On reduced-gravity flow through sills

Abstract

The hydraulic flow of a reduced-gravity fluid of non-negative potential vorticity through a sill is considered. It is shown that for any flow with a reversal of current, another, physically realisable, flow exists which is unidirectional and/or resting, and carries more flux than the original flow. Thus only non-negative flows need be considered when examining maximal hydraulic fluxes. Then, for a simple sill (one which slopes downward on the left and upward on the right, looking downstream), it is shown that zero potential vorticity flow, possibly modified by having a region of motionless fluid at its right, carries the maximum flux possible for that sill shape. This makes the calculation of maximal fluxes for a given sill considerably simpler, and examples of parabolic and V-shaped sills are computed.     

ACCURACY OF KINEMATIC WAVE AND DIFFUSION WAVE APPROXIMATIONS FOR TIME-INDEPENDENT FLOW WITH MOMENTUM EXCHANGE INCLUDED

Errors in the kinematic wave and diffusion wave approximation for time-independent (or steady-state) cases of channel flow with momentum exchange included were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors of less than 1% for KF²₀≥7·5 and up to 12% for KF²₀≤0·75 for the upstream boundary condition of zero discharge and finite depth, where K is the kinematic wave number and F₀ is the Froude number. The kinematic wave approximation was reasonably accurate except at the channel boundaries and for small values of KF²₀ (≤1). The accuracy of these approximations was significantly influenced by the downstream boundary condition both in terms of the error magnitude and the segment of the channel reach for which these approximations would be applicable. © 1997 by John Wiley & Sons, Ltd.     

Accuracy of the kinematic wave and diffusion wave approximations for time-independent flows with infiltration included

Errors in the kinematic wave and diffusion wave approximation for time-independent (or steady-state) cases of channel flow with infiltration were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors of less than 1·4% for KF²₀≥7·5, and up to 14% for KF²₀≤0·75 for the upstream boundary condition of zero discharge and finite depth, where K is the kinematic wave number and F₀ is the Froude number. The kinematic wave approximation was reasonably accurate except at the channel boundaries and for small values of KF²₀ (≤1). The accuracy of these approximations was significantly influenced by the downstream boundary, both in terms of the magnitude of the error and the segment of the channel reach for which these approximations would be applicable. © 1998 John Wiley & Sons, Ltd.     

A numerical study of detached shear layers in a rotating sliced cylinder

Abstract

The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ^¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the Rossby number arising from the analysis in the ageostrophic part of the detached shear layer.     

Errors of kinematic wave and diffusion wave approximations for time‐independent flows with infiltration and momentum exchange included

Error equations for kinematic wave and diffusion wave approximations were derived for time‐independent flows on infiltrating planes and channels under one upstream boundary and two downstream boundary conditions: zero flow at the upstream boundary, and critical flow depth and zero depth gradient at the downstream boundary. These equations specify error in the flow hydrograph as a function of space. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors below 2% for values of KF (e.g. KF ≥ 7·5), where K is the kinematic wave number and F is the Froude number. Even for small values of KF (e.g. KF = 2·5), the errors were typically less than 3%. The accuracy of the diffusive approximation was greatly influenced by the downstream boundary condition. For critical flow depth downstream boundary condition, the error of the kinematic wave approximation was found to be less than 10% for KF ≥ 7·5 and greater than 20% for smaller values of KF. This error increased with strong downstream boundary control. The analytical solution of the diffusion wave approximation is adequate only for small values of K. Copyright © 2005 John Wiley & Sons, Ltd.     

A) A. I. H. S. / A) I. A. S. H.

SUMMARY

Basing their studies on an extended Reynolds principle of similarity, the authors present in this paper a thorough theoretical investigation of the turbulent flow in pipes.

Making appropriate assumptions regarding the distribution of the turbulent apparent viscosity over the entire pipe diameter, the well-known discharge formulas, i. e. the resistance laws, of Prandtl-V. Kármán for the hydraulically smooth and the hydraulically rough region, and of Colebrook for the transition region—the accuracy of these being well established by measurements—can be clearly shown to be correct.

Thus it is also possible to determine the velocity distribution in the region near the wall for the hydraulically smooth region and the transition region, as well as for the hydraulically rough region.

The concept, that the flow with a fully developed roughness effect begins when the laminar boundary layer at the wall disappears can be seen to be in close agreement with the asssumptions made. The limits which were established for the hydraulically rough region and the transition region correspond well with the curve plotted by PRANDTL and the boundary curve of Rouse.     

