Bed-Shear Stress in Turbulent Wave-Current Boundary Layers

An approach for calculating turbulent flows in a wave-current boundary layer over a slowly varying bed is presented. Waves are periodic in time with several harmonics. In this paper, we adopt a time invariant eddy viscosity model, in which the eddy viscosity is linearly proportional to the distance from the bed. The boundary-layer flow field is solved analytically in terms of Fourier components. The approach allows fast computations and can be easily included in a phase resolving wave propagation model. As a part of the results, bottom shear stress and the spatial variation of the boundary layer thickness are also obtained. Present results compare well with experimental data and can explain the asymmetries in the bottom shear stress under sawtooth shaped waves.     

Optimal configuration of blast furnace slag runner to reduce fluid flow stresses at wall using mathematical modelling

Abstract

A three-dimensional mathematical model was developed to predict the wall shear stresses due to flow of liquid slag in slag runner of 'G' blast furnace of Tata Steel under different conditions. The liquid slag flow in the slag runner was considered to be turbulent and incompressible. The model was developed for single phase, steady state and isothermal conditions. To this end, the Navier Stokes equations along with continuity and turbulence equations (standard k–? model) were simultaneously solved with appropriate boundary conditions at the associated physical boundaries of the calculation domain. Several configurations were numerically assessed with respect to reduced shear stresses on the wall of the slag runner to select the best one. Due to accelerating flow the operating heights of liquid slag (density 2800 kg m^–3 at 1500°C) within the slag runner for different configurations were estimated with the help of Bernoulli's and continuity equations and fixed before the computation. The different configurations comprised of three segments with different parameters of either elevation or radius of curvature. Relatively high shear stresses were numerically predicted at the joint area of second and third segments of the slag runner for all the configurations. The radius of curvature was found as the dominant factor to reduce the shear stress at the joint region.     

Modeling of a Rough-Wall Oscillatory Boundary Layer Using Two-Equation Turbulence Models

The standard k?ω turbulence model and two versions of blended k?ω/k?ε models have been used to study the characteristics of a one-dimensional oscillatory boundary layer on a rough surface. The wall boundary condition for the specific dissipation rate of turbulent kinetic energy at the wall is specified in terms of a function based on wall roughness. A detailed comparison has been made for mean velocity, turbulent kinetic energy, Reynolds stress, and wall shear stress with the available experimental data. The three models predict the above properties reasonably well. In particular, the prediction of turbulent kinetic energy for the rough case by the blended models is much better than that for smooth oscillatory boundary layers as reported in previous studies. As a result of the present study, the use of one of the blended models in calculating the sediment transport in coastal environments may be recommended.     

Hydrodynamic Behavior of Asymmetric Oscillatory Boundary Layers at Low Reynolds Numbers

An experimental and numerical study has been carried out to study the wave boundary layers under asymmetric waves. The experiments were conducted in an oscillating tunnel using a simple mechanical system to generate an asymmetric oscillatory motion similar to cnoidal waves. The velocities were measured by laser Doppler velocimetry and the bottom shear stress was calculated from the cross-stream velocity profile. A low Reynolds number k–ε model was used to predict the hydrodynamic properties of the cnoidal wave boundary layers. After validating the model with the experimental data, a series of numerical experiments were carried out to study the transitional behavior of these boundary layers by virtue of friction factor and phase difference between mean free-stream velocity and bottom shear stress. Finally a stability diagram was drawn to demarcate the laminar, transition, and fully turbulent regimes using the numerical results. The present study would be useful for the hydraulic and coastal engineers interested in calculating bottom shear stress in order to compute the sediment transport in coastal environments.     

Particle Image Velocimetry Measurements within a Laboratory-Generated Swash Zone

A particle image velocimetry (PIV) technique is used to make vertically resolved two-dimensional measurements in swash zone flows, which are notoriously recalcitrant to quantitative measurement. The PIV implementation directs the light sheet into the measurement region from beneath the beach thus avoiding issues of free surface diffraction effects. Fluorescent particles and an optical filter are used to ensure that only particles, and not bubbles or free surface anomalies, are imaged. The spatially and temporally resolved velocity fields measured in a plunging and spilling wave-driven swash zone are used to investigate the boundary layer structure of the mean and turbulent quantities as well as the phase evolution of the bed shear stress, near-bed turbulent kinetic energy, and the dissipation. Results suggest that vertical structure in spilling and plunging wave forced swash zones are similar. The uprush phase is dominated by bore-generated and bore-advected turbulence, which evolves analogously to grid turbulence, while the downrush phase is ultimately dominated by boundary layer generated turbulence, which compares well near-bed with classic flat plate boundary layer theory.     

