Numerical simulation of flows around two circular cylinders by mesh‐free least square‐based finite difference methods

In this paper, the mesh‐free least square‐based finite difference (MLSFD) method is applied to numerically study the flow field around two circular cylinders arranged in side‐by‐side and tandem configurations. For each configuration, various geometrical arrangements are considered, in order to reveal the different flow regimes characterized by the gap between the two cylinders. In this work, the flow simulations are carried out in the low Reynolds number range, that is, Re=100 and 200. Instantaneous vorticity contours and streamlines around the two cylinders are used as the visualization aids. Some flow parameters such as Strouhal number, drag and lift coefficients calculated from the solution are provided and quantitatively compared with those provided by other researchers. Copyright © 2006 John Wiley & Sons, Ltd.     

A high‐order scheme for the incompressible Navier–Stokes equations with open boundary condition

We present a numerical scheme to solve the incompressible Navier–Stokes equations with open boundary condition. After replacing the incompressibility constraint by the pressure Poisson equation, the key is how to give an appropriate boundary condition for the pressure Poisson equation. We propose a new boundary condition for the pressure on the open boundary. Some numerical experiments are presented to verify the accuracy and stability of scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of flapping‐wing insect hovering flight at unsteady flow

A computational fluid dynamics (CFD) analysis was conducted to study the unsteady aerodynamics of a virtual flying bumblebee during hovering flight. The integrated geometry of bumblebee was established to define the shape of a three‐dimensional virtual bumblebee model with beating its wings, accurately mimicking the three‐dimensional movements of wings during hovering flight. The kinematics data of wings documented from the measurement to the bumblebee in normal hovering flight aided by the high‐speed video. The Navier–Stokes equations are solved numerically. The solution provides the flow and pressure fields, from which the aerodynamic forces and vorticity wake structure are obtained. Insights into the unsteady aerodynamic force generation process are gained from the force and flow‐structure information. The CFD analysis has established an overall understanding of the viscous and unsteady flow around the virtual flying bumblebee and of the time course of instantaneous force production, which reveals that hovering flight is dominated by the unsteady aerodynamics of both the instantaneous dynamics and also the past history of the wing. A coherent leading‐edge vortex with axial flow and the attached wingtip vortex and trailing edge vortex were detected. The leading edge vortex, wing tip vortex and trailing edge vortex, which caused by the pressure difference between the upper and the lower surface of wings. The axial flow, which include the spanwise flow and chordwise flow, is derived from the spanwise pressure gradient and chordwise pressure gradient, will stabilize the vortex and gives it a characteristic spiral conical shape. Copyright © 2006 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.     

A comparison of time integration methods in an unsteady low‐Reynolds‐number flow

This paper describes three different time integration methods for unsteady incompressible Navier–Stokes equations. Explicit Euler and fractional‐step Adams–Bashford methods are compared with an implicit three‐level method based on a steady‐state SIMPLE method. The implicit solver employs a dual time stepping and an iteration within the time step. The spatial discretization is based on a co‐located finite‐volume technique. The influence of the convergence limits and the time‐step size on the accuracy of the predictions are studied. The efficiency of the different solvers is compared in a vortex‐shedding flow over a cylinder in the Reynolds number range of 100–1600. A high‐Reynolds‐number flow over a biconvex airfoil profile is also computed. The computations are performed in two dimensions. At the low‐Reynolds‐number range the explicit methods appear to be faster by a factor from 5 to 10. In the high‐Reynolds‐number case, the explicit Adams–Bashford method and the implicit method appear to be approximately equally fast while yielding similar results. Copyright © 2002 John Wiley & Sons, Ltd.     

