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Geophysical flows of practical interest encompass turbulent boundary layer flows. The velocity profile in turbulent flows is generally described by a log- or a power-law applicable to certain zones of the boundary layer, or by wall-wake law for the entire zone of the boundary layer. In this study, a novel theory is proposed from which the power-law velocity profile is obtained for the turbulent boundary layer flow. The new power-law profile is based on the conservation of mass and the skin friction within the boundary layer. From the proposed theory, analytical expressions for the power-law velocity profile are presented, and their Reynolds-number dependency is highlighted. The velocity profile, skin friction coefficient and boundary layer thickness obtained from the proposed theory are validated by the reliable experimental data for zero-pressure gradient turbulent boundary layers. The expressions for Reynolds shear stress and eddy viscosity distributions across the boundary layer are also obtained and validated by the experimental data. © 2011 Institute of Geophysics.

World University root node

Department of Civil & Infrastructure Engineering

Acta Geophysica

Power-law velocity profile in turbulent boundary layers: An integral reynolds-number dependent solution

This experimental study is devoted to quantification of the near-bed turbulence characteristics at an entrainment threshold of noncohesive sediments. Near the bed, the departure in the distributions of the observed time-averaged streamwise velocity from the logarithmic law is more for immobile beds than for entrainment-threshold beds. In the Reynolds shear stress distributions, a damping that occurs near the bed for sediment entrainment is higher than that for immobile beds. Quadrant analysis reveals that in the near-bed flow zone, ejections and sweeps on immobile beds cancel each other, giving rise to the outward interactions, whereas sweeps are the dominant mechanism toward sediment entrainment. The bursting duration for entrainment-threshold beds is smaller than that for immobile beds. On the other hand, the bursting frequency for entrainment-threshold beds is larger than that for immobile beds. The third-order correlations indicate that during sediment entrainment, a streamwise acceleration associated with a downward flux and advection of streamwise Reynolds normal stress is prevalent. The streamwise and the downward vertical fluxes of turbulent kinetic energy (TKE) increase with sediment entrainment. The TKE budget proves that for sediment entrainment, the pressure energy diffusion changes drastically to a negative magnitude, indicating a gain in turbulence production. © 2011 American Society of Civil Engineers.

Fulltext

Journal of Hydraulic Engineering

Near-bed turbulence characteristics at the entrainment threshold of sediment beds

Universal probability density functions (PDFs) of two-dimensional turbulent velocity fluctuations, Reynolds shear stress and conditional Reynolds shear stresses in flows over smooth and rough beds are obtained using a Gram-Charlier series expansion based on the exponential distribution. To include skewness and kurtosis, the series is truncated up to moments of Order 4. The distributions of PDFs obtained theoretically and from the experimental data are in agreement. The conditional Reynolds shear stresses related to the ejections and sweeps are well represented by the exponential distribution, but those related to the outward and inward interactions depart from the theoretical distributions. © 2010 International Association for Hydro-Environment Engineering and Research.

Journal of Hydraulic Research

Universal probability distributions of turbulence in open channel flows

The paper presents a study on the influence of gravity on the incipient motion and the bed-load transport of sediment. The computation of critical bed-shear stress is revisited considering the balance of forces (hydrodynamic forces and submerged self-weight) acting on a solitary sediment particle lying on an arbitrary sloping bed. Modified effective bed-shear stress and the corresponding critical bed-shear stress, which are defined to assess the incipient motion of sediment in the direction of resultant force, are applied for the estimation of bed-load transport rate in the direction of resultant force. The sediment transport induced by the gravitational force, which is oblique to the direction of the drag force induced by flow, is incorporated into the bed-load transport equation. This modified model provides a reasonable prediction of the critical bed-shear stress and the bed-load transport rate. The model is validated by experimental data. It can be applied to steep slopes and can also avoid the problem of singularity that arises in numerically calculation of sediment transport rate. Additionally, the vectorial transport rate obtained in the model calculation can be implemented in a numerical simulation of channel bed evolution. © 2010 ASCE.

