Convergence properties of high-Reynolds-number separated flow calculations

The convergence properties of an iterative solution technique for the Reduced Navier–Stokes equations are examined for two-dimensional steady subsonic flow over bump and trough geometries. Techniques for decreasing the sensitivity to the initial pressure approximation, for fine meshes in particular, are investigated. They are shown to improve the robustness of the relaxation process and to decrease the computational work required to obtain a converged solution. A semi-coarsening multigrid technique that has previously been found to be particularly advantageous for high-Reynolds-number (Re) flows with flow separation and with highly stretched surface-normal grids is applied herein to further accelerate convergence. Solutions are obtained for the laminar flow over a trough that is more severe than has been considered to date. Sufficient axial grid refinement in this case leads to a shock-like reattachment and, for sufficiently large Re, to a local ‘divergence’ of the numerical computations. This ‘laminar flow breakdown’ appears to be related to an instability associated with high-frequency fine-grid modes that are not resolvable with the present modelling. This behaviour may be indicative of dynamic stall or of incipient transition. The breakdown or instability is shown to be controllable by suitable introduction of transition turbulence models or by laminar flow control, i.e. small amounts of wall suction. This lends further support to the hypothesis that the instability is of a physical rather than numerical character and suggests that full three-dimensional analysis is required to properly capture the flow behaviour. Another inference drawn from this investigation is that there is a need for careful grid refinement studies in high-Re flow computations, since coarser grids may yield oscillation-free solutions that cannot be obtained on finer grids.     

Numerical simulation of viscous flow interaction with an elastic membrane

A numerical fluid–structure interaction model is developed for the analysis of viscous flow over elastic membrane structures. The Navier–Stokes equations are discretized on a moving body‐fitted unstructured triangular grid using the finite volume method, taking into account grid non‐orthogonality, and implementing the SIMPLE algorithm for pressure solution, power law implicit differencing and Rhie–Chow explicit mass flux interpolations. The membrane is discretized as a set of links that coincide with a subset of the fluid mesh edges. A new model is introduced to distribute local and global elastic effects to aid stability of the structure model and damping effects are also included. A pseudo‐structural approach using a balance of mesh edge spring tensions and cell internal pressures controls the motion of fluid mesh nodes based on the displacements of the membrane. Following initial validation, the model is applied to the case of a two‐dimensional membrane pinned at both ends at an angle of attack of 4° to the oncoming flow, at a Reynolds number based on the chord length of 4 × 10³. A series of tests on membranes of different elastic stiffness investigates their unsteady movements over time. The membranes of higher elastic stiffness adopt a stable equilibrium shape, while the membrane of lowest elastic stiffness demonstrates unstable interactions between its inflated shape and the resulting unsteady wake. These unstable effects are shown to be significantly magnified by the flexible nature of the membrane compared with a rigid surface of the same average shape. Copyright © 2007 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.     

Numerical simulation and hydrodynamic visualization of transient viscous flow around an oscillating aerofoil

Unsteady viscous flow around a large-amplitude and high-frequency oscillating aerofoil is examined in this paper by numerical simulation and experimental visualization. The numerical method is based on the combination of a fourth-order Hermitian finite difference scheme for the stream function equation and a classical second-order scheme to solve the vorticity transport equation. Experiments are carried out by a traditional visualization method using solid tracers suspended in water. The comparison between numerical and experimental results is found to be satisfactory. Time evolutions of the flow structure are presented for Reynolds numbers of 3 × 10³ and 10⁴. The influence of the amplitude and frequency of the oscillating motion on the dynamic stall is analysed.     

An efficient algorithm for solving the incompressible fluid flow equations

The present paper reports on a modified pressure implicit predictor corrector type scheme for solving the flow governing equations, in which a consistent formulation is combined with a multi-grid solver for the pressure correction. In addition a parabolic sublayer (PSL) approach for the treatment of the flow in the vicinity of solid walls is critically evaluated in terms of accuracy and computational efficiency. The lid-driven cavity flow is chosen as the test case and results are presented for Reynolds numbers ranging from 100 to 1000. Predictions with the proposed scheme indicate substantial computational savings and fairly good agreement when compared with previous work. The PSL approach reduces the computing time, but with increasing Reynolds numbers the accuracy of the solutions tends to deteriorate.     

Hot flow analysis of swirling sudden-expansion dump combustor

Hot flow of a sudden-expansion dump combustor with swirling is analysed by employing an infinite chemical reaction rate. Turbulence properties are closed using one type of algebraic Reynolds stress model and two types of κ–? model. One of the κ–? models includes a swirling effect modification to the ε-equation. Computations have been performed by the SIMPLE-C algorithm with a power-law scheme. The calculated results of the momentum fields and turbulence quantities for swirling flow are compared with the available experimental data. It is shown that the standard κ–? model gives poor prediction of the mean velocity, particularly the tangential velocity. For the hot flow analysis of a sudden-expansion dump combustor with swirling flow it is suggested that it is necessary to use the modified κ–? model or algebraic Reynolds stress model.     