Suppression of vortex shedding for flow around a circular cylinder using optimal control

Adjoint formulation is employed for the optimal control of flow around a rotating cylinder, governed by the unsteady Navier–Stokes equations. The main objective consists of suppressing Karman vortex shedding in the wake of the cylinder by controlling the angular velocity of the rotating body, which can be constant in time or time‐dependent. Since the numerical control problem is ill‐posed, regularization is employed. An empirical logarithmic law relating the regularization coefficient to the Reynolds number was derived for 60?Re?140. Optimal values of the angular velocity of the cylinder are obtained for Reynolds numbers ranging from Re=60 to Re=1000. The results obtained by the computational optimal control method agree with previously obtained experimental and numerical observations. A significant reduction of the amplitude of the variation of the drag coefficient is obtained for the optimized values of the rotation rate. Copyright © 2002 John Wiley & Sons, Ltd.     

A nodal‐based implementation of a stabilized finite element method for incompressible flow problems

The objective of this paper is twofold. First, a stabilized finite element method (FEM) for the incompressible Navier–Stokes is presented and several numerical experiments are conducted to check its performance. This method is capable of dealing with all the instabilities that the standard Galerkin method presents, namely the pressure instability, the instability arising in convection‐dominated situations and the less popular instabilities found when the Navier–Stokes equations have a dominant Coriolis force or when there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal‐based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements, thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this gives rise to questions regarding the consistency and the conservation properties of the final scheme, which are addressed in this paper. Copyright © 2000 John Wiley & Sons, Ltd.     

Comparative study of implicit and subgrid‐scale model large‐eddy simulation techniques for low‐Reynolds number airfoil applications

A computational study of a high‐fidelity, implicit large‐eddy simulation (ILES) technique with and without the use of the dynamic Smagorinsky subgrid‐scale (SGS) model is conducted to examine the contributions of the SGS model on solutions of transitional flow over the SD7003 airfoil section. ILES without an SGS model has been shown in the past to produce comparable and sometimes favorable results to traditional SGS‐based large‐eddy simulation (LES) when applied to canonical turbulent flows. This paper evaluates the necessity of the SGS model for low‐Reynolds number airfoil applications to affirm the use of ILES without SGS‐modeling for a broader class of problems such as those pertaining to micro air vehicles and low‐pressure turbines. It is determined that the addition of the dynamic Smagorinsky model does not significantly affect the time‐mean flow or statistical quantities measured around the airfoil section for the spatial resolutions and Reynolds numbers examined in this study. Additionally, the robustness and reduced computational cost of ILES without the SGS model demonstrates the attractiveness of ILES as an alternative to traditional LES. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Multigrid solution of the incompressible Navier–Stokes equations for three‐dimensional recirculating flow: coupled and decoupled smoothers compared

A comparison of multigrid methods for solving the incompressible Navier–Stokes equations in three dimensions is presented. The continuous equations are discretised on staggered grids using a second‐order monotonic scheme for the convective terms and implemented in defect correction form. The convergence characteristics of a decoupled method (SIMPLE) are compared with those of the cellwise coupled method (SCGS). The convergence rates obtained for computations of the three‐dimensional lid‐driven cavity problem are found to be very similar to those obtained for computations of the corresponding two‐dimensional problem with comparable grid density. Although the convergence rate of SCGS is thus superior to that of SIMPLE, the decoupled method is found to be more efficient computationally and requires less computing time for a given level of convergence. The linewise implementation of the coupled method (CLGS) is also investigated and shown to be more efficient than SCGS, although the convergence rate and computing time required per cycle are both found to depend on the direction of sweep. The optimal implementation of CLGS is found to be only marginally more effective than SIMPLE, but a change to the structure of the data storage would increase the advantage. Copyright © 1999 John Wiley & Sons, Ltd.     

On numerical errors and turbulence modeling in tip vortex flow prediction

The accuracy of tip vortex flow prediction in the near‐field region is investigated numerically by attempting to quantify the shortcomings of the turbulence models and the flow solver. In particular, some turbulence models can produce a ‘numerical diffusion’ that artificially smears the vortex core. Low‐order finite differencing techniques of the convective and pressure terms of the Navier–Stokes equations and inadequate grid density and distribution can also produce the same adverse effect. The flow over a wing and the near‐wake with the wind tunnel walls included was simulated using 2.5 million grid points. Two subset problems, one using a steady, three‐dimensional analytical vortex, and the other, a vortex obtained from experiment and propagated downstream, were also devised in order to make the study of vortex preservation more tractable. The method of artificial compressibility is used to solve the steady, three‐dimensional, incompressible Navier–Stokes equations. Two one‐equation turbulence models (Baldwin–Barth and Spalart–Allmaras turbulence models), have been used with the production term modified to account for the stabilizing effect of the nearly solid body rotation in the vortex core. Finally, a comparison between the computed results and experiment is presented. Published in 1999 by John Wiley & Sons, Ltd.     

