Numerical simulation of three-dimensional unsteady viscous flow past finite cylinders in an unbounded fluid at low intermediate reynolds numbers

The flow past finite circular cylinders for Reynolds numbers 40 and 70 were simulated by numerical solutions of the incompressible Navier-Stokes equations. A nonuniform cartesian grid was used for the computation. The numerical scheme used was the QUICK scheme. Comparisons with experimental measurements of Jayaweera and Mason show that the results of the simulation are satisfactory. Features of three-dimensional unsteady viscous flow past finite cylinders, such as the pyramidal wake and the three-dimensional von Karmen vortex street, are successfully simulated.This work was supported by U.S. NSF Division of Atmospheric Science, Physical Meteorology Program, Grant ATM-9002299. All correspondence must be addressed to P.K. Wang.     

Global aerodynamic instability of twin cylinders in cross flow

This paper comprises an in-depth physical discussion of the flow-induced vibration of two circular cylinders in view of the time-mean lift force on stationary cylinders and interaction mechanisms. The gap-spacing ratio T/D is varied from 0.1 to 5 and the attack angle α from 0° to 180° where T is the gap width between the cylinders and D is the diameter of a cylinder. Mechanisms of interaction between two cylinders are discussed based on time-mean lift, fluctuating lift, flow structures and flow-induced responses. The whole regime is classified into seven interaction regimes, i.e., no interaction regime; boundary layer and cylinder interaction regime; shear-layer/wake and cylinder interaction regime; shear-layer and shear-layer interaction regime; vortex and cylinder interaction regime; vortex and shear-layer interaction regime; and vortex and vortex interaction regime. Though a single non-interfering circular cylinder does not correspond to a galloping following quasi-steady galloping theory, two circular cylinders experience violent galloping vibration due to shear-layer/wake and cylinder interaction as well as boundary layer and cylinder interaction. A larger magnitude of fluctuating lift communicates to a larger amplitude vortex excitation.     

Numerical simulation of viscous oscillatory flow past four cylinders in square arrangement

The results of a numerical study of the viscous oscillating flow past four circular cylinders, for a constant frequency parameter equal to 50 and KC ranging between 0.2 and 10, are presented. The cylinders were placed on the vertices of a square, two sides of which were perpendicular and two parallel to the oncoming flow, for pitch ratios, P/D, ranging between 2 and 5. The finite-element method was employed for the solution of the Navier-Stokes equations, in the formulation where the stream function and the vorticity are the field variables, whereas the pressure distribution throughout the computational domain was obtained from the solution of Poisson’s equation. When the Keulegan-Carpenter number is lower than 4, the flow remains symmetrical with respect to the horizontal axis of symmetry of the solution domain and periodic at consecutive cycles. As KC increases to 4, the flow becomes aperiodic in different cycles, although symmetry with respect to the horizontal central line of the domain is preserved. For KC equal to 5, asymmetries appear intermittently in the flow, which are eventually amplified as KC increases still further. These asymmetries, in association with the aperiodicity of flow in different cycles, lead to an almost chaotic configuration as KC grows larger. For characteristic cases the flow pattern and the time histories of the in-line and transverse forces exerted on the cylinders are presented. The mean transverse forces acting on the cylinders, the r.m.s. values of the in-line and transverse forces, together with the drag and inertia coefficients of the in-line force, were evaluated for each pitch ratio in the range of Keulegan-Carpenter numbers examined and are presented in diagrams.     

Numerical modeling of a viscous incompressible unsteady separated flow past a rotating cylinder

For studying unsteady flow past a rotating circular cylinder the Navier-Stokes equations are used. The numerical algorithm is based on an artificial-compressibility method, an implicit three-layer second-order scheme with subiterations with respect to time, a third-order scheme with splitting of the flux vectors for the convective terms, and a central-difference scheme for integrating the viscous terms. The calculated velocity profiles, the vorticity fields, the Strouhal numbers, the distribution of the pressure and friction coefficients over the cylinder surface, and the coefficients of the drag and lift forces for the laminar flow regime are analyzed.     

Finite element solution of flow between eccentric cylinders with viscous dissipation

A computational study of viscous flow between two eccentrically rotating cylinders is presented in which the effect of viscous dissipation is taken into account. The space discretization is based on piecewise linear finite elements with velocity stabilization, while the method of characteristics is used for time integration. Numerical results illustrate the efficiency of the adopted approach.     

Numerical method for unsteady viscous hydrodynamical problem with free boundaries

A finite element method for the transient incompressible Navier–Stokes equations with the ability to handle multiple free boundaries is presented. Problems of liquid–liquid type are treated by solving two coupled Navier–Stokes problems for two separate phases. The possibility to solve problems of liquid–gas, liquid–liquid–gas or liquid–liquid–liquid type is demonstrated too. Surface tension effects are included at deformable interfaces. The method is of Lagrangian type with mesh redefinition. A predictor-corrector scheme is used to compute the position of the deformable interface with automatic control of its accuracy and smoothness. The method is provided with an automatic choice of the time integration step and an optional spline filtration of the truncation error at the free surface. In order to show the accuracy of the method, tests and comparisons are presented. Numerical examples include motion of bubbles and multiple drops.     

