Lateral circulation in a stratified open channel on a 120° bend

Authors


Corresponding author: N. Williamson, School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia. (nicholas.williamson@sydney.edu.au)

Abstract

[1] We present direct numerical simulation results for both isothermal and density-stratified turbulent flow in an open channel into and around a 120° bend with a bulk Reynolds number of 7500 and Prandtl number of 1.5. The bend is sharp with a radius-to-channel breadth ratio of 1.5. The bulk Richardson number for the stratified flow is 2.4 based on overall channel depth. The gradient Richardson number (Rig) varies between 10 and 20 at the entrance to θ ≈ 60°, where θ is the angular location. Above θ ≈ 60 − 120, Rig≈ 1. In isothermal flow, the well-known helical flow structure is observed. In stratified conditions, the vertical variation in relative strength of the outward-directed baroclinic pressure gradient and the centrifugal acceleration leads to a more complex circulation structure. In the near bed region and immediately above the interface, the centrifugal acceleration is greater, driving flow radially inward, while just below the density interface the baroclinic pressure gradient is greater, leading to outward-directed flow. This produces a four layer circulation structure with potentially significant implications for sediment erosion and transport. Additionally, this produces a complex dynamic at the density interface where the shear orientation varies through approximately 200° over the mixing layer depth.

1. Introduction

[2] In rivers and estuarine channels, stable density stratification can occur as a result of surface heating or saline intrusions from either ground water or ocean inflow. The flow within the channels can be strongly modified by the presence of this density gradient, producing changes in the bottom shear stress, cross-stream circulation, and mixing behavior. Many of these behaviors are presently not well understood and limit the applicability of flow-forecasting models.

[3] One important question is how changes in stratification affect the channel flow around bends. On a channel bend, depth-varying streamwise velocity produces a depth-varying centrifugal force, which is relatively stronger than the opposing (inwards directed) barotropic pressure gradient (BTPG) near the channel base. This induces cross-stream secondary circulation, which redistributes the momentum laterally [Rozovskii, 1957; Blanckaert and Vriend, 2004; Blanckaert, 2010]. Stratification can increase vertical shear in the streamwise velocity profile and produce a similarly more pronounced variation in the centrifugal forcing and circulation strength. The lateral circulation in this case tends to raise the lower dense fluid toward the inside of the bend, setting up a gravity-driven, baroclinic restoring (outward-directed) force. The relative contribution of these forces together with bottom friction depends on flow and geometric conditions and is the subject of continuing research [Chant and Wilson, 1997; Seim and Gregg, 1997; Lacy and Monismith, 2001; Lacy et al., 2003; Chant and Wilson, 2002; Nidzieko et al., 2009]. A recent estuarine field study [Nidzieko et al., 2009] has shown that the interaction of the secondary circulation and the restoring baroclinic pressure gradient (BCPG) in stratified flow can lead to a more complex three layer circulation pattern. Furthermore, in meander bends, the streamwise variation in channel curvature produces a similar streamwise variation in centrifugal forcing. The response of the baroclinic restoring force can lag the changes in centrifugal forcing, producing a downstream adjustment and potentially a spanwise seiche [Chant and Wilson, 1997; Lacy and Monismith, 2001; Chant and Wilson, 2002; Nidzieko et al., 2009]. Asymmetric mixing caused by asymmetric bathymetry or other factors can also affect lateral circulation. Cheng et al. [2009] performed a numerical study of a real estuary channel, with actual estuary bathymetry and reported a dual circulation cell behavior after a bend under strongly stratified flow at maximum ebb. In this study, a lateral density gradient was observed to be produced by lateral asymmetry of mixing between the thalweg and the shoal. The resulting baroclinic restoring force contributed to this dual circulation behavior.

[4] A second question is how the more complex flow in a stratified channel bend affects mixing across a density interface. To date, there has been relatively little study of stratified shear layers subjected to vertically varying lateral forcing and shear. The streamwise variation in the large-scale motion in the channel suggests that the mixing behavior would be similarly affected by the change in forcing around the bend. At present, little is known about how the unsteady flow structure or mixing characteristics are affected by the dominant flow mechanisms and how they vary with spatial location and how it might affect modeling parameters such as entrainment rate or eddy viscosity.