Sediment transport on arbitrary slopes: Simplified model

Double-averaged turbulence characteristics in a uniform flow over a gravel bed are studied. The log-law for the velocity is preserved above the interfacial sublayer, while the velocity within that sublayer follows a polynomial law. The Reynolds shear stress distribution within the roughness sublayer has a considerable damping. A quadrant analysis suggests that the sweeps become the prevalent mechanism within the interfacial sublayer. Within the interfacial sublayer, the quadrant plots of the form-induced velocities form a tilted pseudo-elliptical shape, suggesting that the fluid undergoes a cyclic process of upward-streamwise deceleration and downward-streamwise acceleration. An upward-downstream flux of the turbulent kinetic energy is prevalent over the entire flow depth, while a downward-upstream flux corresponding to sweep events is the mechanism within the interfacial sublayer. The energy budget indicates that the turbulent production, dissipation and diffusion, having attained their peaks, reduce sharply within the interfacial sublayer, and the pressure energy diffusion changes drastically to a negative magnitude below the virtual bed level. © 2010 International Association for Hydro-Environment Engineering and Research.

Double-averaging turbulence characteristics in flows over a gravel bed

A distinguished universal aspect of wall-bounded turbulent flows is Theodore von Kármán's universal logarithmic law (log-law) of the wall, describing the time-averaged streamwise velocity variation with distance from the wall. The log-law has a universal von Kármán constant κ governing the velocity gradient. There remain a number of cases of the non-universality of κ in fluvial streams. In particular, it behaves as a variable in flows with low submergence, or if there is bed- and suspended-load transport. This research focuses on the non-universality of κ raising various issues and inviting future research directions. © 2010 International Association for Hydro-Environment Engineering and Research.

Non-universality of von Kármán's κ in fluvial streams

The influence of seepage (lateral flow) on the turbulence characteristics in free-surface flows over an immobile rough boundary is investigated. Steady flows having zero-pressure gradient over an immobile rough boundary created by uniform gravels of 4.1 mm in size were simulated experimentally with injection (upward seepage) and suction (downward seepage) applied through the boundary. A Vectrino (acoustic Doppler velocimeter) was used to measure the instantaneous velocities, which are analyzed to explore second- and third-order correlations, turbulent kinetic energy, turbulent energy budget, and conditional Reynolds shear stresses. It is observed that the second-order correlations decrease in presence of injection and increase in suction. The turbulent diffusivity and mixing length increase in presence of injection and decrease in suction. The third-order correlations suggest that the ejections are prevalent over the entire flow depth. The near-boundary flow is significantly influenced by the existence of upward seepage, which is manifested by a reduction in streamwise flux and the vertical advection of streamwise Reynolds normal stress. In addition, the upward flux and the streamwise advection of vertical Reynolds normal stress are also affected. The streamwise flux of turbulent kinetic is found to migrate upstream, while the vertical flux of turbulent kinetic energy is transported upward. The fluxes increase in presence of injection and decrease in suction. Energy budget evidences a lag between the turbulent dissipation and production and an opposing trend in the turbulent and pressure energy diffusions. A quadrant analysis for the conditional Reynolds shear stresses reveals that the ejection and sweep events are the primary contributions toward the total Reynolds shear stress production, with ejections dominating over the entire flow depth. The effect of seepage is shown to affect the magnitude of such events. However, in case of sweeps, this phenomenon is the opposite. The mean-time of occurrence of ejections and that of sweeps in suction are more persistent than those in no-seepage and injection. © 2010 ASCE.

