Assessing the predictive capabilities of isotropic, eddy viscosity Reynolds-averaged turbulence models in a natural-like meandering channel

Authors

  • Seokkoo Kang,

    1. St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Twin Cities,Minneapolis, Minnesota,USA
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  • Fotis Sotiropoulos

    Corresponding author
    1. St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Twin Cities,Minneapolis, Minnesota,USA
      Corresponding author: F. Sotiropoulos, St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Twin Cities, 2 Third Ave. SE, Minneapolis, MN 55414, USA. (fotis@umn.edu)
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Corresponding author: F. Sotiropoulos, St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Twin Cities, 2 Third Ave. SE, Minneapolis, MN 55414, USA. (fotis@umn.edu)

Abstract

[1] The predictive capabilities of an isotropic, eddy viscosity turbulence model for closing the unsteady Reynolds-averaged Navier-Stokes (RANS) equations are systematically investigated by simulating turbulent flow through a field-scale meandering channel and comparing the computed results with the large-eddy simulation (LES) of the same flow recently reported by Kang and Sotiropoulos (2011). To facilitate the comparison of the two turbulence models, both RANS simulation and LES are carried on exactly the same grid with the same numerical method. The comparisons show that while the RANS model captures the curvature-driven secondary flow within the bend, it fails completely to predict other key flow features in the channel, which are predicted by the LES and also observed in flow visualization experiments. These features include the inner and outer bank shear layers, the outer bank secondary cell, and the inner bank horizontal recirculation zone. By analyzing the results of the LES, we conclusively show that flow features not predicted by the RANS calculation are located in regions of the flow with high levels of turbulence anisotropy. The extent of these regions and, consequently, the degree of disagreement between the RANS and LES predictions are shown to depend on the stream geometry and the flow rate. Our results underscore the major challenges confronting the computationally expedient, isotropic RANS models, which are widely used today in three-dimensional hydrodynamic and morphodynamic simulations.

1. Introduction

[2] During recent years, three-dimensional (3-D) computational fluid dynamics (CFD) modeling has emerged as a powerful tool for studying complex turbulent flows in man-made and natural waterways. Such models have been applied to simulate flows in simple curved open channels [Wilson et al., 2003; Van Balen et al., 2009, 2010a, 2010b; Stoesser et al., 2010; Constantinescu et al., 2011] and real-life meandering streams or rivers [Ferguson et al., 2003; Rodriguez et al., 2004; Kang et al., 2011; Kang and Sotiropoulos, 2011]. Available CFD models for open channel and river flows do not solve directly the unsteady, three-dimensional Navier-Stokes equations because carrying out direct numerical simulation (DNS) for real-life waterways at high Reynolds number is not feasible because of the excessive computational cost. For that, all existing CFD models solve either temporally averaged or spatially filtered forms of the Navier-Stokes equations, which give rise to Reynolds-averaged Navier-Stokes (RANS) and large-eddy simulation (LES) models, respectively. As a result of the averaging or filtering operations, additional terms, such as the Reynolds stress tensor (for RANS models) or the subgrid stress (SGS) tensor (for LES), appear in the resulting equations, which need to be closed by an appropriate turbulence closure model. In RANS models the eddy viscosity concept is widely employed to model the Reynolds stress tensor, which assumes that the anisotropic part of the Reynolds stress tensor is linearly proportional to the time-averaged strain rate tensor [Pope, 2000; Durbin and Pettersson Reif, 2001]. This is the so-called Boussinesq approximation, which is inherently based on the assumption that turbulence is locally isotropic. Consequently, RANS models based on the Boussinesq approximation are often referred to as isotropic RANS models. Examples of such models include, among others, the k- inline image model [Launder and Sharma, 1974], the k- inline image model [Wilcox, 1993], the shear stress transport (SST) model [Menter et al., 2003], etc. In LES models (see, e.g., Sagaut [1988]), on the other hand, the Smagorinsky model [Smagorinsky, 1963] has been widely employed to model the SGS tensor. The Smagorinsky model is also based on the eddy viscosity model, which assumes that the SGS tensor is linearly proportional to the instantaneous, resolved strain rate tensor.

[3] In spite of the fact that in both isotropic RANS and LES models an isotropic eddy viscosity model is employed, the role of the eddy viscosity is fundamentally different in the two approaches. Namely, in RANS models the eddy viscosity accounts for all scales of turbulent eddies, except perhaps very large scale periodic unsteadiness which can be captured by unsteady RANS models. In LES models, on the other hand, scales of motion larger than the grid spacing are directly resolved by the simulation and only subgrid-scale eddies are modeled. As a result in flows in which large-scale organized coherent structures are responsible for producing most of the Reynolds stresses, as is often the case in natural waterways, LES with sufficient resolution can resolve directly most turbulence in the flow and only a relatively small portion of the energy needs to be handled by the SGS model. A direct consequence of this fundamental difference between LES and RANS simulation is that as the grid is refined the former converges to a DNS while the latter converges to a grid-independent time-averaged RANS solution [Sagaut, 1988; Pope, 2000].