Periodic Diffusional Mass Transfer near Sediment/Water Interface: Theory

Time-variable (periodic) flow over a lake bed, and the associated boundary layer development, have the potential to control or at least influence rates of mass transfer across the sediment/water interface. An analysis for instantaneous and time averaged flux of a material across the sediment/water interface for infinite supply in the water and infinite sink in the sediment is presented. The water flow above the interface is characterized by the shear velocity (U?) which is a periodic function of time with a maximum amplitude of (U?0) as may be typical of an internal seiche (internal standing wave) motion in a density stratified lake. The relationship between the shear velocity on the lake bed and the wind shear on the lake surface is illustrated for an extremely simplified two-layered lake of constant depth. For a less restrictive analysis, shear velocities on a lake bed have to be obtained either from field measurements or from a three-dimensional lake circulation model driven by atmospheric forcing including wind. Smaller and wind-sheltered lakes will have lower (U?0) and periodicities (T). The response of the diffusive boundary layer was related to the period of the periodic motion (T), Schmidt number (Sc), and shear velocity (U?). The vertical diffusive flux at the sediment/water interface was expressed by a Sherwood number (Sh), either instantaneous or time averaged. The mean Sherwood number (Shave) varies with shear velocity of the wave motion over the sediment bed, Schmidt number (Sc) and the period (T) due to the response of the diffusive boundary layer to the time variable water velocity. Effective diffusive boundary layers develop only at low shear velocities. Where they do, maximum and minimum boundary layer thickness depends on all three independent variables (T, Sc, and U?0). The diffusive boundary layer strongly affects sediment/water mass transfer, i.e., Sherwood numbers. Mass transfer averaged over a period can be substantially less than that produced by steady-state flow at the same U?0 and Sc. At Sc = 500, typical for dissolved oxygen, the mass transfer ratio can be reduced to 60% of steady state, depending on the internal wave period (T).     

Boundary Shear Distribution in Straight Ducts and Open Channels

A semianalytical model was developed to predict boundary shear distribution in straight, noncircular ducts and open channels. The model was developed using a simplified streamwise vorticity equation, which involves only secondary Reynolds stress terms. These terms are representative of transverse turbulence anisotropy and nonhomogeneity. Transverse anisotropy is modeled using a universal function. Shear stresses are incorporated into the model by applying the momentum transfer model. An empirical model was employed to calculate the effect of the channel boundary on shear stresses. The final equation was applied to calculate boundary shear distribution in triangular ducts and trapezoidal open channels. The model predictions were well correlated with experimental data.     

Large-Eddy Simulation of Turbulent Flow over a Fixed Two-Dimensional Dune

Turbulent open-channel flow over a two-dimensional dune is studied using an established large-eddy simulation code. The free surface is approximated as a shear free boundary. Turbulence statistics and instantaneous flow structures are examined. Numerical results from two computational grids agree with each other, and are also in good agreement with recently obtained experimental data. The mean velocity profiles show significant changes along the dune and there is no region that conforms to the standard law-of-the-wall. Profiles of the Reynolds stresses show distinct peaks marking the shear layer that originates from flow separation at the dune crest. Secondary peaks found further from the dune are ascribed to the shear layer over the upstream dune. Details of the separated flow and development of the flow after reattachment are well predicted. Quadrant analysis of the Reynolds shear stress shows that turbulent ejections dominate the near-wall motions. Complex water surface flow structures are visualized.     