Journal of Engineering Mechanics

Turbulence characteristics in flows subjected to boundary injection and suction

During the past decades, several investigations were conducted to assess the adequacy of countermeasures against local scour at bridge piers. The countermeasures can be broadly classified into two categories: (1) flow-altering and (2) bed-armouring countermeasures. Flow-altering countermeasures, as treated herein, can be classified into four main groups based on their shapes and performances. A comprehensive review of the up-to-date studies on various types of flow-altering countermeasures is presented, placing special emphasis on recently proposed methods while also revisiting studies of former methods. Experimental conditions of different studies under which the tests were performed are critically examined. A preliminary evaluation of the previously conducted experiments indicates that the results of several tests were influenced by side-wall, sediment size, flow shallowness and temporal effects. Experimental results on the combinations of different countermeasures, including their limitations and difficulties in field applications, are also discussed. Finally, suggestions are presented for further research on selected countermeasures. © 2010 International Association for Hydro-Environment Engineering and Research.

Flow-altering countermeasures against scour at bridge piers: A review

Wooden debris structures are commonly used in stream restoration to alter sediment transport and flow distributions to improve fish habitats in degraded streams. Various wooden structures were tested in a laboratory flume to determine their ability to change flow patterns, resulting in local scour. A parametric study was made to predict scour hole depth. The results indicate that the underminer structure should cover the full channel width and divert the whole flow below the structure. Maximum scour depth was obtained with a single structure perpendicular to the flow direction and overall scour depth was governed by structural height and sediment particle size distribution. With increasing structure angle with respect to flow, the area and volume of the scour hole increase, whereas the scouring depth decreases. Coarser sediment size reduces scour depth for both single and consecutive structures. The optimum underminer spacing ratio was found to be 2 to 3 times the channel width. © 2010 International Association for Hydro-Environment Engineering and Research.

Effect and design of an underminer structure

Live-bed scour at a vertical-wall abutment is experimentally investigated with specific attention paid to the conceptual issues concerning the temporal development of local scour phenomenon. First explored are the time scales for the initial rising phase of the time variation of scour depth. An appropriate identification of such scales and of their normalizing parameters makes it possible to recognize a quantitative dependency of nondimensional time scales on flow intensity. Second, the time scales for the subsequent fluctuations around a mean equilibrium value are considered. Experimental results indicate that the quasiperiodical fluctuations of scour depths do not always correspond to those of bed forms. A conceptual model is outlined to explain this aspect. © 2010 ASCE.

Temporal scales for live-bed scour at abutments

This paper presents an experimental study on turbulent flow characteristics in submerged plane wall jets subjected to injection (upward seepage) and suction (downward seepage) from the wall. The vertical distributions of time-averaged velocity components, turbulence intensity components and Reynolds shear stress at different horizontal distances are presented. The horizontal distributions of wall shear stress determined from the Reynolds shear stress profiles are also furnished. The flow field exhibits a decay of the jet velocity over a horizontal distance. The wall shear stress and the rate of decay of the jet velocity increase in the presence of injection and decrease with suction. Based on the two-dimensional Reynolds-averaged Navier-Stokes equations of a steady turbulent flow, the velocity and Reynolds shear stress distributions in the fully developed zone subjected to no seepage, injection and suction are theoretically computed. The response of the turbulent flow characteristics to injection and suction is analysed from the point of view of similarity characteristics, growth of the length scale and decay of the velocity and turbulence characteristics scales. The significant observation is that the velocity, Reynolds shear stress and turbulence intensities in the fully developed zone are reasonably similar under both injection and suction on applying the appropriate scaling laws. An analysis of the third-order moments of velocity fluctuations reveals that the inner layer of the jet is associated with the arrival of low-speed fluid streaks causing an effect of retardation. On the other hand, the upper layer of the jet is associated with the arrival of high-speed fluid streaks causing an effect of acceleration. Injection influences the near-wall distributions of the third-order moments by increasing the upward turbulent advection of the streamwise Reynolds normal stress. In contrast, suction influences the near-wall distributions of the third-order moments by increasing the downward turbulent advection of the streamwise Reynolds normal stress. Also, injection and suction change the vertical turbulent flux of the vertical Reynolds normal stress in a similar way. The streamwise turbulent energy flux travels towards the jet origin within the jet layer, while it travels away from the origin within the inner layer of the circulatory flow. The turbulent energy budget suggests that the turbulent and pressure energy diffusions oppose each other, and the turbulent dissipation lags the turbulent production. The quadrant analysis of velocity fluctuations reveals that the inward and outward interactions are the primary contributions to the Reynolds shear stress production in the inner and outer layers of the jet, respectively. However, injection induces feeble ejections in the vicinity of the wall. © 2010 Cambridge University Press.