[4] Isotropic RANS models have been popular engineering simulation tools for resolving complex flow and sediment transport in rivers mainly because of their computational expedience. For instance, such models have been employed to predict flows in large rivers [Sinha et al., 1998; Ferguson et al., 2003; Rodriguez et al., 2004; Ge and Sotiropoulos, 2005], flows in curved open channels [Wilson et al., 2003; Stoesser et al., 2010; Kang et al., 2011], sediment transport in curved open channels [Wu et al., 2000; Olsen, 2003; Khosronejad et al., 2011] and to model delta morphodynamics [Edmonds and Slingerland, 2010]. In spite of these numerous applications, the limitation of the RANS models in predictions of turbulent flows in real-life waterways has yet to be extensively investigated.

[5] There exist few previous studies which attempted to compare the performance of isotropic RANS models and LES for predicting flows in simple curved channels. Stoesser et al. [2010], for instance, simulated flow in a flat-bed circular bend using LES and isotropic RANS models. Their RANS simulation results were in good agreement with the LES results and were able to predict the curvature-driven secondary flow cell as well as the outer bank secondary cell, which has been linked to turbulence anisotropy effects [Blanckaert and Vriend, 2004; Van Balen et al., 2009; Kang and Sotiropoulos, 2011]. However, Stoesser et al. [2010] reported that the RANS model fails to predict the persistence of the outer bank cell (OBC) until the exit of the bend. Van Balen et al. [2010a] simulated flow in a circular bend using the isotropic RANS model and LES. They showed very good agreement between the mean velocity profiles predicted by the RANS and LES models. They further argued that the good agreement is because in the flow they considered there is a strong tendency toward isotropy of the turbulence stresses in the bend and turbulence related momentum transport does not play a major role in the transport of mean momentum. They also concluded that assumptions made within either the filtering (LES) or the statistical (RANS) framework are of marginal influence and a simple modeling of the turbulence effects suffices. Van Balen et al. [2010b], on the other hand, compared the RANS simulation and the LES for flow in a different circular bend and showed that the RANS computation cannot predict the OBC and the internal shear layer near the inner bank. However, they did not provide clear explanations on why the RANS results are poorer than the LES results.

[6] It is evident from the above literature review that only few studies have been reported thus far aimed at systematically investigating the predictive capabilities or RANS and LES and that the results of these studies are often inconclusive and/or contradictory to each other. Furthermore, no study has been reported in the literature seeking to compare RANS and LES models for a natural meandering channel. In this study, we seek to contribute to clarifying the predictive capabilities of isotropic RANS models in simulations of turbulent flow in a natural-like meandering channel. We employ the SST model of Menter et al. [2003] to close the RANS equations [see Kang et al., [2011] and apply it in unsteady RANS mode to simulate bankfull flow in the field-scale meandering stream at the St. Anthony Falls Laboratory (SAFL) Outdoor StreamLab (OSL) [Kang and Sotiropoulos, 2011]. The RANS simulations are compared with the LES and experimental measurements of the same flow previously reported by Kang and Sotiropoulos [2011]. To eliminate numerical uncertainties and enable a meaningful comparison of the two turbulence modeling strategies, both simulations were carried out under exactly the same conditions with same number of grid nodes and time step. The LES reported by Kang and Sotiropoulos [2011] revealed a series of complex flow phenomena occurring in the meander bend, which include the curvature-driven secondary flow, the anisotropy driven outer bank secondary flow, horizontal recirculation (or dead) zones along the inner bank, and the outer and inner bank shear layers. In this paper we will scrutinize the ability of the RANS model to capture these phenomena and also identify the physical reasons for any modeling shortcomings. We will also discuss the implications of our findings in the context of applying isotropic RANS models to hydrodynamic and morphodynamic simulations of natural waterways. We first briefly describe the computational models we employ in section 2 and subsequently describe the experimental facility (OSL) we use as the test case and the computational parameters in section 3. This is followed by comparisons of the numerical solutions of the LES and RANS simulation in section 4, and the discussion of the flow visualization experiments in section 5. In section 6 we discuss the physical reasons for the discrepancies between the results of the two models, and finally in section 7 we summarize the key conclusions of this work.

2. Governing Equations and Numerical Method

[7] We solve numerically the unsteady RANS (URANS) and continuity equations closed with the SST turbulence model [Menter et al., 2003] formulated in generalized curvilinear coordinates. The governing equations are described in detail by Kang et al. [2011] and are omitted here for the sake of brevity.