Numerical simulations of laminar flow over a 3D backward-facing step

A numerical investigation of laminar flow over a three-dimensional backward-facing step is presented with comparisons with detailed experimental data, available in the literature, serving to validate the numerical results. The continuity constraint method, implemented via a finite element weak statement, was employed to solve the unsteady three-dimensional Navier–Stokes equations for incompressible laminar isothermal flow. Two-dimensional numerical simulations of this step geometry underestimate the experimentally determined extent of the primary separation region for Reynolds numbers Re greater than 400. It has been postulated that this disagreement between physical and computational experiments is due to the onset of three-dimensional flow near Re ≈ 400. This paper presents a full three-dimensional simulation of the step geometry for 100⩽ Re⩽ 800 and correctly predicts the primary reattachment lengths, thus confirming the influence of three-dimensionality. Previous numerical studies have discussed possible instability modes which could induce a sudden onset of three-dimensional flow at certain critical Reynolds numbers. The current study explores the influence of the sidewall on the development of three-dimensional flow for Re greater than 400. Of particular interest is the characterization of three-dimensional vortices in the primary separation region immediately downstream of the step. The complex interaction of a wall jet, located at the step plane near the sidewall, with the mainstream flow reveals a mechanism for the increasing penetration (with increasing Reynolds number) of three-dimensional flow structures into a region of essentially two-dimensional flow near the midplane of the channel. The character and extent of the sidewall-induced flow are investigated for 100⩽Re⩽ 800. © 1997 John Wiley & Sons, Ltd.     

Boundary element analysis of driven cavity flow for low and moderate Reynolds numbers

A boundary element method for steady two‐dimensional low‐to‐moderate‐Reynolds number flows of incompressible fluids, using primitive variables, is presented. The velocity gradients in the Navier–Stokes equations are evaluated using the alternatives of upwind and central finite difference approximations, and derivatives of finite element shape functions. A direct iterative scheme is used to cope with the non‐linear character of the integral equations. In order to achieve convergence, an underrelaxation technique is employed at relatively high Reynolds numbers. Driven cavity flow in a square domain is considered to validate the proposed method by comparison with other published data. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder

This paper presents a numerical study of a two‐dimensional time‐dependent flow around a cylinder. Its main objective is to provide accurate reference values for the maximal drag and lift coefficient at the cylinder and for the pressure difference between the front and the back of the cylinder at the final time. In addition, the accuracy of these values obtained with different time stepping schemes and different finite element methods is studied. Copyright © 2004 John Wiley & Sons, Ltd.     

A simple and efficient error analysis for multi‐step solution of the Navier–Stokes equations

A simple error analysis is used within the context of segregated finite element solution scheme to solve incompressible fluid flow. An error indicator is defined based on the difference between a numerical solution on an original mesh and an approximated solution on a related mesh. This error indicator is based on satisfying the steady‐state momentum equations. The advantages of this error indicator are, simplicity of implementation (post‐processing step), ability to show regions of high and/or low error, and as the indicator approaches zero the solution approaches convergence. Two examples are chosen for solution; first, the lid‐driven cavity problem, followed by the solution of flow over a backward facing step. The solutions are compared to previously published data for validation purposes. It is shown that this rather simple error estimate, when used as a re‐meshing guide, can be very effective in obtaining accurate numerical solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Time‐dependent flow across a step: the slip with friction boundary condition

The paper studies numerically the slip with friction boundary condition in the time‐dependent incompressible Navier–Stokes equations. Numerical tests on two‐ and three‐dimensional channel flows across a step using this boundary condition on the bottom wall are performed. The influence of the friction parameter on the flow field is studied and the results are explained according to the physics of the flow. Due to the stretching and tilting of vortices, the three‐dimensional results differ in many respects from the two‐dimensional ones. Copyright © 2005 John Wiley & Sons, Ltd.     

