A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method

This paper describes and compares two vorticity‐based integral approaches for the solution of the incompressible Navier–Stokes equations. Either a Lagrangian vortex particle method or an Eulerian finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is used to compute the velocity field from a vorticity distribution over a fluid domain. The vorticity boundary condition is improved by the use of an iteration scheme connected with the well‐established panel method. In the early stages of development of flows around an impulsively started circular cylinder, and past an impulsively started foil with varying angles of attack, the computational results obtained by the Lagrangian vortex method are compared with those obtained by the Eulerian finite volume method. The comparison is performed separately for the pressure fields as well. The results obtained by the two methods are in good agreement, and give a better understanding of the vorticity‐based methods. Copyright © 2005 John Wiley & Sons, Ltd.     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

An implementation of the Spalart–Allmaras DES model in an implicit unstructured hybrid finite volume/element solver for incompressible turbulent flow

In this paper, we describe an implicit hybrid finite volume (FV)/element (FE) incompressible Navier–Stokes solver for turbulent flows based on the Spalart–Allmaras detached eddy simulation (SA‐DES). The hybrid FV/FE solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at vertices, making the current solver a staggered‐mesh scheme. The SA‐DES turbulence equation is solved after the velocity and the pressure fields have been updated at the end of each time step. The same matrix‐free FV method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity, which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA‐DES model in a hybrid flow solver. This paper will address important implementation issues for high‐Reynolds number flows where highly stretched elements are typically used. In addition, some aspects of implementing the SA‐DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flow past a flat plate and a high‐Reynolds number flow around a high angle‐of‐attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite volume solution of the Navier–Stokes equations in velocity–vorticity formulation

For the incompressible Navier–Stokes equations, vorticity‐based formulations have many attractive features over primitive‐variable velocity–pressure formulations. However, some features interfere with the use of the numerical methods based on the vorticity formulations, one of them being the lack of a boundary conditions on vorticity. In this paper, a novel approach is presented to solve the velocity–vorticity integro‐differential formulations. The general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so‐called generalized Biot–Savart formula combined with a fast summation algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well‐known fractional step approaches are used to solve the vorticity transport equation. The paper describes in detail how we accurately impose no normal‐flow and no tangential‐flow boundary conditions. We impose a no‐flux boundary condition on solid objects by the introduction of a proper amount of vorticity at wall. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential‐flow boundary condition. As application examples, the impulsively started flows through a flat plate and a circular cylinder are computed using the method. The present results are compared with the analytical solution and other numerical results and show good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

A 3D second‐order accurate projection‐based Finite Volume code on non‐staggered,non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows

It is well known that exact projection methods (EPM) on non‐staggered grids suffer for the presence of non‐solenoidal spurious modes. Hence, a formulation for simulating time‐dependent incompressible flows while allowing the discrete continuity equation to be satisfied up to machine‐accuracy, by using a Finite Volume‐based second‐order accurate projection method on non‐staggered and non‐uniform 3D grids, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by first solving an elliptic equation on a compact stencil that is by performing a standard approximate projection method (APM). In such a way, three sets of divergence‐free normal‐to‐face velocities can be computed. Then, a second elliptic equation for a scalar field is derived by prescribing that its additional discrete gradient ensures the continuity constraint based on the adopted linear interpolation of the velocity. Characteristics of the double projection method (DPM) are illustrated in details and stability and accuracy of the method are addressed. The resulting numerical scheme is then applied to laminar buoyancy‐driven flows and is proved to be stable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.     

Numerical solutions for impulsively started and decelerated viscous flow past a circular cylinder

A finite difference study of the unsteady two-dimensional flow past a circular cylinder has been conducted using vorticity and streamfunction as the dependent variables. The two cases considered were impulsively started and decelerated flows. The impulsively started problem was considered to validate the method and has yielded results which agree quite closely with existing results from both calculations and experiments. The decelerated flow analysis produced results which can be explained in terms of induced velocity effects from existing wake vortices for both suddenly stopped and uniformly decelerated flows.     

Flow past an impulsively started vertical plate with variable surface temperature and mass flux

Finite-difference solution of the transient natural convection flow of an incompressible viscous fluid past an impulsively started semi-infinite vertical plate with variable surface temperature and mass flux is presented here. The Velocity profiles are compared with exact solution and are found to be in good agreement. The steady-state velocity, temperature and concentration profiles are shown graphically. It is observed that there is a rise in the velocity due to the presence of a mass diffusion. The local as well as average skin-friction, Nusselt number and Sherwood number are shown graphically. Received on 27 May 1998     

A multiblock operator‐splitting algorithm for unsteady flows at all speeds in complex geometries

This paper describes a non‐iterative operator‐splitting algorithm for computing all‐speed flows in complex geometries. A pressure‐based algorithm is adopted as the base, in which pressure, instead of density, is a primary variable, thus allowing for a unified formulation for all Mach numbers. The focus is on adapting the method for (a) flows at all speeds, and (b) multiblock, non‐orthogonal, body‐fitted grids for very complex geometries. Key features of the formulation include special treatment of mass fluxes at control volume interfaces to avoid pressure–velocity decoupling for incompressible (low Mach number limit) flows and to provide robust pressure–velocity–density coupling for compressible (high‐speed) flows. The method is shown to be robust for all Mach number regimes for both steady and unsteady flows; it is found to be stable for CFL numbers of order ten, allowing large time steps to be taken for steady flows. Enhancements to the method which allow for stable solutions to be obtained on non‐orthogonal grids are also discussed. The method is found to be very reliable even in complex engineering applications such as unsteady rotor–stator interactions in turbulent, all‐speed turbomachinery flows. Copyright © 2004 John Wiley & Sons, Ltd.     

