Strong Flow Separation Research Articles

Assessment of RANS turbulence models based on the cell-based smoothed finite element model for prediction of turbulent flow

There is a growing body of literature that recognizes the importance of Smoothed Finite Element Method (S-FEM) in computational fluid dynamics (CFD) fields and, to a lesser extent, in complex turbulent flow problems. This study evaluates the performance of Reynolds-averaged Navier-Stokes (RANS) turbulence models within the S-FEM framework for predicting incompressible turbulent flows. Our assessment of three turbulence models based on the cell-based S-FEM (CS-FEM) is convincingly supported by testing on three flow problems. It is found that the CS-FEM exhibits superior mesh robustness compared to the Finite Volume Method (FVM) and achieves higher computational accuracy than the Finite Element Method (FEM). Notably, the CS-FEM combined with the standard k-epsilon model (CS-FEM-SKE) and the realizable k-epsilon model (CS-FEM-RKE) demonstrate robust performance in handling severely distorted meshes, with CS-FEM-RKE outperforming in regions of strong flow separation and convection. The Spalart-Allmaras model with CS-FEM (CS-FEM-SA) offers faster computational speed but shows poor mesh robustness. The hexcore mesh based on CS-FEM-RKE is employed to evaluate the aerodynamic performance of High-speed train (HST), resulting in enhanced computational efficiency. The outcomes show good agreement with other numerical studies and experimental data. Overall, it also highlights the latent capability of CS-FEM in solving complex engineering problems.

Flow pattern regime and unsteady characteristics of ventilated cavitating flow around the axisymmetric bodies with different headforms

This paper presents an experimental investigation of the flow pattern regime and unsteady characteristics of ventilated cavities with different headform shapes. The test model consists of a removable headform with three different forebodies (the conical, the blunt, and the hemispherical) and a common cylinder rear body. Experiments are conducted in a closed-loop cavitation tunnel. First, the flow pattern regimes on the ventilated cavity for different headforms are discussed in detail, and the dimensions of several flow patterns are measured. The results show that the cavity dimension and the regime are strongly dependent on the headform shape, and all typical flow patterns are introduced by schematic illustrations. Second, the ventilation hysteresis that happened during the flow pattern transition is pointed out. A quantitative gas leakage model is employed to explain the cause of hysteresis and flow pattern transition, and the results show the Strouhal number for different headforms is approximately ∼0.21. However, the blunt presented stronger gas leakage with a large constant parameter, K = 0.80. Finally, the unsteady characteristics of the ventilated cavity around different tested headforms are involved through descriptions of developments of the recirculating vortex and transparent cavity. In addition, an estimated cavitation number is applied to investigate the unsteady characteristics, and the maximum cavitation number and the strongest characteristic are obtained by the blunt headform due to the large drag coefficient and strong flow separation.

Interference effect on flow features and aerodynamic mechanism over the roof between two tall buildings

Interference effect of an upwind building on flow features and wind power potential over roof of principal building was investigated through numerical simulations. Flow features were examined at different wind directions and spacing ratios (d/B = 2, 3, 4, 5, and 6), and associated aerodynamic mechanisms were analyzed. Turbulence intensity was less affected by interfering building than mean velocity. d/B exerted a more important impact on mean velocity at θ = 0° and 180° than those at oblique wind directions. The isolated building had large power potential at leeward locations due to strong flow separation at θ = 0° compared with principal building. The largest skew angle βmax fluctuated greatly with roof locations (from −15° to 80°) of isolated building at θ = 0°. In comparison, βmax over principal building changed slightly with roof locations (from −16° to 3°) due to smooth flow separation and reattachment at θ = 0°. Effect of d/B on the largest wind energy and hub height of roof-mounted wind turbines depended on θ. Recommendations for the siting of HAWTs on roofs were finally provided.

