Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers

Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.     

Effect of tube spacing on the vortex shedding characteristics of laminar flow past an inline tube array: A numerical study

The effect of tube spacing on the vortex shedding characteristics and fluctuating forces in an inline cylinder array is studied numerically. The examined Reynolds number is 100 and the flow is laminar. The numerical methodology and the code employed to solve the Navier-Stokes and continuity equations in an unstructured finite volume grid are validated for the case of flow past two tandem cylinders at four spacings. Computations are then performed for a six-row inline tube bank for eight pitch-to-diameter ratios, s, ranging from 2.1 to 4. At the smallest spacing examined (s = 2.1) there are five stagnant and symmetric recirculation zones and weak vortex shedding activity occurs only behind the last cylinder. As s increases, the symmetry of the recirculation zones breaks leading to vortex shedding and this process progressively moves upstream, so that for s = 4 there is clear shedding from every row. For any given spacing, the shedding frequency behind each cylinder is the same. A critical spacing range between 3.0 and 3.6 is identified at which the mean drag as well as the rms lift and drag coefficients for the last three cylinders attain maximum values. Further increase to s = 4 leads to significant decrease in the force statistics and increase in the Strouhal number. It was found that at the critical spacing there is 180° phase difference in the shedding cycle between successive cylinders and the vortices travel a distance twice the tube spacing within one period of shedding.     

Large eddy simulation of turbulent flow past a square cylinder confined in a channel

Turbulent flow past a square cylinder confined in a channel is numerically investigated by large eddy simulation (LES). The main objectives of this study are to extensively verify the experimental results of Nakagawa et al. [Exp. Fluids 27(3) (1999) 284] by LES and to identify the features of flows past a square cylinder confined in a channel in comparison with the conventional one in an infinite domain. The LES results obtained are in excellent agreement with the experiment both qualitatively and quantitatively. The well-known Kármán vortex shedding is observed. However, the vortices shed from the cylinder are significantly affected by the presence of the plates; mean drag and fluctuation of lift force increase significantly. Furthermore, periodic and alternating vortex-rollups are observed in the vicinity of the plates. The rolled-up vortex is convected downstream together with the corresponding Kármán vortex; they form a counter-rotating vortex pair. It is also revealed that the cylinder greatly enhances mixing process of the flow.     

Surface effects on transient three-dimensional flows around rotating spheres at moderate Reynolds numbers

Transient wake flow patterns and dynamic forces acting on a rotating spherical particle with non-uniform surface blowing are studied numerically for Reynolds numbers up to 300 and dimensionless angular velocities up to Ω=1. This range of Reynolds numbers includes the three distinct wake regimes i.e., the steady axisymmetric, the steady non-symmetrical and the unsteady with vortex shedding. The Navier–Stokes equations for an incompressible viscous flow are solved by a finite volume method in a three-dimensional, time accurate manner. An interesting feature associated with particle rotation and surface blowing is that they can affect the near wake structure in such a way that unsteady three-dimensional wake flow with vortex shedding develops at lower Reynolds numbers as compared to flow over a solid sphere in the absence of these effects and thus, vortex shedding occurs even at Re=200. Global properties, such as the lift and drag coefficients, and the Strouhal number are also significantly affected. It is shown that the present data for the average lift and drag coefficients correlate well with:

C_L/(1+Ω)^3.6=0.11

C_D(1+20V_S)^0.2/(1+Ω)^Re/1000=24(1+Re^2/3/6)/Re

where V_S is the average surface blowing velocity normalized by the free stream velocity.     

