Laminar flow in a square duct of strong curvature

Abstract

Calculated values of the three velocity components and measured values of the longitudinal component are reported for the flow of water in a 90° bend of 40 x 40mm cross-section; the bend had a mean radius of 92mm and was located downstream of a 1[sdot ]8m and upstream of a 1[sdot ]2m straight section. The experiments were carried out at a Reynolds number, based on the hydraulic diameter and bulk velocity, of 790 (corresponding to a Dean number of 368). Flow visualization was used to identify qualitatively the characteristics of the flow and laser-Doppler anemometry to quantify the velocity field. The results confirm and quantify that the location of maximum velocity moves from the centre of the duct towards the outer wall and, in the 90° plane, is located around 85% of the duct width from the inner wall. Secondary velocities up to 65% of the bulk longitudinal velocity were calculated and small regions of recirculation, close to the outer corners of the duct and in the upstream region, were also observed.The calculated results were obtained by solving the Navier–Stokes equations in cylindrical co-ordinates. They are shown to exhibit the same trends as the experiments and to be in reasonable quantitative agreement even though the number of node points used to discretize the flow for the finite-difference solution of the differential equations was limited by available computer time and storage. The region of recirculation observed experimentally is confirmed by the calculations. The magnitude of the various terms in the equations is examined to determine the extent to which the details of the flow can be represented by reduced forms of the Navier–Stokes equations. The implications of the use of so-called ‘partially parabolic’ equations and of potential- and rotational-flow analysis of an ideal fluid are quantified.

Numerical computation of buoyancy-induced recirculation in curved square duct laminar flow

“…The calculation scheme and its testing have already been described in detail in [3,4]. Its extension and validation for predicting forced convection heat transfer in curved duct flows may be found in [5].…”

“…Solution of the system of finite difference transport equations with appropriately differenced boundary conditions is achieved by means of a cyclic series of predictor-corrector operations as described in [3,4]. Briefly, the method .…”

“…A review of experimental, analytical and numerical studies up to 1975 is given in [1]. Examples of analytical and numerical· studies for the laminar floW regime are'given ;n, [2][3][4][5][6], ,and for the turbulent flow regime (using two-equation turbulence models) i,n [7][8][9]. While the laminar flow cases have yielded to numerical prediction and are currently limited mainly by cost considerations dictated by computational time and storage requirements, calculations of corresponding Motions driven by buoyant forces arise ducts with curvature the criterion is given turbulent flows are less accurate [9].…”

“…In principle, the accuracy of such computations in the laminar flow regime for an incompressible fluid are limited only by the nature of the equations solved (parabolic, semi-elliptic or elliptic)l and the error incurred lFor flows in ducts of mild curvature wherein longitudinal and cross-stream pressure variation can be decoupled, calculation schemes based on parabolic forms of transport equations [6] (boundary layer equations) may be used. For stronger curvature it is necessary to account more exactly for ellipticity in the pressure field [12], or resort to semi-elliptic or fully elliptic calculation schemes which allow for the direct determination of pressure [3][4][5]8, 9]~-.…”

“…The choice of conditfons was dictated by the availability in [3]of corresponding measurements and calculations of this flow in the absence of thermal effects.…”