Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel computation of viscous incompressible flows using Godunov‐projection method on overlapping grids

The Godunov‐projection method is implemented on a system of overlapping structured grids for solving the time‐dependent incompressible Navier–Stokes equations. This projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The Godunov procedure is applied to estimate the non‐linear convective term in order to provide a robust discretization of this terms at high Reynolds number. In order to obtain the pressure field, a separate procedure is applied in this modified Godunov‐projection method, where the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain, as they offer the flexibility of simplifying the grid generation around complex geometrical domains. This combination of projection method and overlapping grid is also parallelized and reasonable parallel efficiency is achieved. Numerical results are presented to demonstrate the performance of this combination of the Godunov‐projection method and the overlapping grid. Copyright © 2002 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

An adaptive cut‐cell method for environmental fluid mechanics

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable‐density, and within arbitrarily complex and possibly anisotropic domains. The method is new because in this context we couple the embedded‐boundary (or cut‐cell) method for complex geometry with block‐structured adaptive mesh refinement (AMR) while maintaining conservation and second‐order accuracy. The accurate simulation of variable‐density fluids necessitates special care in formulating projection methods. This variable‐density formulation is well known for incompressible flows in unit‐aspect ratio domains, without AMR, and without complex geometry, but here we carefully present a new method that addresses the intersection of these issues. The methodology is based on a second‐order‐accurate projection method with high‐order‐accurate Godunov finite‐differencing, including slope limiting and a stable differencing of the nonlinear convection terms. The finite‐volume AMR discretizations are based on two‐way flux matching at refinement boundaries to obtain a conservative method that is second‐order accurate in solution error. The control volumes are formed by the intersection of the irregular embedded boundary with Cartesian grid cells. Unlike typical discretization methods, these control volumes naturally fit within parallelizable, disjoint‐block data structures, and permit dynamic AMR coarsening and refinement as the simulation progresses. We present two‐ and three‐dimensional numerical examples to illustrate the accuracy of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Vorticity–velocity formulation of the 3D Navier–Stokes equations in cylindrical co‐ordinates

A finite difference method is presented for solving the 3D Navier–Stokes equations in vorticity–velocity form. The method involves solving the vorticity transport equations in ‘curl‐form’ along with a set of Cauchy–Riemann type equations for the velocity. The equations are formulated in cylindrical co‐ordinates and discretized using a staggered grid arrangement. The discretized Cauchy–Riemann type equations are overdetermined and their solution is accomplished by employing a conjugate gradient method on the normal equations. The vorticity transport equations are solved in time using a semi‐implicit Crank–Nicolson/Adams–Bashforth scheme combined with a second‐order accurate spatial discretization scheme. Special emphasis is put on the treatment of the polar singularity. Numerical results of axisymmetric as well as non‐axisymmetric flows in a pipe and in a closed cylinder are presented. Comparison with measurements are carried out for the axisymmetric flow cases. Copyright © 2003 John Wiley & Sons, Ltd.     

Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth‐order compact formulation for the steady 2‐D incompressible Navier–Stokes equations is presented. The formulation is in the same form of the Navier–Stokes equations such that any numerical method that solve the Navier–Stokes equations can easily be applied to this fourth‐order compact formulation. In particular, in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601 × 601. Using this formulation, the steady 2‐D incompressible flow in a driven cavity is solved up to Reynolds number with Re = 20 000 fourth‐order spatial accuracy. Detailed solutions are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Moving immersed boundary method

A novel implicit immersed boundary method of high accuracy and efficiency is presented for the simulation of incompressible viscous flow over complex stationary or moving solid boundaries. A boundary force is often introduced in many immersed boundary methods to mimic the presence of solid boundary, such that the overall simulation can be performed on a simple Cartesian grid. The current method inherits this idea and considers the boundary force as a Lagrange multiplier to enforce the no‐slip constraint at the solid boundary, instead of applying constitutional relations for rigid bodies. Hence excessive constraint on the time step is circumvented, and the time step only depends on the discretization of fluid Navier‐Stokes equations, like the CFL condition in present work. To determine the boundary force, an additional moving force equation is derived. The dimension of this derived system is proportional to the number of Lagrangian points describing the solid boundaries, which makes the method very suitable for moving boundary problems since the time for matrix update and system solving is not significant. The force coefficient matrix is made symmetric and positive definite so that the conjugate gradient method can solve the system quickly. The proposed immersed boundary method is incorporated into the fluid solver with a second‐order accurate projection method as a plug‐in. The overall scheme is handled under an efficient fractional step framework, namely, prediction, forcing, and projection. Various simulations are performed to validate current method, and the results compare well with previous experimental and numerical studies.     