[5] Our focus in this work is to examine how the large-scale flow is modified by stratification and how the mixing layer develops in these flows. We examine a relatively severe case where the bend is very sharp, the buoyancy forces are strong, and the stratification has a sharp two layer density profile. Such a flow might be expected where an adjoining water source provides an influx of cooler water or where saline water purging from a hypersaline deep river pool is flushed into the main channel. Such river pools are found in some conditions in inland Australian river systems [Kirkpatrick and Armfield, 2005; Gillam, 2010; Kirkpatrick et al., 2012]. The dissipation length and mixing behavior of these events are of significant interest to environmental managers of these river systems.

[6] Recently, numerical simulations of Navier-Stokes equations, using either direct numerical simulations (DNS) or large-eddy simulations (LES) with a subgrid scale model for the turbulence closure, have been performed for nonstratified open channel flow on straight channels [Pan and Banerjee, 1995] and on bends with regular [van Balen et al., 2009, 2010a] and irregular bathymetry [van Balen et al., 2010b; Constantinescu et al., 2011]. Comparisons of these simulations with laboratory-scale experiments have shown them to be effective in these types of flows.van Balen et al. [2009, 2010a] used LES of regular rectangular curved open channels to investigate the mechanisms that drive the outer bank circulation cell found in some curved channels. van Balen et al. [2009] found that the cell is a result of turbulence aniostropy and centrifugal forces, adding to the observations made in earlier experimental studies [Blanckaert and Vriend, 2004, 2005]. We are unaware of the use of DNS or LES simulations to examine these flows where density stratification is present.

[7] We use a DNS of fully turbulent strongly stratified flow in an idealized open-channel flow with rectangular cross section and a 120° bend. A second DNS of nonstratified flow is performed under the same conditions to provide a reference case for the flow. The results are used to describe how a shallow dense intrusion can affect the spatial variation in flow structure and the spatial variation in the forces which drive circulation in the radial momentum budget. We also examine the effect of lateral circulation on mixing behavior and the spatial variation in mixing behavior. Insection 2, the simulation and numerical method are described, in section 3, the mean flow behavior is described. In section 3.1, the dynamics of the mixing behavior are described, and in section 3.2, the mechanisms that drive lateral circulation are presented. In section 4, the conclusions are presented.

2. Governing Equations

[8] In our DNS, we solve the three-dimensional Navier-Stokes equations for an incompressible fluid with the Oberbeck-Boussinesq approximation for buoyancy. The scalar field is not solved for the nonstratified flow, which we refer to hereafter as isothermal flow. The equations for the conservation of mass, momentum, and energy can be written as

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and

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respectively. The governing nondimensional parameters are the bulk Reynolds number inline image, Prandtl number inline image, and the bulk Richardson number inline image, where inline image is the reduced gravity. The gravitational acceleration vector (aligned with the negative y direction) is given by g and β is the coefficient of thermal expansion. ν and α are the kinematic viscosity and scalar diffusivity of the fluid. The equations are made nondimensional by the bulk channel velocity Ub, the channel depth D, and the temperature difference at the inlet inline image, where inline image and inline image are the temperatures of the lower cold layer and upper hot layer, respectively. The resulting nondimensional velocity, length, time, pressure, and temperature are denoted by lowercase and given as inline image, inline image, inline image, inline image, and inline image. Dimensional values are uppercase.

[9] The equations are solved using ALE, an unstructured finite volume solver described in detail by Norris et al. [2011]. The code uses a cell-centered colocated storage arrangement for flow variables, with cell-face velocities calculated using the Rhie-Chow momentum interpolation. We use a structured orthogonal mesh that is cylindrical in the curved sections of the channel. The spatial derivatives are discretized using second-order central finite differences, except for the scalar advective term which uses second-order central finite differences with the ULTRA flux limiter. The flux limiter prevents regions of nonphysical negative temperature field occurrence, which can cause the solution to become unstable. The Adams-Bashforth time advancement scheme is used for the nonlinear terms and Crank-Nicolson for the time advancement of the diffusive terms [Ferziger and Perić, 2002]. The pressure correction equation is solved using a stabilized biconjugate gradient solver with an incomplete Cholesky factorization preconditioner [Ferziger and Perić, 2002]. The momentum and temperature equations are solved using a Jacobi solver. The code was benchmarked against published data sets for turbulent channel flow at inline image [Kozuka et al., 2009], and was found to accurately reproduce the mean field and Reynolds stresses at the same resolution as that of the reference data set.

