Analytical solutions for water flow passing over a vegetal area

This study is aimed at investigating the vertical velocity profile of flow passing over a vegetal area by an analytical approach. The soil ground is considered as pervious and thus non-zero velocity at the ground surface can be estimated. The soil and vegetation layers are regarded as homogeneous and isotropic porous media. Therefore the solution of the flow can be obtained by applying the theory of turbulent flow and Biot’s theory of poroelasticity after dividing the flow field into three layers: homogenous water, vegetation and pervious soil. The velocity distribution is compared with the experimental data of [Rowiński PM, Kubrak J. A mixing-length model for predicting vertical velocity distribution on flows through emergent vegetation. J Hydrol Sci 2002;47(6):893–904] to show its validity. In addition, five dimensionless parameters denoting the variation of slope, permeability of soil, Reynolds stress, density of vegetation, and relative height of vegetation are proposed to reveal their effects on the surface water flow. The analytical solutions of flow velocity can also be simplified into simpler expressions to describe the flow passing over a non-vegetated area.     

INTERNATIONALE GLAZIOLOGISCHE GRÖNLANDEXPEDITION, 1957–1959

Abstract

Equations are derived for vapor transport within a constant flux layer of the atmosphere. The physical model is based on the Reynolds analogy for fully turbulent flow, so that vapor and momentum flux are assumed similar. The shear stress is deduced from analysis of the wind profile by means of equations available for non-neutral conditions. Empirical expressions are proposed using the friction velocity rather than the wind velocity at a given level in order to take account of the stability of the air.     

The wave-induced boundary layer under long internal waves

The boundary layer formed under the footprint of an internal solitary wave is studied by numerical simulation for waves of depression in a two-layer model of the density stratification. The inviscid outer flow, in the perspective of boundary-layer theory, is based on an exact solution for the long wave-phase speed, yielding a family of fully nonlinear solitary wave solutions of the extended Korteweg–de Vries equation. The wave-induced boundary layer corresponding to this outer flow is then studied by means of simulation employing the Reynolds-averaged Navier–Stokes (RANS) formulation coupled with a turbulence closure model validated for wall-bounded flows. Boundary-layer characteristics are computed for an extensive range of environmental conditions and wave amplitudes. Boundary-layer transition, identified by monitoring the eddy viscosity, is correlated in terms of a boundary-layer Reynolds number. The frictional drag is evaluated for laminar, transitional, and turbulent cases, and correlations are presented for the friction coefficient plus relevant measures of the boundary-layer thickness.     

Effect of the oceanic lithosphere velocity on free convection in the asthenosphere beneath mid-ocean ridges

The paper presents results obtained in experiments on a horizontal layer heated from below in its central part and cooled from above; the layer models the oceanic asthenosphere. Flow velocity and temperature profiles are measured and the flow structure under boundary layer conditions is determined (at Rayleigh numbers Ra > 5 × 10⁵). The flow in the core of a plane horizontal layer heated laterally and cooled from above develops under conditions of a constant temperature gradient averaged over the layer thickness. The flow core is modeled by a horizontal layer with a moving upper boundary and with adiabatic bounding surfaces under conditions of a constant horizontal gradient of temperature. Exact solutions of free convection equations are found for this model in the Boussinesq approximation. Model results are compared with experimental data. Temperature and flow velocity ranges are determined for the boundary layer regime. Based on the experimental flow velocity profiles, an expression is found for the flow velocity profile in a horizontal layer with a mobile upper boundary heated laterally and cooled from above. Free convection velocity profiles are obtained for the asthenosphere beneath a mid-ocean ridge (MOR) with a mobile lithosphere. An expression is obtained for the tangential stress at the top of the asthenosphere beneath an MOR and the total friction force produced by the asthenospheric flow at the asthenosphere-lithosphere boundary is determined.     