[8] The numerical method is that developed by Kang et al. [2011] and is capable of carrying out URANS simulation and LES of turbulent flows in natural waterways. The method employs the Curvilinear Immersed Boundary (CURVIB) [Ge and Sotiropoulos, 2007] for handling the arbitrary geometric complexity of natural waterways coupled with a wall model for reconstructing the velocity field at grid nodes near solid boundaries in turbulent flow simulations [Kang et al., 2011]. The governing equations are discretized with second-order accurate, three-point central finite differences on a hybrid staggered/nonstaggered grid [Gilmanov and Sotiropoulos, 2005; Ge and Sotiropoulos, 2007] and integrated in time using an efficient fractional step algorithm [Ge and Sotiropoulos, 2007]. Matrix-free Krylov-based solvers are used to solve the momentum and pressure equations and algebraic multigrid is employed to accelerate the convergence of the pressure equation [Kang et al., 2011]. For more details about the numerical method and the implementation of the RANS and LES models the reader is referred to [Kang et al., 2011].

3. Test Cases and Computational Details

3.1. The SAFL Outdoor StreamLab

[9] The Outdoor StreamLab (OSL) is the field-scale experimental facility located at the University of Minnesota St. Anthony Falls Laboratory in Minneapolis, Minnesota, United States. The OSL is an approximately 50 m long, 3 m wide, and 0.3 m deep (at bankfull flow) sand bed meandering stream channel with a pool-riffle sequence located in a 40 m by 20 m basin. The bed materials in the riffles are gravels ranging from 10 to 15 cm in size and are large enough to withstand the maximum bed shear stress at all times, such that the riffle topography is mostly fixed. The bed in the pool was initially constructed with a flat sand bed and was allowed to evolve naturally toward a quasi-equilibrium state, with point bars and deep pool regions along the inner and outer banks, respectively. The wavelength of the meander is approximately 25 m, and the channel was laid out as a sine-generated curve with moderate sinuosity (1.3). The stream bank is protected by the vegetation along with the coconut fiber bank stabilization matting and is practically fixed. Bed and water surface topography in the OSL were collected on a 1 cm horizontal grid at submillimeter vertical accuracy using a laser scanner mounted to a separate channel-spanning portable carriage. The bed topography reconstructed from the measured data is shown in Figure 1. Mean water surface elevation measurements were sampled at 50 Hz over a centimeter-scale spaced grid using an ultrasonic distance sensor. More details of this experimental facility are given by Kang et al. [2011] and Kang and Sotiropoulos [2011].

Figure 1.

The St. Anthony Falls Laboratory Outdoor StreamLab bathymetry obtained from high-resolution measurements. The contour levels denote bed elevations, and the symbols and numbers mark velocity measurement locations. The arrow indicates the flow direction.

3.2. Computational Details

[10] In this study we carry out RANS simulation of the bankfull flow in the OSL using the SST model [Menter et al., 2003]. The same flow was simulated by Kang and Sotiropoulos [2011] using LES. We select the SST model [Menter et al., 2003] instead of the k- inline image model to close the RANS equations because the former performed better than the latter in simulating the base flow of the OSL [Kang et al., 2011]. Kang et al. [2011] showed that the k- inline image model yielded turbulence kinetic energy levels in the riffle zones of the OSL nearly 1 order of magnitude higher than those measured and calculated with LES and the SST model. They found that this was due to the inherent limitation of two-equation, isotropic eddy viscosity models in flows with regions of flow stagnation, and showed that the SST model did not suffer from such a problem as it incorporates a limiter in the turbulence kinetic energy production term.

[11] The bankfull condition flow rate in the OSL is Q = 2.85 × 10−1 m3 s−1, which yields Reynolds and Froude numbers on the basis of depth and velocity at the inlet, approximately equal to inline image and inline image, respectively. The computational grid used for the RANS simulation is identical to that used in the LES of Kang and Sotiropoulos [2011]. Namely, the background curvilinear grid consists of inline image grid points in the streamwise, transverse and vertical directions, respectively, and the total number of grid points is about 49 million. The grid spacing in the streamwise, transverse and vertical directions is approximately 2.0 cm, 1.5 cm and 6 mm, respectively. The fixed, time-averaged bed topography and measured mean sloping water surface are prescribed during the computation in exactly the same manner as in the LES of Kang and Sotiropoulos [2011]. In the LES of Kang and Sotiropoulos [2011], we fed instantaneous velocity fields obtained from a separate LES of fully developed flow in a straight open channel with the same cross section as that of the inlet OSL cross section at the same Reynolds number. In the present RANS simulation, the time-averaged inlet velocity obtained from the LES of fully developed flow and constant free stream values of turbulence quantities (k and inline image) are specified at the inlet boundary, and a zero-gradient Neumann-type boundary condition is applied for the velocity and turbulence quantities at the outlet boundary. We note that while the inlet velocity boundary conditions are different in the two simulations (steady in RANS versus unsteady in LES), they are consistent with the respective turbulence modeling approach. The RANS simulation is carried out in a time-accurate (URANS) manner in order to capture any large scale unsteady flow that may exist. Note that even though both the spatial and temporal resolution we employ in our simulation is clearly much too fine for a RANS calculation, we employ it herein in order to ensure that the RANS and LES results can be compared free of any numerical uncertainties. To further facilitate a conclusive comparison of the two turbulence modeling strategies we also handle bed roughness in the RANS simulation in exactly the same manner as in the LES. Namely no roughness model is employed in the wall model used in the RANS simulation but rather roughness elements up to 10 cm in size in the riffle regions are resolved directly using a sufficiently fine computational grid. The details of this approach are described by Kang et al. [2011].