[10] The simulation domain is an open rectangular channel with a short initial straight section of length inline image, followed by a 120° bend and then a second straight section of length inline image. The channel width is inline imageand the bend radius-to-width ratio is inline image, where the radius is taken from the center of the channel ( inline image). A schematic diagram is given in Figure 1.

Figure 1.

Schematic of the simulation domain.

[11] The upstream streamwise boundary is the inlet. Here, the velocity field is played back from a record obtained in a previous DNS simulation of periodic open straight channel flow with the same bulk Reynolds number. In this way, the flow entering the domain is a fully developed turbulent isothermal open-channel flow. At the inlet, the temperature field is set as the step function given inequation (4). The initial nondimensional thermal mixing layer thickness is therefore inline image.

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[12] The downstream streamwise boundary is “open,” where streamwise gradients in velocity components and temperature are set to zero. In the spanwise (r) direction, the domain is bounded by no-slip side walls. The bottom boundary (y= 0) is also a no-slip wall. The top boundary (y = 1) approximates a free surface with a slip wall condition, i.e., inline image, inline image and inline image.

[13] In streamwise (s), wall-normal (y), and spanwise (r) directions, the grid has inline image nodes with inline image and inline image. In the streamwise direction, inline image at inline image from the inlet to the start of the bend. On the bend inline image, where inline image is the angular location on the bend, the grid is linearly stretched between inline image over inline image with the transform to a cylindrical coordinate system. Following the bend, in the second straight section, the grid is expanded in the streamwise direction with a constant 5% growth rate.

[14] The region of interest in this study is the section on the bend. The primary purpose of the initial straight section (with inline image) is to allow the flow to adjust to the suddenly imposed density gradient and any effects of the downstream bend. The intention of the second straight region after the bend is to provide a buffer region, which ensures that the outlet does not interfere with the upstream flow.

[15] A time step of inline image was used with the Courant number ranging between inline image. Once the flow was fully developed, the simulations were advanced for a further nondimensional simulation time of t = 70, approximately six complete turnovers of fluid in the domain. Flow statistics were collected during this time at inline image intervals.

[16] The geometric ratios of inline image and inline image have been chosen to give a very sharp bend with strong helical streamwise circulation. The nominal bulk Richardson number of inline image is not high but we show later that the more relevant gradient Richardson number varies considerably through the bend, so both very stable regions and regions with strong mixing are observed. The bulk Reynolds number is given, the arbitrary value of Re = 7500, to ensure the flow is fully turbulent. We show in section 3 that the wall shear stress is high throughout both channel simulations. In most rivers and estuaries, the scalar diffusivity is low resulting in inline image for thermal diffusivity in water and inline image for salinity diffusivity. In this study, we have arbitrarily taken inline image to limit the resolution requirements of the simulations as the Batchelor scale inline image, where ηis the Kolmogorov length scale. For the stratified flow, calculations performed a posteriori show that in the near-wall region, over inline image, inline image. Over the remainder of the channel depth, inline image.

3. Results

[17] In this study, we primarily examine the flow in the bend and so use a cylindrical coordinate system with angular location θ, radial location r, and vertical location y. The velocity components are therefore inline image. In the bend, xs is also used to denote the distance from the start of the bend along the channel center at inline image. The start of the channel is then located at inline image, the start of the bend at inline image, the end of the bend at inline image, and the channel exit at inline image. In the stratified flow, we refer to the upper nonstratified region as the “overflow” and the more dense lower region as the “intrusion” hereafter.

[18] We examine the spatial development of the flow and the lateral circulation in both simulations in Figures 24. Here time-averaged quantities are plotted, which are denoted by an overbar. InFigure 2, contours of the time-averaged streamwise velocity field inline image are shown at locations around the bend over inline image with unscaled velocity vectors overlaid. The contour of inline image is also given as an indication of intrusion height. We refer to this reference height hereafter as inline image. Also shown on the figure are shaded contours of the mean streamwise component of vorticity ωθ, where

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Figure 2.

Contours of mean streamwise vorticity inline image (shading) together with velocity vectors at inline image (a, f), 30 (b, g), 60 (c, h), 90 (d, i), and 115 (e, j) degrees, where (a–e) is the stratified flow and (f–j) the isothermal flow result. Thin lines show inline image at 1.2, 1.0, and 0.8 as indicated on the plots. Thick line shows inline image. Velocity vectors are not to scale and show direction only.

