8.2.1 Review of previous computations

In this section, we review the numerical schemes used by recent high-resolution computations; such as those of Anderson & Reider [3], Kruse & Fischer [120], Koumoutsakos & Leonard [117], and Wu, Wu, Ma, & Wu [249], among others. A review of earlier computations has been given by, for example, Lecointe & Piquet [123].

Anderson & Reider [3] use a finite difference scheme to compute the flow at Reynolds numbers $Re=1,000$, 3,000 and 9,500. The vorticity equation and the Poisson equation for streamfunction are used. Their scheme is fourth-order accurate in space and time. A fourth-order Runge-Kutta scheme is used for time stepping. To maintain fourth-order convergence in time a smoothed initial vorticity distribution is used. Anderson [7] obtains the boundary condition for the vorticity from an integro-differential equation, which he derives by enforcing both the no-slip condition and its time derivative. At the outer boundary of the domain, the convection term in the vorticity equation is discretized using upwind differencing; the diffusion term is discretized using zero vorticity flux across the boundary. The streamfunction equation is solved using the domain decomposition method of Anderson [8] for accurate handling of the velocity boundary condition at large distances away from the cylinder surface. At the interface of the domains, continuity of the velocity is enforced by an iterative procedure. For the computation of the flow over the cylinder at $Re=9,500$, the interface is a circle of radius 1.5 times that of the cylinder and it divides the flow domain into two annular regions. A $2048\times256$ mesh which is uniformly spaced in circumferential and radial directions respectively, is used in the inner annular region; the outer annular region is handled analytically using solutions of the Laplace equation. They present data for streamlines, isovorticity contours, drag coefficient, surface pressure, and surface vorticity.

Wu, Wu, Ma & Wu [249] use a finite difference scheme to compute the flow at Reynolds number $Re=9,500$. The vorticity equation and the Poisson equation for the stream function are used. A viscous splitting of the vorticity equation is employed. For the convection of vorticity, the second-order upwind differencing scheme proposed by B. P. Leonard [127] and the second-order TVD Runge-Kutta time stepping scheme proposed by Shu & Osher [209] are used. For diffusion, the Peaceman-Rachford ADI factorization with central difference spatial discretization and Crank-Nicholson time discretization is used. For the boundary condition for the vorticity on the cylinder surface, the vorticity flux obtained from the momentum equation equation in the circumferential direction is used. The pressure gradient term in that momentum equation is found by iteration. The streamfunction at the outer boundary of the computational domain is obtained from the potential flow velocity due to the translation of the cylinder; they find that this simplified implementation requires the outer boundary to be located as far away as possible from the cylinder surface to obtain accurate results. In fact, their computed radial velocity along the rear symmetry axis and the drag coefficient are significantly different depending on how far away the outer boundary is located. In the $Re=9,500$ computational results used for comparison in the following sections, the outer boundary is a circle of radius 20 times that of the cylinder. The computational grid consists of $512\times900$ mesh points which are uniformly spaced in the circumferential and exponentially stretched in the radial directions respectively; and the time step $\Delta t=0.0025$. They present computational data for streamlines, radial velocity along the rear symmetry axis, drag coefficient, slip velocity, and surface vorticity.

Hakizumwami [105] uses a finite difference method to compute the flow at $Re=3,000$ and 9,500; flow symmetry is assumed. The vorticity equation and the Poisson equation for the streamfunction are used. The vorticity equation is discretized using a second order central difference scheme. For time stepping, the second order Adam-Bashforth scheme is used for $Re=3,000$ and the fourth order Runge-Kutta scheme for $Re=9,500$. A Fourier transform is applied in the tangential direction for the Poisson equation and the resulting equations are discretized using a second order central difference scheme. A boundary condition for the vorticity on the surface is obtained, to second order accuracy, using the streamfunction equation and the no-slip condition. At the outer boundary of the computational domain, the vorticity is set to zero and the velocity is taken to be the potential flow due to uniform translation of the cylinder. The grid consists of $120\times128$ mesh points for the $Re=3,000$ and $300\times128$ for the $Re=9,500$; the mesh points are uniformly spaced in the circumferential direction and exponentially stretched in the radial direction. The outer boundary is a circle of radius 5 times that of the cylinder. The time step $\Delta t=0.01$. The computational data for streamlines, radial velocity along the rear symmetry axis, and surface vorticity are presented.

Loc [133] uses a finite difference scheme to compute the flow at Reynolds numbers $Re=300$, 550 and 1,000; flow symmetry is assumed. The vorticity equation and the Poisson equation for the streamfunction are used. The vorticity equation is discretized using a second order accurate scheme. The Poisson equation is discretized using a compact fourth order accurate scheme. The boundary condition for vorticity on the surface is obtained to second order accuracy using the streamfunction and the no-slip condition. At the outer boundary of the computational domain, the vorticity is set to zero and the velocity was taken to be the potential flow due to uniform translation of the cylinder. The grid consists of $41\times41$ mesh points for $Re=300$, $61\times61$ mesh points for $Re=550$ and $81\times41$ mesh points for $Re=1,000$; the mesh points are uniformly spaced in the circumferential direction and exponentially stretched in the radially direction. The time step $\Delta t$ is 0.05 for $Re=300$, 0.033 for $Re=550$ and 0.025 for $Re=1,000$. The outer boundary is a circle of radius 20 times that of the cylinder. The computational data for streamlines, radial velocity along the rear symmetry axis, drag coefficient, and surface vorticity are presented.