A Numerical Study on the Influence of Transverse Grooves on the Aerodynamic Performance of Micro Air Vehicles Airfoils

Micro Air Vehicles (MAVs) airfoils usually operate at low Reynolds number conditions, where viscous drag will consume a large amount of propulsion power. Due to the small dimensions, many drag reduction methods have failed, resulting in limited current research. To develop an effective method of reducing viscous drag, transverse grooves were placed on the surface of MAVs airfoils in this study, and a numerical investigation was implemented to uncover the corresponding flow control law as well as the mechanism. Research has shown that transverse grooves have an impact on the drag and lift of airfoils. For drag, properly sized transverse grooves have the effect of reducing drag, but under high adverse pressure gradients or when the continuous arrangement of grooves is excessive, the optimal drag reduction effect achieved by the grooves is weakened, and even the drag increases due to the significant increase in pressure difference. In severe cases, it may also cause strong flow separation, which is not conducive to MAV flight. For lift, the boundary vortex in the groove has the ability to reduce the static pressure near the groove. However, high adverse pressure gradients or too many grooves will thicken the boundary layer and increase the blockage effect, resulting in a large static pressure on the grooved side of the airfoil (with an increase in drag). From the perspective of circulation, the static pressure changes on the suction and pressure surfaces have opposite effects on lift. Considering the comprehensive aerodynamic performance of the airfoil, we designed a high lift-to-drag ratio airfoil with grooves, which increased the lift-to-drag ratio by 33.747% compared to the smooth airfoil. Based on the conclusions, we proposed preliminary design criteria for grooved airfoils, providing guidance for subsequent research and applications.

Role of Shape and Kinematics in the Hydrodynamics of a Fish-like Oscillating Hydrofoil

In the present two-dimensional numerical study, we investigate the roles of geometrical parameters of a hydrofoil (shape/curvature of the leading and trailing edges and thickness) and kinematic parameters (phase difference between heave and pitch (ϕ)) on the propulsive performance of different-shaped hydrofoils oscillating at maximum angles of attack up to αmax=30∘. The study was carried out at a fixed non-dimensional maximum heave to chord ratio h∘/C=0.75, Strouhal number St=0.25, and Reynolds number Re=5000. Our findings reveal that hydrofoil performance and stability improve with leading and trailing edge curvatures but decline as thickness increases. By analyzing the near-wake structure, we establish that even minimal flow separation increases power consumption while moderate flow separation enhances thrust. Over the range of different-shaped hydrofoils at different αmax and ϕ, maximum propulsion efficiency occurs for those parameters for which there is a small degree of flow separation but with no roll-up of a separating vortex. In comparison, maximum thrust generation occurs when there is a moderately strong flow separation but without induction of a significant amount of fluid around the trailing edge. These insights offer valuable knowledge for understanding fish propulsion efficiency and have applications in designing autonomous underwater vehicles (AUVs) and micro-air vehicles (MAVs).

Validation of computational fluid dynamics simulations for determining pressure loss coefficients of ventilation components

Ventilation systems include a variety of components for which necessary pressure loss data is often unavailable. Computational fluid dynamics simulations could substitute for expensive measurements, but validation simulations with suitable data are crucial to assess model uncertainties. Existing CFD validation studies either did not focus specifically on pressure losses, only covered few components, or did not include recent developments in turbulence modelling. In the present work, 33 bends, 4 gates and 2 tees were simulated using a consistent approach. Computational fluid dynamics simulations were validated with published data: rectangular high-edge and wide-edge bends from the experimental dataset of Sprenger, gates and diverging tees from the SMACMA guide. The considered flows cover important basic flow phenomena: deflection, splitting and flow separation. The 39 components were simulated with three turbulence models at 14 Reynolds numbers. The simulations predicted pressure loss coefficients accurately for various components. Cases with strong flow separation regions were most challenging. The model prediction uncertainty was assessed by carrying out simulations with three selected turbulence models. As in the experimental data from Sprenger, the simulations showed a distinct dependence of pressure loss coefficients on the Reynolds number for bends. In contrast, for abrupt deflections and flow separation at sharp edges, the Reynolds number dependency was minor. Technical pressure loss data of ductwork components is needed for the dimensioning, optimisation, and energy assessment of ventilation systems. The present validation study assesses the present state of the art of CFD simulations to determine pressure loss coefficients and the resulting prediction uncertainties.

