The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.

A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry

Part I presents the statistical theory of turbulence, and Part 2 the coherent structures in open-channel flows and boundary layers. The book is intended for advanced students and researchers in hydraulic research, fluid mechanics, environmental sciences and related disciplines. References Index.

Turbulence in open-channel flows

Life in Moving Fluids—The physical biology of flow

Bed shear stress is a fundamental variable in river studies to link ﬂow conditions to sediment transport. It is, however, difﬁcult to estimate this variable accurately, particularly in complex ﬂow ﬁelds. This study compares shear stress estimated from the log proﬁle, drag, Reynolds and turbulent kinetic energy (TKE) approaches in a laboratory ﬂume in a simple boundary layer, over plexiglas and over sand, and in a complex ﬂow ﬁeld around deﬂectors. Results show that in a simple boundary layer, the log proﬁle estimate is always the highest. Over plexiglas, the TKE estimate was the second largest with a value 30 per cent less than the log estimate. However, over sand, the TKE estimate did not show the expected increase in shear stress. In a simple boundary layer, the Reynolds shear stress seems the most appropriate method, particularly the extrapolated value at the bed obtained from a turbulent proﬁle. In a complex ﬂow ﬁeld around deﬂectors, the TKE method provided the best estimate of shear stress as it is not affected by local streamline variations and it takes into account the increased streamwise turbulent ﬂuctuations close to the deﬂectors. It is suggested that when single-point measurements are used to estimate shear stress, the instrument should be positioned close to 0·1 of the ﬂow depth, which corresponds to the peak value height in proﬁles of Reynolds and TKE shear stress. Copyright © 2004 John Wiley & Sons, Ltd.

Comparing Different Methods of Bed Shear Stress Estimates in Simple and Complex Flow Fields

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

Flow over a developed bottom topography in a bend has been investigated experimentally. The measuring section is in the outer-bank half of the cross section at 60° into the bend. Spatial distributions of the mean velocities, turbulent stresses, and mean-flow and turbulent kinetic energy are presented. The cross-sectional motion contains two cells of circulation: besides the classical helical motion (center-region cell), a weaker counterrotating cell (outer-bank cell) is observed in the corner formed by the outer bank and the water surface. The downstream velocity in the outer half-section is higher than the one in straight uniform flow; the core of maximum velocities is found close to the separation between both circulation cells, well below the water surface. The turbulence structure in a bend is different from that in a straight flow, most notably in a reduction of the turbulent activity toward the outer bank. Both the outer-bank cell and reduced turbulent activity have a protective effect on the outer ...

Mean Flow and Turbulence in Open-Channel Bend

[1] Phenomena concerning flow, morphology, and water quality in rivers are often investigated by means of a depth-integrated flow model, coupled to a sediment transport model and a water quality model. In such depth−integrated models, the vertical structure of the flow is represented by a closure submodel, which mainly has to account for the secondary circulation, which (1) redistributes by advection the flow, the boundary shear stresses, the sediment transport, and dissolved and suspended matter, (2) causes the direction of the bed shear stress to deviate from the direction of the depth-averaged velocity and thereby influences the bed topography, and (3) gives rise to additional friction losses as compared with straight uniform flow. The commonly used linear closure submodels are shown to fail in reproducing essential features in moderately to strongly curved flow, because they neglect the feedback between the downstream velocity and the secondary circulation. A nonlinear closure submodel taking this feedback into account is proposed and shown to yield results that compare well with experimental data, even for very strongly curved flow. The feedback effects turn out to be controlled almost exclusively by a single parameter, which enables their parameterization in a relatively simple way. This control parameter also helps to objectively distinguish weak, moderate, and strongly curved flows. The proposed closure submodel has the potential of improving the performance of depth-integrated flow-sediment-water quality models without much extra computational effort.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2003WR002068

Nonlinear modeling of mean flow redistribution in curved open channels

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

Confluences with relatively low discharge and momentum flux ratios where a small steep tributary with a high supply of poorly sorted sediment joins a large, low-gradient main channel commonly occur in nature, but they have not yet been investigated. Measurements of the three-dimensional velocity field, turbulence, sediment transport, bed material grain size and morphology are reported in a laboratory setting that is representative of confluences on the Upper Rhone River, Switzerland. The difference between the low-flow depth in the steep tributary and the higher flow depth in the main channel creates a marked bed discordance in the tributary zone. Due to this bed discordance, the tributary flow penetrates into the main channel mainly in the upper part of the water column, whereas the main-channel flow is hardly hindered by the tributary in the lower part of the water column, giving rise to a two-layer flow structure in the confluence zone. In confluences with high supply of coarse sediment from the tributary, the development of a deposition bar downstream from the confluence reduces the flow area and causes flow acceleration that contributes to an increase in sediment transport capacity. The sediment supplied by the tributary is mainly sorted and transported on the face of the bar by the near-bed flow originating from the main channel. The sediment transport capacity is further increased by the three-dimensionality of the flow, which is characterized by maximum velocities occurring near the bed, and by a considerable increase in turbulent kinetic energy generated in the shear layer at the interface of the flows originating from the main channel and the tributary. A conceptual model is proposed for the hydro-morpho-sedimentary processes, and compared to existing conceptual models for confluences with different characteristics.

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Koen Jacques Ferdinand Blanckaert

Papers

Secondary Flow in Sharp Open-channel Bends

Mean Flow and Turbulence in Open-Channel Bend

Nonlinear modeling of mean flow redistribution in curved open channels

Momentum Transport in Sharp Open-Channel Bends

Flow and sediment dynamics in channel confluences