Dissipation of Turbulent Kinetic Energy in a Tundish by Inhibitors with Different Designs

Turbulent flow of liquid steel and its control is studied using different geometries of turbulence inhibitors. Four designs of turbulence inhibitors were characterized through experiments of tracer injection in a water model and mathematical simulations using the Reynolds Stress Model (RSM) of turbulence. Inhibitor geometries included octagonal‐regular, octagonal‐irregular, pentagonal and squared. A layer of silicon oil was used to model the behaviour of tundish flux during steel flow. Fluid flows in a tundish using these geometries were compared with that in a bare tundish. Experimental and simulation results indicate that the flow in a bare tundish and a tundish using turbulence inhibitors open large areas of oil close to the ladle shroud due to strong shear stresses at the water‐oil interface with the exception of the squared inhibitor. Oil layer opening phenomena are explained by the high gradient of the dissipation rate of turbulent kinetic energy. Using the squared inhibitor the kinetic energy reports a high gradient from the tundish floor to the free bath surface as compared with other geometries.     

Prediction of hemolysis in turbulent shear orifice flow

This study proposes a method of predicting hemolysis induced by turbulent shear stress (Reynolds stress) in a simplified orifice pipe flow. In developing centrifugal blood pumps, there has been a serious problem with hemolysis at the impeller or casing edge; because of flow separation and turbulence in these regions. In the present study, hemolysis caused by turbulent shear stress must occur at high shear stress levels in regions near the edge of an orifice pipe flow. We have computed turbulent shear flow using the low-Reynolds number k-epsilon model. We found that the computed turbulent shear stress near the edge was several hundreds times that of the laminar shear stress (molecular shear stress). The peak turbulent shear stress is much greater than that obtained in conventional hemolysis testing using a viscometer apparatus. Thus, these high turbulent shear stresses should not be ignored in estimating hemolysis in this blood flow. Using an integrated power by shear force, it is optimal to determine the threshold of the turbulent shear stress by comparing computed stress levels with those of hemolysis experiments or pipe orifice blood flow.     

Rough Wall Boundary Layer on Plates in Open Channels

Experiments were performed to measure the characteristics of a turbulent boundary layer developing on a rough surface placed in an open channel flow at close proximity to the free surface. Streamwise velocity measurements were made with a one-component laser Doppler velocimeter system at the top of the spherical roughness elements. Measurements at three stations downstream of the plate leading edge show the growth of the boundary layer on the rough wall and its interaction with the exterior open-channel flow and the free surface. Resorting to the turbulence profile provides an alternative definition of the boundary layer thickness. The near-wall flow follows the well-known logarithmic law with a shift due to roughness. In the outer layer, there are two opposing effects: the free surface tends to decrease the wake component while the roughness tends to increase it. The streamwise turbulence intensity is affected by the shear and turbulence in the exterior flow, the effect of the free surface being greater than that of wall roughness.     

Case Study: Modeling of Sediment Transport and Wind-Wave Impact in Lake Okeechobee

There are increasing demands for reliable engineering tools for sediment modeling and water resources management. The Lake Okeechobee environmental model (LOEM), which was calibrated and verified to simulate sediment resuspension and transport in Lake Okeechobee, Florida, is a dependable tool to meet those demands. The LOEM contains 2,126 horizontal grid cells and 5 vertical layers. The primary hydrodynamic and sediment transport driving forces are wind waves, surface wind stresses, and inflows/outflows. The LOEM was calibrated and verified, using two sets of observed data from May 16 to June 13, 1989 and January 17 to March 3, 2000, respectively. The model results indicate that sediment solids are resuspended primarily by wind–wave action and transported by lake circulation. The strong relationship between significant wave height and suspended sediment concentration in the lake indicates that sediment resuspension is primarily driven by wind-induced waves. To simulate this sediment resuspension, the processes of wind–wave- and current-induced bottom shear stresses on the lake bed were added to the LOEM. Once resuspended, the suspended sediment is transported to different areas of the lake by wind-induced currents. The importance of wind-wave, currents, and their interactions to sediment transport is included and discussed. By using the comprehensive data set for model calibration and verification, the LOEM model is proven to be a useful tool to water sources management in the lake.     