Fourth‐ and tenth‐order compact finite difference solutions of perturbed circular vortex flows

In this study, high‐order compact finite difference calculations are reported for 2D unsteady incompressible circular vortex flow in primitive variable formulation. The fourth‐order Runge–Kutta temporal discretization is used together with fourth‐ or tenth‐order compact spatial discretization. Dependent on the perturbation initially imposed, the solutions display a tripole, triangular or square vortex. The comparison of the predictions with the detailed spectral calculations of Kloosterziel and Carnevale (J. Fluid Mech. 1999; 388 :217–257) shows that the vorticity fields are very well captured. The spectral resolution of the present method was quantified from the decomposition of the vorticity distribution in its azimuthal components and compared with reported spectral results. Using identical grid resolution to the reference results yields negligible differences in the main features of the flow. The perturbation amplitude and its first harmonic are virtually identical to the reference results for both fourth‐ or tenth‐order spatial discretization, as theoretically expected but seldom a posteriori verified. The differences between the two spatial discretizations appear only for coarser grids, favouring the tenth‐order discretization. Copyright © 2005 John Wiley & Sons, Ltd.     

Fourth‐order accurate compact‐difference discretization method for Helmholtz and incompressible Navier–Stokes equations

A fourth‐order accurate solution method for the three‐dimensional Helmholtz equations is described that is based on a compact finite‐difference stencil for the Laplace operator. Similar discretization methods for the Poisson equation have been presented by various researchers for Dirichlet boundary conditions. Here, the complicated issue of imposing Neumann boundary conditions is described in detail. The method is then applied to model Helmholtz problems to verify the accuracy of the discretization method. The implementation of the solution method is also described. The Helmholtz solver is used as the basis for a fourth‐order accurate solver for the incompressible Navier–Stokes equations. Numerical results obtained with this Navier–Stokes solver for the temporal evolution of a three‐dimensional instability in a counter‐rotating vortex pair are discussed. The time‐accurate Navier–Stokes simulations show the resolving properties of the developed discretization method and the correct prediction of the initial growth rate of the three‐dimensional instability in the vortex pair. Copyright © 2004 John Wiley & Sons, Ltd.     

SIMPLE‐type preconditioners for cell‐centered,colocated finite volume discretization of incompressible Reynolds‐averaged Navier–Stokes equations

This paper contains a comparison of four SIMPLE‐type methods used as solver and as preconditioner for the iterative solution of the (Reynolds‐averaged) Navier–Stokes equations, discretized with a finite volume method for cell‐centered, colocated variables on unstructured grids. A matrix‐free implementation is presented, and special attention is given to the treatment of the stabilization matrix to maintain a compact stencil suitable for unstructured grids. We find SIMPLER preconditioning to be robust and efficient for academic test cases and industrial test cases. Compared with the classical SIMPLE solver, SIMPLER preconditioning reduces the number of nonlinear iterations by a factor 5–20 and the CPU time by a factor 2–5 depending on the case. The flow around a ship hull at Reynolds number 2E9, for example, on a grid with cell aspect ratio up to 1:1E6, can be computed in 3 instead of 15 h.Copyright © 2012 John Wiley & Sons, Ltd.     

Solving high Reynolds‐number viscous flows by the general BEM and domain decomposition method

In this paper, the domain decomposition method (DDM) and the general boundary element method (GBEM) are applied to solve the laminar viscous flow in a driven square cavity, governed by the exact Navier–Stokes equations. The convergent numerical results at high Reynolds number Re = 7500 are obtained. We find that the DDM can considerably improve the efficiency of the GBEM, and that the combination of the domain decomposition techniques and the parallel computation can further greatly improve the efficiency of the GBEM. This verifies the great potential of the GBEM for strongly non‐linear problems in science and engineering. Copyright © 2004 John Wiley & Sons, Ltd.