Large eddy simulation of pulsating flow over a circular cylinder at subcritical Reynolds number

The pulsating cross-flow over a single circular cylinder at the subcritical Reynolds number Re_D = 2580 is studied with the large eddy simulation (LES) technique using the standard Smagorinsky model as well as a dynamic model in which the test filtered quantities are evaluated through a truncated Taylor series expansion. The filtered equations are discretised using the finite volume method in an unstructured, collocated grid arrangement with a second-order accurate method, in both space and time. The predictions are compared against very detailed experiments for mean velocities and Reynolds stresses that were performed in a duct of cross-section 72 mm × 72 mm using the PIV technique. The effects of mesh refinement close to the cylinder as well as of subgrid scale model are also examined. The numerical predictions are in very good agreement with the measurements in terms of mean as well as turbulence quantities. The instantaneous flow patterns of the flow field are examined and the effect of the external flow pulsation on the wake characteristics such as vortex formation length, vortex strength, Strouhal number as well as the lift and drag coefficients is quantified. The vortex formation length is decreased while the mean drag, as well as the rms values of the drag and lift coefficients increase significantly under pulsating flow conditions. The performance of the LES technique is analysed in the light of the wake characteristics.     

On the use of incomplete sensitivities for feedback control of laminar vortex shedding

A feedback control method based on incomplete sensitivities and gradient evaluation by complex variable method is proposed and applied to the problem of the control of the laminar vortex shedding past a circular cylinder. This procedure results in a low-cost control algorithm, which does not require to compute the gradient of the Navier–Stokes solution with respect to the controllers. For the sake of usability for practical applications, realistic sensors and actuators are used. Validation simulations aiming at controlling the drag of the cylinder are presented and compared with previous published results, proving the efficiency of the proposed method. Application to the control of the cylinder lift is also shown.     

Numerical analysis and computations of a high accuracy time relaxation fluid flow model

We present a numerical study based on continuous finite element analysis for a time relaxation regularization of Navier–Stokes equations. This regularization is based on filtering and deconvolution. We study the convergence of the regularized equations using a fully discretized filter and deconvolution algorithm. Velocity and pressure error estimates and the L ₂ Aubin–Nitsche lift technique are proved for the equilibrium problem, and this analysis is accompanied by the velocity error estimate for the time-dependent problem, too. Thus, optimal error estimates in L ₂ and H ¹ norms are derived and followed by their computational verification. Also, computational results of the vortex street are presented for the two-dimensional cylinder benchmark flow problem. Maximum drag and lift coefficients and difference in pressure between the front and back of the cylinder at the final time were investigated as well, showing that the time relaxation regularization can attain the benchmark values.     

Flow characteristics of two rotating side-by-side circular cylinder

This paper presents a numerical investigation of the characteristics of the two-dimensional laminar flow around two rotating circular cylinders in side-by-side arrangements. In order to consider the combined effects of the rotation and the spacing between two cylinders on the flow, numerical simulations are performed at a various range of absolute rotational speeds (|α|?2) for four different gap spacings of 3, 1.5, 0.7 and 0.2 at Reynolds number of 100 showing the typical two-dimensional vortex shedding. As |α| increases, the flow changes its condition from periodic to steady after a critical rotational speed, which depends on the gap spacing. In the cases of gap spacings of 3 and 0.2, the wake keeps the same pattern, until flow reaches the steady state. However, for the gap spacings of 1.5 and 0.7, the wake patterns change in the unsteady regimes. For the cases in which the flow is unsteady, the Strouhal number strongly depends on the gap. For a fixed gap spacing, the variation of the Strouhal number is significant when the wake pattern is changed according to the rotational speed. Regardless of the gap spacing, as |α| increases, the lift increases and the drag decreases. Quantitative information about the flow variables such as the pressure coefficient and wall vorticity distributions on the cylinders is highlighted.     

A numerical study of flow past a cylinder with cross flow and inline oscillation

Two-dimensional fluid flow around an oscillating circular cylinder is studied numerically at different values of oscillation frequency and amplitude. A novel finite element method which uses discretization along the characteristic line is used for simulation. The solver is coupled to a mesh movement scheme using the Arbitrary Lagrangian-Eulerian (ALE) formulation to account for body motion in the flow field. Two cases of cylinder motion have been studied, cross flow and inline oscillation. In both cases, occurrence of lock on is investigated and the bounds of the lock on region are determined. A comparison of the numerical results with the experimental data indicates that 2D simulation is valid up to Re = 300. Beyond that, 3D effects appear. By using flow visualization, effect of a cylinder oscillation on the flow field and wake pattern has been studied. Also, variation of the mean drag coefficient against the oscillation parameters is discussed. The numerical results are in good agreement with the experimental data available in the literature.     