Boundary Shear Stress in Spatially Varied Flow with Increasing Discharge

The distribution of the wall shear stress on the bed and sidewalls of an open channel receiving lateral inflow was obtained from experimental measurements of the distribution of the velocity in the viscous sublayer using a laser doppler velocimeter. The experiments were conducted in a 0.4 m wide by 7.5 m long flume. Lateral inflow was provided into the channel from above via sets of nozzles positioned toward the downstream end of the flume. Lateral inflow was provided over a length of 1.9 m. The results indicate that the local boundary shear stresses are significantly influenced by lateral inflow. The significant variation occurs near and around the region where the lateral inflow enters the channel. At various measurement positions along the lateral inflow zone, mean boundary, mean wall, and mean bed shear stresses were obtained and compared. The results indicate that the mean boundary shear stresses increase from the upstream to the downstream ends of the lateral inflow zone. The results also indicate that the mean bed shear stress is always greater than the mean wall shear stress, which are approximately 30–60% of the mean bed shear stress. The friction factor in the Darcy–Weisbach equation was obtained from both the mean boundary shear stress and from the equation describing the water surface elevation in an open channel receiving lateral inflow (equation for spatially varied flow with increasing discharge). The results indicate that the estimated friction factors from the latter approach are significantly larger. Also, the estimated friction factors from both approaches are higher than the values predicted from the Blasius equation which describes the friction factor for wide uniform open channel flows. They were also higher than values predicted from the Keulegan equation, which is an empirically derived equation for flow in roof drainage gutters. The study highlights the deficiencies in the existing equations used to predict friction factors for spatially varied flow and that further research is required to explore the distribution of boundary shear stress in an open channel receiving lateral inflow.     

LES and RANS Studies of Oscillating Flows over Flat Plate

Oscillatory flows over a flat plate are studied by using Large Eddy Simulation (LES) and Reynolds-Average Navier-Stokes (RANS) methods. A dynamic subgrid scale (SGS) model is employed in LES, while the Saffman's turbulence model in RANS. The mean velocity profile, the turbulence intensity, and the wall shear stress are computed and compared with earlier experimental and numerical works. The results indicate that the flow behaviors are quite different during the accelerating and decelerating phases of the oscillating cycle. The transition from laminar to turbulent is also investigated as a function of the Reynolds number, R, which represents the square of the ratio of the oscillation amplitude at free stream to the thickness of the Stokes layer at the plate. The present results both from LES and RANS show that the transition occurs in the range of 5 × 104 < R < 5 × 105. The evolution of the flow structure in the Stokes layer during the transition from laminar to turbulent is clearly demonstrated from the numerical results. The friction coefficient of the amplitude of oscillatory surface shear stress varies as R?0.5 with a phase angle of 45° in laminar regime and transition to R?0.23 with a phase angle of about 10° in turbulence regime. These variations in the surface shear stress with the Reynolds number are in excellent agreement with the earlier experimental results of Kamphuis and the numerical results of Blondeaux. The excellent agreement between the LES and RANS demonstrated that Saffman's turbulence model, as originally intended by Saffman, is applicable for unsteady flows.     

Experiments on Unsteady Turbulent Pipe Flow

The paper explores the time-wise evolution of selected turbulence parameters during gravity-driven flow establishment of incompressible fluids in rigid circular pipes. Two initial conditions are considered: flow starting from rest, passing through laminar-to-turbulent transition, and terminating in a turbulent steady state; and transient flow between two turbulent steady states. It is found that, in the second case, the properties considered, i.e., local temporal mean velocity and its transverse distribution, axial turbulence intensity, and wall shear stress, are monotonically increasing with time. However, for flow starting from rest, all properties are strongly affected by the development of turbulence. In particular, at the critical moment when laminar-to-turbulent transition is complete, the wall shear stress changes abruptly from one to the other, identifying wall shear stress as a very sensitive indicator of criticality.     

Response of Velocity and Turbulence to Sudden Change of Bed Roughness in Open-Channel Flow

The experimental study shows how an open-channel flow would respond to a sudden change (from smooth to rough) in bed roughness. Using a two-dimensional acoustic Doppler velocimeter and a laser Doppler velocimeter, the velocity, turbulent intensities, and Reynolds stress profiles at different locations along a laboratory flume were measured. Additionally, the water surface profile was also measured using a capacitance-type wave height meter. The experimental data show the formation of an internal boundary layer as a result of the step change in bed roughness. The data show that this boundary layer grows much more rapidly than that formed in close-conduit flows. The results also show that the equivalent bed roughness, bed-shear stress, turbulent intensities, and Reynolds stress change gradually over a transitional region, although the bed roughness changes abruptly. The behavior is different from that observed in close-conduit flows, where an overshooting property—which describes the ability of the bed-shear stress to attain a high-peak value over the section with the larger roughness, was reported. A possible reason for the difference is the variation of the water surface profile when an open-channel flow is subjected to a sudden change in bed roughness.     

