Abstract: A compact high-order finite difference method is developed for discretization of the unsteady vorticity transport equations (VTE) for 2-D incompressible flow. The algorithm is based on the flux reconstruction method of Huynh [1, 2], extended to evaluate a Poisson equation for the streamfunction, which enforces the kinematic relationship between the velocity and vorticity fields while satisfying the continuity equation. Unlike other finite difference discretizations of the VTE, where the wall vorticity is approximated by finite differencing the wall-normal gradient of the streamfunction [3], the new method applies a Neumann boundary condition for the diffusion step such that it cancels the slip velocity resulting from the solution of the Poisson equation for the streamfunction. In this paper, the formulation for the high-order solver will be detailed and benchmark results presented, for example, for flow in a lid-driven cavity and over a circular cylinder at various Reynolds numbers.Formulation:&nbsp;In the present method the computational domain consists of unstructured, curved quadrilateral meshes, with vorticity, streamfunction, and velocity in each element discretized using a [KxK] tensor product of piecewise continuous 1-D (Lagrange) polynomials, where K is the number of collocation points&nbsp;in each direction. The VTE are solved on topologically rectangular meshes using standard mapping. Communication between neighboring elements is established for both convective and diffusive fluxes using Huynh’s recipes for flux reconstruction [1, 2]. In this work, simple averaging of the neighboring fluxes is applied at the interfaces of the elements (as well as Roe’s flux limiter for convection). Further, the&nbsp;&nbsp;collocation points in each direction are set at the Gauss-Legendre nodes, and flux reconstruction applied using right Radau polynomial of order K; at the interfaces, a Legendre polynomial is used, which increases the order of accuracy of the method (for odd values of K) by one without incurring additional CPU cost.&nbsp;Briefly, one computational timestep of the method involves the following steps:&nbsp;(1)&nbsp;Solve the Poisson equation for the streamfunction with the proper Dirichlet boundary conditions to enforce the normal flux boundary condition for velocity.&nbsp;(2)&nbsp;The ensuant wall-normal gradient of the streamfunction is a slip velocity, uslip, which is rendered null via&nbsp;a Neumann (vorticity)&nbsp;boundary condition&nbsp;for diffusion.&nbsp;(3)&nbsp;Solve the VTE&nbsp;using the prescribed velocity (and induced vorticity) boundary conditions. Both 2nd- and 4th-order Runge-Kutta methods are incorporated for explicit time integration of the problem.Results:&nbsp;Solution of the Poisson equation for the streamfunction&nbsp;being a key contribution, we first show a benchmark for the streamfunction and its post-processed gradient (i.e., velocity). We demonstrate that the developed method follows theoretical convergence rates for both the variable and its gradient, perfectly; the flux reconstructed gradient maintains the same convergence order as the variable itself.We also demonstrate excellent benchmarks using the example of 2-D lid-driven cavity flow at various Reynolds numbers up to Re = 10K, for which benchmark data is available [3]. At Re = 5K, the number of degrees of freedom (DOF) using our method (uniform mesh distribution and K = 3 in each direction) is 962&nbsp;vs&nbsp;2572&nbsp;by that of Ghia et al [3], which is a&nbsp;7-fold drop&nbsp;in the number of resources required. Further, using a non-uniform mesh and K = 4 (clustered toward the boundaries), the DOF by our method is only 322, which is a&nbsp;factor of 64.5 reduction&nbsp;in the required CPU resources.In addition to the cavity flow problem, the final paper will include results from flow over a circular cylinder at various Reynolds numbers up to ReD&nbsp;= 9,500, demonstrating significant advantages of the method.[1]&nbsp;Huynh, H.T.,&nbsp;AIAA 2007-4079, pp. 1-43. Miami, FL, June 25-28, 2007.&nbsp;[2]&nbsp;Huynh, H.T.,&nbsp;AIAA 2009-403, pp. 1-34. Orlando, FL, January 5-8, 2009.[3] Ghia, U., et al,&nbsp;J Comput Phys, Vol. 48, pp. 387-411,1982.&nbsp;

A High-Order Flux Reconstruction Method for 2-D Vorticity Transport

Paper Proceedings: https://asmedigitalcollection.asme.org/FEDSM/FEDSM2021/volume/85284

ASME 2021 Fluids Engineering Division Summer Meeting (FEDSM2021)

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