4. Comparisons of the LES and RANS Results

[12] As we mentioned above, the RANS simulation was carried out in a time accurate manner. The solution, however, ultimately converged to steady state. In this section we compare the so obtained steady state RANS solution with the statistically converged, time-averaged flow field obtained from the LES of Kang and Sotiropoulos [2011].

[13] We first compare the LES and RANS computations with the measurement at the cross section shown in Figure 1. This cross section is located inside the bend, where the flow patterns are complicated by the three-dimensional secondary motion, and the velocity data were collected using an acoustic Doppler Velocimetry (see Kang and Sotiropoulos [2011] for details). We compare the computed and measured mean streamwise and transverse velocity and turbulence kinetic energy (TKE) profiles at this cross section in Figure 2. As seen, both the LES and RANS models predict the mean velocities and TKE reasonably well, and the difference of the two calculations are marginal. From this plot, one might thus conclude that the performance of the two models is equally good. In fact, one could even suggest from these comparisons that in certain areas (see, for example, the streamwise velocity profile at location 1) the RANS model outperforms the LES as its results appear to be in somewhat better agreement with the measurements. It is important to keep in mind, however, that in complex, three-dimensional open channel flows great caution needs to exercised when attempting to derive conclusions about the predictive capabilities of a turbulence model using as only metric the level of agreement between the model predictions and sparse point measurements. As we will subsequently show, the three-dimensional flow patterns predicted by the LES and RANS models are fundamentally different.

Figure 2.

Comparisons of mean streamwise velocity, mean transverse velocity, and turbulence kinetic energy (TKE) with the measurements at the cross section near the bend apex (solid lines, large-eddy simulation (LES); dotted lines, Reynolds-averaged Navier-Stokes (RANS); symbols, measurements). The measurement locations are shown in Figure 1. The horizontal dashed and solid lines denote the locations of the free surface and bed, respectively.

[14] The calculated mean streamwise velocity contours at the water surface obtained by LES and RANS models are compared with each other in Figures 3a and 3b, respectively. The LES result (Figure 3a) shows that the high-velocity region from the first riffle becomes narrow as it enters the pool to form a strong jet-like flow structure. Near the apex of the bend this jet-like flow structure breaks down and impinges on the outer stream bank diffusing laterally before it encounters the second riffle. Another distinctive feature of the LES solution is the formation of a recirculation zone along the inner bank of the meander bend, which is marked by the pocket of negative streamwise velocities (in blue color). The RANS simulation (Figure 3b), on the other hand, fails to predict the sharp jet-like flow structure as well as the impinging of the high-velocity core on the outer stream bank. In sharp contrast with the LES result, the RANS solution yields a laterally diffused high-velocity region with nearly constant width along the entire bend. In addition, the horizontal recirculation zone along the inner bank of the bend present in the LES solution is not captured at all in the RANS calculation.

Figure 3.

Comparisons of (a and b) the mean streamwise velocity contours and (c and d) turbulence kinetic energy contours superimposed with streamlines at the water surface computed with (left) LES [Kang and Sotiropoulos, 2011] and (right) RANS models for the bankfull flow. The arrow indicates the flow direction.

[15] In Figures 3c and 3d we plot the TKE contours superimposed with mean streamlines at the water surface computed by LES and RANS models, respectively. The mean streamlines predicted by the LES (Figure 3c) clearly show the existence of the inner bank recirculation region, which is also found in the mean streamwise velocity plot (Figure 3a). Moreover, the LES predicts the presence of the two thin shear layers marked by high levels of TKE located along the inner and outer banks of the bend, which are the inner bank shear layer (IBSL) and the outer bank shear layer (OBSL) previously reported by Kang and Sotiropoulos [2011]. As seen in Figures 3a and 3c, the IBSL and OBSL delineate the jet-like, high-velocity region that develops within the bend, which suggests that these flow features are intimately connected. The physical mechanisms giving rise to these intricate flow features have been discussed in detail by Kang and Sotiropoulos [2011]. Here it suffices to mention that the OBSL arises because of the lateral flow convergence at the water surface and the associated downward flow toward the bed induced by the subsequently discussed complex and highly three-dimensional secondary flow patterns within the bend (see also Figure 5a). The IBSL, on the other hand, is the direct consequence of the inner bank recirculation zone. In stark contrast with the LES solution, the mean flow predicted by the RANS calculation (Figure 3d) exhibits neither the two shear layers (IBSL and OBSL) nor the horizontal recirculation zone. Instead, the RANS flow field exhibits a thin shear layer with high levels of TKE positioned very close to the outer bank. The lateral convergence of water surface streamlines that dominates the LES flow patterns is entirely absent in the RANS solution, which suggests that the two flow fields are fundamentally different.

