Rough Turbulent Flow Research Articles

INITIAL MOTION OF STREAMBEDS COMPOSED OF COARSE UNIFORM SEDIMENTS.

Results of an experimental study of the initial motion of coarse uniform sediments are presented in this paper. The experiments were conducted in an 8 m long, 0·3 m wide by 0·3 m deep tilting flume. Natural sediments were used and they ranged in size from 1 mm to 14 mm. The purpose of the experiments was to determine flow conditions associated with the initiation of bed sediment motion. To eliminate the subjectivity in defining the beginning of streambed movement, the concept of critical-bed state is linked to the intensity of sediment motion. The experiments reveal that critical-bed shear-stress for sediment motion depends not only on the grain size, but also on the bed slope. The latter fact is explained by the effect of relative depth (depth to grain-size ratio) on overall flow resistance. It is also shown that the value of critical dimensionless bed shear-stress is not constant for rough turbulent flow, as is usually assumed. The same conclusion follows from the measurements of turbulence characteristics near the surface of the bed. A revised Shields diagram relating critical stress, grain Reynolds number and bed slope is derived for different intensities of sediment motion. Finally, a diagram relating critical flow depth, bed slope and grain size is presented in a form suitable for use by design engineers.

Initial motion of streambeds composed of coarse uniform sediments

Results of an experimental study of the initial motion of coarse uniform sediments are presented in this paper. The experiments were conducted in an 8 m long, 0·3 m wide by 0·3 m deep tilting flume. Natural sediments were used and they ranged in size from 1 mm to 14 mm. The purpose of the experiments was to determine flow conditions associated with the initiation of bed sediment motion. To eliminate the subjectivity in defining the beginning of streambed movement, the concept of critical-bed state is linked to the intensity of sediment motion. The experiments reveal that critical-bed shear-stress for sediment motion depends not only on the grain size, but also on the bed slope. The latter fact is explained by the effect of relative depth (depth to grain-size ratio) on overall flow resistance. It is also shown that the value of critical dimensionless bed shear-stress is not constant for rough turbulent flow, as is usually assumed. The same conclusion follows from the measurements of turbulence characteristics near the surface of the bed. A revised Shields diagram relating critical stress, grain Reynolds number and bed slope is derived for different intensities of sediment motion. Finally, a diagram relating critical flow depth, bed slope and grain size is presented in a form suitable for use by design engineers.

Turbulent Open-Channel Flow in Circular Corrugated Culverts

Circular corrugated pipes are used as culverts at road crossings, and in many regions in Canada, the United States, and other countries, they have to be designed to allow fish passage during periods of fish migration. In many of these culverts, if the velocities are too large to allow fish passage, some type of baffles need to be installed to reduce the velocities. This paper presents the results of a laboratory study of the velocity field in turbulent open-channel flow in a circular corrugated pipe of diameter 0.622 m for three slopes of 0.55%, 1.14%, and 2.55% and a range of discharges from 30 l/s to 200 l/s. The Manning n was found to be equal to 0.023. Velocities were relatively small in some portion of the flow near the boundary of the pipe, and these low velocity regions may be useful for fish passage upstream. In the region of fully developed flow, in the central vertical plane, the longitudinal velocity was described by the Prandtl equation for rough turbulent flow, with a dip in the velocity profiles near the water surface. The velocity profiles in the noncentral planes were also described by the Prandtl equation for rough turbulent flow, but with a significant dip in the upper part of the flow. An empirical method was devised to describe the geometrical and kinematical properties of this velocity dip. The general findings of this study were also found to be valid for flow in a large corrugated pipe of diameter of 4.27 m with two slopes of 0.14% and 1.42%.

The Role of Rheology in the Pipelining of Mineral Slurries

This paper combines new experimental evidence and pipeline flow models to challenge widely held views in the literature. These include the existence of a yield stress and its determination, the existence of shear-thinning at high shear rates, and the continuum approximation which is routinely applied to these slurries. The implications of these critical points for laminar, laminar/turbulent transition and turbulent flow are investigated. It is shown that interpretation of rheological data at low and high shear rates is crucial for flow modelling in all three types of flow. It is concluded that the yield pseudoplastic model should be used for mineral slurries and low shear rate phenomena may be ignored. A high shear rate asymptote was not found or deemed necessary. The yield stress is fundamentally important for the laminar/turbulent transition and results in the critical velocity becoming independent of pipe diameter. Accurate rheology is important in smooth turbulent flow. Rough turbulent flow requires a particle roughness effect and is independent of rheology.

