The wake region in the steady low-Reynolds-number flow past a cylinder

Summary Evidently the flow field defined in the title contains three rather than two discernable regimes. The first is the vicinity of the cylinder where the flow is viscosity dominated. The second is known as the outer field. There space variations are milder, viscosity is much less effective and the influence of the uniform parallel stream is dominant. In this regime the stream function representing the obstacle's disturbance is governed by the fourth-order Oseen equation. However, evidently its recorded solution fails over the half-plane downstream of the cylinder's axis. It is the author's contention that this failure reflects the existence of a third regime-the wake-where space variations are sharp but only in the transverse direction. To obtain a solution for the entire flow field an additional low-Reynolds-number expansion is constructed. It is matched with the well-known ones prevailing in the inner and outer regimes.     

Stokes flow past a smooth cylinder

Summary Ranger's solution of the two-dimensional Stokes flow past a smooth body is analyzed in detail. The body can be made convex or concave depending on the choice of its two parameters and concavity is found to be necessary but not sufficient for separation to occur. A relationship exists between the formation of a Stokesian wake and the curvature at the concave end of the body. Particular attention is given to the case when the body degenerates to a circular arc. The two-dimensional Stokes flow past a circular arc is strikingly similar to the axisymmetric Stokes flow past a spherical cap.     

Steady magnetohydrodynamic flow past a circular cylinder

The steady two-dimensional, viscous, electrically conducting flow around a circular cylinder is investigated. The flow and magnetic field are uniform and parallel at large distances from the cylinder. The equations and boundary conditions are derived for arbitrary values of R, R_mand β, where R is the Reynolds number, R_m the magnetic Reynolds number and β, the ratio of the square of the Alfvén speed to the square of the main stream speed. Because of the large number of parameters involved, the numerical solution is restricted to R = 40, R_m = 1 and infinity and 0 ? β ? 4. Also the cylinder is taken to be a perfect conductor, this avoids having to compute the magnetic field within the cylinder. The numerical computations for the non-magnetic case, i.e. β = 0, are presented and are found to be in good agreement with existing results. The effect of increasing the strength of the magnetic field (i.e. increasing β) on the drag coefficient, the size and position of the standing vortex and the increasing effect of the upstream propagation of disturbances are examined.     

Steady viscous flow past a tapered cylinder

The three-dimensional nature of the viscous flow past a linearly tapered circular cylinder is examined at low Reynolds numbers. The numerical solution of the unsteady Navier–Stokes equations converges to a steady state. The primary flow in planes perpendicular to the cylinder axis is practically indistinguishable from the two-dimensional flow past a uniform cylinder. A secondary spanwise flow is observed in the stagnation zone going from the wide end towards the narrow end, whereas a secondary motion on the rear side goes in the opposite direction. In spite of this secondary flow, the length of the separation zone varies linearly with the local Reynolds number.     

Uniform shear flow past a circular cylinder

Summary This paper solves the two-dimensional problem of the unsteady shear flow of a viscous incompressible fluid past a circular cylinder. The flow is calculated using two methods. The first takes the form of a double series solution where an expansion is carried out in powers of the time, t, and in powers of the parameter where R is the Reynolds number. This approximate analytical solution is valid for small times following the start of the motion and for large Reynolds numbers. The second method involves a spectral-finite difference procedure for numerically integrating the full Navier-Stokes equations expressed in terms of a stream function and vorticity. Our results demonstrate that for small times and moderately large R the two methods of solution are in good agreement. Results are presented for a wide range of Reynolds numbers. Comparisons with previous studies are also carried out and discussed.     

Stability of viscous flow past a circular cylinder

A spectral method which employs trigonometric functions and Chebyshev polynomials is used to compute the steady, incompressible laminar flow past a circular cylinder. Linear stability methods are used to formulate a pair of decoupled generalized eigenvalue problems for the growth of symmetric and asymmetric (about the dividing streamline) perturbations. We show that, while the symmetric disturbances are stable, the asymmetric perturbations become unstable at a Reynolds number about 40 with a Strouhal number about 0.12. The critical conditions are found to depend on the size of the computational domain in a manner similar to that observed in the laboratory.     

Steady flow of a power-law fluid past a cylinder

Summary Considered in this paper is the two-dimensional steady flow of a power-law fluid past a stationary circular cylinder. The governing nonlinear equations, expressed in terms of a stream function and vorticity, were solved by finite differences for Reynolds numbers (based on the radius of the cylinder)R=5,20, 40 for various power-law indices,n. Parameters such as the drag coefficient, separation angle, wake length and critical Reynolds number are presented and contrasted with those of a Newtonian fluid (n=1) to illustrate the non-Newtonian effects. For a given-Reynolds number a consistent behaviour withn was observed in the parameters for the ranges considered. The results obtained for the Newtonian case agree well with documented results.     

