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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Leonard, A., Wray, A.: A new numerical method for the simulation of three-dimensional flow in a pipe. Lecture Notes in Physics, vol. 170, (E. Krause, ed.), Springer, p. 335 (1982)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, U.K. (2010)
Dalecky, YuL, Fomin, S.V.: Measures and Differential Equations in Infinite-Dimensional Space. Kluwer Academic Publ, Dordrecht (1991)
Rogallo, R.S.: Numerical Experiments in Homogeneous Turbulence (1981). NASA-TM-81315
Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publ. Inc., Mineola, New York (2001)
Kollmann, W.: Simulation of vorticity dominated flows using a hybrid approach: I formulation. Comput. Fluids 36, 1638–1647 (2007)
Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences (1996)
Schlichting, H.: Boundary Layer Theory. McGraw-Hill (1987)
Panton, R.L.: Incompressible Flow. Wiley, New York, 2nd ed (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kollmann, W. (2019). Appendix C: Scalar and Vector Bases for Periodic Pipe Flow. In: Navier-Stokes Turbulence. Springer, Cham. https://doi.org/10.1007/978-3-030-31869-7_25
Download citation
DOI: https://doi.org/10.1007/978-3-030-31869-7_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-31868-0
Online ISBN: 978-3-030-31869-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)