Friction Factor of Meandering Flows

An expression for the friction factor of rough turbulent meandering flows is suggested: flow cross section is rectangular, the plan shape of the meandering channel is sine-generated. This expression gives the value of the friction factor as a function of position (which is determined by, among other factors, the local channel curvature) and channel sinuosity. The validity of the present expression is tested against results of laboratory measurements. The measurements were carried out in two meandering channels, one typifying “small” sinuosity, the other “large” sinuosity. It is found that the vertically averaged flows in these two channels exhibit two distinctly different (“in-” and “out-going”) flow patterns, depending on whether the channel sinuosity is “large” or “small.” These two radically different flow pictures cannot be supplied by the vertically averaged equations of motion if they are solved for a constant friction factor. The present consideration of the friction factor as a function of positio...

Characteristics of Skimming Flow over Stepped Spillways

This paper presents the results of a laboratory study on the characteristics of fully developed skimming flow in a large model of a stepped spillway for two slopes, for a range of discharges with yc/h in the range of 0.7–4.4. Fully developed aerated flow on a stepped spillway can be divided into lower and upper regions, similar to those for self-aerated flow in steep chutes. The air concentration distributions in these two regions agree with the equations developed by Straub and Anderson for flow in steep chutes. It was found that the depth at which the air concentration is equal to 90% can be considered as the depth of aerated flow on stepped spillways. In the lower region, the velocity profiles were described by the Karman-Prandtl equation for rough turbulent flow when an equivalent bed roughness was used. A correlation was developed for the skin friction coefficient to predict the Reynolds shear stress at the virtual bed of the stepped spillway. It was found that the relative energy loss in the stepped spillway is in the range of 48–63%. It was also found that the mean air concentration on a stepped spillway is larger than that in a corresponding chute.

Modelling wave–current interactions in rough turbulent bottom boundary layers

The aim of the present paper is to explain some of the differences between previously published analytical and numerical models of combined wave and current bottom boundary layer flow. To this end, the Grant and Madsen (1979) model for wave–current, rough turbulent flow is modified to include both first and second harmonic time variations in the eddy viscosity ( K). The functional form of the coefficients controlling the amount of time variation is established by analysing the numerical model results of Davies (1990). The addition of time variation in K reduces the strong non-linearity exhibited by the mean stress in the original Grant and Madsen model for current dominated cases, and reproduces the veering of the current predicted by numerical turbulence closure models.

Statistical development and validation of discharge equations for natural channels

Although the Manning equation is widely accepted as the empirical flow law for rough turbulent open-channel flow, using the equation in practical situations such as slope-area computations is fraught with uncertainty because of the difficulty in specifying the value of the reach resistance, Manning's n. Riggs (1976, J. Res. US Geol. Surv., 4: 285–291) found that n was correlated with water-surface slope, and proposed a multiple-regression equation that obviates the need for estimating n in slope-area estimates of discharge. Because his relation was developed from a relatively small sample ( N = 62), had potential flaws owing to multicollinearity, and was not thoroughly validated, we used an expanded data base ( N = 520) and objective methods to develop a new relation for the same purpose: Q = 1.564 A 1.173 R 0.400 S −0.0543log S where Q is discharge (m 3 s −1), A is cross-sectional area (m 2), R is hydraulic radius (m), and S is water-surface slope. We validated Rigg's model and our model using 100 measurements not included in model development and found that both give similar results. Riggs's model is somewhat better in terms of actual (m 3 s −1) error, but ours is better in terms of relative (log Q) error. We conclude that either Riggs's or our model can be used in place of Manning's equation in slope-area computations, but that our model is preferable because it has less bias, minimizes multicollinearity, and performs better when applied to discharge changes in individual reaches. We also found that our model performs better than those of Jarrett (1984, J. Hydraul. Eng., 110: 1519–1539) or Riggs in the range of applicability of Jarrett's equation (0.15 m ≤ R ≤ 2.13 m; 0.002 ≤ S ≤ 0.052). Both Riggs's and our models significantly overestimate Q in flows satisfying both the following conditions: Q < 3 m 3s −1 and Froude number less than 0.2. For other in-bank flows in relatively straight reaches, our model can be recommended for use in slope-area computations and other applications of the Chézy or Manning equations over a wide range of channel sizes (0.41 m 2 ≤ A ≤ 8520 m 2) and slopes (0.00001 ≤ S ≤ 0.0418), thus obviating the difficulty of a priori determination of the resistance factor.