Time-dependent solutions of the viscous incompressible flow past a circular cylinder by the method of series truncation

Semi-analytic solutions of the Navier-Stokes equations are calculated for two-dimensional, symmetrical, viscous incompressible flow past a circular cylinder. The stream and vorticity functions are expanded in the finite Fourier series and then substituted in the Navier-Stokes equations. This led to a system of coupled parabolic partial differential equations which are solved numerically. More terms of the series are required as Reynolds number increases and the present calculations were terminated at Reynolds number 600 with 60 terms of Fourier series. The results are compared with similar calculations and experimental data for Reynolds numbers 60, 100, 200, 500, 550 and 600. At the termination of the calculations for Reynolds numbers 60 and 100, the separation angle, the wake length, the drag coefficient, and the vorticity distributions around the surface were very close to their steady-state values. A secondary vortex appeared on the surface of the cylinder in the case of Reynolds numbers 500, 550 and 600. The wake length, the drag coefficient and the separation angle differ slightly at a given instant in the case of Reynolds numbers 500, 550 and 600.     

A coarse-grid projection method for accelerating incompressible MHD flow simulations

Coarse grid projection (CGP) is a multiresolution technique for accelerating numerical calculations associated with a set of nonlinear evolutionary equations along with stiff Poisson’s equations. In this article, we use CGP for the first time to speed up incompressible magnetohydrodynamics (MHD) flow simulations. Accordingly, we solve the nonlinear advection–diffusion equation on a fine mesh, while we execute the electric potential Poisson equation on the corresponding coarsened mesh. Mapping operators connect two grids together. A pressure correction scheme is used to enforce the incompressibility constrain. The study of incompressible flow past a circular cylinder in the presence of Lorentz force is selected as a benchmark problem with a fixed Reynolds number but various Stuart numbers. We consider two different situations. First, we only apply CGP to the electric potential Poisson equation. Second, we apply CGP to the pressure Poisson equation as well. The maximum speed-up factors achieved here are approximately 3 and 23, respectively, for the first and second situations. For the both situations, we examine the accuracy of velocity and vorticity fields as well as the lift and drag coefficients. In general, the results obtained by CGP are in an excellent to reasonable range of accuracy. The CGP results are significantly more accurate compared to the numerical simulations of the advection–diffusion and electric potential Poisson equations on pure coarse scale grids.

     

Numerical simulation of an oscillating cylinder in a cross-flow at low Reynolds number: Forced and free oscillations

A numerical simulation of the flow past a circular cylinder which is able to oscillate transversely to the incident stream is presented in this paper for a fixed Reynolds number equal to 100. The 2D Navier-Stokes equations are solved by a finite volume method with an industrial CFD code in which a coupling procedure has been implemented in order to obtain the cylinder displacement. A preliminary work is first conducted for a fixed cylinder to check the wake characteristics for Reynolds numbers smaller than 150 in the laminar regime. The Strouhal frequency f_S and the aerodynamic coefficients are thus controlled among other parameters. Simulations are then performed with forced oscillations characterized by the frequency ratio F = f₀/f_S, where f₀ is the forced oscillation frequency, and by the adimensional amplitude A. The wake characteristics are analyzed using the time series of the fluctuating aerodynamic coefficients and their power spectral densities (PSD). The frequency content is then linked to the shape of the phase portraits and to the vortex shedding mode. By choosing interesting couples (A, F), different vortex shedding modes have been observed, which are similar to those of the Williamson-Roshko map. A second batch of simulations involving free vibrations (so-called vortex-induced vibrations or VIV) is finally carried out. Oscillations of the cylinder are now directly induced by the vortex shedding process in the wake and therefore, the time integration of the motion is realized by an explicit staggered algorithm which provides the cylinder displacement according to the aerodynamic charges exerted on the cylinder wall. Amplitude and frequency response of the cylinder are thus investigated over a wide range of reduced velocities to observe the different phenomena at stake. In particular, the vortex shedding modes have also been related to the frequency response observed and our results at Re = 100 show a very good agreement with other studies using different numerical approaches.     