[19] In Figure 3, the vertical profiles of inline image and inline image are shown at radial locations inline image between inline image. The intrusion depth at these locations is again indicated by the line of inline image. In Figure 4, the vertical profiles of inline image and inline image are shown at inline image at intervals around the bend with inline image plotted against θ location.

Figure 3.

Thin lines indicate vertical profiles of mean radial velocity inline image at inline image at inline image (a), 30 (b), 60 (c), 90 (d), and 115 (e) degrees for isothermal simulation (dashed line) and stratified flow (solid line). Thick line indicates inline image.

Figure 4.

Vertical profiles of mean streamwise velocity inline image for isothermal simulation (thin dashed line) and stratified flow (thin solid line) and inline image(dash-dotted line) at inline image and inline image. The overlaid thick line indicates inline image with respect to θ (on the lower abscissa).

[20] Figures 2a and 2f depict the stratified and isothermal flows, respectively, as they enter the bend at inline image. The density interface is nearly horizontal (Figure 2a), and in both cases, the radial velocity is redistributing streamwise velocity toward the inner wall of the bend (Figure 3a). At the base of the channel, the radial velocity of both flows is inward directed. A small circulation region appears near the outer side wall at the surface. There is little difference between the streamwise velocity profiles of the two flows at this location (Figure 4).

[21] At inline image (Figures 2b and 2g), differences emerge between the two flows. In the isothermal flow, a channel-centered lateral circulation cell covers most of the channel. The radial velocity is inward directed over most of the channel for inline image. Above inline image, the flow is outward directed, forming a two layer circulation cell denoted “A” in Figure 2g. Near the outer side wall at the surface, a second small circulation cell is observed with the opposite rotation, denoted “B” in the figure. This second cell, often termed the “outer-bank cell,” has been the subject of a number of recent studies in nonstratified flow [Blanckaert and Vriend, 2004, 2005; van Balen et al., 2009]. van Balen et al. [2009] showed it to be a result of the interaction between the spatial distribution of turbulence stresses and centrifugal forcing. The same outer circulation cell is also observed in the stratified flow, with the radial velocity profiles of the two flows closely aligned over this region ( inline image at inline image; Figures 3a–3c).

[22] The streamwise velocity is redistributed further toward the inner side wall in both cases, with the shift being more pronounced in the stratified flow. The radial velocity is inward directed at the near-bed region in both the isothermal flow and stratified flow but this flow is much weaker in the stratified flow ( inline image in Figure 3b). This lateral motion raises the density interface toward the inner side wall. It is clear from the contours of vorticity in Figure 2b that in the stratified flow there is a region of opposing ( inline image) circulation through the interface.

[23] This behavior is clearer at inline image, in the contour plots in Figure 2c, and in the velocity profiles in Figure 3c. At the base of the channel there is a thin region, inline image, where the radial velocity is inward directed in both simulations. In this region, the isothermal velocity peak is nearly five times that of the maximum in the stratified flow. In the stratified flow between inline image and the density interface, there is a region of outward flow. The stratified and isothermal flows are both inward directed immediately above the interface and then outward directed near the surface. In this way, in the stratified flow, two circulation cells have developed in the center of the channel with the same rotation. Separating them is a thin layer in the interface with the opposite circulation sense. These cells are strong with a maximum velocity of inline image at inline image.

[24] From inline image, the outer bank circulation cells become increasingly prominent in both the stratified and isothermal flows. At inline image and inline image, the circulation cell extends over half of the cavity width in the stratified flow (Figures 2d and 2e). In the isothermal flow, the cell extends down to inline image compared with inline image for the stratified flow.

[25] In isothermal flows, Blanckaert and Graf [2004] showed that advective transport by both the outer bank cell and the main channel circulation cell has the effect of reducing the streamwise velocity near the surface and accelerating it toward the lower region of the channel. This effect can be seen in Figure 4, where the streamwise velocity profiles are flattened in both stratified and isothermal flows. In the isothermal flow, the location of maximum velocity occurs in the lower half of the channel for inline image (Figures 2h–2j), as seen by Blanckaert and Graf [2004] and Blanckaert and Vriend [2004].