Journal of Fluid Mechanics

Submerged wall jets subjected to injection and suction from the wall

This study presents the turbulent flow field in submerged plane wall-jets on horizontal fully rough walls detected by a Vectrino velocimeter. For the comparison between the fully rough and smooth submerged wall-jets, the smooth submerged wall-jet case was also revisited. The two-dimensional Reynolds averaged boundary layer equations of a steady turbulent flow are analyzed to determine the velocity and Reynolds shear stress profiles in the fully developed zone of smooth and fully rough submerged wall-jets. The response of the turbulent flow characteristics in submerged wall-jets to wall roughness is examined from the point of view of similarity characteristics, growth of the length scale, and decay of the velocity and turbulence characteristics scales; and compared with the response of those to smooth and transitionally rough walls. The significant observation is that with an appropriate scaling, the velocity, Reynolds shear stress and turbulence intensities in the fully developed zone of fully rough submerged wall-jets are reasonably similar. The rate of decay of jet-velocity on fully rough walls is greater than that on smooth wall, but it is less than that on transitionally rough walls due to the presence of roughness sub-layer on the fully rough walls. Analysis of the third-order moments of velocity fluctuations reveals that the jet-layer is associated with the arrival of low-speed fluid parcel causing an effect of retardation; while the inner-layer of circulatory flow is associated with the arrival of high-speed fluid parcel causing an effect of acceleration. Wall roughness influences the peak values (both positive and negative) of the third-order moments increasing them significantly. Thus, on fully rough walls, the arrival processes of low- and high-speed fluid parcels become stronger in the jet-layer and the inner-layer of circulatory flow, respectively. Also, the peak values of streamwise and vertical flux of turbulent kinetic energy increase significantly in the presence of fully rough walls. © 2010 International Association of Hydro-environment Engineering and Research, Asia Pacific Division.

Journal of Hydro-Environment Research

Fully rough submerged plane wall-jets

This paper presents the results of a laboratory flume experimental study on the interaction of bank vegetation and gravel bed on the flow velocity (primarily on the location of the maximum velocity, Umax) and the Reynolds stress distributions. The results reveal that the dip of the maximum velocity below the water surface is up to 35% of flow depth and the difference between Umax and the velocity at the water surface is considerable in the presence of vegetation on the walls. The zone of the log-law varies from y/h=2 up to 15 percent of flow depth and it does not depend on distance from the wall. Deviation of the velocity profile in the outer layer over a gravel bed with vegetation cover on the walls is much larger than the case of flow over a gravel bed without vegetation cover on the walls. The presence of vegetation on the walls changes uniform flow to non-uniform flow. This fact can be explained by considering the nonlinear Reynolds stress distribution and location of maximum velocity in each profile at different distances across the flume. The Reynolds stress distributions at the distance 0.02 m from the wall have negative values and away from the wall, they change the sign taking positive values with specific convex form with apex in higher location. Average of von Karman constant κ for this study is equal to 0.16. Based on κ=0.16, the methods of Clauser and the Reynolds stress are compatible for determination of shear velocity. © 2009 International Research and Training Centre on Erosion and Sedimentation and the World Association for Sedimentation and Erosion Research.

