INSTABILITY AND TRANSITIONS OF FLOW IN A CURVED SQUARE DUCT - THE DEVELOPMENT OF 2 PAIRS OF DEAN VORTICES

Steady developing flow of an incompressible Newtonian fluid in a curve d duct of square cross-section (the Dean problem) is investigated both experimentally and numerically. This study is a continuation of the w ork by Bara, Nandakumar & Masliyah (1992) and is focused on flow rates between Dn = 200 and Dn = 600 (Dn = Re/(R/a)(1/2), where Re is the Re ynolds number, R is the radius of curvature of the duct and a is the d uct dimension; the curvature ratio, R/a, is 15.1). Numerical simulatio ns based on the steady three-dimensional Navier-Stokes equations predi ct the development of a 6-cell secondary flow pattern above a Dean num ber of 350. The 6-cell state consists of two large Ekman vortices and two pairs of small Dean vortices near the outer wall that result from the primary instability that is of centrifugal nature. The 6-cell flow state develops near theta = 80 degrees and breaks down symmetrically into a 2-cell flow pattern. The apparatus used to verify the simulatio ns had a duct dimension of 1.27 cm and a streamwise length of 270 degr ees. At a Dean number of 453, different velocity profiles of the 6-cel l flow state at theta = 90 degrees and spanwise profiles of the stream wise velocity at every 20 degrees were measured using a laser-Doppler anemometer. All measured velocity profiles, as well as flow visualizat ion of secondary flow patterns, are in very good agreement with the si mulations, indicating that the parabolized Navier-Stokes equations giv e an accurate description of the flow. Based on the similarity with bo undary layer flow over a concave wall (the Gortler problem), it is sug gested that the transition to the 6-cell flow state is the result of a decreasing spanwise wavelength of the Dean vortices with increasing f low rate. A numerical stability analysis shows that the 6-cell flow st ate is unconditionally unstable. This is the first time that detailed experiments and simulations of the development of a 6-cell flow state are reported.