[26] From inline image, the streamwise velocity is redistributed radially across the channel in both flows as seen in the contours of inline image in Figures 2e and 2d and 2i and 2j. In the stratified flow, the intrusion flows radially outward, and the density interface is almost horizontal as it leaves the bend.

[27] The isothermal simulation presented in this study is qualitatively similar to the results presented in other laboratory studies. van Balen et al. [2010a] examined regular open channel flow on a 193° bend with inline image. They examined three ratios of width to depth W/D, 12.0, 8.2, and 6.3. Increasing the depth (decreasing W/D ratio) was found to increase the strength of both the channel center circulation cell and the outer bank cell over the start of the channel. In the present study, a similarly strong curvature is used ( inline image), but our depth ratio is less than half that of the deepest case in van Balen et al. [2010a] at inline image. The prominence of the outer-bank cell in both the stratified and isothermal flows, as seen by the vorticity plots inFigure 2, could then be attributed to the small aspect ratio we examine. In Figure 5, we plot the mean circulation Γ, for the circulation cells observed in both the stratified and isothermal flow, where Γ is calculated as

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where Acis the cross-sectional area of the cells. The circulation in the outer bank cells increases from inline image and is almost constant over inline image for both stratified and isothermal flows, whereas van Balen et al. [2010a]found that the circulation strength of the outer-bank cell reaches its peak at approximately inline image for their cases with W/D 8.2 and 6.3. The work of van Balen et al. [2010a]suggests that these structures will be less significant in full-scale river or estuaries whereW/D is inline image. Figure 5shows that the channel-centered circulation cell is more strongly affected by stratification with the circulation strength reduced compared with the isothermal flow. The circulation strength of the cell at the density interface ( inline image region) is also plotted on Figure 5 together with the circulation strength of the cell in the intrusion ( inline image region). The circulation of these cells increases over inline image, reaches a maximum value between inline image, and then is reduced toward the bend exit. We recall this result in section 3.2.

Figure 5.

Mean streamwise circulation Γ of the outer bank cell (solid line), the main channel cell (dashed line), the cell in the density interface (dashed dotted line), and the cell in the lower intrusion (dotted line), shown with angular location. Thick lines indicate isothermal flow and thin lines indicate stratified flow.

[28] The estuarine field study of Nidzieko et al. [2009] provides a benchmark for the stratified flow simulation. In that study, the authors reported a three layered circulation structure in an estuary bend under conditions of strong stratification. Above the pycnocline, the authors observed the same two layer circulation behaviour, as noted in our stratified simulation. Below the pycnocline, they observed the third layer in which the radial velocities were oriented in the same direction as those at the surface. In our study, we observe a fourth layer where the lateral velocity reverses again and is oriented toward the inside of the bend, the same direction as that for nonstratified flow at the channel bed.

[29] The fourth layer we observe is thin, existing over inline image, and the mean radial velocity in this layer is also very weak, particularly at the exit of the bend ( inline image). In the field conditions reported by Nidzieko et al. [2009], this corresponds to several tens of centimeters above the bed. The lowest measurement point in that study was 1 m above the bed, raising the possibility that such a region was not observed because of its small extent. (We are grateful to N.J. Nidzieko for making this point.) The consistent observations of the mean circulation behavior in our numerical simulations and the field study over the region where both measurements coincide, notwithstanding the variation in bathymetry and scale of the two flows, supports this argument.

[30] This result has important implications for sediment transport and entrainment behavior. It implies that in stratified flow, near the bed, the lateral component of the erosion forces act in the same direction as for nonstratified flow, but with reduced strength. The ultimate sediment transport balance across the channel is a more complex issue as the radial flow just below the interface is outward directed; however, this is beyond the scope of this study. In Figure 6, we plot the contours of mean bottom wall shear stress inline image, which includes both radial and streamwise components and the associated vector scaled with magnitude. Figure 6 underlines the observation that throughout the channel bend, in both the stratified (Figure 6a) and isothermal flows (Figure 6b), the wall shear stress acts toward the inner side wall. The magnitude of the shear stress is higher, and the radial component is more significant in the isothermal flow. In the stratified flow, the variation from straight open channel flow through the bend produces very little modification to the mean wall shear stress. The friction Reynolds number inline image, based on friction velocity inline image, varies between inline image, high enough to ensure the flow is turbulent through the length of the channel.

Figure 6.