Application of a pseudo-viscous method to the calculation of the steady supersonic flow past a waisted body

A finite difference method for solving mixed initial and boundary value problems governed by hyperbolic partial differential equations is described. The method has been developed specifically to calculate the flow field associated with any arbitrary two-dimensional or axi-symmetric body placed in a uniform supersonic stream. In order to preserve accuracy in the neighborhood of the body surface, the equations governing the motion are formulated so that

• 1 Streamlines form one system of co-ordinate curves.
• 2 Two of the equations of motion correspond to characteristic relations.

The boundary condition at the body is applied by omitting the equation associated with the characteristic curve reaching the body surface from inside. The method presented is a pseudo-viscous method and flow fields which include shock waves can be treated with ease. The method has been used to calculate the flow past a particular axi-symmetric waisted body. The shape of this body is such that a secondary internal shock wave is generated by coalescence of Mach lines behind the bow shock wave. The results obtained are compared with theoretical and experimental results obtained by other workers. The investigation has been carried out as part of a programme of work needed to validate the pseudo-viscous method used.     

The initial oscillatory flow past a circular cylinder

This paper presents results obtained from an initial approximation for the flow around a circular cylinder in two-dimensional oscillating flow. The analysis is developed in terms of the scalar vorticity and stream function. An expansion in powers of time from the start of the motion is obtained using an exact analysis which extends the results of boundary-layer theory by taking into account corrections for finite Reynolds number. The time development of the physical properties of the flow are determined both by means of analytical expressions and by an accurate numerical procedure. The surface pressure, drag and surface vorticity are calculated and various estimates of the time of separation and the distance moved in this time are obtained. The phenomenon of steady streaming is not considered in this paper since the time of validity of the expansions is small. The agreement between the analytical and numerical results at small times is excellent.     

Shear flow past a square cylinder near a wall

A numerical study on the wake behind a square cylinder placed parallel to a wall has been made. The cylinder is considered to be within the boundary layer of the wall, so that the outside flow is taken to be due to uniform shear. Flow has been investigated in the laminar Reynolds number (based on the cylinder height) range. The interaction of wall boundary layer on the vortex shedding at Reynolds number up to 1400.0 has been investigated for cylinder to wall gap height 0.5 and 0.25 times the cylinder height. The gap flow between the cylinder and wall during a period of vortex shedding has been obtained. The governing unsteady Navier–Stokes equations are discretised through the finite volume method on staggered grid system. An algorithm SIMPLE has been used to compute the discretised equations iteratively. Our results show that at the gap height 0.5 times the cylinder height the vortex shedding occurs at a Strouhal number greater than for an isolated cylinder. Vortex shedding suppression occurs and wake becomes steady up to a certain value of Reynolds number at gap height 0.25 time the cylinder height. At higher Reynolds number the formation of a single row of negative vortices behind the cylinder when it is in close proximity to wall is noteworthy. Due to the shear, the drag experienced by the cylinder is found to decrease with the reduction of gap height.     

A time-space decomposition method for calculating the nearfield pressure generated by a pulsed circular piston

A time-space decomposition approach is derived for numerical calculations of the transient nearfield pressure generated by a circular piston. Time-space decomposition analytically separates the temporal and spatial components of a rapidly converging single integral expression, thereby converting transient nearfield pressure calculations into the superposition of a small number of fast-converging spatial integrals that are weighted by time-dependent factors. Results indicate that, for the same peak error value, time-space decomposition is at least one or two orders of magnitude faster than the Rayleigh-Sommerfeld integral, the Schoch integral, the Field II program, and the DREAM program. Time-space decomposition is also faster than methods that directly calculate the impulse response by at least a factor of 3 for a 10% peak error and by a factor of 17 for a 1% peak error. The results show that, for a specified maximum error value, time-space decomposition is significantly faster than the impulse response and other analytical integrals evaluated for computations of transient nearfield pressures generated by circular pistons.     

Wall effects on separation of flow past a circular cylinder

Abstract

The nature of the flow past a circular cylinder has been a classical problem, raising many questions concerning the various wake phenomena that have been observed. This study focused on the critical Reynolds number for the onset of the recirculation region for a flow restricted in a channel. The influence of the bounded walls is examined. The trend is that larger critical Reynolds numbers are accomplished with larger values of blockage ratio (defined as the ratio of cylinder diameter to the channel width). Furthermore, as the blockage ratio tends to zero, the trend seems to imply that the critical Reynolds number approaches the experimental value for flows in an unbounded domain.     

