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 Reу300 and proper orthogonal decomposition is used to extract the empirical eigenfunctions at Reϭ430, 750, 1050, and Prϭ0.71. 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 Reϭ430, 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 Prϭ7.1 is briefly discussed.
The present investigation deals with the developflU!nl of low-order represeniations of transitional free convection in a vertical channel with discrete heaters. The governing equations are sowed using a spectral element method. Proper orlhogollll1 decomposuion (POD) is applied to extract the most energetic eigenfunctions (and lhe related spatWlemporaJ structures) from time-dependen: numerical solutions of the full model equations at a Groshof number higher than the critical value. Using the computed eigenfunctions in a truncated series expansion, reconstruction of Ihe original flow and temperature fields is aehieved in an optimal way. It is found that almost aUthe flow and temperature fluctuation energy is captured by the first six eigenmodes. A low-dimensional set of nonlinear ordinary differential equations tha: describes the dynamics of lhe flow and temperature fields is also derived. It is found that low-order models based on retaining at least four eigenmodes for eaeh field predict stable, self-sustained oscillations with correct amplitude and frequency.
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