Near-Wall Reconstruction of Higher-Order Moments and Length Scales using the POD

Abstract

An analysis of the near-wall behavior of the proper orthogonal decomposition (POD) eigenfunctions (see Lumley 1967) derived from direct numerical simulation (DNS) of channel flow of Dinavahi and Zang (1993) is performed. This work is driven in part by a desire to include structure in turbulence models. Although this particular application is in the near wall region (because of availability of the data base), the ideas discussed here are applicable in free shear flows as well. Consistent with previous studies, a low order multi-mode reconstruction of the kinetic energy and Reynolds shear stress suffices. A similar reconstruction of the isotropic dissipation rate is shown to be insufficient, however. An analysis is performed of the multi-mode composition of the dissipation rate in the near-wall region, and it is shown that a significant number of higher-order modes are required to achieve the correct asymptotic consistency in the near-wall region. In an attempt to avoid this problem, a length scale definition is proposed in terms of an integration of the two-point correlation tensor which factors in the presence of the wall. The wall is accounted for by only integrating out to 2y⁺and not over the entire domain. Viscous and inviscid estimates for the dissipation rate are used in the near-wall and core regions, respectively, in conjunction with this length scale representation, to obtain an estimate of the dissipation rate throughout the domain. The resulting dissipation rate exhibits the proper behavior near the wall and in the inertial layer. Higher-order POD mode estimates of the length scale are computed and compared to that computed using the full correlation tensor from the DNS results of Dinavahi and Zang (1993).