[16] To further underscore the differences between the RANS and LES solutions, we compare in Figure 4 the respective mean 3-D streamlines near the apex of the bend. The high-elevation regions along the inner bank of the bend (in red colors) denote the point bar, which formed naturally during the experiment. The LES solution (Figure 4a) predicts the existence of a tornado-like, horizontal recirculation eddy with its center positioned directly above the crest of the point bar. The velocity in the horizontal recirculation (or dead) zone is very low as seen in Figure 3a, which enhances the deposition of fine sediments inside the recirculation zone and contributes to the formation of the point bar. The RANS calculation (Figure 4b), however, does not predict the existence of the recirculation zone and yields nearly straight 3-D mean streamlines passing above the point bar, a flow pattern that is neither consistent with nor can explain the formation of point bar.

Figure 4.

Comparisons of the mean 3-D streamlines near the bend calculated by (a) LES [Kang and Sotiropoulos, 2011] and (b) RANS models for the bankfull flow. The streambed is colored with bed elevation contours. The arrow indicates the flow direction.

[17] To compare the secondary flow patterns predicted by the RANS and LES models, we plot mean streamlines at cross sections near the apex of the bend superimposed with contours of mean streamwise velocity. As discussed extensively by Kang and Sotiropoulos [2011], the LES (Figure 5a) predicts the formation of two counterrotating secondary cells, namely the center region cell (CRC) and the outer bank cell (OBC). The CRC is driven by the curvature-induced imbalance between the centrifugal force and the transverse pressure gradient near the bed [Humphrey et al., 1981; Johannesson and Parker, 1989]. The OBC, on the other hand, is generated by the combined effect of the turbulence anisotropy and the centrifugal force [Blanckaert and Vriend, 2004; Van Balen et al., 2009; Kang and Sotiropoulos, 2011]. As evidenced by Figure 5b, the RANS simulation only predicts the CRC but fails completely to capture the OBC. As shown by Kang and Sotiropoulos [2011], the lateral collision of the OBC and CRC at the water surface gives rise to the formation of the line of convergence in the mean streamlines at the water surface, which coincides with the OBSL as shown in Figure 3c. Hence, the results we have presented so far collectively suggest that the failure of the RANS model to predict the OBSL (see Figure 3d) can be attributed to its inability to predict the OBC. Another important flow phenomenon captured by the LES (see Figure 5a) is the plunging of the high-velocity core toward the bed near the outer bank of the bend. This phenomenon is related to the streamwise momentum redistribution due to the existence of the OBC [Kang and Sotiropoulos, 2011]. The displacement of the high-velocity core toward the streambed locally increases the bed shear stress and contributes to the channel deepening in the outer part of the meander bend. The RANS flow field (Figure 5b), however, does not exhibit the plunging of the high-velocity core toward the bed. This is to be expected, however, since the RANS model fails completely to capture the OBC.

Figure 5.

Comparisons of the mean two-dimensional streamlines and streamwise velocity contours at cross sections near the bend calculated by (a) LES [Kang and Sotiropoulos, 2011] and (b) RANS models for the bankfull flow. The arrow indicates the flow direction.

5. Comparisons With Flow Visualization Experiments

[18] In this section, we elucidate the water surface flow patterns in the stream by carrying out two flow visualization experiments. The goal of these experiments is to show that the inner bank recirculation zone and the line of flow convergence at the water surface observed in the LES flow field are also present in the OSL experimental facility. The flow visualization experiments were carried out for the same flow conditions as those for which the mean velocity and turbulence statistics measurements were obtained by manually introducing floating confetti at various locations on the water surface and recording its motion using a digital camera.

[19] In the first visualization experiment, floating confetti was released all at once along the inner bank of the bend in the region where the LES revealed the presence of the inner bank recirculation zone (see Figure 4a). The motion of the confetti in this region was subsequently recorded for 70 s, a time interval which is well over two flow-through times of the stream. Figure 6 shows the snapshots taken during the last 10 s of the recorded interval, which is between 60 and 70 s after the confetti release. It is seen that even after 60 s, which approximately corresponds to two flow-through times of the stream, a considerable amount of confetti still remains trapped near the inner bank of the bend within the same region where the confetti was initially released. To further illustrate the persistence and spatial extent of the region where tracers are trapped in the experimental stream, in Figure 7 we superimpose all recorded images shown in Figure 6. It should be noted that the experimental images, which depict instantaneous streak lines, cannot be directly compared with the mean flow streamlines plotted in Figures 3 and 4. Nevertheless, Figures 6 and 7 clearly show that confetti tracers are trapped for long times within the same region along the inner bank where the LES reveals the presence of recirculating flow. As already discussed in section 4, the RANS model completely fails to predict this significant flow feature (see Figure 4b).