International Journal of Sediment Research

Influence of bank vegetation and gravel bed on velocity and Reynolds stress distributions

Flow characteristics around a circular cylinder positioned near a plane boundary (on which laminar boundary layer flow develops in the absence of circular cylinder), are investigated for Reynolds numbers R ranging from 7.8 × 102 to 1.15 × 104. Particle image velocimetry and fiber laser Doppler velocimetry were used to measure the velocity fields and velocity time histories, respectively. Flow structures are particularly revealed using flow visualization technique at R=7.8 × 102 for gap ratios G/D (where G is the net gap between the surface of circular cylinder and the plane boundary), varying from 0 to 4. Based on the experimental results, the variation of Strouhal number of shedding vortex (or eddy) with G/D, the mechanism of vortex shedding suppression, and the streamwise velocity profiles of the upper shear layers and gap flows for small G/D are all discussed. Although the regular, alternate vortex shedding is suppressed for G/D&lt;0.5, the periodicity could be detected due to the vortex (or eddy) shedding from the upper shear layer of the circular cylinder. Gap flow switching randomly is found and first put forward to be the main reason of multipeak or broadband spectral characteristics of the shedding event at a certain small gap ratio. It is also found that the streamwise velocity profiles of the upper shear layer, where periodic shedding eddies originate, exhibit well-behaved similarity. In addition, a unique similarity of mean streamwise velocity profiles of the gap flows is demonstrated for G/D≤0.3. For R &lt; 4 × 103, the S increases as G/D decreases to its maximum around G/D ≅ 0.5 and then decreases as G/D decreases. For R ≥ 4 × 103, although most of the previous studies indicate that the S is insensitive to G/D, the present study shows that S still increases as G/D decreases but the variations of S are in a small range (i.e., 0.18≤S≤0.22). © 2009 ASCE.

Flow characteristics around a circular cylinder placed horizontally above a plane boundary

The paper presents a design method to determine the maximum equilibrium scour depth at piers embedded in a sand-bed overlain by a thin armor-layer of gravels using the experimental data of Dey and Raikar and those of Ettema. The proposed equation is in terms of empirical relationships, termed K-factors, which account for the effects of flow depth, pier shape, flow intensity, bed sediment size and armor gravel size on scour depth for individual cases of scour holes as identified by Dey and Raikar. These K-factors are determined by fitting envelope curves to the experimental data. The estimated largest possible scour depths that can occur at circular and square piers with an armor-layer are 3.15 and 3.47 times the pier width, respectively. © 2009 Korean Society of Civil Engineers and Springer-Verlag Berlin Heidelberg GmbH.

KSCE Journal of Civil Engineering

Maximum scour depth at piers in armor-beds

The one-dimensional (1D) momentum equation for free surface flow over curved boundaries has so far been traditionally developed integrating the Reynolds equations across the flow section in a boundary-fitted system of coordinates, such as flows over spillways and round-crested weirs. The formulation of the Reynolds equation was performed in a Cartesian coordinate system instead of boundary-fitted coordinates. Consequently, the traditional free surface flow equations do not strictly apply to the flow over curved boundaries. In this study, a novel generalized (1D) momentum equation for steady free surface flow over curved boundaries is developed by considering a modified formulation of the boundary-fitted coordinates. In an application of the present model, the singular-point of a round crested weir is found to be closer to the crest than as predicted by the traditional theory. A generalized expression for the pressure distribution is also proposed. © 2009 International Association of Hydraulic Engineering and Research.

One-dimensional channel flow equations with curvature revisited

Steady state suspended-load of sediment transported in flow over erodible beds usually is treated by the advection-diffusion approach, though in recent years, it is being treated as a two-phase flow phenomenon incorporating kinetics of sediment particles. Among the advection-diffusion approaches, Rouse's (1937) equation is the well-known, although a number of researchers in later periods have attempted to improve it by modifying the mixing length concept taking into account other aspects. In this paper, the advection-diffusion approach and associated logarithmic law of flow velocity are revisited. It is concluded from the logarithmic law that the Reynolds shear stress is a linear function of height above the bed, which reduces to bed shear stress in the case of a long horizontal channel. As a consequence, it is shown that the volumetric concentration of sediment is best approximated by the sum of two power laws of height above the bed. An equation is derived for the suspended-load transport rate in terms of elementary functions. © 2009 International Research and Training Centre on Erosion and Sedimentation and the World Association for Sedimentation and Erosion Research.