CFD Simulation of Centrifugal Pump with Different Impeller Blade Trailing Edges

The centrifugal pump is one of the most widely used types of power machinery in the field of ship and ocean engineering, and the shape of the impeller blade trailing edge has an important influence on their performance. To reveal the mechanism of the effect of different trailing edges on external performance, the internal flow of 16 types of impeller blade trailing edges of a centrifugal pump, consisting of Bezier trailing edges, rounding on the pressure side, cutting on the suction side, and the original trailing edge is studied by numerical simulation. The reverse flow, shaft power, and energy loss distribution in the impeller and diffuser along the streamwise direction are analyzed by calculating them on each micro control body sliced from the fluid domain. The entropy production theory and Ω-vortex identification method are used to display the magnitude and location of energy loss and the vortex structure. Finally, a static structural analysis of the impeller with different trailing edges is performed. The results show that different impeller trailing edges can clearly affect the efficiency of the pump, i.e., the thinner the trailing edge, the higher the efficiency, with the thickest model reducing efficiency by 5.71% and the thinnest model increasing efficiency by 0.59% compared to the original one. Changing the shape of the impeller trailing edge has a great influence on the reverse flow, shaft power, and energy loss near the impeller trailing edge and diffuser inlet but has little influence on the leading part of the impeller. The distribution of local entropy production rate, energy loss, and reverse flow along the streamwise direction shows similar rules, with a local maximum near the leading edge of the impeller due to the impact effect, and a global maximum near the impeller trailing edge resulting from strong flow separation and high vortex strength due to the jet-wake flow. Thinning the impeller trailing edge and smoothing its connection with the blade can reduce the vortex strength and entropy production near the impeller trailing edge and diffuser inlet, improve the flow pattern, and reduce energy loss, thus improving the pump efficiency. In all models, the maximum equivalent stress is less than 6.5 MPa and the maximum total deformation is less than 0.065mm. The results are helpful for a deeper understanding of the complex flow mechanism of the centrifugal pump with different blade trailing edges.

A Numerical Performance Analysis of a Rim-Driven Turbine in Real Flow Conditions

The tidal turbines represent a new frontier for extracting energy from tides source. Despite the technology being mature, new solutions aimed at improving performance, reliability with reduced environmental impact, manufacturing and installation costs are currently under investigation. The Rim-driven turbine (abbreviated as RDT) was recently proposed. A RDT resembles a ducted turbine (abbreviated as DT), as both contain blades and a duct. The present study aims at investigating the detail performance and flow field of a RDT in a real flow based on the China Zhaitang Island’s tidal current data. To show the difference between the RDT and DT, simulations are also performed on the corresponding DT. It is found that the power and thrust for the two configurations exhibit time-periodic behavior that is consistent with the wave frequency. At axial flow, the fluctuation amplitude on the power and thrust increase with the increase of tip speed ratio. The RDT has higher power output when operating at lower tip speed ratio and has a potential reduction in flow resistance and disturbance with respect to the DT. At yawed flow, the fluctuation amplitude on the power and thrust decrease with the increase of yaw angle. The RDT has less capable of compensating the effect of yawed inflow in reducing the power than the DT at larger yaw angle. In addition, the power and thrust generates micro-amplitude fluctuation integrated into the main waveform, which the frequency is consistent with the turbine rotation frequency. The wake characteristics analysis reveals that the yawed flow field is more turbulent, and the two configurations suffer strong unsteady flow separation along the whole span. Strong interactions are observed between the rotor’s main wake and the duct’s upper wake. The yaw angle primarily determines the downstream wake deflection direction and significantly changes the wake shape and vortex structures. Meanwhile, the wake flow is found to recover more quickly at larger yaw angle. Besides, due to the open-center of RDT, a part free-stream flow is allowed to travel through and forms an obvious high velocity zone. The presence of open-center of RDT has avoided the low velocity zone, improved the wake structure and accelerated wakes recover, which seems to give an advantageous effect in operating a RDT.

Wind tunnel experiment on rectangular-shaped arch-supported membrane structures

This paper highlights the investigation of wind tunnel experiment on rectangular-shaped arch-supported membrane structures. Series of wind tunnel experiments are carried out to record the time history of pressure under different wind directions for both enclosed and open structures. In order to present the comparative analysis, contours and graphs for mean and peak wind pressure coefficients are plotted considering different parameters such as rise-span ratio, wind direction, and boundary wall. Study shows that under oblique wind direction, strong flow separation occurs in the windward arch edge which creates larger suction pressure coefficient on the immediate downside of windward arch. Based on area and intensity of negative pressure coefficient, 45⁰ is the most critical wind direction. Influence of rise-span ratio is noticeable. When rise-span ratio increases from 0.2 to 0.35, pressure coefficient increases significantly, however when rise-span ratio further increases to 0.5, increase in pressure coefficient is marginal. Moreover, in order to ease the engineering design application, roof surface is divided into different zones and assigned with mean and peak wind pressure coefficient.