Three-Dimensional Numerical Simulation of Compound Channel Flows

A three-dimensional numerical study is presented for the calculation of turbulent flow in compound channels. The flow simulations are performed by solving the three-dimensional Reynolds-averaged continuity and Navier–Stokes equations with the k?ε turbulence model for steady-state flow. The flow equations are solved numerically with a general-purpose finite-volume code. The results are compared with the experimental data obtained from the UK Flood Channel Facility. The simulated distributions of primary velocity, bed shear stress, turbulent kinetic energy, and Reynolds stresses are used to investigate the accuracy of the model prediction. The results show that, using an estimated roughness height, the primary velocity distributions and the bed shear stress are predicted reasonably well for inbank flows in channels of high aspect ratio (width/depth ≥ 10) and for high overbank flows with values of the relative flow depth greater than 0.25.     

Turbulent Boundary Layer over Symmetric Bodies with Rigid and Flexible Surfaces

This paper presents the results of an investigation concerning the development of a turbulent boundary layer over a 2D symmetrical aerofoil and a 3D axisymmetric body with rigid and flexible surfaces. The experimental work included detailed measurements of the mean velocity profiles, pressure distribution, and drag force. The thin shear layer equations were solved numerically using a modified turbulence model to obtain the characteristics of the turbulent boundary layer. The results of this study indicate a significant difference between the characteristics of flow over rigid surfaces and those of flow over flexible surfaces of the same geometry. The mean velocity of flow in the case of flexible surfaces is smaller than the corresponding velocity of flow in the case of a rigid surface for a major part of the boundary layer. The boundary layer thicknesses are consistently higher on flexible surfaces than those on the corresponding rigid surfaces. Furthermore, in the case of flexible surfaces, drag reduction was always observed. The amount of reduction was seen to be systematically dependent on the characteristics of the flexible surface.     

Apparent Roughness in Wave–Current Flow: Implication for Coastal Studies

High turbulence intensities generated by waves in the wave bottom boundary layer affect the mean current velocity and should be taken into account for calculation of currents in the presence of waves. This influence of the wave-induced turbulence on the mean current can be schematized by introducing an “apparent” bed roughness, which is larger than the physical bottom roughness. Apparent bed roughness is defined here as roughness that provides the same depth-mean velocity for current alone configuration as for the wave–current flow. A one-dimensional vertical “k–l” turbulence closure model that allows detailed time dependent flow modeling has been applied for apparent roughness computations. The domain of variable parameters is chosen according to the Israeli near-shore conditions. An approximate expression for apparent bed roughness calculations as a function of wave and current parameters based on this turbulence closure model is derived. Simulation of flow patterns on the Tel Aviv coast using the three-dimensional Costal and Marine Engineering Research Institute flow model and implementing apparent roughness maps, calculated by the approximate expression, has been performed.     

Two-Dimensional Simulation Model of Sediment Removal and Flow in Rectangular Sedimentation Basin

A steady, two-dimensional numerical model was created to study the hydrodynamics of a rectangular sedimentation basin under turbulent conditions. The strip integral method was used to formulate the flow equations, using a forward marching scheme for solving the governing partial differential equations of continuity, momentum, advection–diffusion, turbulent kinetic energy, and its dissipation. In this way the flow equations were converted to a set of ordinary differential equations (ODEs) in terms of the key physical parameters. These parameters, along with a set of shape functions, describe flow variables including the velocity, the concentration of suspended sediments, and both the kinetic energy and its dissipation rate. Four Gaussian distributions were investigated, one corresponding to each flow parameter. In order to calculate the turbulent shear stresses, a two-equation turbulence model (i.e., k-ε model) was used. A fourth order Runge–Kutta method numerically integrates the set of ODEs. Simulation results were compared with experimental data, and close agreement (generally within 5–10%) was observed.