Suspended load in flows on erodible bed

Based on the Reynolds averaged Navier-Stokes (RANS) equations and the time-averaged continuity equation, a theory of turbulent shear flow over an undulating sand bed is developed addressing the instability criterion of plane sand beds in free-surface flows leading to the formation of sand waves. In the analysis, the integration of RANS equations leads to generalized Saint Venant equations, in which the time-averaged streamwise velocity is characterized by a power law obtained from turbulence closure, treating the curvilinear streamlines by the Boussinesq approximation. As a consequence, the modified pressure distribution has a departure from the traditionally linear hydrostatic pressure distribution. The instability analysis of a plane sand bed yields the curves of the Froude number versus nondimensional wave number, determining an instability zone for which at lower Froude numbers (less than 0.8), the plane bed becomes unstable with the formation of dunes; whereas at higher Froude numbers, the plane bed becomes unstable with the formation of standing waves and antidunes. For higher Froude numbers, the experimental data for antidunes lie within the unstable zone; while for lower Froude numbers, the same is found for dunes with some experimental scatter. © 2009 The American Physical Society.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Reynolds averaged theory of turbulent shear flows over undulating beds and formation of sand waves

Experiments on clear-water scour below underwater pipelines (initially laid on the sediment bed) in uniform and nonuniform sediments under steady flow were conducted. Equilibrium scour profiles were modeled by a cubic polynomial. The experimental results are examined to describe the influence of various parameters on equilibrium scour depth. The equilibrium scour depth ds increases with increase in approach flow depth h for shallow flow depths, becoming independent of higher flow depths when h/b&gt;5, where b=pipe diameter. However, the curves of scour depth versus sediment size d and Froude number Fb have a maximum value of ds/b=1.65 at b/d=27 and Fb=0.6. The influence of sediment gradation on scour depth is prominent for nonuniform sediments, which reduce scour depth to a large extent due to the formation of armor layer within the scour hole. The influence of different shaped cross sections of pipes on the scour depth was investigated, where the shape factors for circular, 45° (diagonal facing) and 90° (side facing) square pipes obtained as 1, 1.29, and 1.91, respectively. Using the data of scour depths at different times, the time variation of scour depth is scaled by an exponential law, where the nondimensional time scale increases sharply with increase in Froude number characterized by the pipe diameter. In addition, clear-water scour below circular pipelines laid on a thinly armored sand bed (the sand bed is overlain by a thin armor layer of gravels) was experimentally studied. Depending on the pipe diameter, armor gravel, and bed-sand sizes, three cases of scour holes were recognized. The comparison of the experimental data reveals that the scour depth below a pipeline with an armor layer under limiting stability of the surface particles (approach flow velocity nearly equaling critical velocity for surface particles) is greater than that without armor layer for the same sand bed, if the secondary armoring formed within the scour hole is scattered. In contrast, the scour depth with an armor layer is less than that without armor layer for the same sand bed, when the scour hole is shielded by the secondary armor layer. © 2008 ASCE.

Clear-water scour below underwater pipelines under steady flow

Results of an experimental study on clear-water scour at submerged cylindrical obstacles (circular cylinders) in uniform bed sediments under steady flow are presented. The scour depths at submerged circular cylinders are compared with the scour depths at corresponding unsubmerged cylinders (extended above the free surface of flow) of the same diameters under similar flow and bed sediment conditions. The scour depth decreases with an increase in submergence ratio. A submergence factor is introduced to determine the scour depth at a submerged cylinder from the information of the scour depth at an unsubmerged cylinder of the same diameter. In addition, the flow fields along the upstream vertical plane of symmetry of unsubmerged and submerged cylinders are presented through vector plots, which reveal that the dimension and strength of the horseshoe vortex decreases with an increase in submergence ratio. The horseshoe vortex circulations, which decrease with an increase in submergence ratio, are computed from the vorticity contours by using the Stokes theorem. © ASCE 2008.