Reverse currents in fast magnetic reconnection

Abstract

It is suggested that reverse currents seen in recent numerical reconnection experiments (Biskamp, 1986; Forbes and Priest, 1983) are caused by the choice of outflow boundary conditions. The specification of the normal velocity at the outflow boundary may result in a mismatch in velocity at the diffusion region which is manifested as a spike of reverse current.     

A three‐dimensional numerical investigation of the idealized planetary boundary layer

Abstract

The nonlinear equations of motion are integrated numerically in time for a region of x‐y‐z space of volume 3h × h × h, where h turns out to be a height slightly above the level where the wind first attains the geostrophic flow direction. Only the ideal case is treated of a horizontal lower boundary, neutral stability, horizontal homogeneity of all dependent mean variables except the mean pressure, and statistically steady state. The resulting flow patterns are turbulent and the eddies transport required amounts of momentum vertically.

Topics which are investigated include the relative directions of stress, wind shear and wind; differences in Ekman wind spirals for the neutral numerical case and a stable atmospheric case; profiles of dimensionless turbulence statistics; effect of allowing the mean density to be either constant or to decrease with height; effect of the wind direction or latitude upon the turbulence intensities; and characteristic structure of the eddies in the planetary boundary layer.     

Stability and stable modes of coastal fronts

Abstract

The stability of a single layer, geostrophic front of zero potential vorticity bounded by a vertical coast (wall) is investigated by means of a Rayleigh integral. It is proved that the flow of the density-driven current is stable at all wavenumbers provided the mean velocity of basic flow exceeds fL (where f is the Coriolis parameter and L is the distance between the wall and the free streamline). The frequency of the stable long waves is either zero or super-inertial.     

Effects of multi-annual climate variability on the hydrodynamic evolution (1833 to present) in a shallow aquifer system in northern Belgium

Abstract

Using a groundwater flow model and long historical meteorological time series data, the evolution of the groundwater flow regime in a multi-layered groundwater flow basin in northern Belgium during the last one and a half centuries (since 1833) is reconstructed. Model output parameters such as piezometric levels, depth to water table, seepage fluxes in the valleys and calculated baseflow to the river system are presented and inter-annual and decadal variations are evaluated against seasonal fluctuations. The main time-varying boundary condition in the model is the aquifer recharge which was estimated using the method of Thornthwaite and Mather based on precipitation and temperature data. The model does not take into account changes in boundary conditions due to changes in land use (deforestation, drainage of cultivated land) or groundwater exploitation. Variations in model output parameters are therefore only due to climatological forcing. Only the natural non-exploited state of the aquifer is considered. Although few historical piezometric measurements are available to verify model output, the results give an indication of the natural hydrodynamic variations on a time scale of decades.

Citation Van Camp, M., Coetsiers, M., Martens, K. & Walraevens, K. (2010) Effects of multi-annual climate variability on the hydrodynamic evolution (1833 to present) in a shallow aquifer system in northern Belgium. Hydrol. Sci. J. 55(5), 763–779.     

Laboratory experiments in a baroclinic annulus with heating and cooling on the horizontal boundaries

Abstract

Experiments have been performed in a cylindrical annulus with horizontal temperature gradients imposed upon the horizontal boundaries and in which the vertical depth was smaller than the width of the annulus. Qualitative observations were made by the use of small, suspended, reflective flakes in the liquid (water).

Four basic regimes of flow were observed: (1) axisymmetric flow, (2) deep cellular convection, (3) boundary layer convective rolls, and (4) baroclinic waves. In some cases there was a mix of baroclinic and convective instabilities present. As a “mean” interior Richardson number was decreased from a value greater than unity to one less than zero, axisymmetric baroclinic instability of the Solberg type was never observed. Rather, the transition was from non-axisymmetric baroclinic waves, to a mix of baroclinic and convective instability, to irregular cellular convection.     

The seepage flow through a free boundary aquifer into a river / L'écoulement d'infiltration d'un aquifère a frontière libre vers une rivière

ABSTRACT

This paper presents a model of the groundwater flow into a river from an aquifer beneath the river. The mathematical problem is to solve Laplace's equation with a free boundary and the solution procedure uses a variational inequality which leads to an approximate solution using finite differences. The method can be used to provide for example, inflow conditions in river modelling calculations.     