Loc & Bouard [134] use the finite difference scheme of Loc [133] to compute the flow at $Re=3,000$ and $Re=9,500$. The boundary condition for the vorticity on the surface is obtained, to second or third order accuracy, using the streamfunction equation and the no-slip condition. At the outer boundary of the computational domain, a simplified vorticity equation is used. The grid consists of $141\times101$ mesh points for $Re=3,000$, and $301\times101$ mesh points for $Re=9,500$; the mesh points are uniformly spaced in the circumferential direction and exponentially stretched in the radially direction. A time step $\Delta t=0.02$ is used. The outer boundary is a circle of radius 5 times that of the cylinder. They present computational data for streamlines, radial velocity along the rear symmetry axis, wake length, and surface vorticity.

Kruse & Fischer use a spectral element method to solve the Navier-Stokes equations in terms of velocity and pressure. The computational grid consists of 6112 spectral elements. There are 10 nodes along each dimension in every element. They assume flow symmetry in their computation. For their computations at $Re=9,500$, the outer boundary is a circle of radius 20 times that of the cylinder.

Chang & Chern [40] use a hybrid vortex method to compute the flow at Reynolds numbers $Re=300$, 550, 1,000, 3,000, 9,500, 20,000, $10^5$ and $10^6$. The vorticity equation and the Poisson equation for streamfunction are used. A viscous splitting of the vorticity equation is employed. In this hybrid vortex method, a mesh is used to solve both the streamfunction equation and the diffusion equation arising from the viscous splitting. The vortex-in-cell scheme proposed by Christiansen [56] is used to obtain the vorticity at the mesh points from the circulation of the vortices. The streamfunction at the outer boundary of the computational domain is obtained by assuming the flow to be uniform there. The vorticity on the cylinder surface is obtained by applying the streamfunction equation on the boundary together with the no-slip condition. For most of their computations, the mesh has $128\times200$ points that are uniformly spaced in the circumferential direction and exponentially stretched in the radial direction. The outer boundary is a circle of radius 25 times that of the cylinder. The time step $\Delta t=0.02$. They presented computational data for streamlines, closed wake (recirculation region) size parameters, ``separation" point (defined as the position where the streamline leaves the cylinder surface, not true separation), radial velocity along the rear symmetry axis, drag coefficient, surface pressure, and surface vorticity.

Koumoutsakos & Leonard [117] used a vortex method to compute the flow at Reynolds numbers $Re=40$, 550, 1,000, 3,000 and 9,500. No flow symmetry was assumed. A viscous splitting of the vorticity equation was used (compare subsection 3.2). The velocity field is computed from the circulation of the vortices or ``particles" using a fast scheme based on that of Greengard and Rokhlin [97]. A second-order Adam-Bashforth time stepping is used to convect the particles. The diffusion is performed using the Particle Strength Exchange scheme discussed in subsection 1.3.3. Koumoutsakos, Leonard & Pépin [118] handle the no-slip boundary condition using a vortex sheet on the cylinder surface; the vortex sheet is allowed to diffuse, leading to a vorticity flux that modifies the strength of the particles near the cylinder surface. In the computations, the uniformity of the particle distribution is periodically restored by remeshing every few time steps. For $Re=9,500$, a time step $\Delta t=0.01$ is used. For this computation the number of particles was about 350,000 at time $t=3.00$. They presented computational data for streamlines, vorticity field, circulation, drag coefficient, position of zero wall shear, surface vorticity, and surface vorticity flux.

Earlier, Pépin [170] also used the Particle Strength Exchange scheme to compute the flow at $Re=550$, 3,000 and 9,500; he assumed flow symmetry. He used the same fast velocity summation and time discretization as Koumoutsakos & Leonard [117]. For $Re=550$ a time step $\Delta t=0.03$ was used; the number of particles at $t=6.0$ was about 50,000. For $Re=3,000$ a time step $\Delta t=0.0275$ was used; the number of particles at $t=5.0$ was about 76,000. For $Re=9,500$ a time step $\Delta t=0.018$ was used; the number of particles at $t=3.25$ was about 80,000. The remeshing was performed typically once every six time steps. He presented computational data for streamlines, closed wake (recirculation region) size parameters, radial velocity along the rear symmetry axis, vorticity field, drag coefficient, and ``separation point" (defined as the position where the streamline leaves the cylinder surface, not true separation).

Cheer [42] uses the random walk method to simulate the flow at Reynolds numbers $Re=3,000$, and 9,500. She uses about 900 sheet vortices and vortex blobs; flow symmetry is imposed by reflecting the vortex elements about the symmetry line. An Euler time stepping scheme is used and the size of the time step is $\Delta t=0.03$. For both the Reynolds numbers, she presents computational data for streamlines, closed wake (recirculation region) size parameters, radial velocity along the rear symmetry axis, and ``separation point" (defined as the position where the streamline leaves the cylinder surface, not true separation).

Smith & Stansby [215] use the random walk to compute the flow for Reynolds numbers $Re=250$, 1,000, 10,000, and $10^5$. For convection of the vortices they use Christiansen's [56] vortex-in-cell method. The mesh consists of $129\times200$ points uniformly spaced in the circumferential direction and exponentially stretched in the radial direction. The outer boundary is a circle of radius 25 times that of the cylinder. The time step $\Delta t=0.02$. At each time step, one to six new vortices are created from each mesh point on the cylinder surface. They present computational data for streamlines, radial velocity along the rear symmetry axis, drag coefficient, ``separation point" (defined as the position where the streamline leaves the cylinder surface, not true separation), surface pressure, and surface vorticity.

Van Dommelen [231,237] uses a random walk method to compute the flow at Reynolds number $Re=550$ and 10,000. The velocity of the vortices are computed using the fast algorithm of Van Dommelen & Rundensteiner [233]. A second order Runge-Kutta method (Heun's method) is used for time stepping. A time step $\Delta t=0.025$ is used. There are about 7,500 vortices at $t=6.00$ for $Re=550$ and 25,000 vortices for $Re=10,000$ at $t=3.00$.