Macroscale surface roughness and frictional resistance in overland flow

The hydraulics of overland flow on rough granular surfaces can be modelled and evaluated using the inundation ratio rather than the flow Reynolds number, as the primary dimensionless group determining the flow behaviour. The inundation ratio describes the average degree of submergence of the surface roughness and is used to distinguish three flow regimes representing partially inundated, marginally inundated and well-inundated surfaces. A heuristic physical model for the flow hydraulics in each regime demonstrates that the three states of flow are characterized by very different functional dependencies of frictional resistance on the scaled depth of flow. At partial inundation, flow resistance is associated with the drag force derived from individual roughness and therefore increases with depth and percentage cover. At marginal inundation, the size of the roughness elements relative to the depth of flow controls the degree of vertical mixing in the flow so that frictional resistance tends to decrease very rapidly with increasing depth of flow. Well-inundated flows are described using rough turbulent flow hydraulics previously developed for open channel flows. These flows exhibit a much more gradual decrease in frictional resistance with increasing depth than that observed during marginal inundation. A data set compiled from previously published studies of overland flow hydraulics is used to assess the functional dependence of frictional resistance on inundation ratio over a wide range of flow conditions. The data confirm the non-monotonic dependence predicted by the model and support the differentiation of three flow regimes based on the inundation ratio. Although the percentage cover and the surface slope may be of importance in addition to the inundation ratio in the partially and marginally inundated regimes, the Reynolds number appears to be of significance only in describing well-inundated flows at low to moderate Reynolds numbers. As these latter conditions are quite rare in natural environments, the inundation ratio rather than the Reynolds number should be used as the primary dimensionless group when evaluating the hydraulics of overland flow on rough surfaces. © 1997 by John Wiley & Sons, Ltd.

Macroscale surface roughness and frictional resistance in overland flow

The hydraulics of overland flow on rough granular surfaces can be modelled and evaluated using the inundation ratio rather than the flow Reynolds number, as the primary dimensionless group determining the flow behaviour. The inundation ratio describes the average degree of submergence of the surface roughness and is used to distinguish three flow regimes representing partially inundated, marginally inundated and well-inundated surfaces. A heuristic physical model for the flow hydraulics in each regime demonstrates that the three states of flow are characterized by very different functional dependencies of frictional resistance on the scaled depth of flow. At partial inundation, flow resistance is associated with the drag force derived from individual roughness and therefore increases with depth and percentage cover. At marginal inundation, the size of the roughness elements relative to the depth of flow controls the degree of vertical mixing in the flow so that frictional resistance tends to decrease very rapidly with increasing depth of flow. Well-inundated flows are described using rough turbulent flow hydraulics previously developed for open channel flows. These flows exhibit a much more gradual decrease in frictional resistance with increasing depth than that observed during marginal inundation. A data set compiled from previously published studies of overland flow hydraulics is used to assess the functional dependence of frictional resistance on inundation ratio over a wide range of flow conditions. The data confirm the non-monotonic dependence predicted by the model and support the differentiation of three flow regimes based on the inundation ratio. Although the percentage cover and the surface slope may be of importance in addition to the inundation ratio in the partially and marginally inundated regimes, the Reynolds number appears to be of significance only in describing well-inundated flows at low to moderate Reynolds numbers. As these latter conditions are quite rare in natural environments, the inundation ratio rather than the Reynolds number should be used as the primary dimensionless group when evaluating the hydraulics of overland flow on rough surfaces. © 1997 by John Wiley & Sons, Ltd.

