Appendix C: Scalar and Vector Bases for Periodic Pipe Flow

Abstract

Bases are constructed for separable Hilbert spaces relevant for the solutions of the Navier–Stokes pdes. In addition, analytic test functions are constructed for the purpose of evaluating representations of scalar and vector fields with respect to bases defined over a compact domain \\(\\mathcal{D}\\), the straight cylinder being an explicit example. Scalar and vector basis functions for the phase space \\(\\Omega \\) (realizations of a turbulent flow) and the test function space \\(\\mathcal{N}_p\\) (argument functions of the characteristic functional) plus analytic functions, for the purpose of testing numerically the convergence properties of the bases, are constructed using cylindrical coordinates suitable for the periodic flow through straight pipes with circular cross section.