No-slip Bottom Boundary Condition Research Articles

Effects of the bottom boundary condition in numerical investigations of dense water cascading on a slope

The flow of dense water along continental slopes is considered. There is a large literature on the topic based on observations and laboratory experiments. In addition, there are many analytical and numerical studies of dense water flows. In particular, there is a sequence of numerical investigations using the dynamics of overflow mixing and entrainment (DOME) setup. In these papers, the sensitivity of the solutions to numerical parameters such as grid size and numerical viscosity coefficients and to the choices of methods and models is investigated. In earlier DOME studies, three different bottom boundary conditions and a range of vertical grid sizes are applied. In other parts of the literature on numerical studies of oceanic gravity currents, there are statements that appear to contradict choices made on bottom boundary conditions in some of the DOME papers. In the present study, we therefore address the effects of the bottom boundary condition and vertical resolution in numerical investigations of dense water cascading on a slope. The main finding of the present paper is that it is feasible to capture the bottom Ekman layer dynamics adequately and cost efficiently by using a terrain-following model system using a quadratic drag law with a drag coefficient computed to give near-bottom velocity profiles in agreement with the logarithmic law of the wall. Many studies of dense water flows are performed with a quadratic bottom drag law and a constant drag coefficient. It is shown that when using this bottom boundary condition, Ekman drainage will not be adequately represented. In other studies of gravity flow, a no-slip bottom boundary condition is applied. With no-slip and a very fine resolution near the seabed, the solutions are essentially equal to the solutions obtained with a quadratic drag law and a drag coefficient computed to produce velocity profiles matching the logarithmic law of the wall. However, with coarser resolution near the seabed, there may be a substantial artificial blocking effect when using no-slip.

LES and RANS simulation of wind- and wave-forced oceanic turbulent boundary layers in shallow water with wall modeling

Abstract Large-eddy simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) simulation of wind and wave forced oceanic turbulence in unstratified shallow water are performed in order to investigate the influence of wall modeling on results. The LES is also used to investigate the dependence of results on downwind and crosswind lengths of the computational domain, representative of a shallow shelf coastal ocean (10-to-30 m deep) unaffected by lateral boundaries. In an unstratified water column, wind and surface gravity-wave forcing generate Langmuir turbulence with largest scales consisting of full-depth Langmuir circulation (LC), counter rotating vortices aligned in the direction of the wind. The LC is known to enhance vertical mixing of momentum throughout the bulk flow, inducing a deviation of the near-bottom mean downwind velocity from the classical log law. In RANS simulations, a traditional wall treatment based on the assumption that the flow satisfies the classical law of the wall is not able to lead to a good representation of the deviation from the log law caused by LC without fine-tuning of one of the coefficients in the law of the wall. An alternative RANS approach is developed via a recent wall treatment presented in [1] based on weak imposition of the no-slip bottom boundary condition, leading to an accurate representation of the log-law deviation induced by the turbulence without having to tune input coefficients. From that standpoint, this weak enforcement of bottom boundary conditions is more robust and better suited for predictive modeling of flows with full-depth LC. In the case of LES, it is found that the deviation of the downwind velocity from the log law caused by the LC is well-resolved without the need to fine tune coefficients in traditional wall treatments. Finally, the use of wall modeling facilitates simulations with expanded domains over horizontal directions (downwind and crosswind). These expanded domains allow resolution of multiple interacting full-depth Langmuir cells. It is found that the horizontal size of the domain does not greatly impact depth profiles of turbulence statistics such as mean velocity and root-mean-square of velocity.

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.