A computational study of vortex-airfoil interaction using high-order finite difference methods

Simulations of the interaction between a vortex and a NACA0012 airfoil are performed with a stable, high-order accurate (in space and time), multi-block finite difference solver for the compressible Navier-Stokes equations.We begin by computing a benchmark test case to validate the code. Next, the flow with steady inflow conditions are computed on several different grids. The resolution of the boundary layer as well as the amount of the artificial dissipation is studied to establish the necessary resolution requirements. We propose an accuracy test based on the weak imposition of the boundary conditions that does not require a grid refinement.Finally, we compute the vortex-airfoil interaction and calculate the lift and drag coefficients. It is shown that the viscous terms add the effect of detailed small scale structures to the lift and drag coefficients.     

Incompressible flow calculations with a consistent physical interpolation finite volume approach

The computation of incompressible three-dimensional viscous flow is discussed. A new physically consistent method is presented for the reconstruction for velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes allows the use of a cell-centered grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The method is validated on several benchmark problems which include steady laminar flow predictions on a two-dimensional cartesian (lid driven 2D cavity) or curvilinear grid (circular cylinder problem at Re = 40), unsteady three-dimensional laminar flow predictions on a cartesian grid (parallelopipedic lid driven cavity) and unsteady two-dimensional turbulent flow predictions on a curvilinear grid (vortex shedding past a square cylinder at Re = 22,000).     

Control of flow using genetic algorithm for a circular cylinder executing rotary oscillation

We propose here a new approach to optimally control incompressible viscous flow past a circular cylinder for drag minimization by rotary oscillation. The flow at Re = 15000 is simulated by solving 2D Navier-Stokes equations in stream function-vorticity formulation. High accuracy compact scheme for space discretization and four stage Runge-Kutta scheme for time integration makes such simulation possible. While numerical solution for this flow field has been reported using a fast viscous-vortex method, to our knowledge, this has not been done at such a high Reynolds number by computing the Navier-Stokes equation before. The importance of scale resolution, aliasing problem and preservation of physical dispersion relation for such vortical flows of the used high accuracy schemes [Sengupta TK. Fundamentals of computational fluid dynamics. Hyderabad, India: University Press; 2004] is highlighted.For the dynamic problem, a novel genetic algorithm (GA) based optimization technique has been adopted, where solutions of Navier-Stokes equations are obtained using small time-horizons at every step of the optimization process, called a GA generation. Then the objective functions is evaluated that is followed by GA determined improvement of the decision variables. This procedure of time advancement can also be adopted to control such flows experimentally, as one obtains time-accurate solution of the Navier-Stokes equation subject to discrete changes of decision variables. The objective function - the time-averaged drag - is optimized using a real-coded genetic algorithm [Deb K. Multi-objective optimization using evolutionary algorithms. Chichester, UK: Wiley; 2001] for the two decision variables, the maximum rotation rate and the forcing frequency of the rotary oscillation. Various approaches to optimal decision variables have been explored for the purpose of drag reduction and the collection of results are self-consistent and furthermore match well with the experimental values reported in [Tokumaru PT, Dimotakis PE. Rotary oscillation control of a cylinder wake. J Fluid Mech 1991;224:77].     

