Spectral representation and filtering of incompressible flow

Several approaches exist to functionally project fluid velocity field data; proper orthogonal decomposition, dynamic mode decomposition, radial basis functions and smoothing kernels, spectral filtering by Fourier representation and wavelet transforms are a few examples. Often this projection is done to de-noise or filter measured velocity field data, yet for many applications the necessity of an incompressible projection is important. In this talk, we present a spectral representation of multi-dimensional vector fields that directly addresses incompressibility. The spectral representation of the fluid flow is obtained by the Galerkin projection of the flow to a class of solenoidal and orthogonal eigenmodes that satisfy the flow boundary conditions. We demonstrate that these eigenmodes span the functional space of solenoidal vector fields. Using these modes, flow field data can be filtered using a truncated series of modes where incompressibility and boundary conditions are preserved. We provide the upper bound of the approximation error and convergence rate, and demonstrate results using example data.