Numerical predictions for unsteady viscous flow past an array of cylinders

The unsteady two-dimensional flow around an array of circular cylinders submerged in a uniform onset flow is analyzed. The fluid is taken to be viscous and incompressible. The array of cylinders consists of two horizontal rows extending to infinity in the upstream and downstream directions. The center-to-center distance between adjacent cylinders is a constant. The Biot-Savart law of induced velocities is used to determine the velocity field due to the free vorticity in the surrounding fluid and the bound vorticity distributed on the surface of each cylinder. The bound vorticity is needed to enforce the no-penetration condition and to account for the production of free vorticity in the solid surfaces. It is governed by a Fredholm integral equation of the second kind. This equation is solved by numerical techniques. The transport of free vorticity in the flow field is governed by the vorticity transport equation. This equation is discretized for a control volume and is solved numerically. Advantage is taken of spatially periodic boundary conditions in the flow direction. This reduces the computational domain to a rectangular region surrounding a single circular cylinder, but necessitates use of a non-orthogonal grid. In order to test the numerical techniques, the simpler case of unsteady flow over a single circular cylinder at various Reynolds numbers if first considered. Results compare favorably with previous experimental and numerical data. Three cases for Reynolds numbers of 10², 10³, and 10⁴ are presented for the array of cylinders. The center-to-center distance is fixed at three diameters. The time development of constant vorticity contours as well as drag, lift, and moment coefficients are shown for each Reynolds number. The motion of stagnation and separation points with time is also given. It is found that the drag for a cylinder in the array may be as low as five percent of that for flow over a single cylinder at the same Reynolds number.