Early stages of an impulsively started unsteady laminar flow past tapered trapezoidal cylinders

Characteristics of the developing recirculation region behind a tapered trapezoidal cylinder and its interaction with the separating shear layer from the leading edges were studied numerically for an impulsively started laminar flow. An unsteady stream function–vorticity formulation was used. The Reynolds numbers considered range from 25 to 1000. Pressure contours, surface pressure coefficient, wake length and drag coefficient were studied through the streamline flow field. Main flow and subflow regimes were identified by an analysis of the evolution of the flow characteristics. It was found that typically, for a given trapezoidal cylinder, flow starts with no separation. As time advances, the symmetrical standing zone of recirculation develops aft of the trapezoidal cylinder. The rate of growth in width, length and structure of the aft end eddies depends on the Reynolds number. In time, separated flow from the leading edges of the trapezoidal cylinder also develops and forms growing separation bubbles on the upper and lower inclined surfaces of the trapezoidal cylinder. As time advances, the separation bubbles on the upper and lower inclined surfaces of the cylinder grow towards the downstream regions and eventually merge with the swelling symmetrical eddies aft of the cylinder. This merging of the flows creates a complex flow regime with a disturbed tertiary flow zone near the merging junction. Eventually, depending on the Reynolds number and the tapered angle of the trapezoidal cylinder, the flow develops into a specific category of symmetrical standing recirculatory flow with its own distinct characteristics. Comparisons with the available results of other investigators showed very good agreement. © 1998 John Wiley & Sons, Ltd.     

Numerical prediction of transonic viscous separated flow past an airfoil

A method for the numerical treatment of transonic viscous flows past an airfoil has been developed allowing for calculation of both attached flows and flows with large separation regions including shock-induced separation. The method accounts for the entropy correction and the presence of vorticity behind the shock. The method is based on a zonal approach. The computation of the separated-flow regimes is provided by a quasi-simultaneous method of viscous—inviscid interaction. A computer code, called VISTRAN, is developed on the basis of this method and extensive comparisons of computed results with experimental data are made.     

Numerical simulation of viscous flow around unrestrained cylinders

A 2-D analysis is made for the dynamic interactions between viscous flow and one or more circular cylinders. The cylinder is free to respond to the fluid excitation and its motions are part of the solution. The numerical procedure is based on the finite volume discretization of the Navier–Stokes equations on adaptive tri-tree grids which are unstructured and nonorthogonal. Both a fully implicit scheme and a semi-implicit scheme in the time domain have been used for the momentum equations, while the pressure correction method based on the SIMPLE technique is adopted to satisfy the continuity equation. A new upwind method is developed for the triangular and unstructured mesh, which requires information only from two neighbouring cells but is of order of accuracy higher than linear. A new procedure is also introduced to deal with the nonorthogonal term. The pressure on the body surface required in solving the momentum equation is obtained through the Poisson equation in the local cell. Results including flow field, pressure distribution and force are provided for fixed single and multiple cylinders and for an unrestrained cylinder in steady incoming flow with Reynolds numbers at 200 and 500 and in unsteady flow with Keulegan–Carpenter numbers at 5 and 10.     

Numerical investigation of forced convection heat transfer in unsteady flow past a row of square cylinders

This paper presents the unsteady laminar forced convection heat transfer from a row of five isothermal square cylinders placed in a side-by-side arrangement at a Reynolds number of 150. The numerical simulations are performed using a finite volume code based on the PISO algorithm in a collocated grid system. Special attention is paid to investigate the effect of the spacing between the cylinders on the overall transport processes for the separation ratios (spacing to size ratio) between 0.2 and 10. No significant interaction between the wakes is observed for spacing greater than four times the diameter at this Reynolds number. However, at smaller spacing, the wakes interact in a complicated manner resulting different thermo-hydrodynamic regimes. The vortex structures and isotherm patterns obtained are systematically presented and discussed for different separation ratios. In addition, the mean and instantaneous drag and lift coefficients, mean and local Nusselt number and Strouhal number are determined and discussed for various separation ratios. A new correlation is derived for mean Nusselt number as a function of separation ratio for such flows.     

Finite-element analysis of fluid flow and heat transfer for staggered bundles of cylinders in cross flow

The two-dimensional Navier-Stokes equations and the energy equation governing steady laminar incompressible flow are solved by a penalty finite-element model for flow across finite depth, five-row deep, staggered bundles of cylinders. Pitch to diameter ratios of 1·5 and 2·0 are considered for cylinders in equilateral triangular and square arrangements. Reynolds numbers studied range from 100 to 400, and a Prandtl number of 0·7 is used. Velocity vector fields, streamline patterns, vorticity, pressure and temperature contours, local and average Nusselt numbers, pressure and shear stress distributions around the cylinder walls and drag coefficients are presented. The results obtained agree well with available experimental and numerical data.     