A direct boundary integral equation formulation for the Oseen flow past a two-dimensional cylinder of arbitrary cross-section

Summary A novel boundary integral formulation is presented for the direct solution of the classical problem of slow flow past a two-dimensional cylinder of arbitrary cross section in an unbounded viscous medium, the equations of motion having first been linearised by the Oseen approximation. It is shown how the governing partial differential equations of motion, together with the no-slip boundary conditions on the cylinder, may be reformulated as a pair of coupled integral equations of the second kind, which may be manipulated further to yield the lift and drag coefficients explicitly, as well as flow characteristics anywhere in the flowfield.The present formulation requires a non-iterative numerical solution procedure which is applicable to low Reynolds number flows. The method is not restricted in its ability to deal with complicated cylinder geometries, as the discretisation of only the cylinder surface is required.Results of the present method are shown to be in good agreement with those of previous analytical and numerical investigations.With 2 Figures     

A boundary element method for steady incompressible thermoviscous flow

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modelling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.     

A domain decomposition method for concurrent coupling of multiscale models

Motivated by atomistic‐to‐continuum coupling, we consider a fine‐scale problem defined on a small region embedded in a much larger coarse‐scale domain and propose an efficient solution technique on the basis of the domain decomposition framework. Specifically, we develop a nonoverlapping Schwarz method with two important features: (i) the use of an efficient approximation of the Dirichlet‐to‐Neumann map for the interface conditions; and (ii) the utilization of the inherent scale separation in the solution. The paper includes a detailed formulation of the proposed interface condition, along with the illustration of its effectiveness by using simple but representative numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of initial flow past a rotating and translating circular cylinder

Abstract

Matched asymptotic expansion is carried out for initial flow past an impulsively started rotating and translating circular cylinder. The study expands upon previous similar analysis to include the effect of rotation with specific emphasis on the evaluation of the lift, drag, moment coefficients and the pressure distribution along the cylinder surface. Three different approaches to evaluate the force, moment coefficients and surface pressure are shown to require different orders of asymptotic expansion as far as leading behaviors are concerned. Some indication of the present analysis about how to numerically implement these different approaches is also discussed.     

基于区域分解的圆锥壳-圆柱壳-圆锥壳组合结构自由振动

提出一种分区广义变分和最小二乘加权残值区域分解法来分析圆锥壳-圆柱壳-圆锥壳组合结构的自由振动。首先将组合结构分解为圆柱壳、圆锥壳子结构,为获取组合壳体的高阶振动特性,进一步将圆柱壳、圆锥壳子结构分解为圆柱壳段和圆锥壳段。采用分区广义变分和最小二乘加权残值法将各壳段分区界面上的位移和转角协调方程引入到组合壳体的势能泛函中,使组合壳体的振动分析问题,归结为在满足分区界面位移和转角协调条件下的无约束泛函变分问题。圆柱壳段及圆锥壳段位移变量的周向和轴向(或母线方向)分量分别以Fourier级数和Chebyshev多项式展开。将区域分解法计算出的组合壳体振动频率与有限元软件ANSYS结果进行对比发现,两者非常吻合,验证了区域分解方法的收敛性和计算精度。     

Steady flow past an elliptic cylinder inclined to the stream

The properties of steady two-dimensional flow past an elliptic cylinder inclined to the oncoming stream are investigated for small to moderate values of the Reynolds number for which good accuracy can be assured. The solutions are based on a numerical method of solution of the Navier-Stokes equations for incompressible fluids which ensures that all the correct conditions of the problem are satisfied. In particular, the solution is carried out in such a way that the vorticity decays rapidly enough at large distances from the cylinder for the lift and drag on the cylinder to be finite. Results are presented for the variation of lift, drag and streamline patterns with inclination and Reynolds number. Two elliptic cylinders (based on their minor-to-major axes ratio) are considered. For an elliptic cylinder with minor-to-major axes ratio 1:5, results are obtained for Reynolds numbers up to 40 and inclination varying from zero to 90°. Streamline plots for these results show a development of the solution from asymmetric flow at zero inclination (with no separation), through asymmetric flows with increasing inclination (with either no separation, separation with a single recirculating region, or separation with two recirculatory regions) to the symmetric flow at 90° incidence (with two counter rotating vortices). Of interest are asymmetric steady-state results which contain two recirculatory regions trailing the cylinder, one attached and one unattached to the cylinder. Results are also obtained for a second elliptic cylinder with minor-to-major axes ratio 1:10 at Reynolds numbers 15 and 30, inclination 45°. These results are found to be in good agreement with corresponding unsteady results taken to long times (which are tending to a steady state).     

A domain decomposition implementation of the SIMPLE method with PVM

A parallel implementation of a finite volume method for the solution of the Navier-Stokes equations on a distributed computing environment through Parallel Virtual Machine (PVM) is reported. The numerical method is implicit and is based on the SIMPLE algorithm in which the system of equations is discretised using a hybrid scheme. An Alternative Direction Implicit (ADI) scheme, and the Thomas tri-diagonal solver are used to solve the algebraic equations. The parallelization of the program is implemented by a domain decomposition strategy on MIMD parallel architectures using PVM platform. The program was tested for laminar flow in a cavity. The parallelisation strategy and performance are discussed. It is concluded that the efficiency is strongly dependent on the grid size, block numbers and the number of processors. Different strategies to improve the computational efficiency are proposed.