A finite element formulation satisfying the discrete geometric conservation law based on averaged Jacobians

In this article, a new methodology for developing discrete geometric conservation law (DGCL) compliant formulations is presented. It is carried out in the context of the finite element method for general advective–diffusive systems on moving domains using an ALE scheme. There is an extensive literature about the impact of DGCL compliance on the stability and precision of time integration methods. In those articles, it has been proved that satisfying the DGCL is a necessary and sufficient condition for any ALE scheme to maintain on moving grids the nonlinear stability properties of its fixed‐grid counterpart. However, only a few works proposed a methodology for obtaining a compliant scheme. In this work, a DGCL compliant scheme based on an averaged ALE Jacobians formulation is obtained. This new formulation is applied to the θ family of time integration methods. In addition, an extension to the three‐point backward difference formula is given. With the aim to validate the averaged ALE Jacobians formulation, a set of numerical tests are performed. These tests include 2D and 3D diffusion problems with different mesh movements and the 2D compressible Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

An implicit edge‐based ALE method for the incompressible Navier–Stokes equations

A new finite volume method for the incompressible Navier–Stokes equations, expressed in arbitrary Lagrangian–Eulerian (ALE) form, is presented. The method uses a staggered storage arrangement for the pressure and velocity variables and adopts an edge‐based data structure and assembly procedure which is valid for arbitrary n‐sided polygonal meshes. Edge formulas are presented for assembling the ALE form of the momentum and pressure equations. An implicit multi‐stage time integrator is constructed that is geometrically conservative to the precision of the arithmetic used in the computation. The method is shown to be second‐order‐accurate in time and space for general time‐dependent polygonal meshes. The method is first evaluated using several well‐known unsteady incompressible Navier–Stokes problems before being applied to a periodically forced aeroelastic problem and a transient free surface problem. Published in 2003 by John Wiley & Sons, Ltd.     

A second look at the role of the fast Fourier transform as an elliptic solver

A fast cosine transform (FCT) is coupled with a tridiagonal solver for the purpose of solving the Poisson equation on irregular and non‐uniform rectangular staggered grids. This kind of solution is required for the pressure field during the simulation of the incompressible Navier–Stokes equations when using the projection method. A new technique using the FCT–tridiagonal solver is derived for the cases where the boundaries of the flow regime do not coincide with the boundaries of the computational domain and for non‐uniform grids. The technique is based on an iterative procedure where a defect equation is solved in every iteration, followed by a relaxation procedure. The method is investigated analytically and numerically to show that the solution converges as a geometric series. The method is further investigated for the effects of the relative size of the rigid body, the grid stretching, size and aspect ratio. The new solver is incorporated with the direct numerical simulation (DNS) and large eddy simulation (LES) techniques to simulate the flows around a backward‐facing step and a 3D rectangular obstacle, yielding results that qualitatively compare well with known results. Copyright © 2005 John Wiley & Sons, Ltd.     

On the geometric conservation law in transient flow calculations on deforming domains

This note revisits the derivation of the ALE form of the incompressible Navier–Stokes equations in order to retain insight into the nature of geometric conservation. It is shown that the flow equations can be written such that time derivatives of integrals over moving domains are avoided prior to discretization. The geometric conservation law is introduced into the equations and the resulting formulation is discretized in time and space without loss of stability and accuracy compared to the fixed grid version. There is no need for temporal averaging remaining. The formulation applies equally to different time integration schemes within a finite element context. Copyright © 2005 John Wiley & Sons, Ltd.     

A comparative study of truly incompressible and weakly compressible SPH methods for free surface incompressible flows

In this paper, the performance of the incompressible SPH (ISPH) method and an improved weakly compressible SPH (IWCSPH) method for free surface incompressible flows are compared and analyzed. In both methods, the Navier–Stokes equations are solved, and no artificial viscosity is used. The ISPH algorithm in this paper is based on the classical SPH projection method with common treatments on solid boundaries and free surfaces. The IWCSPH model includes some advanced corrective algorithms in density approximation and solid boundary treatment (SBT). In density approximation, the moving least squares (MLS) approach is applied to re‐initialize density every several steps to obtain smoother and more stable pressure fields. An improved coupled dynamic SBT algorithm is implemented to obtain stable pressure values near solid wall areas and, thus, to minimize possible numerical oscillations brought in by the solid boundaries. Three representative numerical examples, including a benchmark test for hydrostatic pressure, a dam breaking problem and a liquid sloshing problem, are comparatively analyzed with ISPH and IWCSPH. It is demonstrated that the present IWCSPH is more attractive than ISPH in modeling free surface incompressible flows as it is more accurate and more stable with comparable or even less computational efforts. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

