Navier-Stokes Turbulence, 1st ed. 2019
Theory and Analysis

The book serves as a core text for graduate courses in advanced fluid mechanics and applied science. It consists of two parts. The first provides an introduction and general theory of fully developed turbulence, where treatment of turbulence is based on the linear functional equation derived by E. Hopf governing the characteristic functional that determines the statistical properties of a turbulent flow. In this section, Professor Kollmann explains how the theory is built on divergence free Schauder bases for the phase space of the turbulent flow and the space of argument vector fields for the characteristic functional. Subsequent chapters are devoted to mapping methods, homogeneous turbulence based upon the hypotheses of Kolmogorov and Onsager, intermittency, structural features of turbulent shear flows and their recognition.

 

Introduction.- Navier-Stokes equations.- Basic properties of turbulent flows.- Flow domains and bases.- Phase and test function spaces.- Probability measure and characteristic functional.- Functional differential equations.- Characteristic functionals for incompressible turbulent flows.- Fdes for the characteristic functionals.- Solution of Hopf type equations in the spatial description.- The role of the pressure .- Properties and construction of Mappings.- M(): Single scalar in homogeneous turbulence.- M(N): Mappings for velocity-scalar and position-scalar Pdfs.- Integral transforms and spectra.- Intermittency.- Equilibrium theory of Kolmogorov and Onsager.- Homogeneous turbulence.- Length and time scales.- The structure of turbulent ows.- Wall-bounded turbulent ows.- The limit of in_nite Reynolds number for incompressible uids.- Appendix A: Mathematical tools.- Appendix B: Example for a measure on a ball in Hilbert space.- Appendix C: Scalar and vector bases for periodic pipe ow.- Modi_ed Jacobi polynomials Pa;b.- n (r).- Orthonormalisation of the modi_ed polynomials Pa;b.- n (r).- Test function space Np: Scalar _elds.- (i) Bases for the test function space Np.- Function spaces: Vector _elds.- (i) Construction of a general vector basis.- (ii) Construction of a solenoidal vector basis.- Gram-Schmidt orthonormalisation.- Appendix D: Green's function for periodic pipe ow.- .- Leray version of the Navier-Stokes pdes.- Appendix E: Semi-empirical treatment of simple wall-bounded ows.- Appendix F: Solutions to problems.- Bibliography.

Dr. Wolfgang Kollmann is Professor, Emeritus in the Mechanical and Aerospace Engineering Department, University of California, Davis.

Outlines fundamental difficulties and presents several approaches for their analysis and solution Emphasizes mathematical treatment of turbulent flows and methods for their computation Reinforces concepts presented with problems to illustrate the theory and to introduce particular examples of turbulent flows such as periodic pipe flow Includes several versions of the Hopf equation derived in spatial/Eulerian and material/Lagrangean description