Interface between turbulent flows above and within rough porous walls

This paper explores the concept of a macroscopic boundary between turbulent flows above and within rough permeable walls. The macroscopic boundary and the associated conditions for macroscopic flow variables have been thoroughly investigated for laminar, but not for turbulent flows. The literature on laminar flows follows two main conceptual models of the boundary: sharp boundary with step changes in macroscopic variables and gradual boundary with smooth changes of variables. The former approach is usually associated with the two-domain simulation models and the latter one with the single-domain models. This paper presents the derivation of the step conditions for velocity and shear stress at the macroscopic boundary between turbulent boundary layer and turbulent porous media flows. The physical meaning of the main terms in the shear stress condition is discussed in order to clarify the relationship between two-domain and single-domain simulation models.     

Experimental and theoretical determination of concentration profiles and influence of particle characteristics in blowing snow

Snow drift transport may cause avalanches on the roads during the periods of snowfall and strong wind. To better understand the factors influencing transport we have developed a theoretical model. This model is based on the boundary layer theory, where the particle mass conservation is considered. Assuming that the saturation is reached, the concentration profile can be represented by a negative exponential law. By means of this analysis, the influence of particle characteristics is explored through the roles of threshold friction velocity and fall velocity. Using fluid mechanics laws, an analysis of the concentration profile resulting from the effect of the wind on a particle bed was also developed. For several velocities of flow and for different kinds of particles an experimental determination of the concentration profile was achieved. We used a laser visualisation and image processing technique to carry out these experiments. The obtained results fit with the values predicted by the theoretical model.     

Time-averaged sub-layer concentration profiles in oscillatory sheet flows

Although thin on the order of several to tens of millimeters, sheet flows normally comprise a lower pick-up sub-layer and an upper contact-load sub-layer, separated at the bed level (z = 0). The time-averaged concentration profile in the pick-up sub-layer shows a ‘convex upward’ curvature, but ‘concave upward’ shape characterizes the time-averaged concentration profile in the contact-load sub-layer. The time-dependent concentration in the contact-load sub-layer is approximately in-phase with the free stream flow velocity, whereas it is nearly in anti-phase with the free stream flow velocity in the pick-up sub-layer. Two distinct analytical expressions of the time-averaged concentration profiles for the respective sub-layers are proposed. The expressions are validated with detailed observation datasets collected in the Groβer Wellenkanal (GWK) prototype wave flume in Hannover, Germany. The agreement between the predicted and the measured values is excellent. Interparticle collisions in the pick-up sub-layer and convective lifting processes associated with vortex shedding in the contact-load sub-layer are considered responsible for the opposite curvatures and in-phase/anti-phase concentration variations. Both transitional boundary and reference concentration are also elaborated.     

Reinvestigation on mixing length in an open channel turbulent flow

The present study proposes a model on vertical distribution of streamwise velocity in an open channel turbulent flow through a newly proposed mixing length, which is derived for both clear water and sediment-laden turbulent flows. The analysis is based on a theoretical consideration which explores the effect of density stratification on the streamwise velocity profile. The derivation of mixing length makes use of the diffusion equation where both the sediment diffusivity and momentum diffusivity are taken as a function of height from the channel bed. The damping factor present in the mixing length of sediment-fluid mixture contains velocity and concentration gradients. This factor is capable of describing the dip-phenomenon of velocity distribution. From the existing experimental data of velocity, the mixing length data are calculated. The pattern shows that mixing length increases from bed to the dip-position, having a larger value at dip-position and then decreases up to the water surface with a zero value thereat. The present model agrees well with these data sets and this behavior cannot be described by any other existing model. Finally, the proposed mixing length model is applied to find the velocity distribution in wide and narrow open channels. The derived velocity distribution is compared with laboratory channel data of velocity, and the comparison shows good agreement.     

On turbulent diffusion from a steady line source in the surface layer under unstable stratification

Summary Solution of the diffusion equation is obtained for a linear source in the boundary layer with unstable stratification, when the Monin-Obukhov similarity theory is used for the wind profile and coefficient of vertical turbulent exchange. On the basis of the solution obtained numerically calculations are made.     