Figure 6.

Snapshots between 60 and 70 s after release of the confetti. The flow direction is from right to left.

Figure 7.

A merge of the four snapshot images in Figures 6. Confetti is concentrated near the recirculation zone.

[20] In the second experiment, confetti is manually spread continuously across the entire width of the stream at section located upstream of the meander bend within the riffle. Throughout this experiment, special care was taken to ensure that the confetti is uniformly fed across the entire channel cross section. Figure 8 shows the snapshots taken in the interval between 18 and 30 s form the start of the experiment. As seen, even though the confetti is initially distributed uniformly across the channel it rapidly converges laterally to form a thin layer near the outer meander bend as is advected by the flow through the bend. Furthermore, this thin layer of flow convergence remains coherent and clearly visible throughout the entire meander bend until the particles reach the next riffle. Figure 8 clearly establishes the fact that essentially all water surface flow that originates within the riffle converges to form a narrow coherent streak near the outer bank. This feature in the experimental flow is very similar to the jet-like flow structure observed in the LES (Figure 3a) and very much in agreement with the existence of the flow convergence line shown in the mean surface streamlines computed by LES in Figure 3c. As discussed extensively by Kang and Sotiropoulos [2011], this line of convergence is a very important feature of the flow as it marks the region where the two counterrotating secondary cells collide at the water surface. To further reinforce the failure of the RANS model to capture this strong convergence of the water surface flow, we plot in Figure 9 calculated (with RANS and LES) mean streamlines at the water surface originating from exactly the same line of initial conditions located within the riffle. Once again, we should caution that the calculated mean streamlines are obviously not the same as the instantaneous streak lines shown in Figure 8. This notwithstanding, however, the persistence of the coherent streak of confetti near the outer bank of the bend in the experimental flow makes a strong case that a line of convergence does exist in the mean in this region of the flow as also revealed by the LES in Figure 9a. In the RANS prediction, on the other hand, the mean streamlines originating from the riffle are distributed throughout the cross section of the meander bend. Some streamlines in the RANS flow field are seen in Figure 9b to cluster very close to the outer bank of the stream in the apex region. This is consistent with the fact that the RANS solution exhibits only the CRC (see Figure 5b), which directs the water surface flow from the inner to the outer bank. In the LES and visualized flow fields, however, the presence of the counterrotating outer bank cell prevents the flow from accumulating along the outer bank and causes the line of convergence observed in Figure 9a and the coherent streak seen in the visualization experiments to form at some distance away from the outer bank.

Figure 8.

Snapshot images showing the trajectories of floating confetti in the stream at between 18 and 30 s after the beginning of the release. The flow direction is from right to left.

Figure 9.

Mean surface streamlines predicted by (a) LES and (b) RANS calculations. All streamlines start from the inlet.

6. Discussion

[21] Although both the LES and RANS model results agree reasonably well with the measurements when compared in terms of few vertical profiles at the cross section of the bend apex (Figure 2), three-dimensional velocity and turbulence kinetic energy plots show significant differences between the predictions of the two models. This finding clearly demonstrates that few measurements in a cross section can neither capture the complexity of the flow in natural streams nor be used to adequately assess the predictive capabilities of various turbulence models. The major differences between the flow fields predicted by the LES and RANS model are summarized as follows. While the RANS simulation captures the existence of the CRC, it fails to capture essentially all other flow features predicted by the LES and also revealed by flow visualization experiments. Namely, the RANS flow field does not exhibit: the jet-like mean flow structure impinging on the outer bank downstream of the riffle; the horizontal recirculation region formed above the point bar near the inner bank; the IBSL delineating the outer edge of the recirculation zone; and the OBSL formed along the line of lateral flow convergence at the water surface where the two counterrotating secondary cells collide. At first glance, the large discrepancies between the two solutions constitute a rather unexpected result since Kang et al. [2011], who recently simulated the flow in the same stream but at a lower discharge (Q = 4.45 × 10−2 m3 s−1) and flow depth using the same LES and SST RANS models employed herein, reported good overall agreement between the two models. More specifically, the mean velocity and turbulence kinetic energy profiles predicted by the two models were in good agreement with each other and the experimental measurements [Kang et al., 2011].

[22] Therefore, our results are consistent with those of previous studies aimed at assessing the relative performance of RANS and LES models in curved open channel flow simulations [Van Balen et al., 2010a, 2010b; Stoesser et al., 2010] in the sense that all such studies are inconclusive and yield results that are inconsistent and highly case specific. For instance, simulations of two different curved open channels using the same RANS and LES models reported by Van Balen et al. [2010a], [2010b] revealed good and poor performance of the RANS model, respectively.

