A MODELING APPROACH TO TRANSITIONAL CHANNEL FLOW R. A. SAHAN, H. GUNES and A. LIAKOPOULOS Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015-3085, U.S.A. (Received 19 June 1995) AbstractÐLow-dimensional dynamical models of transitional ¯ow in a periodically grooved channel are numerically obtained. The governing partial dierential equations (continuity and Navier±Stokes equations) with appropriate boundary conditions are solved by a spectral element method for Reynolds number Re = 430. The method of empirical eigenfunctions (proper orthogonal decomposition) is then used to extract the most energetic velocity eigenmodes, enabling us to represent the velocity ®eld in an optimal way. The eigenfunctions enable us to identify the spatio-temporal (coherent) structures of the ¯ow as travelling waves, and to explain the related ¯ow dynamics. Using the computed eigenfunctions as basis functions in a truncated series representation of the velocity ®eld, low-dimensional models are obtained by Galerkin projection. The reduced systems, consisting of few non-linear ordinary dierential equations, are solved using a fourth-order Runge±Kutta method. It is found that the temporal evol- ution of the most energetic modes calculated using the reduced models are in good agreement with the full model results. For the modes of lesser energy, low-dimensional models predict typically slightly lar- ger amplitude oscillations than the full model. For the slightly supercritical ¯ow at hand, reduced models require at least four modes (capturing about 99% of the total ¯ow energy). This is the smallest set of modes capable of predicting stable, self-sustained oscillations with correct amplitude and fre- quency. POD-based low-dimensional dynamical models considerably reduce the computational time and power required to simulate transitional open ¯ow systems. # 1998 Elsevier Science Ltd. All rights reserved 1. Introduction 2. Full model equations 2.1. Formulation of the problem 2.2. Method of solution 3. Development of low-order models 4. Results and discussion 4.1. Eigenvalues 4.2. Eigenfunctions and spatio-temporal (coherent) structures 4.3. Reconstruction of the ¯ow ®eld variables and optimization of the mode retaining process 4.4. Low-dimensional model predictions 5. Concluding remarks Bibliographic List 1. INTRODUCTION A large number of engineering applications deal with ¯ow in periodically repeated con®gur- ations such as ¯ow over grooves and ¯ow in grooved channels [1±6]. The grooved channel geo- metry, shown in Fig. 1, is encountered in many low-speed applications such as cooling of electronic devices and circuit boards [5,6], and in ¯ow and heat transfer in compact heat exchan- gers [4]. In small size systems with moderate Reynolds number, Re, transport enhancement is important and is achieved by mixing due to hydrodynamic instabilities [6]. The grooved channel geometry represents one such con®guration characterized by wall bounded ¯ow with separation. The ¯uid ¯ow in the channel can be divided into two parts: the bypass and the groove regions (see Fig. 2). The ¯ow patterns in the bypass and groove regions dier greatly. Approximately parallel ¯ow structure is present in the bypass region which is separated from the recirculating zones within the grooved regions by shear layers [5,6]. The shear layer partitioning the ¯ow regions does not permit convective exchange of ¯uid. The combination of the above factors results in dierences in convective exchange of ¯uid between the regions. The overall ¯ow Computers & Fluids Vol. 27, No. 1, pp. 121±136, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0045-7930/98 $19.00 + 0.00 PII: S0045-7930(97)00016-9 121