Secondary currents are a characteristic feature of flow in open-channel bends. Besides the classical helical motion (centre-region cell), a weaker and smaller counter-rotating circulation cell (outer-bank cell) is often observed near the outer bank, which is believed to play an important role in bank erosion processes. The mechanisms underlying the circulation cells, especially the outer-bank cell, are still poorly understood, and their numerical simulation still poses problems, not least due to lack of detailed experimental data. The research reported herein provides detailed experimental data on both circulation cells in an open-channel bend such as found in nature. Furthermore, the underlying dynamics are investigated by simultaneously analysing the vorticity equation and the kinetic energy transfer between the mean flow and the turbulence. This shows that turbulence plays a minor role in the generation of the centre-region cell, which is mainly due to the centrifugal force. By accounting for the feedback between the downstream velocity profile and the centre-region cell, a strongly simplified vorticity balance is shown to yield accurate predictions of the velocities in the centre region. For strong curvatures, however, a fully threedimensional flow description is required. Due to the non-monotonic velocity profiles, the centrifugal force favours the outer-bank cell. Moreover, terms related to the anisotropy of the cross-stream turbulence, induced by boundary proximity, are of the same order of magnitude and mainly enhance the outer-bank cell. Both mechanisms strengthen each other. The occurrence of the outer-bank cell is shown to be not just due to flow instability, like in the case of curved laminar flow, but also to kinetic energy input from turbulence.

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Secondary Flow in Sharp Open-channel Bends

Numerical path following

Flow in open-channel bends is characterized by cross-stream circulation, which redistributes the velocity and the boundary shear stress and thereby shapes the characteristic bed topography. Besides a center-region cell, classical helical motion, a weaker counterrotating outer-bank cell often exists. In spite of its engineering importance, the mechanisms underlying distributions of the velocity and the boundary shear stress in open-channel bends, and especially the role of both circulation cells, are not yet fully understood. In order to investigate these mechanisms, an evaluation is made of the various terms in the momentum equations based on the data measured, which gave the following results. The outer-bank cell forms a buffer layer that protects the outer bank from any influence of the center-region cell and keeps the core of maximum velocity a distance from the bank. Advective momentum transport by the center-region cell is a dominant mechanism; it significantly contributes to the observed outward shift of the downstream velocity and the bed shear stress and to flattening of the vertical profiles of the velocity. This important advective momentum redistribution has to be included in the depth-integrated flow models often used in engineering practice. Commonly used linear models overpredict the effects of the center-region cell. Based on results of the analysis of experimental data, these models are extended by accounting for the feedback between the center-region cell and the downstream velocity. The nonlinear model obtained clearly reveals the mechanisms of the center-region cell and its advective momentum transport. An analysis of nonlinear model results confirms and complements the analysis of experimental data. A true quasithree-dimensional flow model is obtained by coupling this nonlinear model to depth-integrated flow models, thus providing an engineering tool for morphodynamical investigations.

Momentum Transport in Sharp Open-Channel Bends

An experimental and numerical study investigating the flow development and fully developed flows of an incompressible Newtonian fluid in a curved duct of square cross section with a curvature ratio of 15.1 is presented. Numerical simulations of flow development from a specified inlet profile were performed using a parabolized form of the steady three-dimensional Navier-Stokes equations. No symmetry conditions were imposed. In general there was good agreement between the numerical predictions of the developing axial velocity profiles and LDV measurements. In addition, for computational expediency, the two-dimensional solution structure was calculated by imposing fully developed conditions together with symmetry conditions along the horizontal duct centreline.Laser-Doppler measurements of axial velocity and flow visualization at Dean number Dn = 125, 137 and 150, revealed a steady and symmetric two-vortex flow at Dn = 125, and a steady and symmetric four-vortex flow at both Dn = 137 and 150 (Dn = Re/(R/a)½, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension). Axial velocity measurements showed that the four-vortex flow at Dn = 150 developed to the solution predicted by the two-dimensional numerical simulation. However, the four-vortex flow at Dn = 137 was still developing when the flow had reached the end of the 240° axial length of the duct. A numerical investigation for Dean numbers in the range of 50 to 175 revealed that at the limit point of the two-cell to four-cell transition the development length appeared to be infinite, and thereafter decreased for increasing Dean numbers. The behaviour of decreasing development length of the four-vortex flow with increasing Dean number has not been reported previously.Using a symmetrically positioned pin at θ = 5° to induce the four-cell flows, the two-dimensional solution structure for Dn [les ] 150 was experimentally observed for the first time. Experiments were consistent with the prediction by Winters (1987) that four-vortex flows are stable to symmetric perturbations, but unstable to asymmetric perturbations. Experimental and numerical investigations suggested that, when perturbed asymmetrically, the four-vortex flow might evolve to flows with sustained spatial oscillations farther downstream.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112092003100

An experimental and numerical study of the Dean problem : flow development towards two-dimensional multiple solutions

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

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The numerical analysis of bifurcation problems with application to fluid mechanics

The spiralling flow within a curved tube of rectangular cross-section normally forms two distinct cells in the plane of the cross-section, but a transition to a different, four-cell flow is known to occur at a critical value of the axial pressure gradient. This paper shows that, for a square cross-section, the transition is a result of a complex structure of multiple, symmetric and asymmetric solutions. The structure is revealed by solving extended systems of equations for steady-state, fully developed, laminar flow, in a finite-element approximation, to locate exactly the positions of singular points. Continuation methods are used to trace the paths of these points as the aspect ratio and radius of curvature vary. For a square cross-section, multiple, symmetric flows of both two and four cells are found in two distinct ranges of axial pressure gradient q. In addition, a pair of asymmetric solutions arise from a symmetry-breaking bifurcation point on the primary flow branch. The paths of limit points and symmetry-breaking bifurcation points are obtained as the aspect ratio γ varies, and a transcritical bifurcation point is found at γ = 1.43. For larger aspect ratios the secondary four-cell branch is disconnected from that of the primary two-cell flow. The bifurcation set in the two parameters q and γ has a complex structure, with a number of higher-order singularities; in particular, the path of limit points is found to cross in the manner of an unfolded swallowtail catastrophe. A stability analysis shows that all multiple solutions except the two-cell type are unstable with respect to either symmetric or antisymmetric perturbations. For values of the aspect ratio less than 1.43 there is a range of axial pressure gradients for which there is no stable flow. The critical Dean numbers of the singular points are found to vary strongly at small values of radius of curvature β, but the bifurcation set remains qualitatively the same. Previous work is interpreted in the light of the present results.

Keith H. Winters

Papers

A bifurcation study of laminar flow in a curved tube of rectangular cross- section