[23] Before discussing the cause of such inconsistent performance of RANS models, let us revisit the Boussinesq approximation on which such models rely. This assumption implies a locally isotropic structure of turbulence is not valid for complex shear flows with extra rates of strains [Bradshaw, 1973] for which anisotropy effects tend to become important. Hence, it is reasonable to hypothesize that the case-specific performance of RANS models could be attributed to different levels of turbulence anisotropy among various test cases. This hypothesis is supported by the fact that Kang and Sotiropoulos [2011] showed in their LES of bankfull flow in the OSL that flow features such as the OBSL, IBSL, OBC and converging streamlines at the water surface are all associated with high levels of turbulence anisotropy. All these are exactly the flow features that could not be resolved by the RANS simulation in this work. It is important to note that turbulence anisotropy is the result of three-dimensionality of the mean flow and in particular the existence of the so-called extra mean strain rates, which largely depend on stream bathymetry, bed roughness, meander sinuosity, and flow parameters such as the Reynolds and Froude numbers. Consequently the inconsistent performance of RANS models in various open channel flow cases could be explained by differences among these cases that affect the level of turbulence anisotropy in each case. It is worth noting that for the flow case for which Van Balen et al. [2010a] obtained good results with their RANS model they reported a strong tendency toward local turbulence isotropy, which would support our overall hypothesis.

[24] To verify our argument that geometric and flow conditions can alter the levels of turbulence anisotropy, we quantify anisotropy levels by plotting contours of the second invariant of the anisotropy defined as follows [Lumley, 1978]:

display math

with

display math

where and inline image denotes velocity fluctuations in the ith direction, k is the turbulence kinetic energy, inline image is the Kronecker delta, and the overbar denotes the temporal (or Reynolds) averaging. Note that as discussed by Kang and Sotiropoulos [2011] inline image is a scalar quantity that is invariant to coordinate system transformation and as such it is not sensitive to the specific coordinate system selected to express the velocity field and calculate the components of the Reynolds stress tensor. In addition, the absolute value of this quantity becomes large in regions where anisotropy becomes important. In Figures 10a and 10b we plot the contours of inline image for the bankfull and base flows of the OSL, obtained from the LES of Kang and Sotiropoulos [2011] and Kang et al. [2011], respectively. As seen in Figure 10a, for the bankfull flow case pockets with locally high turbulence anisotropy are found along the IBSL and OBSL, which are associated with the recirculation zone and the lateral collision of the OBC and CRC, respectively. For the base flow case (Figure 10b), on the other hand, turbulence anisotropy is significantly weaker within the bend, which supports our underlying hypothesis about the good overall performance of the RANS model for this case [Kang et al., 2011]. Therefore, the anisotropy plots in Figure 10 essentially confirm our argument that the geometric and flow conditions can significantly change the local levels of turbulence anisotropy and thus result in highly case-specific performance of isotropic RANS models.

Figure 10.

Contours of the anisotropy measure inline image for (a) the bankfull flow [Kang and Sotiropoulos, 2011] and (b) base flow [Kang et al., 2011] conditions predicted by the LES at the several streamwise cross sections. The arrow indicates the flow direction.

[25] Although we only considered a fixed bed stream in this study, it is important to note that our findings suggest that the turbulence anisotropy levels in a channel can significantly affect the predictive capabilities of morphodynamic models based on isotropic RANS turbulence models, which have been widely employed for predicting flow and sediment transport in curved open channels [Wu et al., 2000; Olsen, 2003; Khosronejad et al., 2011]. More specifically, we could speculate that applying a coupled isotropic RANS hydromorphodynamic model to predict the OSL streambed bathymetry under bankfull flow conditions is likely to yield physically incorrect results. In particular, since the RANS model cannot capture the inner bank recirculation region, the resulting flow field that drives sediment transport and deposition patterns lacks the key mechanism that causes fine sediments to be trapped and deposit along the inner bank of the bend to form the point bar. Moreover, the location and elevation of the channel thalweg throughout the meander bend may be wrongly predicted because isotropic RANS models are not able to capture OBC and its interaction with the CRC. As we already discussed, it is the interaction between these two secondary flow cells that leads to phenomena that contribute to channel deepening such as the collision of the two cells at the water surface (Figure 3c), the so-induced down flow, and the associated plunging of the high-velocity core toward the bed along the inner bank of the bend (Figure 5a).