Wave boundary layers in rough turbulent flow

A one-layer time-invariant eddy viscosity model is specified to develop a mathematical model for describing the essential features of the turbulent wave boundary layer over a rough bed. The functional form of the eddy viscosity is evaluated based on a modified one-equation turbulence model in which the eddy viscosity varies in time and space. The present eddy viscosity model simplifies much of the mathematical complexity in many existing models. Predictions from the present model have been compared favorably with a wide range of experimental data. It is found that the eddy viscosity model adopted in the present study is physically reasonable.

On the Estimation of the Time Required for Emptying Cylindrical Reservoirs Connected to Pipes

The problem investigated in this paper is the accurate evaluation of the time required for emptying a reservoir of constant cross-sectional area in which a pipe is fixed.In the analysis, a friction coefficient equation is considered which represents the different flow regimes on a Moody diagram satisfactorily. The approximate equation for estimating the time of emptying a reservoir, considering fully rough turbulent flow is derived and discussed. A computer program for estimating the accurate and approximate times of emptying reservoirs is presented. On the basis of the computer program a number of graphs are provided and the factors affecting the above estimation are studied. It is found that in some cases the estimated approximate time is in error by more than 25%.

Application of neural networks in stratified flow stability analysis

A feed-forward back-propagation-type neural network was used to predict the flow conditions when interfacial mixing in stratified estuaries commences. This was achieved by training the network to extrapolate data from laboratory experiments performed over many years by several researchers. Before this training could be carried out, however, many decisions concerning the size of the network required and its training parameters had to be made. These decisions were made on the basis of successfully training a similar stratified flow condition, that of thermal wedges downstream of a power plant's outlet, where the theoretical solution is known. Finally, these results were compared with an approximate stability equation utilizing results from inviscid flow theory, rough turbulent flow theory, and laboratory experiments on interfacial friction. Although the agreement was not exact, it was close enough to predict what the stability conditions in real estuaries should be. This prediction was verified with the only prototype data available, that from three fjords, which agreed with both the neural network and theoretical results.

Bottom shear stress in the surf zone

To investigate the bottom shear stress in the surf zone, detailed laboratory measurements were made of the free surface elevations and velocities for the case of regular waves spilling on a rough, impermeable 1:35 slope. The velocity profiles were measured at several vertical lines in the cross‐shore direction to include the shoaling region seaward of breaking, the break point, the transition region, and the inner surf zone. Each vertical line included measuring points at a fraction of the grain height above the rough, fixed bottom. A logarithmic layer was found to exist in the bottom boundary layer for most of the phases over a wave period seaward of the break point and in the surf zone. A regression analysis was used at each phase to estimate the shear velocity and bottom roughness from the phase‐averaged horizontal velocities in the lower portion of the bottom boundary layer. The bottom friction factor was estimated from a quadratic friction equation based on the measured horizontal velocity above the bottom boundary layer together with the estimated shear velocity. The quadratic friction equation with the fitted friction factor was shown to predict the temporal variation of the bottom shear stress within a factor of 2. The bottom roughness estimated from the grain size assuming rough turbulent flow was shown to agree qualitatively with the measured values. The cross‐shore variation of the friction factor estimated from an empirical formula developed for nonbreaking waves was shown to agree within a factor of 2 of the measured values.

Turbulent boundary-layer flow over fixed aerodynamically rough two-dimensional sinusoidal waves

Results from a wind tunnel study of aerodynamically rough turbulent boundary-layer flow over a sinusoidal surface are presented. The waves had a maximum slope (ak) of 0.5 and two surface roughnesses were used. For the relatively rough surface the flow separated in the wave troughs while for the relatively smooth surface it generally remained attached. Over the relatively smooth-surfaced waves an organized secondary flow developed, consisting of vortex pairs of a scale comparable to the boundary-layer depth and aligned with the mean flow. Large-eddy simulation studies model the flows well and provide supporting evidence for the existence of this secondary flow.