Integral vorticity transport method on an overset grid

We present an overset grid method for solution of the integro‐differential vorticity–velocity formulation of the Navier–Stokes equations for two‐dimensional, incompressible flow. The method uses a body‐fitted inner grid, on which vorticity is evolved semi‐implicitly, and a Cartesian outer grid with explicit vorticity evolution. The Biot–Savart integral is solved using an adaptive, optimized multipole acceleration method. The Biot–Savart integration is performed over all inner grid cells, over all ‘active cells’ of the outer grid that lie entirely outside of the inner grid, and over sub‐elements of a set of ‘overhanging’ cells of the outer grid that overlap part of the inner grid. A novel method is developed using a level‐set distance function to rapidly and easily partition the overhanging grid cells, which is essential for the Biot–Savart integration in order to avoid double‐counting vorticity in the overhanging region. A similar decomposition into outer, inner and overhanging cells is used in solving for pressure using a boundary‐element formulation, which requires evaluation of an integral over the vorticity field using a method similar to that used for the Biot–Savart integral. The new overset grid method is applied to flow past stationary and moving bodies in two dimensions and found to agree well with prior experimental and numerical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Two‐dimensional compact finite difference immersed boundary method

We present a compact finite differences method for the calculation of two‐dimensional viscous flows in biological fluid dynamics applications. This is achieved by using body‐forces that allow for the imposition of boundary conditions in an immersed moving boundary that does not coincide with the computational grid. The unsteady, incompressible Navier–Stokes equations are solved in a Cartesian staggered grid with fourth‐order Runge–Kutta temporal discretization and fourth‐order compact schemes for spatial discretization, used to achieve highly accurate calculations. Special attention is given to the interpolation schemes on the boundary of the immersed body. The accuracy of the immersed boundary solver is verified through grid convergence studies. Validation of the method is done by comparison with reference experimental results. In order to demonstrate the application of the method, 2D small insect hovering flight is calculated and compared with available experimental and computational results. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid immersed boundary and material point method for simulating 3D fluid–structure interaction problems

A numerical method is developed for solving the 3D, unsteady, incompressible Navier–Stokes equations in curvilinear coordinates containing immersed boundaries (IBs) of arbitrary geometrical complexity moving and deforming under forces acting on the body. Since simulations of flow in complex geometries with deformable surfaces require special treatment, the present approach combines a hybrid immersed boundary method (HIBM) for handling complex moving boundaries and a material point method (MPM) for resolving structural stresses and movement. This combined HIBM & MPM approach is presented as an effective approach for solving fluid–structure interaction (FSI) problems. In the HIBM, a curvilinear grid is defined and the variable values at grid points adjacent to a boundary are forced or interpolated to satisfy the boundary conditions. The MPM is used for solving the equations of solid structure and communicates with the fluid through appropriate interface‐boundary conditions. The governing flow equations are discretized on a non‐staggered grid layout using second‐order accurate finite‐difference formulas. The discrete equations are integrated in time via a second‐order accurate dual time stepping, artificial compressibility scheme. Unstructured, triangular meshes are employed to discretize the complex surface of the IBs. The nodes of the surface mesh constitute a set of Lagrangian control points used for tracking the motion of the flexible body. The equations of the solid body are integrated in time via the MPM. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at stationary curvilinear grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The influence of the fluid on the body is defined through pressure and shear stresses acting on the surface of the body. The HIBM & MPM approach is validated for FSI problems by solving for a falling rigid and flexible sphere in a fluid‐filled channel. The behavior of a capsule in a shear flow was also examined. Agreement with the published results is excellent. Copyright © 2007 John Wiley & Sons, Ltd.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

Finite‐element simulation of incompressible fluid flow in an elastic vessel

Finite‐element simulation was performed to predict the incompressible Navier–Stokes flow in a domain, partly bounded by an elastic vessel, which is allowed to vary with time. Besides satisfying the physical conservation laws, both surface and the volume conservation laws are satisfied at the discrete level for ensuring the balance between physical and geometrical variables. Several problems which are amenable to analytical solutions were tested for validating the method. The simulated results are observed to agree favourably with analytical solutions. Having verified the applicability of the finite‐element code to problems involving moving grids, we consider an incompressible fluid flow bounded by rigid and elastic vessel walls. Our emphasis was placed on the validation of the formulation developed within the moving‐grid framework. Copyright © 2003 John Wiley & Sons, Ltd.     

Applications of the artificial compressibility method for turbulent open channel flows

A three‐dimensional (3‐D) numerical method for solving the Navier–Stokes equations with a standard k–ε turbulence model is presented. In order to couple pressure with velocity directly, the pressure is divided into hydrostatic and hydrodynamic parts and the artificial compressibility method (ACM) is employed for the hydrodynamic pressure. By introducing a pseudo‐time derivative of the hydrodynamic pressure into the continuity equation, the incompressible Navier–Stokes equations are changed from elliptic‐parabolic to hyperbolic‐parabolic equations. In this paper, a third‐order monotone upstream‐centred scheme for conservation laws (MUSCL) method is used for the hyperbolic equations. A system of discrete equations is solved implicitly using the lower–upper symmetric Gauss–Seidel (LU‐SGS) method. This newly developed numerical method is validated against experimental data with good agreement. Copyright © 2005 John Wiley & Sons, Ltd.