Development of a hybrid finite volume/element solver for incompressible flows

In this paper, we report our development of an implicit hybrid flow solver for the incompressible Navier–Stokes equations. The methodology is based on the pressure correction or projection method. A fractional step approach is used to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free implicit cell‐centred finite volume method. The Poisson equation derived from the fractional step approach is solved by the node‐based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centres and the auxiliary variable at cell vertices, making the current solver a staggered‐mesh scheme. Numerical examples demonstrate the performance of the resulting hybrid scheme, such as the correct temporal convergence rates for both velocity and pressure, absence of unphysical pressure boundary layer, good convergence in steady‐state simulations and capability in predicting accurate drag, lift and Strouhal number in the flow around a circular cylinder. Copyright © 2007 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.     

A second‐order time accurate finite element method for quasi‐incompressible viscous flows

A finite element method for quasi‐incompressible viscous flows is presented. An equation for pressure is derived from a second‐order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second‐order time accurate scheme and a least‐squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second‐order accurate method and our previous first‐order accurate formulation is shown. The method is also demonstrated in the computation of the leaky‐lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 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.     

A promising boundary element formulation for three‐dimensional viscous flow

In this paper, a new set of boundary‐domain integral equations is derived from the continuity and momentum equations for three‐dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex‐variable technique is used to compute the divergence of velocity for internal points, while the traction‐recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of high‐Reynolds number flow around circular cylinders by a three‐step FEM–BEM model

An innovative computational model, developed to simulate high‐Reynolds number flow past circular cylinders in two‐dimensional incompressible viscous flows in external flow fields is described in this paper. The model, based on transient Navier–Stokes equations, can solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The pressure is assumed to be zero at infinite boundary and the external flow field is simulated using a direct boundary element method (BEM) by solving a pressure Poisson equation. A three‐step finite element method (FEM) is used to solve the momentum equations of the flow. The present model is applied to simulate high‐Reynolds number flow past a single circular cylinder and flow past two cylinders in which one acts as a control cylinder. The simulation results are compared with experimental data and other numerical models and are found to be feasible and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐accurate intermediate boundary conditions for large eddy simulations based on projection methods

Projection methods are among the most adopted procedures for solving the Navier–Stokes equations system for incompressible flows. In order to simplify the numerical procedures, the pressure–velocity de‐coupling is often obtained by adopting a fractional time‐step method. In a specific formulation, suitable for the incompressible flows equations, it is based on a formal decomposition of the momentum equation, which is related to the Helmholtz–Hodge Decomposition theorem of a vector field in a finite domain. Owing to the continuity constraint also in large eddy simulation of turbulence, as happens for laminar solutions, the filtered pressure characterizes itself only as a Lagrange multiplier, not a thermodynamic state variable. The paper illustrates the implications of adopting such procedures when the decoupling is performed onto the filtered equations system. This task is particularly complicated by the discretization of the time integral of the sub‐grid scale tensor. A new proposal for developing time‐accurate and congruent intermediate boundary conditions is addressed. Several tests for periodic and non‐periodic channel flows are presented. This study follows and completes the previous ones reported in (Int. J. Numer. Methods Fluids 2003; 42, 43 ). Copyright © 2005 John Wiley & Sons, Ltd.     

A numerical study of projection algorithms in the finite element simulation of three‐dimensional viscous incompressible flow

In this work a comparative study of two versions of the projection algorithm used either for time integration or as an iterative method to solve the three‐dimensional incompressible Navier–Stokes equations is presented. It is also shown that these projection algorithms combined with the finite element method are particularly suited for the treatment of outflow boundary conditions in the context of external flows. This assertion is illustrated by means of some numerical examples in which five types of boundary conditions are compared. The scheme is applied to simulate the flow past a cylinder clamped on two fixed parallel solid walls. Comparison with experimental data available for this problem shows good agreement of the velocity and pressure fields, both computed with continuous piecewise linear elements. Copyright © 1999 John Wiley & Sons, Ltd.     

On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions

This paper is concerned with the analysis of the Helmholtz–Hodge decomposition theorem since it plays a fundamental role in the projection methods that are adopted in the numerical solution of the Navier–Stokes equations for incompressible flows. The paper highlights the role of the orthogonal decomposition of a vector field in a bounded domain when general boundary conditions are in effect. In fact, even if Fractional Time‐Step Methods are standard procedures for de‐coupling the pressure gradient and the velocity field, many problems are encountered in performing the decoupling with higher accuracy. Since the problem of determining a unique and orthogonal decomposition requires only one boundary condition to be well posed, thus either the normal or the tangential ones, result exactly imposed at the end of the projection. Numerical errors are introduced in terms of both the pressure and the velocity but the orthogonality of decomposition guarantees that the former does not contribute to affect the accuracy of the latter. Moreover, it is shown that depending on the meaning of the vector to be decomposed, i.e. acceleration or velocity, the true orthogonal projector can be defined only when suitable boundary conditions are verified. Conversely, it is shown that when the decomposition results non‐orthogonal, the velocity accuracy suffers of other errors. The issue on the resulting accuracy order of the procedure is clearly addressed by means of several accuracy studies and a strategy for improving it is proposed. This paper follows and integrates the issues reported in Iannelli and Denaro (Int. J. Numer. Meth. Fluids 2003; 42 : 399–437). Copyright © 2003 John Wiley & Sons, Ltd.