Numerical modelling of aeolian erosion over a surface with non‐uniformly distributed roughness elements

The present study is focused on the analysis of the mean wall friction velocity on a surface including roughness elements exposed to a turbulent boundary layer. These roughness elements represent non‐erodible particles over an erodible surface of an agglomeration of granular material on industrial sites. A first study has proposed a formulation that describes the evolution of the friction velocity as a function of geometrical parameters and cover rate with different uniform roughness distributions. The present simulations deal with non‐uniform distributions of particles with a random sampling of diameters, heights, positions and arrangements. The evolution (relative to geometrical parameters of the roughness elements) of the friction velocity for several non‐uniform distributions of roughness elements was analysed by the equation proposed in the literature and compared to the results obtained with the numerical simulations. This comparison showed very good agreement. Thus, the formulation developed for uniform particles was found also to be valid for a larger spectrum of particles noted on industrial sites. The present work aims also to investigate in detail the fluid mechanics over several roughness particles. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust classification for the joint velocity‐intermittency structure of turbulent flow over fixed and mobile bedforms

Two datasets of turbulence velocities collected over different bedform types under contrasting experimental conditions show similarity in terms of velocity‐intermittency characteristics and suggest a universality to the velocity‐intermittency structure for flow over bedforms. One dataset was obtained by sampling flow over static bedforms in different locations, and the other was based on a static position but mobile bedforms. A flow classification based on the velocity‐intermittency behaviour is shown to reveal some differences from that based on an analysis of Reynolds stresses, boundary layer correlation and turbulent kinetic energy. This may be attributed to the intermittency variable, which captures the local effect of individual turbulent flow structures. Locations in the wake region or the outer layer of the flow are both shown to have a velocity‐intermittency behaviour that departs from that for idealized wakes or outer layer flow because of the superposition of localized flow structures generated by bedforms. The combined effect of this yields a velocity‐intermittency structure unique to bedform flow. The use of a time series of a single velocity component highlights the potential power of our approach for field, numerical and laboratory studies. The further validation of the velocity‐intermittency method for non‐idealized flows undertaken here suggests that this technique can be used for flow classification purposes in geomorphology, hydraulics, meteorology and environmental fluid mechanics. © 2014 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.     

BED SHEAR IN EVOLVING SCOUR AT A CYLINDER:A THEORETICAL APPROACH

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A new technique for the calculation of aeration development zones and nonaerated zones in turbulent flows on the spillway faces of dams and in prismatic flumes and its field verification

A new technique for calculating turbulent aerated and nonaerated flows on spillway faces of dams and in prismatic flumes is developed and validated against field data. The technique is based on original relationships and hypotheses proposed by the author and allowing one to identify the position in a watercourse of the section where aeration appears and the section of its stabilization and to calculate, within these boundaries, the distribution of the mean velocity of liquid phase, averaged aeration factors, and the aeration depths corresponding to the conventional upper boundary of flow section with the local value of the aeration factor a _loc = 0.99.     

Three-layer model for vertical velocity distribution in open channel flow with submerged rigid vegetation

Based on the detailed laboratory experiments and theoretical analysis, a new three-layer model is proposed to predict the vertical velocity distribution in an open channel flow with submerged vegetation. The time averaged velocity and turbulence behaviour of a steady uniform flow with fully submerged artificial rigid vegetation was measured using a 3D Micro ADV, and the vertical distribution of velocity and Reynolds shear stress at different vegetation height, vegetation density and measuring positions were obtained. The results show that the velocity profile consists of three hydrodynamic regimes (i.e. the upper non-vegetated layer, the outer and bottom layer within vegetation); accordingly different methods had been adopted to describe the vertical velocity distribution. For the upper non-vegetated layer, a modified mixing length theory combined with the concept of ‘the new vegetation boundary layer’ was adopted, and an analytical model was presented to predict the vertical velocity distribution in this region. For the bottom layer within vegetation, the depth average velocity was obtained by numerically solving the momentum equations. For the upper layer within vegetation, the analytical solution was presented by expressing the shear stress as a formula fitted to the experimental data. Finally, the analytical predictions of the vertical velocity over the whole flow depth were compared with the results obtained by other researchers, and the good agreement proved that the three-layer model can be used to predict the velocity distribution of the open channel flow with submerged rigid vegetation.     