Computation of unsteady viscous flow and aeroacoustic noise of cross flow fans

Time-accurate viscous flow solutions are sought for the prediction of unsteady flow characteristics and associated aeroacoustic blade tonal noise of a cross flow fan. The two-dimensional incompressible Navier-Stokes equations in a moving coordinate are time-accurately solved by an unstructured finite-volume method on triangular meshes, and a sliding mesh technique is utilized at the interface between the domain rotating with blades and the stationary one for allowing the unsteady interactions. An accuracy assessment of the present method is made by comparing the fan performances with experimental data for a rotational speed at 1000 rpm and the Reynolds number 5300 based on blade tip speed and chord length. With the computed unsteady viscous flow solutions, sound pressure is predicted using Curle’s equation and narrow-band noise characteristics of three impellers with a uniform and two random pitch (type-A and -B) blades are compared by their sound pressure level spectra. Also, the frequency modulations of the blade passing frequency noise by random pitch fans are discussed.     

Computations of optimal cylinder flow control in weakly conductive fluids

A nonlinear adjoint-based optimal control approach of cylinder wake by electromagnetic force has been investigated numerically in the paper. A cost functional representing the balance of the regulated quantities with different weights and interaction parameter N (Lorentz force) has been constituted, where the regulated quantities related with flow and force are taken as targets of regulation and the Lorentz force, (as interaction parameter N), is taken as a control input. Based on the cost functional and Navier-Stokes equations, the corresponding adjoint equations have been derived and the sensitivity of the cost functional is found to be a simple function of the adjoint stream function in the adjoint field. For the different regulations, the forms of optimal control rules are similar while the adjoint equations are different. The receding-horizon predictive control setting is employed to discuss the optimal control problems. Under the action of optimal N(t), the flow separation is suppressed fully, so that the oscillations of drag and lift are suppressed and the total drag coefficient decreases dramatically. For the different regulations, the control effects have some differences due to the different values of optimal inputs corresponding to the different adjoint flow fields.     

Numerical study of Taylor-Couette flow with an axial flow

The flow between two concentric cylinders with the inner one rotating and with an imposed pressure-driven axial flow is studied using numerical simulation. This study considers the identical flow geometry and flow parameters as in the experiments of Wereley and Lueptow [Phys. Fluids 11 (12) (1999) 3637], where particle image velocimetry measurements were carried out to obtain detailed velocity fields in a meridional plane of the annulus. The objectives of this investigation are to numerically verify the experimental results of Wereley and Lueptow and to further study detailed flow fields and bifurcations related to Taylor-Couette flow with an imposed axial flow. The vortices in various flow regimes such as non-wavy laminar vortex, wavy vortex, non-wavy helical vortex, helical wavy vortex and random wavy vortex are all consistently reproduced with their experiments. It is demonstrated that ‘shift-and-reflect’ symmetry holds in Taylor-Couette flow without an imposed axial flow. In case of Taylor-Couette flow with an imposed axial flow, one can find that the shift-and-reflect symmetry is roughly valid for the remaining velocity field after subtracting the annular Poiseuille flow. The axial flow stabilizes the flow field and decreases the torque required by rotating the inner cylinder at a given speed. Growth rate of the flow instability is defined and used in predicting the type of the vortices. The velocity vector fields obtained also reveal the same vortex characteristics as found in the experiments of Wereley and Lueptow.     

Spectral simulations of an unsteady compressible flow past a circular cylinder

An unsteady compressible viscous wake flow past a circular cylinder has been successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for a periodic problem is introduced. We have further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It has been shown that the solution of a model problem has exponential growth in time if an ‘improper’ boundary conditions procedure is used. A characteristic boundary conditions, which is based on the characteristics of the Euler equations of gas dynamics, has been derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data.     

Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integro-differential formulation

A numerical method for the solution of the Navier-Stokes equations is developed using an integro-differential formulation of the equations. The method permits the actual computation to be confined to the viscous region of the flow and offers a drastic reduction in the number of data points required in the numerical procedure. The integro-differential formulation is presented along with discussion of the kinetic and kinematic aspects of the problem and the interplay between the two aspects. Results for several parallel flow problems and for the flow past a circular cyliner are presented. For the circular cylinder, it is shown that the introductions of a splitter plate behind the cylinder suppresses vortex shedding.