Investigation on the flow mechanism of nacelle airframe interaction for podded blended wing body transport

For the flow interaction between the podded engine and the airframe of blended wing body configuration(BWB), taking the 300 seats class BWB civil transport NPU-BWB-300 designed by Northwestern Polytechnical University as the research object, the influence of the podded engines on the BWB airframe at typical high and low speed conditions were investigated by CFD method, and the airframe-nacelle interference mechanism was revealed. The results indicate that the podded engines mainly affect the high speed performance of BWB, but have little effect on the low speed performance. The flow interaction between the airframe and the nacelle at high speed condition is serious when podded the engines, which leads to strong shock wave and flow separation. The flow mechanism of the above-mentioned interaction is as follows: firstly, the large supersonic region and shock wave on the nacelle external surface interferes with airframe surface flow seriously, which induces shock wave and flow separation; secondly, a convergent-divergent channel is formed between the airframe and the nacelle, resulting in the "throat" effect, which produces shock wave and flow separation.

Influence of guide vane profile on the flow energy dissipation in a reversible pump-turbine at pump mode

Pumped storage hydro (PSH) is a mature and widely used energy storage technique. The hydraulic dissipation of PSH will lead to the decline of power station efficiency. Therefore, it is important to find the reasons flow energy dissipation (FED). The adjustment of the flow channel will cause the change of flow energy dissipation. Guide vane is an important component of pump turbine, which has an influence on the problems of FED. In this paper, the FED in pump turbine with two guide vane profiles are analyzed. The combination of experimental method and numerical simulation method is adopted. Through experiments, the influence of two guide vane profiles on performance is clarified, and the method to judge the severity of Q-H instability is proposed. The FED under different working conditions were analyzed by entropy production. When the flow rate is at design point, the wake and local small flow separation causes FED. When the flow rate is large, FED is caused by the strong wake and flow separation in the runner. When the flow is small, the complex flow pattern in guide vane and back-flow cause high FED. The guide vane profiles have a great influence on FED. The match of blade leading edge and flow direction and flow control ability of guide vane affect the flow state caused the variation of FED.

Influence of the inlet boundary layer on non-axisymmetric endwall contouring effects in a linear compressor cascade

The modern aero-engine compressors are commonly designed to be highly loaded which can introduce strong three-dimensional flow separation, and effective flow control is of vital importance in improving compressor performance. Non-axisymmetric endwall contouring nowadays is conducted on axial compressors to control the flow separation. In this paper, a contoured endwall is designed based on a linear CDA compressor cascade. The genetic algorithm and ANN surrogate model are used in the optimization process of the hub contouring, based on RANS simulations. Three inlet boundary layers which follow the fully turbulent boundary layer assumption are generated with different thickness, and the influence mechanism is numerically investigated. The optimal endwall with streamwise concave near the suction side provides great performance improvement at all inlet boundary layers at the aerodynamic design point. The endwall flow near the suction side is accelerated by the contoured endwall with cross-flow pressure gradient altered, suppressing the corner separation. The main pressure loss is closely related to one streamwise vortex that blocks the endwall flow. A thicker inlet boundary layer can induce early forming and increased strength on the streamwise vortex, enhancing the cross-flow movement and its interaction with the blade surface flow. In addition, the pressure loss near endwall in the inlet boundary layer is found more dominant on the endwall contouring effect than the boundary layer thickness.

High-Fidelity Simulation of Turbulent Flow Past Gaussian Bump

A spanwise-periodic computation of a turbulent flow past a Gaussian bump is performed in the form of a hybrid direct numerical simulation and wall-resolved large-eddy simulation. A fourth-order spatially accurate flow solver is employed to perform the simulation, using 10.2 billion grid points for a Reynolds number of 170,000 based on the bump height. The key findings from the simulation are reported in the acceleration and deceleration flow regions associated with the bump shape. Significant anisotropy in the normal Reynolds stresses, along both the wall-normal and streamwise directions, is observed within the acceleration region. The ratio between the Reynolds shear stress and turbulent kinetic energy in that region also experiences significant deviations from the norms of a zero pressure gradient turbulent boundary layer. The chosen Reynolds number generates strong flow separation in the adverse pressure gradient region, which is in contrast with a previous simulation at half the Reynolds number that only indicated incipient separation. An internal layer generated in the acceleration region evolves into a free shear layer that develops in the deceleration region and separates. Proper modeling of this inner layer appears crucial to predict the flow separation. Surface curvature effects on the attached flow development are also discussed.