Comment on ‘generalized potential flow theory and direct calculation of velocities applied to the numerical solution of the Navier–Stokes and the Boussinesq equations’

In a recent paper a generalized potential flow theory and its application to the solution of the Navier–Stokes equation are developed.¹ The purpose of this comment is to show that the analysis presented in that paper is in general not correct. We note that the theoretical development of Reference 1 is in fact an extension—although not cited—of some work first done by Hawthorne for steady inviscid flow.² Hawthorne's solution is correct, and his analysis, which we briefly describe, provides a useful introduction to this note.     

Numerical simulation of time‐dependent hydrodynamic removal of a contaminated fluid from a cavity

The time‐dependent hydrodynamic removal of a contaminated fluid from a rectangular cavity on the floor of a duct is analysed numerically. Laminar duct flows are considered for Reynolds numbers of 50 and 1600 where the characteristic length is the duct height. Two cases are considered where: (1) the fluid density in the cavity is the same as that for the duct fluid and (2) the cavity fluid has a higher density than the duct fluid but the two fluids are miscible. The flow is solved by a numerical solution of the time‐dependent Navier–Stokes equations. Attention is focused on the convective transport of contaminated fluid out from the cavity and the effect of duct flow velocity profile on the cleaning process. Passive markers are used in the numerical simulation for the purpose of identifying the contaminated cavity fluid. The results show that the flow patterns in the cavity are influenced by the type of duct flow. From a cleaning perspective, the results suggest that it is easier for the duct flow to penetrate a cavity and to remove contaminated cavity fluid when the duct flow is of the Poiseuille type and the aspect ratio is large. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of pipe flow transition. Part I. Direct numerical simulation

We have developed an accurate hybrid finite-difference code for the simulation of unsteady incompressible pipe flow. The numerical scheme uses compact finite differences of at least eighth-order accuracy for the axial coordinate, and Chebyshev and Fourier polynomials for the radial and azimuthal coordinates, respectively. Boundary conditions for the incompressible flow are enforced using an influence-matrix technique, and the Poisson equation for pressure is solved using a fast direct method. The code has been used to simulate and analyze the spatial transition process in developed laminar pipe flow at a Reynolds number of Re=2350. Results of the simulation are compared to experimental data given by Han, Tumin and Wygnanski [18]. PACS 47.11.+j, 47.20.Ft, 47.27.Cn     

An improved Vorticity–Potential method for three-dimensional duct flow simulations

An improved Vorticity–Potential method is presented for the numerical solutions of three-dimensional duct flow problems. The solution procedure requires first a potential solution. Then the viscous effects are added through the vorticity transport equation. By using body-fitted coordinates, the method is applied to simulate the incompressible laminar flows in a square elbow and in a twisted square elbow.     

Global pressure relaxation for laminar two-dimensional internal flow

This study extends the reduced Navier–Stokes (RNS) global pressure relaxation procedure developed by Rubin and co-workers for external flow to internal flow applications. The streamwise pressure gradient is split into a backward-differenced or initial value component, as in boundary layer marching, and a forward-differenced or boundary value component that represents the elliptic downstream effects. The streamwise convection terms are upwind-differenced and all other streamwise derivatives are backward-differenced. We thus obtain a standard boundary layer marching technique imbedded in a conventional line relaxation technique. For compressible flow the pressure iteration determines the interior flow interation as well as the inlet mass flux that is consistent with the outflow pressure boundary condition. Results have been computed for incompressible flow in both rectangular and curved channels, and for subsonic compressible flow in the simulation of an aerofoil in a wind tunnel. Converged solutions were obtained over a range of Reynolds numbers generating small to moderately large separation bubbles.     

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.     

Numerical simulation of axisymmetric turbulent flow in combustors and diffusers

Numerical studies of turbulent flow in an axisymmetric 45° expansion combustor and bifurcated diffuser are presented. The Navier-Stokes equations incorporating a k–? model were solved in a non-orthogonal curvillinear co-ordinate system. A zonal grid method, wherein the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. However, treatment of the zonal boundaries required special handling. Boundary overlap and interpolating techniques were used and an adjustment of the flow variables was required to assure conservation of mass flux. Three finite differencing methods—hybrid, quadratic upwind and skew upwind—were used to represent the convection terms. Results were compared with existing experimental data. In general, good agreement between predicted and measured values was obtained.     

Numerical simulation of orbitally shaken viscous fluids with free surface

Orbitally shaken bioreactors are an emerging alternative to stirred‐tank bioreactors for large‐scale mammalian cell culture, but their fluid dynamics is still not well defined. Among the theoretical and practical issues that remain to be resolved, the characterization of the liquid free surface during orbital shaking remains a major challenge because it is an essential aspect of gas transfer and mixing in these reactors. To simulate the fluid behavior and the free surface shape, we developed a numerical method based on the finite element framework. We found that the large density ratio between the liquid and the gas phases induced unphysical results for the free surface shape. We therefore devised a new pressure correction scheme to deal with large density ratios. The simulations operated with this new scheme gave values of wave amplitude similar to the ones measured experimentally. These simulations were used to calculate the shear stress and to study the mixing principle in orbitally shaken bioreactors. Copyright © 2012 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.     

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.