An experimental study of Stewartson layer separation

Abstract

The separation of sidewall boundary layers in a rotating annulus of homogeneous fluid is studied experimentally. The flow is driven by a differentially rotating lid, and a very small uniform slope of the bottom causes a weak mountain vortex pair to form in the interior, away from the sidewalls. A necessary condition for aerodynamic separation of the sidewall boundary layers is derived and compared with the experimental results. The laboratory flow separates for parameters that are just slightly more inviscid than those required by the necessary condition for the existence of adverse pressure gradients at the wall. As the bottom friction is decreased further, the flow becomes unsteady and chaotic. The most interesting aspect of this problem is that chaotic interior behavior, associated with the separated boundary layer, is observed for parameter values for which the interior topographically forced flow is, by itself, essentially linear.     

Macroturbulent coherent structures in an ice‐covered river flow using a pulse‐coherent acoustic Doppler profiler

The aim of this work is to compare macroturbulent coherent structures (MCS) geometry and organization between ice covered and open channel flow conditions. Velocity profiles were obtained using a Pulse‐Coherent Acoustic Doppler Profiler in both open channel and ice‐covered conditions. The friction imposed by the ice cover results in parabolic shaped velocity profiles. Reynolds stresses in the streamwise (u) and vertical (v) components of the flow show positive values near the channel bed and negative values near the ice cover, with two distinctive boundary layers with specific turbulent signatures. Vertically aligned stripes of coherent flow motions were revealed from statistics applied to space‐time matrices of flow velocities. In open channel conditions, the macroturbulent structures extended over the entire depth of the flow whereas they were discontinued and nested close to the boundary walls in ice‐covered conditions. The size of MCS is consequently reduced in scale under an ice cover. The average streamwise length scale is reduced from 2.5 to 0.4Y (u) and from 1.5 to 0.4Y (v) where Y is the flow depth. In open channel conditions, the vertical extent of MCS covers the entire flow depth, whereas the vertical extent was in the range 0.58Y–1Y (u) and 0.81Y–1Y (v) in ice‐covered conditions. Under an ice cover, each boundary wall generates its own set of MCS that compete with each other in the outer region of the flow, enhancing mixing and promoting the dissipation of coherent structures. Copyright © 2012 John Wiley & Sons, Ltd.     

A mathematical model of polythermal glaciers and ice sheets

Abstract

A mathematical model of the flow and temperature distribution of polythermal glaciers or ice sheets is deduced. Cold ice is treated as a non-linear viscous heat conducting fluid, while temperate ice is regarded as a binary mixture of ice and water. The simplest mixture concept with two balance laws of mass but only one balance law of momentum and energy is proposed. The field equations for the ice and water content and the boundary conditions which must hold at the free surface, at the ice-water interface, at the cold-temperature transition surface and at the rock-bed are deduced. In particular it is shown that an earlier formulation of polythermal ice due to Fowler and Larson (1978) is inconsistent. No boundary value problems are solved as the emphasis is on the physical motivation and justification of the principles.     

Inertial effects in an oceanic circulation model

Abstract

The flow in a mechanically driven thin barotropic rotating fluid system is analysed. The linear theory of Baker and Robinson (1969) is modified and extended into the non-linear regime.

An internal parameter, the “local Rossby number”, is indicative of the onset of nonlinear effects. If this parameter is 0(1) then inertial effects are as important as Coriolis accelerations in the interior of the transport-turning western boundary layer and both of its Ekman layers. The inertial effects in the Ekman layers, ignored in previous explorations of non-linear wind driven oceanic circulation, are retained here and calculated using an approximation of the Oseen type. The circulation problem is reduced to a system of scalar equations in only two independent variables; the system is valid for non-small local Rossby number provided only that the approximate total vorticity is positive.

To complete the solution for small Rossby number a boundary condition for the inertially induced transport is needed. It is found by examining the dynamics controlling this additional transport from the western boundary layer as the transport recirculates through the rest of the ocean basin. The strong constraint of total recirculation within the western boundary layer (zero net inertial transport) is derived.

The calculated primary inertial effects are in agreement with the observations of the laboratory model of Baker and Robinson (1969).

The analysis indicates the extent to which three-dimensional non-linear circulation can be reduced to a two dimensional problem.     

Investigation of the groundwater modelling component of the Integrated Water Flow Model (IWFM)

ABSTRACT

The Integrated Water Flow Model (IWFM), developed by the California Department of Water Resources, is an integrated hydrological model that simulates key flow processes including groundwater flows, streamflow, stream–aquifer interactions, rainfall–runoff and infiltration. It also simulates the agricultural water demand as a function of soil, crop and climatic characteristics, as well as irrigation practices, and allows the user to meet these demands through pumping and stream diversions. This study investigates the modelling performance of the groundwater module of IWFM using several hypothetical test problems that cover a wide range of settings and boundary conditions, by comparing the simulation results with analytical solutions, field and laboratory observations, or with results from MODFLOW outputs. The comparisons demonstrate that IWFM is capable of simulating various hydrological processes reliably.