Contours of mean shear on the channel bottom wall inline image, including both streamwise and radial components, for (a) stratified flow and (b) isothermal flow. Scaled vectors of shear are overlaid. Scale is given in the figures.

[31] We also note at this point that a related observation is found in the numerical study by Cheng et al. [2009], where it was reported that a four layer circulation cell was observed, similar to that reported here, under strongly stratified flow at maximum ebb. The lateral density gradient in this case was produced by lateral asymmetry of mixing between the thalweg and the shoal. This suggests that while the geometry considered in the present study is idealized, the strong vertical variation in lateral velocity and the accompanying variation in shear observed here may be generated by other mechanisms than those documented in the present study.

3.1. Interface Dynamics

[32] The mixing behavior at the density interface in the stratified flow is complex owing to the streamwise development of the flow and variation in shear magnitude and orientation vertically through the interface. To delineate between these regimes, we use the local gradient Richardson number, Rig given here as

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[33] In Figure 7, Rig is plotted with height at five locations between inline image at inline image. On the same plots, the mean temperature inline image is given together with the orientation of mean shear, γ, with respect to the streamwise direction, where

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Figure 7.

Vertical profiles Rig (solid line), γin degrees (dash-dotted line), and inline image (dashed line) at inline image, at inline image. Arrows indicate plot θ locations on the lower axis. The overlaid thick solid horizontal line indicates inline image with respect to θ (on the lower abscissa).

[34] The vertical location inline image with respect to θ is overlaid on the plots for reference. At the inlet and in the bend before inline image, inline image so the radial shear is small. Over this initial developing region, the interface is sharp and Rig is very large, so the stratification is very stable. Over inline image, the lateral circulation cells affect the mixing layer, and inline image varies strongly with y as shown in Figure 3b–3e. In the lower and upper region of the mixing layer, the radial component of shear is oriented toward the outer side wall, with inline image. In the center region, near inline image the orientation is toward the inner side wall with inline image. Between the center layer and the two outer layers, γ has an orientation of 0°. Here inline image (see Figure 3), so shear is due only to the streamwise component. At these locations, there are local peaks in Rig.

[35] These peaks in Rig coincide with the extremities of the mixing layer where inline image and inline image. Between the locally high values of Rig is a local minimum point of Rig, which also approximately coincides with inline image and where shear has its maximum orientation toward the inner side wall, i.e., a highly negative value of γ. Vertically through this mixing layer, γ undergoes a total change in orientation of 200°. At inline image, inline image; just above this at inline image inline image while just above this at inline image, the orientation is back to inline image. This behavior is observed through the length of the cavity for inline image.

[36] In Figure 8, Rig is plotted together with γ, against streamwise location, at the height inline image and radius inline image. The gradient Richardson number is shown with and without the radial component included. The radial component of the shear is significant and greater than the streamwise shear component through most of the bend. The angle γ given in Figure 8 indicates the shear is oriented toward the inner wall from inline image to the exit at inline image. From inline image ( inline image) inline image at the location of inline image.

Figure 8.

Rig (Solid line) with streamwise distance at inline image and inline image. The dashed line indicates Rig without the radial velocity gradient included. Insert plot is expanded view of values at inline image. The dash-dotted line indicates the angle of shear,γ, shown on the right axis in degrees.

[37] Strang and Fernando [2001] used extensive experimental results to define regimes of mixing behavior as a function of Rigfor density-stratified flow. For inline image, the authors reported that entrainment through the density interface is dominated by Kelvin-Helmholtz (K-H hereafter) billows. Over inline image, both K-H and asymmetric wave instabilities were reported to exist and over inline image, asymmetric waves dominate. The critical gradient Richardson number above which K-H instabilities do not occur is inline image. At the center of the density interface, inline image, inline image and the flow is in the stable mixing regime until inline image ( inline image), after which the reduced Rigindicates the flow is susceptible to K-H billows.

[38] The effect of the variation in Rig through the bend can be seen in Figure 9, where the instantaneous flow field is shown at inline image,and 90° for one instant in time. In Figure 9, the velocity vectors are given in each plot and the shading indicates φ magnitude. The contour lines for inline image, and 0.95 are shown to visualize the interface deformation.

Figure 9.

Instantaneous flow visualization at inline image. Thick line indicates inline image.