Bayesian model evaluation of three k–ω turbulence models for hypersonic shock wave–boundary layer interaction flows

Shock wave–boundary layer interaction (SWBLI) is one of the most prevalent challenges in the field of fluid mechanics. Excessive interaction may lead to strong flow separation, which is an important factor causing the deterioration of aircraft performance. Reynolds averaged Navier–Stokes (RANS) models remain the primary methods for predicting flow separation for engineering applications. RANS simulation results typically have uncertainty owing to the drawbacks of the model structure and the variability of the parameters. Therefore, evaluation of a turbulence model based on nominal predictions may be insufficient for reflecting its actual simulation capabilities. In this study, the Bayesian method is employed to re-evaluate the simulation capabilities of three k–ω models for hypersonic SWBLI flows: the original k–ω model and two k–ω models with compressibility corrections (Wilcox's and Zeman's corrections). This is achieved by perturbing the model parameters and combining with experimental data. Bayesian model evaluation based on different datasets is performed using the embedded model error method, and the three turbulence models are compared from three aspects: posterior mean of the quantities of interests, model error, and Bayesian evidence. The results show that the Bayesian calibration can effectively improve the predictive abilities of the three models for wall pressure and wall heat flux. Different from the large differences in the nominal results, the performance of the two calibrated k–ω models with compressibility corrections are very close and superior to that of the original model. Moreover, the evaluation results of this study are also proven to be applicable to other similar flows.

Investigations of active boundary layer control using DDES

The root area of a wind turbine rotor is usually constructed by very thick airfoils with a relative thickness of more than 35%. An airfoil at that thickness is characterized with strong flow separation. To solve this issue, vortex generators can be used as an active control in order to stabilize the airflow and improve the aerodynamic performance, consequently. Boundary layer control however, investigated employing numerical techniques, strongly depend on the employed method as well as the mesh especially due to a weak vortex conservation in the RANS model. Using the method of Delayed detached eddy simulation (DDES) the solution can benefit from effort-efficient boundary layer modeling as well as LES vortex resolving apart from the surface. The present studies aims to investigate suitable numerical approach mesh dependencies in comparison to experimental data, for application to thick airfoil in future studies.

Time-resolved 3D imaging of two-phase fluid flow inside a steel fuel injector using synchrotron X-ray tomography

The multiphase flow inside a diesel injection nozzle is imaged using synchrotron X-rays from the Advanced Photon Source at Argonne National Laboratory. Through acquisitions performed at several viewing angles and subsequent tomographic reconstruction, in-situ 3D visualization is achieved for the first time inside a steel injector at engine-like operating conditions. The morphology of the internal flow reveals strong flow separation and vapor-filled cavities (cavitation), the degree of which correlates with the nozzle’s asymmetric inlet corner profile. Micron-scale surface features, which are artifacts of manufacturing, are shown to influence the morphology of the resulting liquid-gas interface. The data obtained at 0.1 ms time resolution exposes transient flow features and the flow development timescales are shown to be correlated with in-situ imaging of the fuel injector’s hydraulically-actuated valve (needle). As more than 98.5% of the X-ray photon flux is attenuated within the steel injector body itself, we are posed with a unique challenge for imaging the flow within. Time-resolved imaging under these low-light conditions is achieved by exploiting both the refractive and absorptive properties of X-ray photons. The data-processing strategy converted these images with a signal-to-noise ratio of ~ 10 into a meaningful dataset for understanding internal flow and cavitation in a nozzle of diameter 200 μm enclosed within 1–2 millimeters of steel.

Tornado-Induced and Straight-Line Wind Loads on a Low-Rise Building With Consideration of Internal Pressure

Tornado and straight-line wind events are often discussed and compared in terms of their intensity, e.g., maximum wind speed, however it is unclear to what extent tornado-induced and straight-line wind-induced wind loads are equivalent even for the same nominal intensity. This lack of understanding inhibits both tornado design philosophies and policies, and communication of tornado risk to the public. This study directly compares existing wind tunnel databases of tornado-induced and straight-line wind-induced pressures, for a similar building model, to evaluate to what extent the induced surface pressures on a typical building differ. The existing datasets used in the study are enhanced with a numerical internal pressure model to facilitate the comparison across a range of opening configurations that would be common in typical buildings. The analysis finds that differences are most pronounced in the overall distribution of pressures across the building surface, and in the magnitudes of pressures in regions of strong flow separation. However, overall the magnitudes of the peak tornado-induced pressures are reasonably similar to straight-line wind-induced pressures, with tornado-induced pressures on average 13% higher than equivalent straight-line wind-induced pressures. Ultimately, this study demonstrates a framework for such comparisons, while recognizing key sources of uncertainty and further research needs.