Problem of transient turbulent convection and a semiempirical profile of turbulent heat exchange coefficient in freshwater bodies

A problem of convective cooling of both freshwater and saltwater lake is formulated. A transient one-dimensional turbulent model with variable layer depth is proposed to describe the process of freezing. The dependence of layer depth on time is determined by a universal equation describing the propagation of Deardorff convective layer. A combination of the main form of equations of Kolmogorov–Obukhov theory and available experimental data is shown to allow the construction of a universal profile of turbulent exchange coefficient, depending on the height.     

Planetary Boundary Layer Flows in the Atmosphere and the Ocean at the Equator

—The boundary layer flows created by the frictional dissipation of the wind speed at the surface in the atmosphere and by surface wind stress in the ocean at the equator and in the equatorial region, are obtained by taking the influence of the surface friction on the zonal velocity as being balanced by vertical transport for the long-term mean flow and by a corresponding time variation for time-dependent flow fields. Solutions are expressed in terms of the velocities in zonal and vertical directions and the divergence of the horizontal current in the two media. It is found that under the ever present easterly flow in the lower atmosphere, the boundary layer flow in the atmosphere is convergence and ascending motion in the lower troposphere, and divergence at the surface and uplift in ocean, and in reverse directions for the westerly flow. Similar results are obtained for time-dependent wind fields and they give way to the steady asymptotic solutions when the period of the variation exceeds 10 months.     

Numerical Modeling of Wind-Induced Flows in Stratified Water Bodies

Two-dimensional (in the vertical plane) wind-induced flows in no-flow-through reservoirs are considered. A numerical algorithm in the flow function–vortex variables is proposed based on the equations of slow stratified flows in the Boussinesq and boundary layer approximations with variable coefficient of vertical turbulent exchange. An analytical solution is given for a simplified problem.     

Boundary layer dissipation and the nonlinear interaction of equatorial baroclinic and barotropic rossby waves

Two-layer equatorial primitive equations for the free troposphere in the presence of a thin atmospheric boundary layer and thermal dissipation are developed here. An asymptotic theory for the resonant nonlinear interaction of long equatorial baroclinic and barotropic Rossby waves is derived in the presence of such dissipation. In this model, a self-consistent asymptotic derivation establishes that boundary layer flows are generated by meridional pressure gradients in the lower troposphere and give rise to degenerate equatorial Ekman friction. That is to say, the asymptotic model has the property that the dissipation matrix has one eigenvalue which is nearly zero: therefore the dynamics rapidly dissipates flows with pressure at the base of the troposphere and creates barotropic/baroclinic spin up/spin down. The simplified asymptotic equations for the amplitudes of the dissipative equatorial barotropic and baroclinic waves are studied by linear theory and integrated numerically. The results indicate that although the dissipation slightly weakens the tropics to midlatitude connection, strong localized wave packets are nonetheless able to exchange energy between barotropic and baroclinic waves on intraseasonal timescales in the presence of baroclinic mean shear. Interesting dissipation balanced wave-mean flow states are discovered through numerical simulations. In general, the boundary layer dissipation is very efficient for flows in which the barotropic and baroclinic components are of the same sign at the base of the free troposphere whereas the boundary layer dissipation is less efficient for flows whose barotropic and baroclinic components are of opposite sign at the base of the free troposphere.     

Flows within the Ekman layer during spin-up of a thermally stratified fluid

Abstract

Finite-difference numerical solutions were obtained to present the flow and temperature field details within the transient Ekman layer during spin-up of a thermally stratified fluid in a cylinder. This complements the earlier studies on stratified spin-up which examined the flows in the interior core region. As the stratification increases, the following changes in the flow field are noticeable. The radial velocity in the Ekman layer decreases in magnitude. The azimuthal flows adjust smoothly from the interior region to the endwall boundary, and the Ekman layer in the azimuthal flow field fades. Vertical motions are inhibited, resulting in a weakened Ekman pumping. The axial vorticity field behaves similarly to the azimuthal flows. The temperature deviation from the equilibrium profile decreases, and the heat transfer flux from the endwall to the fluid decreases. The thickness of the thermal layer is larger than the velocity layer thickness. Illustrative comparisons of the relative sizes of the terms in the governing equations are conducted in order to assess the stratification effect in the adjustment process of the fluid.     

INHERENT DISTRIBUTION LAW OF TURBULENT INTENSITY IN SEDIMENT-LADEN FLOW

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