[39] At inline image (with inline image), the overturns in the density interface are not seen. In the right of Figure 9a, large eddies in the overflow can be seen impinging on the interface and advecting the diffuse fluid from the upper part of the mixing layer into the overflow. At inline image inline image at inline image. Intermittent oscillations move through the interface, as can be seen in the contours of φ. At inline image, with the same mean gradient Richardson number as at inline image, overturns in the density interface observed in the interface at the location inline image and inline image, as shown in Figure 9c.

[40] In Figure 10, the instantaneous Rig, ur, and φ profiles are shown at two locations at inline image. In Figure 10a, the profile coincides with an overturn in the density interface at inline image and inline image ( inline image), as indicated in Figure 9c by the line A. At this location, indicated by the arrow in Figure 10a, Rig is negative and the shear is oriented toward the outer side wall. The line in Figure 10b coincides with the line B in Figure 9c. Here there is a density overturn at the center of the density interface at inline image ( inline image), with the shear oriented toward the inner side wall. In both locations, the inflection in the velocity profile and characteristic rolling up of the density interface is indicative of K-H billows [Strang and Fernando, 2001; Klaassen and Peltier, 1989]. In Figures 10a and 10b, the time-average values forRig are overlaid. At inline image, the Rig indicates that the flow is very stable in the mean sense with inline imageso the presence of K-H billows is unexpected but can be explained by local transient reductions inRig, as seen in Figure 10a.

Figure 10.

The instantaneous (black lines) and mean (red lines) radial velocity (thin solid lines), Rig (dashed lines) and φ at (a) inline image and (b) inline image at inline image with reference to locations “A” and “B” indicated in Figure 9c.

3.2. Radial Momentum Budget

[41] The radial momentum budget is used here to demonstrate how the dominant mechanisms that drive the flow vary with θ in the stratified flow. The budget terms are written here as

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[42] The first set of terms in the balance are the acceleration terms, which act to redistribute radial momentum in the channel. The radial (Ar) and vertical (Ay) acceleration terms are generally reported to be smaller than the streamwise ( inline image) acceleration term. The difference between the total radial pressure gradient inline image and the BCPG is referred to as BTPG. The BCPG is set up in the intrusion if it is disturbed and can be seen as a restoring force. The BTPG opposes the centrifugal acceleration (CA) in the bend. The barotropic pressure and the centrifugal acceleration terms are the dominant terms through most of the channel, but their sum has the same order of magnitude as the other leading terms in the balance. We primarily refer to the combination of these terms hereafter, the logic being that it is the difference between these terms which drives circulation. We refer to this sum as the “reduced centrifugal force” (RCA), where inline image. The turbulent diffusion terms (TS) and the viscous stress (VS) are frictional or drag terms which oppose motion. The dominant terms in this balance are plotted in Figures 11a and 11b for the stratified and isothermal flows.

Figure 11.

Vertical profiles of the dominant radial momentum budget terms at inline image at inline image for (a) stratified flow and (b) isothermal flow. The balance terms are, with reference to equation (9): VS (thin solid line), BCPG (thick solid line), inline image (thick dashed line), inline image(thick dash-dotted line), andAr(thin dash-dotted line). TS andAy are small so omitted. In (a), the thick horizontal line indicates the depth where inline image.

[43] In the overflow region of the stratified flow simulation, as indicated by the region above inline image in Figure 11a, the flow is dominated by an inviscid balance with inline image. At the entrance and exit of the bend, the streamwise variation of lateral momentum is large as indicated by the inline image term. The overflow is accelerated outward, as indicated by inline image, from inline image and again from inline image but inward from inline image.

[44] In the intrusion, the BCPG, the streamwise advective term, and the reduced centrifugal acceleration term are the leading order terms. Over most of the intrusion depth the BCPG term is larger than RCA for inline image and from inline image so the inline image term acts to accelerate the intrusion toward the outside wall. Between inline image, RCA is greater.

[45] This result indicates that the overflow quickly responds to the bend, rapidly increasing radial momentum downstream before and after the bend, while the intrusion responds more slowly once the baroclinic restoring force is set up, as suggested by Nidzieko et al. [2009]. At the end of the bend, RCAgoes to zero and the baroclinic term is still high. This may lead to a downstream adjustment with a gravity-driven seiche in the intrusion that has been observed elsewhere [Chant and Wilson, 1997; Lacy and Monismith, 2001; Chant and Wilson, 2002; Nidzieko et al., 2009].

