Reduced dynamical models of nonisothermal transitional grooved-channel flow

Abstract

Reduced dynamical models are derived for transitional flow and heat transfer in a periodically grooved channel. The full governing partial differential equations are solved by a spectral element method. Spontaneously oscillatory solutions are computed for Reynolds number Re⩾300 and proper orthogonal decomposition is used to extract the empirical eigenfunctions at Re=430, 750, 1050, and Pr=0.71. In each case, the organized spatio-temporal structures of the thermofluid system are identified, and their dependence on Reynolds number is discussed. Low-dimensional models are obtained for Re=430, 750, and 1050 using the computed empirical eigenfunctions as basis functions and applying Galerkin’s method. At least four eigenmodes for each field variable are required to predict stable, self-sustained oscillations of correct amplitude at “design” conditions. Retaining more than six eigenmodes may reduce the accuracy of the low-order models due to noise introduced by the low-energy high order eigenmodes. The low-order models successfully describe the dynamical characteristics of the flow for Re close to the design conditions. Far from the design conditions, the reduced models predict quasi-periodic or period-doubling routes to chaos as Re is increased. The case Pr=7.1 is briefly discussed.