Method of accelerating convergence for gas kinetic algorithm based on digital constitutive relation of macroscopic equations

<sec>In the simulation of complex multi-scale flows covering various flow regimes, the computational efficiency of gas kinetic method by which the evolution equation of velocity distribution function is solved directly is the key to engineering applications. In order to accelerate simulation for steady flows, a gas kinetic algorithm accelerated by utilizing the macroscopic conservative equations with a digital constitutive relation is developed. In this algorithm, the contribution of the high-order terms of stress and heat flux in macroscopic conservative equations is determined by the gas kinetic solution. Meanwhile, the solution of the macroscopic conservative equations provides the macroscopic quantities for the equilibrium distribution function in the Boltzmann model equation, where a fully implicit scheme to solve the Boltzmann model equation is developed.</sec> <sec>Extensive validations are performed for the cavity flow, the supersonic flow around the cylinder, and the interactive rarefied flow around two side-by-side cylinders. The results from the above method are in good agreement with the results from the conventional gas kinetic unified algorithm and the direct simulation Monte Carlo method. It can be concluded that the nonlinear constitutive relation of rarefied flow can be well captured by the present method. And the ability of this method to simulate complex flows such as shock wave, strong wall shear and flow separation is demonstrated. Furthermore, the present method has shown to be much faster than the conventional gas kinetic unified algorithm, especially for the low-<i>Kn</i> flows. As the value of <i>Kn</i> increases, the acceleration rate decreases, because the effect of flow convection becomes weak. Meanwhile, more effort is needed to reduce inner loop iterations to improve its efficiency. </sec>

Nacelle-airframe integration design method for blended-wing-body transport with podded engines

Strong shock waves and flow separation often occur during the integration of nacelle and airframe for blended-wing-bodies with podded engines. To address this problem, this paper presents an integration method with numerical simulations. The philosophy of channeling flow and avoiding the throat effect on the nacelle and airframe is established based on the analysis of flow interference in the initial configuration. A parametric integration design method is proposed from two-dimensional plane to three-dimensional space with control mechanisms and selection principles of the key parameters determined by their influences. Results show that strong shock waves and flow separation can be successfully eliminated under the influence of both the reshaped channel and decelerated inflow below the nacelle. Supersonic regions around the nacelle are effectively reduced, concentrating mainly on the lip position. Thus, a significant cruise drag reduction (8.7%) is achieved though the pressure drag of the nacelle increases.

Interaction of Waves with Idealized High‐Relief Bottom Roughness

AbstractConsiderable uncertainty exists about how to represent wave bottom friction in coastal systems like reefs where orbital excursions are similar to roughness element size. Here, interactions between waves and large bottom roughness were investigated using Large Eddy Simulations of oscillatory flow over infinite hemisphere arrays. Wave amplitude, period, and hemisphere spacing were varied to investigate the dependence of kinematics and dynamics on dimensionless parameters. The net effect of topography on the oscillatory flow was assessed using a spatially and phase‐averaged Navier‐Stokes framework. Dynamics depended strongly on Keulegan‐Carpenter number (KC), the ratio of wave orbital excursion to roughness element size. For , flow separation was weak, form drag was small, stress gradients were negligible, and the main effect of topography on the flow was the inertial force associated with acceleration around roughness elements. For , strong flow separation occurred, and both drag and inertial forces were important. Phase‐dependent dispersive stresses were the main mechanism for vertical momentum transfer between the canopy layer and the overlying water column. Friction factors based on the drag force increased with KC for , different from previously proposed empirical curves, but approached these curves for high KC. Friction factors based on the total force decreased with increasing KC and were consistent with previously proposed curves. These results highlight the importance of distinguishing the total force on the bottom, the drag force that removes energy from the flow, and the shear stress above the canopy layer, which were very different for the parameter range in this study.