[46] The isothermal flow behaves similarly to the overflow in the stratified simulation with inline image significant from inline image and inline image and balanced by RCA. The development of the circulation through the channel can be inferred from differences in the adjustment inline image over the height of the channel. At the entrance to the bend in both stratified and isothermal flows ( inline image in Figures 11a and 11b), the vertical variation in the reduced centrifugal force RCA, produced by a vertical variation in streamwise velocity profile results in an increased acceleration ( inline imagemore negative) in the upper half of the domain, resulting in the well-known two layer circulation cell observed in isothermal flow. This can also be seen in the increase in circulation of the main channel cell, as shown inFigure 5 over inline image.

[47] In the stratified flow over inline image, the inline imageterm is relatively weak in the upper region of the overflow and in the near-bed region, but it is significant in the mixing layer, where it balances the BCPG and RCA terms. This indicates that there is a change in circulation downstream driven by the vertical variation of these terms across the interface. In the lower mixing layer, BCPG is larger, while in the upper mixing layer, RCA is larger. The circulation in the lower cell is therefore aided by an excess of the BCPG term over the RCA term and circulation in the overflow is aided by an excess of the RCAover the BCPG term. In the near-bed region, the baroclinic term is consistently smaller than RCAthrough the length of the channel, leading to a thin layer of inward-directed flow. The presence of the baroclinic term does however reduce the net driving force compared with the isothermal flow where this term is not present, and RCA is balanced by increased shear as indicated by VS (Figure 11b). The result of this is the much lower radial velocities observed near the bed in the stratified flow compared with the isothermal flow, as shown in Figures 3b–3e, and the reduced shear stress shown in Figure 6.

4. Conclusions

[48] DNS of isothermal and strongly stratified open channel flow on a sharp 120° bend have been used to examine the effect of stratification on flow structure and development. Stratified flow enters the bend with a similar structure to the isothermal flow but the initial lateral flow raises the density interface at the inside of the bend and sets up the restoring BCPG. The overflow quickly responds to the bend, rapidly increasing radial momentum downstream before and after the bend, while the intrusion responds more slowly once the baroclinic restoring force is set up, as suggested in Nidzieko et al. [2009]. At the exit of the bend, the BCPG is still high suggesting a further downstream adjustment with a gravity-driven seiche in the intrusion, as observed byChant and Wilson [1997], Lacy and Monismith [2001], Chant and Wilson [2002], and Nidzieko et al. [2009].

[49] The baroclinic force and the centrifugal force are in competition over the height of the channel as reported by Nidzieko et al. [2009]. In the upper mixing layer, the centrifugal force is greater, driving inward flow, whereas in the lower mixing layer, the BCPG is stronger than the centrifugal forcing, generating a net outward force. This is consistent with the three layered structure observed by Nidzieko et al. [2009]in a stratified estuary. In the present work, we observe an additional thin layer in the near-bed region where the BCPG is smaller than the centrifugal force producing a net inward force and flow. The additional flow reversal creates a four layer lateral circulation flow structure. This novel observation has potentially significant implications for sediment erosion and lateral transport across the channel, should it be confirmed in the field.

[50] The wall shear stress at the bed was shown to be oriented toward the inner wall throughout the channel in both stratified and isothermal flows. The radial component and overall magnitude of this shear is significantly reduced in the stratified case, where it is opposed by the outward-directed BCPG.

[51] The magnitude and orientation of shear varies sharply across the mixing layer. At the bottom and at the top of the mixing layer, the vertical shear is oriented toward the outer side wall, with γ, the angle of shear with respect to the streamwise coordinate, approximately equal to 65. In the center of the mixing layer, it is oriented toward the inner wall with inline image. The mixing layer in this study is initially sharp with a stable Richardson number. After inline image the Richardson number decreases to inline image, and an interesting mixing dynamic is observed. KH billows are observed in the upper mixing layer and through the center of the layer and have the same rotational sense as the shear at these locations. The result suggests that simple parameterizations of entrainment or flow stability, based only on shear in the streamwise velocity profile, may miss this very important dynamic in the flow.

Acknowledgments

[52] The authors gratefully acknowledge the support of the Victorian Environmental Protection Authority and the Australian Research Council (ARC). The first author was supported by ARC postdoctoral research fellowship DP110103417.