Proper orthogonal decomposition and incompressible flow: An application to particle modeling

The proper orthogonal decomposition (POD) is a reduced-order modeling technique that is used to compactly represent unsteady flows. In this paper, we use the POD to capture the parametric variation of a flow with Reynolds number. We study incompressible, axisymmetric, steady flow over spherical particles at various Reynolds numbers in order to give an alternative to correlation-based approaches for predicting the drag on a sphere. In most previous applications of the POD for reduced-order modeling of incompressible flow, the POD modes typically have only described the velocity field; the pressure field was not directly modeled. Since we are interested in drag, which is dependent on the pressure, we formulate the method to directly include the pressure field of an incompressible flow. The POD modes are then derived from numerical flow solutions obtained using an hp-finite element method. A reduced-order model is created by performing a streamwise-upwind-Petrov–Galerkin (SUPG) projection of the incompressible Navier–Stokes equations onto the space spanned by the POD modes. The SUPG approach is taken because when pressure modes are included the Galerkin method fails to give unique solutions for incompressible flow. This is demonstrated for some simple test cases. An efficient numerical implementation is also developed using a Taylor expansion of the SUPG projection of the Navier–Stokes equations. Finally, values of drag are computed from the reduced-order model. Drag can be calculated to within 1.0% of the direct numerical simulations using only a small number of modes while still retaining all of the essential physics around the particle.