EDITOR M.C. Acreman; ASSOCIATE EDITOR A. Efstratiadis     

Accuracy of kinematic wave and diffusion wave approximations for time-independent flows

Errors in the kinematic wave and diffusion wave approximations for time-independent (or steady-state) cases of channel flow were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors in the range 1–2% for values of KF (? 7.5), where K is the kinematic wave number and F₀ is the Froude number. Even for small values of KF (e.g. KF²₀ = 0.75), the errors were typically less than 15%. The accuracy of the diffusion wave approximation was greatly influenced by the downstream boundary condition. The error of the kinematic wave approximation was found to be less than 13% in the region 0.1 ? x ? 0.95 for KF = 7.5 and was greater than 30% for smaller values of KF (? 0.75). This error increased with strong downstream boundary control.     

Breakdown of boundary layers: (i) on moving surfaces; (ii) in semi-similar unsteady flow; (iii) in fully unsteady flow

Abstract

The breakdown and separation or reattachment of boundary layers adjoining a mainstream are studied in the three related situations (i)-(iii) of the title. For (i) the classical steady boundary layer generally admits a logarithmic singularity in the displacement when breakdown occurs on a downstream-moving surface whereas the corresponding singularity for an upstream-moving surface can be logarithmic or of minus-one-sixth form. Conversely, the breakdown can be delayed to the onset of zero mainstream flow, in which case the displacement singularity is again logarithmic. In certain flows these singularities prove to be removable locally, yielding a breakaway separation or reattachment and including the first known successes of a classical strategy in describing large-scale separation. Other flows, by contrast, require an interactive strategy. Again, even on a fixed surface a breakdown different from Goldstein's can be produced if there is a moving section of surface further upstream. The application to (ii), semi-similar unsteady boundary layers, e.g. near an impulsively started wedge-like trailing edge, then follows readily and predicts analogous forms of singularity. The corresponding singularity in displacement predicted for fully unsteady classical boundary layers, (iii), occurs within a finite time and, like (i) (usually) and (ii), a three-tiered breakdown is involved at first. Subsequently interaction comes into play. Comparisons with numerical and/or earlier work are noted. In all three situations (i)-(iii), although the dynamics involved near breakdown, separation or reattachment are predominantly inviscid, the presence of small viscosity is of significance in enforcing smoothness of the local velocity profiles.     

Terrestrial heat flow in Japan

Terrestrial heat flow, Q=K×ΔT/ΔZ cal/cm² sec has been determined at 51 localities (39 on land and 12 in the sea) in and around the Japanese Islands. The average values of observed heat flow in land and sea are 1.53µ cal/cm²sec and 1.48µcal/cm²sec respectively. These value do not differ greatly from the world’s averages. The outstanding features of the heat flow distribution are as follows:a) High heat flow region (Q>2.0µcal/cm²sec) exists in the Inner Zone of the Honshu Arc. This region of high heat flow is more distinct in the northeastern Japan than in the southwestern Japan.b) The High heat flow region seems to extend, through the Fossa Magna area, down to the Izu-Mariana Arc.c) It is also probable that a similar high heat flow zone exists in the inner side of the Kurile Arc.d) These zones of high heat flow precisely coincide with the zones of the Cenozoic orogeny in the area concerned.e) Far off the coast of the northeastern Japan, the area at about 150° E may be a high heat flow region.f) Low heat flow (Q<1.0µcal/cm²sec) prevails in the Pacific coast side of the northeastern Japan and in the oceanic area directly east of it, including the area of the Japan Trench.g) The region bounded by the above mentioned high and low heat flow regions has heat flow which is more or less normal. Based on these measurements, a « steady state ” temperature distribution in the crust has been calculated for each of the above regions of high, low and intermediate heat flow, and it was found that there is a large temperature differences between the bottom of the crust of the high and low heat flow regions: the temperature at the Moho boundary in the high heat flow regions should be as high as some 800～1000°C (d=27 km), whereas that under the low heat flow region should be only about 200°C (d=23 km). The high general temperature at the Moho under the high heat flow region seems to favor a production of magma in the upper mantle. Calculated Moho temperatures disfavor the hypothesis that the Moho boundary is due to phase transition.