A lattice Boltzmann-finite element model for two-dimensional fluid–structure interaction problems involving shallow waters

doi:10.1016/j.advwatres.2014.01.003

Advances in Water Resources

Volume 65, March 2014, Pages 18-24

https://doi.org/10.1016/j.advwatres.2014.01.003 Get rights and content

Highlights

•
Shallow water equations are solved through the lattice Boltzmann method.
•
Solid bodies are idealized by beam finite elements.
•
The fluid and solid solvers are coupled within a proper strategy.
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The model is validated against the one-dimensional dam break benchmark.
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Water waves consequently to a partial dam break impacts deformable beams.

Abstract

In this paper, a numerical method for the modeling of shallow waters interacting with slender elastic structures is presented. The fluid domain is modeled through the lattice Boltzmann method, while the solid domain is idealized by corotational beam finite elements undergoing large displacements. Structure dynamics is predicted by using the time discontinuous Galerkin method and the fluid–structure interface conditions are handled by the Immersed Boundary method. An explicit coupling strategy to combine the adopted numerical methods is proposed and its effectiveness is tested by computing the error in terms of the energy that is artificially introduced at the fluid–solid interface.

Introduction

Efficient and accurate handling of large-displacement fluid–structure interaction (FSI) is one of the most challenging and practically important topics in contemporary numerical modeling of computational fluid dynamics (CFD), since a wide set of applications can be covered, spanning automotive, mechanics, biomedicine, and biomimetics. From an environmental point of view, an interesting challenge is an accurate solution of shallow water equations, which can be used to predict coastal erosion, wave propagation, tidal flows and even pollutant transport, among the others. As well known, CFD can be studied through a macroscopic classical point view, consisting in the solution of the Navier–Stokes equations governing continuum fluid dynamics. Even if this approach was deeply consolidated in the last century, it is affected by a few draw backs, such as handling complex moving geometries, for example.

In the last decades, the lattice Boltzmann (LB) method [1] arose as an effective alternative to the classical continuum-based CFD. Such method is based on Boltzmann’s equation, which describe the evolution in space and time of a fictitious set of particles by a mesoscopic point of view. Boltzmann’s equation is discretized and solved on an Eulerian grid which is kept fixed during the analysis. The LB method has proved to be an effective method to solve various CFD problems, such as multiphase flows [2], non-Newtonian blood flow [3], [4], [5], [6], large-eddy simulations of turbulent flows [7], and even fluid–structure interaction (FSI) problems [8], [9], [10], among the others. Moreover, such method has been effectively used to solve shallow water equations [11], [12], thus introducing the LB method in coastal engineering.

In this paper, a numerical model for two-dimensional FSI phenomena involving shallow waters is presented. For this purpose, special attention is devoted to account for the presence of an immersed solid. A possible approach consists of the adoption of the so called interpolated bounce-back rule [13], [14]. Even if this approach is widespread, handling a general geometry represents a complex task from an implementation point of view and tracking the boundary position with respect to the lattice grid is computationally intensive, especially upon solid deformation. For this reason, it can be realized that computing the distribution of forces over the solid boundaries, as required by FSI applications, tends to become cumbersome. In addition, within the FSI framework, a proper procedure should be devised in order to tackle lattice nodes that are activated and de-activated due to moving solid boundaries caused by structure deformation. These refill procedures typically violate mass conservation [15], [16]. In opposition to the interpolated bounce-back rule, the Immersed Boundary (IB) method [17] is adopted, according to the approach in [18], [19] and integrated within the coupling strategy. In this way, the implementation is quite general and the algorithm complexity is almost independent from the structure geometry. Moreover, no refill procedure is needed and fluid mass is strictly conserved. Elastic beam finite elements capable of undergoing large displacements are used to capture structure deformation. Space discretization is based on the corotational formulation [20] and time discretization on the discontinuous Galerkin formulation implemented according to [21], [22]. Firstly, the proposed solution procedure is validated. Then, a partial dam break is simulated, with the flow impacting solids. If rigid fixed structures are assumed, consideration about the water height are carried out. On the other hand, if the deformability is accounted for, the interface energy is evaluated and it is used to assess the effectiveness and accuracy of the proposed methodology.

The paper is organized as follows. In Section 2, the governing equations are stated. In Section 3, the numerical methods are recalled, together with the coupling strategy. In Section 4, the validation of the LB method for shallow waters equations and numerical results from a test case are discussed. Finally, in Section 5 some conclusions are drawn.

Section snippets

Governing equations

The fluid domain is governed by the two-dimensional shallow water equations: $\frac{\partial h}{\partial t} + \frac{\partial ({hv}_{χ})}{\partial x_{χ}} = 0,$ $\frac{\partial ({hv}_{χ})}{\partial t} + \frac{\partial ({hv}_{χ} v_{ψ})}{\partial x_{ψ}} + \frac{\partial}{\partial x_{χ}} (\frac{{gh}^{2}}{2}) = ν [\frac{\partial^{2} ({hv}_{χ})}{\partial x_{χ} \partial x_{ψ}}] + F_{χ},$ where h is the water height, t is the time, x is the position, $v_{χ}$ is the depth-averaged velocity in the $χ$ direction, g is the gravitational acceleration, $ν$ is the kinematic viscosity and $F_{χ}$ accounts for an external force acting in the $χ$ direction.

The equations of structure motion read as follows: $EJ \frac{\partial^{4} r_{y}}{\partial x^{4}} + m \frac{\partial^{2} r_{y}}{\partial t^{2}} = - q_{y} (x),$ $EA \frac{\partial^{4} r_{x}}{\partial x^{4}} - m \frac{\partial^{2} r_{x}}{\partial t^{2}} = - q_{x} (x),$ where EJ

Fluid and structure modeling

In this section, the methods selected to describe fluid and structure dynamics are briefly recalled.

Numerical results and discussion

First, a validation of the adopted LB model for shallow waters equations is given. Notice that the validation of the other numerical methods, i.e. the corotational formulation and the FSI coupling algorithm, has been already presented in [16], [28]. Then, a partial dam break is simulated, where the fluid flow impacts structures which are assumed to be first rigid and then deformable.

Conclusions

A numerical model able to simulate two-dimensional FSI phenomena involving shallow waters and deformable structure has been proposed. A multi-relaxation time LB model has been adopted for the fluid domain, which enhances the stability of the algorithm with respect to the single-relaxation time scheme. On the structure side, the TDG method has been adopted within a corotational beam finite element and the IB method has been used to account for the presence of the solid body in the lattice

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A lattice Boltzmann-finite element model for two-dimensional fluid–structure interaction problems involving shallow waters

Highlights

Abstract

Introduction

Section snippets

Governing equations

Fluid and structure modeling

Numerical results and discussion

Conclusions

J Biomech

Comput Math Appl

Int J Heat Fluid Flow

Comput Fluids

Adv Water Res

Comput Fluids

J Comput Phys

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

Phys Rep

Int J Solids Struct

Comput Methods Appl Mech Eng

Lattice Boltzmann methods for multiphase flow simulations across scales

Commun Comput Phys

Analysis of blood flow in deformable vessels via a lattice Boltzmann approach

Int J Mod Phys C