A theoretical investigation of boundary layer flow and bottom shear stress for smooth, transitional, and rough flow under waves

Velocity and shear stress distributions and the relationship between maximum near‐bottom orbital velocity uom and maximum shear velocity u*m in an oscillatory boundary layer are computed for hydraulically smooth, transitional, and rough turbulent flow using unsteady boundary layer theory and a single, continuous expression for eddy viscosity K. Velocity profiles over a half wave cycle calculated using a time‐independent form of K compare favorably with available measured profiles in smooth, transitional, and rough turbulent flows; computed shear stress profiles agree reasonably well with measured stress profiles. Use of a time‐dependent eddy viscosity generally improves the agreement between measured and computed velocity and shear stress near the bed but not in the outer boundary layer. Maximum computed bottom shear stress, however, does not differ significantly from values calculated with a time‐independent K owing to the choice of u*m as the turbulent velocity scale in K. The ratio of maximum orbital velocity to maximum shear velocity is computed as a function of two nondimensional parameters, a Reynolds number Re*=u*mδw/ν and an inverse Rossby number ξ0=ωz0/u*m; δW is wave boundary layer thickness, ω is wave frequency, ν is kinematic viscosity, and z0 is the bottom roughness parameter. These independent variables of the nondimensional unsteady boundary layer equation can be related to the more commonly used wave boundary layer parameters which are expressed in terms of orbital velocity instead of shear velocity. At the fully rough and fully smooth turbulent flow limits, u*m/uom is given by a single curve as a function of ξ0 or Re*, respectively. These curves compare favorably with available measurements and expressions for wave friction factor. The nondimensional equations also yield the dependence of u*m/uom on ξ0 and Re* for transitionally rough turbulent flow.

Application of Neural Networks in Stratified Flow Stability Analysis

A feed-forward back-propagation-type neural network was used to predict the flow conditions when interfacial mixing in stratified estuaries commences. This was achieved by training the network to extrapolate data from laboratory experiments performed over many years by several researchers. Before this training could be carried out, however, many decisions concerning the size of the network required and its training parameters had to be made. These decisions were made on the basis of successfully training a similar stratified flow condition, that of thermal wedges downstream of a power plant's outlet, where the theoretical solution is known. Finally, these results were compared with an approximate stability equation utilizing results from inviscid flow theory, rough turbulent flow theory, and laboratory experiments on interfacial friction. Although the agreement was not exact, it was close enough to predict what the stability conditions in real estuaries should be. This prediction was verified with the only prototype data available, that from three fjords, which agreed with both the neural network and theoretical results.

Bottom friction beneath random waves

The effect of random waves on the bottom friction is studied by assuming that the wave motion is a stationary Gaussian narrow-band random process. The approach is also based on simple explicit friction coefficient formulas for sinusoidal waves. The probability distribution functions of the maximum bottom shear stress for laminar flow as well as smooth turbulent and rough turbulent flow are presented. The maximum bottom shear stress follows the Rayleigh distribution for laminar flow and the Weibull distribution for smooth turbulent and rough turbulent flow. Some characteristic statistical values of the maximum bottom shear stress for the three flow regimes are also given.

Frictional Resistance Of Randomly Oscillating Surfaces

The frictional resistance of randomly oscillating surfaces is studied by assuming that the oscillatory motion is a stationary Gaussian narrow-band random process. The approach is also based on simple explicit frictional resistance formulas for harmonic oscillations. The probability distribution functions of the maximum surface shear stress for laminar flow as well as smooth turbulent and rough turbulent flow are presented. The maximum surface shear stress follows the Rayleigh distribution for laminar flow and the Weibull distribution for smooth turbulent and rough turbulent flow. Examples of the friction coefficients based on the mean values of the maximum surface shear stress for the three flow regimes are also given.

Quasi-steady Property of Bottom Friction in A Wave Boundary Layer

A wave friction coefficient applicable to short and long waves is theoretically derived for laminar and rough turbulent flow. This friction coefficient approaches the conventional wave friction factor for short waves with the increase of water depth and/or with the decrease of wave period. At another limit, it asymptotically approaches the friction law for steady current with the decrease of water depth and/or with the increase of wave period. A criterion is proposed to determine the condition that the bottom friction coefficient for steady flow can be used in place of that for waves.