[26] To further underscore the difficulties one may encounter when applying an isotropic RANS model to carry out morphodynamic simulations of the bankfull case, we plot in Figure 11 the mean boundary shear stress of the bankfull flow calculated by the LES and RANS models. In the CURVIB-wall modeling approach, boundary shear stress is calculated at each immersed boundary node by solving the simplified boundary layer equation using the velocities at the solid surface and the second off-wall node as boundary conditions (see Kang et al. [2011] for details). The boundary shear stress directly at a triangular element of the immersed boundary surface (e.g., channel bed) is interpolated from that at the immersed boundary node closest to the triangular element. As seen, the LES (Figure 11a) predicts streaks of high boundary shear stress along the thalweg of the channel and low shear stress along the inner stream bank near the point bar. The RANS model (Figure 11b), on the other hand, fails to predict the high shear stress zones along the thalweg and yields overall higher boundary shear stress along the inner stream bank compared to the LES. In Figure 12 we compare the computed mean boundary shear stress across the cross section of the bend apex marked in Figure 11. As Figure 12 shows the RANS model predicts almost constant shear stress (approximately 0.4 Pa) across the center region of the cross section, while the LES predicts the peak shear stress (approximately 0.7 Pa) near the thalweg of the channel and a gradual decrease of the shear stress toward the inner and outer banks. Another important difference is that the RANS model predicts higher shear stress in the inner bank region ( inline image), which is probably because the RANS model did not predict the presence of the recirculation region where the mean velocity is low. The difference of the predicted shear stress is largest near the thalweg region, where the RANS model underpredicts the shear stress by approximately 35 percent relative to that of the LES.

Figure 11.

Contours of the mean boundary shear stress calculated by (a) LES [Kang and Sotiropoulos, 2011] and (b) RANS models for the bankfull flow. The arrow indicates the flow direction, and the dashed line across the channel bed indicates the location of the cross section of the bend apex, where the boundary shear stress is compared in Figure 12.

Figure 12.

(a) Channel topography and (b) the calculated mean boundary shear stress (LES by Kang and Sotiropoulos [2011], solid line; RANS model, dash-dotted line) at the cross section of the bend apex (marked in Figure 11) for the bankfull flow.

7. Conclusion

[27] In this paper we carried out numerical simulation of turbulent bankfull flow in the Outdoor StreamLab, which is a field-scale meandering stream, using an isotropic RANS model under the exact same computational conditions as the previously reported LES [Kang and Sotiropoulos, 2011]. The computed results show that although both turbulence models are able to reasonably predict the velocity and TKE profiles measured at several locations, they yield fundamentally different 3-D flow fields. Namely, the isotropic RANS model fails to predict the outer bank shear layer with surface flow convergence, and the inner bank recirculation zone, which are predicted by the LES and also observed in the flow visualization experiments reported in this work. This finding demonstrates that for complex three-dimensional flows as those encountered in natural waterways few point measurements in a cross section can neither capture the complexity of the flow nor be used to adequately assess the predictive capabilities of various turbulence models. Our results also show that the above mentioned two flow phenomena are associated with high level of turbulence anisotropy, which we argue is the cause of the poor performance of the isotropic RANS model. By comparing turbulence anisotropy levels calculated with LES for the same channel but for different discharge, we showed that the local turbulence anisotropy level is a function of geometric and flow conditions. This finding is important as it can be used to reconcile findings from previous computational studies, which showed the performance of isotropic RANS models to be case specific.

[28] Our work has clearly shown that application of isotropic RANS models to simulate turbulent flows in meandering channels without full understanding of the inherent limitations of such models could lead to nonphysical results. This conclusion is especially true in flow cases for which the geometry and flow parameters are such as to induce extensive regions of large turbulence anisotropy, which cannot be handled by isotropic RANS models. The presence of such regions in the flow, however, are difficult if not impossible to determine a priori without detailed measurements or carrying out computationally intensive LES. Therefore, future work should be directed toward identifying and systematically evaluating specific channel geometric and flow parameters that affect turbulence anisotropy to help computational modelers select appropriate turbulence modeling strategies on a case-specific basis.

[29] Finally, we emphasize that our critical assessment of the limitations of RANS models is strictly valid for isotropic eddy viscosity models. Another class of RANS models that is not discussed in this study is the so-called nonlinear eddy viscosity models [Craft et al., 1996; Gatski and Jongen, 2000] or algebraic [Gatski and Speziale, 1993] and full Reynolds stress transport models [Hanjalic and Launder, 1972]. Such models are inherently formulated to resolve the anisotropy of the Reynolds stresses in flows with extra rates of strain and should, at least in principle, be able to mitigate the weaknesses of isotropic RANS models. Further studies on evaluating the predictive capabilities of such models in natural waterways will be required since anisotropic RANS models can provide a more efficient and less computationally intensive alternative to LES.

Acknowledgments

[30] This work was supported by NSF grants EAR-0120914 (as part of the National Center for Earth-Surface Dynamics) and EAR-0738726. Computational resources were provided by the University of Minnesota Supercomputing Institute. The authors wish to thank Anne Lightbody and Craig Hill for providing the bed topography data of the Outdoor StreamLab and are also grateful to Jessica Kozarek and Leonardo Chamorro for their assistance in the flow visualization experiments.