Structure, scaling, and synthesis of proper orthogonal decomposition modes of inhomogeneous turbulence

By examining the proper orthogonal decomposition (POD) of turbulent channel flow and turbulent thermal convection in the inhomogeneous direction of each flow, it is found that the POD eigenfunctions separate into two types of modes having different structure and scaling. For these flows, the modes can be represented in the form of amplitude-modulated and phase-modulated oscillations in the wall-normal direction. The phase functions can be extracted by several means and they contain much of the physics embedded in the two-point spatial correlation function. The phase and envelope functions are analyzed to interpret POD mode behavior. The first several most energetic modes are large-scale and their structure is idiosyncratic relative to the rest of the modes and characteristic of the type of the flow. Outside this group of idiosyncratic modes, the remaining modes are less energetic, smaller scale, and asymptotically self-similar in the sense that the phases scaled using sequency correlate well and approach a single limiting curve with increasing order (mode number). This asymptotic phase function is characteristic of the type of turbulence, i.e., it differs between channel flow and thermal convection, despite it being small-scale and, therefore, presumably insensitive to the form of the boundary conditions. The work of Moser [“Kolmogorov inertial range spectra for inhomogeneous turbulence,” Phys. Fluids 6, 794 (1994)] provides an approximate relationship for the asymptotic phase function in terms of the distribution of viscous dissipation ϵ(y). A more general framework for this relationship is presented in which the idiosyncratic modes determine the energy distribution and the asymptotically self-similar modes determine the distribution of viscous dissipation. The foregoing results are exploited to develop a complete set of orthonormal basis functions that approximate the POD modes and possess convergence properties that are comparable to the optimal convergence of the exact POD and significantly better than those of conventional orthogonal polynomials. This procedure, called “synthetic POD,” uses only knowledge of the distribution of mean dissipation (or equivalently, the asymptotic phase function) without a priori computation of the more complicated two-point correlation functions.