. For the first time, this work uses the so-called derivative-free spectral algorithm for nonlinear equations in the simulation of flows in porous media. The model considered here is the one employed to describe the displacement of miscible compressible fluid in porous media with point sources and sinks, where the density of the fluid mixture varies exponentially with the pressure. This spectral algorithm is a modern method for solving large-scale nonlinear systems, which does not use any explicit information associated with the Jacobin matrix of the considered system, being a Jacobian-free approach. Two dimensional problems are presented, along with numerical results comparing the spectral algorithm to a well-developed Jacobian-free inexact Newton method. The results of this paper show that this modern spectral algorithm is a reliable and efficient method for simulation of compressible flows in porous media.     

Application of NEM in seepage analysis with a free surface

As one kind of meshless methods, the natural element method (NEM) constructs shape functions based on the Voronoi diagrams, and it has advantages of both the conventional meshless method and the finite element method. Since the nodes are independent of the integral mesh, it is more suitable for the analysis of seepage with a free surface than the finite element method. In addition, its shape functions satisfy the Kronecker δ conditions, therefore, its boundary conditions can be dealt with much easily than those of such meshless method as element-free Galerkin method (EFGM). In this paper, the NEM was used in the seepage analysis of dams. The initial free surface was assumed first in the calculations, and the location of the free surface was adjusted according to the calculation results. The examples showed that the natural element method lead to satisfactory results.     

An ISPH scheme for numerical simulation of multiphase flows with complex interfaces and high density ratios

Multiphase problems with high density ratios and complex interfaces deal with numerical instabilities and require accurate considerations for capturing the multiphase interfaces. An Incompressible Smoothed Particle Hydrodynamics (ISPH) scheme is presented to simulate such problems. In order to keep the present scheme simple and stable, well-established formulations are used for discretizing the spatial derivatives and a repulsive force is applied at the multiphase interface between particles of different fluids to maintain the interface sharpness. Special considerations are included to overcome the difficulties to model severe physical discontinuities at the interface and surface tension effects are taken into account. Different particle shifting schemes are also tested for a range of problems. Several two phase flows are investigated and the presented scheme is validated against both analytical and numerical solutions. A detailed study is also carried out on the influence of the repulsive force in an ISPH scheme showing that this simple treatment efficiently enhances the interface capturing features. The comparisons indicate that the proposed scheme is robust and capable of simulating a wide range of multiphase problems with complex interfaces including low to high ratios for density and viscosity.     

Application of lattice Boltzmann method in free surface flow simulation of micro injection molding

The filling flow in micro injection molding was simulated by using the lattice Boltzmann method (LBM). A tracking algorithm for free surface to handle the complex interaction between gas and liquid phases in LBM was used for the free surface advancement. The temperature field in the filling flow is also analyzed by combining the thermal lattice Boltzmann model and the free surface method. To simulate the fluid flow of polymer melt with a high Prandtl number and high viscosity, a modified lattice Boltzmann scheme was adopted by introducing a free parameter in the thermal diffusion equation to overcome the restriction of the thermal relaxation time. The filling flow simulation of micro injection molding was successfully performed in the study.     

Coupled model of surface water flow, sediment transport and morphological evolution

This paper presents a mathematical model coupling water flow and sediment transport dynamics that enables calculating the changing surface morphology through time and space. The model is based on the shallow water equations for flow, conservation of sediment concentration, and empirical functions for bed friction, substrate erosion and deposition. The sediment transport model is a non-capacity formulation whereby erosion and deposition are treated independently and influence the sediment flux by exchanging mass across the bottom boundary of the flow. The resulting hyperbolic system is solved using a finite volume, Godunov-type method with a first-order approximate Riemann solver. The model can be applied both to short time scales, where the flow, sediment transport and morphological evolution are strongly coupled and the rate of bed evolution is comparable to the rate of flow evolution, or to relatively long time scales, where the time scale of bed evolution associated with erosion and/or deposition is slow relative to the response of the flow to the changing surface and, therefore, the classical quasi-steady approximation can be invoked. The model is verified by comparing computed results with documented solutions. The developed model can be used to investigate a variety of problems involving coupled flow and sediment transport including channel initiation and drainage basin evolution associated with overland flow and morphological changes induced by extreme events such as tsunami.     

Regularized shallow water equations and a new method of numerical simulation of the open channel flows

The regularized shallow water (RSW) equation system is proposed together with a central-difference algorithm for its numerical solution. The possibilities of the new computational model are demonstrated for a 1D dam break and Le-Veque’s perturbation tests, and for 2D asymmetric dam break test problems.     

Superposition of local zoom models and simultaneous calibration for 1D–2D shallow water flows

We address the problem of coupling 2D shallow water equations with 1D shallow water equations (St-Venant equations), as applied to river-floodplain flows. Mathematical coupling conditions are derived classicaly from the 3D Navier–Stokes equations by integrating over the vertical wet section, when overflowing occurs. It leads to extra source terms in the 1D equations. Next we assume to be in a variational data assimilation context, then the optimal control process allows to couple both models and assimilate data simultaneously (Joint Assimilation Coupling algorithms). Two different versions of JAC algorithms are presented and compared. In a numerical test case, we superimpose the local 2D model on the 1D global model. The results show the efficiency of the present simultaneous superposition–assimilation approach.     

Investigation of wall bounded flows using SPH and the unified semi-analytical wall boundary conditions

Semi-analytical wall boundary conditions present a mathematically rigorous framework to prescribe the influence of solid walls in smoothed particle hydrodynamics (SPH) for fluid flows. In this paper they are investigated with respect to the skew-adjoint property which implies exact energy conservation. It will be shown that this property holds only in the limit of the continuous SPH approximation, whereas in the discrete SPH formulation it is only approximately true, leading to numerical noise. This noise, interpreted as a form of “turbulence”, is treated using an additional volume diffusion term in the continuity equation which we show is equivalent to an approximate Riemann solver. Subsequently two extensions to the boundary conditions are presented. The first dealing with a variable driving force when imposing a volume flux in a periodic flow and the second showing a generalization of the wall boundary condition to Robin type and arbitrary-order interpolation. Two modifications for free-surface flows are presented for the volume diffusion term as well as the wall boundary condition. In order to validate the theoretical constructs numerical experiments are performed showing that the present volume flux term yields results with an error 5 orders of magnitude smaller then previous methods while the Robin boundary conditions are imposed correctly with an error depending on the order of the approximation. Furthermore, the proposed modifications for free-surface flows improve the behavior at the intersection of free surface and wall as well as prevent free-surface detachment when using the volume diffusion term. Finally, this paper is concluded by a simulation of a dam break over a wedge demonstrating the improvements proposed in this paper.     

Comparison of finite element and pendulum models for simulation of sloshing

The pendulum model is a cost effective tool for the simulation of sloshing. However, the accuracy and applicability of the model has not been well established. In this article, we compare the simulation results obtained from the pendulum model and a more complicated finite element model for sloshing of liquids in tanker trucks. In the pendulum model, we assume that the liquid in the tanker is a point mass oscillating like a frictionless pendulum subjected to an external acceleration. In the finite element model, we solve the full Navier-Stokes equations written for two fluids to obtain the location and motion of the free surface. Stabilized finite element formulations are used in these complex 3D simulations. These finite element formulations are implemented in parallel using the message-passing interface libraries. The numerical example includes the simulation of sloshing in tanker trucks during turning.