A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De ≈ 1.3, for an extensibility parameter of the model of L² = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L² = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De ≈ 4.0–4.5.     

串列双圆柱绕流问题的数值模拟

本文运用有限体积方法,对绕串列放置的双圆柱的二维不可压缩流动进行了数值计算。为研究两圆柱不同间距对圆柱相互作用和尾流特征的影响,选取间距比Ｌ／Ｄ（Ｌ为两圆柱中心间的距离,Ｄ为圆柱直径）在１．５～５．０之间每隔０．５共八个有代表性的间距进行了计算模拟。计算均在Ｒｅ＝２００条件下进行。计算结果表明：对该绕流问题,流动特征在很大程度上取决于间距的大小。且间距存在一临界值,间距比从小于临界值变化到大于临界     

Numerical studies of slow viscous rotating flow past a sphere. III

The flow of steady incompressible viscous fluid rotating about the z-axis with angular velocity ω and moving with velocity u past a sphere of radius a which is kept fixed at the origin is investigated by means of a numerical method for small values of the Reynolds number Re_ω. The Navier–Stokes equations governing the axisymmetric flow can be written as three coupled non-linear partial differential equations for the streamfunction, vorticity and rotational velocity component. Central differences are applied to the partial differential equations for solution by the Peaceman–Rachford ADI method, and the resulting algebraic equations are solved by the ‘method of sweeps’. The results obtained by solving the non-linear partial differential equations are compared with the results obtained by linearizing the equations for very small values of Re_ω. Streamlines are plotted for Ψ = 0·05, 0·2, 0·5 for both linear and non-linear cases. The magnitude of the vorticity vector near the body, i.e. at z = 0·2, is plotted for Re_ω = 0·05, 0·24, 0·5. The correction to the Stokes drag as a result of rotation of the fluid is calculated.     

The flow of a viscous liquid past a doubly periodic row of cylinders

The flow of a liquid in a system of parallel cylinders arranged perpendicularlyto the flow is considered for small Reynolds numbers.     

Periodic solutions of rigid body–viscous flow interaction

This paper describes the work on extending the finite element method to cover interactions between a viscous flow and a moving body. The problem configuration of interest is that of an arbitrarily shaped body undergoing a simple harmonic motion in an otherwise undisturbed incompressible fluid. The finite element modelling is based on a primitive variables representation of the Navier-Stokes equations using curved isoparametric elements. The non-linear boundary conditions on the moving body are obtained using Taylor series expansion to approximate the velocities at the fixed finite element grid points. The method of averaging is used to analyse the resulting periodic motion of the fluid. The stability of the periodic solutions is studied by introducing small perturbations and applying Floquet theory. Numerical results are obtained for several example body shapes and compared with published experimental results. Good agreement is obtained for the basic non-linear phenomenon of steady streaming.     

Computation of turbulent flow past an array of cylinders using a spectral method with Brinkman penalization

A major difficulty in computing engineering flows at high Reynolds number is the need for non-uniform grids adapted to solid boundaries that may be moving or changing shape. These non-uniform grids are expensive to calculate and cannot be used with the most accurate or efficient numerical schemes. We present one solution to this problem: a Brinkman (volume) penalization of the obstacle which allows an efficient pseudo-spectral method to be used to solve the Navier–Stokes equations on a Cartesian grid. Although this is the most severe test of the penalization (due to the global support of the Fourier basis), it is shown that the method still yields reasonable results. We also present an analytical solution of Stokes flow calculated using the penalization which illustrates the error and continuity properties of the approach. Work is currently underway to implement the penalization approach in a wavelet basis.     

Flow of a viscous fluid past an array of spheres

The Stokes flow of a viscous incompressible fluid through a periodic array of impenetrable spheres with linear friction on the boundary is considered. A solution and an expression for the drag are obtained to terms of order c^5/3 compared with unity (c is the volume concentration of the spheres). The proposed algorithm permits solution with any required degree of accuracy. The solution contains as limits the cases of perfect slip and no-slip on the surfaces of the spheres. In the problem with the no-slip condition, an asymptotically exact lower bound for the drag, which is valid for all values of the concentration c, is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 37–44, July–August, 1981.     

The unsteady flow generated by an oscillating wall with transpiration

In this work, an analytical solution for the fluid behavior over flat plates with impulsive and oscillating motions, starting from rest, and with wall transpiration, is presented. The classical solution of this problem is given by Panton [7] and is found to be an especial case of the solution here presented. The analytical solution is obtained without the use of any special transformations, such as Laplace or Fourier transforms. Instead, an extension of the variable separation technique is used together with similarity arguments. A non-dimensional number—the transpiration rate—is used to take into account the injection or suction of fluid at the wall. This parameter is shown to be of great influence on the proposed velocity solution.