Ekman layers in the Southern Ocean: spectral models and observations, vertical viscosity and boundary layer depth

Abstract. Spectral characteristics of the oceanic boundary-layer response to wind stress forcing are assessed by comparing surface drifter observations from the Southern Ocean to a suite of idealized models that parameterize the vertical flux of horizontal momentum using a first-order turbulence closure scheme. The models vary in their representation of vertical viscosity and boundary conditions. Each is used to derive a theoretical transfer function for the spectral linear response of the ocean to wind stress. The transfer functions are evaluated using observational data. The ageostrophic component of near-surface velocity is computed by subtracting altimeter-derived geostrophic velocities from observed drifter velocities (nominally drogued to represent motions at 15-m depth). Then the transfer function is computed to link these ageostrophic velocities to observed wind stresses. The traditional Ekman model, with infinite depth and constant vertical viscosity is among the worst of the models considered in this study. The model that most successfully describes the variability in the drifter data has a shallow layer of depth O(30–50 m), in which the viscosity is constant and O(100–1000 m2 s−1), with a no-slip bottom boundary condition. The second best model has a vertical viscosity with a surface value O(200 m2 s−1), which increases linearly with depth at a rate O(0.1–1 cm s−1) and a no-slip boundary condition at the base of the boundary layer of depth O(103 m). The best model shows little latitudinal or seasonal variability, and there is no obvious link to wind stress or climatological mixed-layer depth. In contrast, in the second best model, the linear coefficient and the boundary layer depth seem to covary with wind stress. The depth of the boundary layer for this model is found to be unphysically large at some latitudes and seasons, possibly a consequence of the inability of Ekman models to remove energy from the system by other means than shear-induced dissipation. However, the Ekman depth scale appears to scale like the climatological mixed-layer depth.

Traveling waves in rotating Rayleigh-Bénard convection.

A combined analytical, numerical, and experimental study of the traveling-wave wall mode in rotating Rayleigh-Bénard convection is presented. No-slip top and bottom boundary conditions are used for the numerical computation of the linear stability, and the coefficients of the linear complex Ginzburg-Landau equation are then computed for various rotation rates. Numerical results for the no-slip boundary conditions are compared with free-slip calculations and with experimental data, and detailed comparison is made at a dimensionless rotation rate Omega=274. It is found that the inclusion of the more realistic no-slip boundary conditions for the top and bottom surfaces brings the numerical linear stability analysis into better agreement with the experimental data compared with results using free-slip top/bottom boundary conditions. Some remaining discrepancies may be accounted for by the finite conductivity of the sidewall boundaries.

The influence of thermochemical convection on the fixity of mantle plumes

A general feature of both isochemical and thermochemical studies of mantle convection is that horizontal plume velocities tend to be smaller than typical convective velocities, however, it is not clear which system leads to a greater fixity of mantle plumes. We perform two- and three-dimensional numerical calculations and compare both thermochemical and isochemical cases with similar convective vigor to determine whether presence of a dense component in the mantle can lead to smaller ratios of horizontal plume velocity to surface velocity. We investigate different viscosity and density contrasts between chemical components in the thermochemical calculations, and we perform isochemical calculations with both free-slip and no-slip bottom boundary conditions. We then compare both visually and quantitatively the results of the thermochemical and isochemical calculations to determine which leads to greater plume fixity. We find that horizontal plume velocities for thermochemical calculations are similar to those from isochemical calculations with no-slip bottom boundary conditions. In addition, we find that plumes tend to be more fixed for isochemical cases with free-slip bottom boundary conditions for two-dimensional calculations, however, in three dimensions, we find that plume fixity is similar to that observed in thermochemical calculations.

The propagation of a gravity current into a linearly stratified fluid

The constant initial speed of propagation (V) of heavy gravity currents, of density ρC, released from behind a lock and along the bottom boundary of a tank containing a linearly stratified fluid has been measured experimentally and calculated numerically. The density difference, bottom to top, of the stratification is (ρb−ρ0) and its intrinsic frequency is N. For a given ratio of the depth of released fluid (h) to total depth (H) it has been found that the dimensionless internal Froude number, Fr = V/NH, is independent of the length of the lock and is a logarithmic function of a parameter R = (ρC−ρ0)/(ρb−ρ0), except at small values of h/H and R close to unity. This parameter, R, is one possible measure of the relative strength of the current (ρC−ρ0) and stratification (ρb−ρ0). The distance propagated by the current before this constant velocity regime ended (Xtr), scaled by h, has been found to be a unique function of Fr for all states tested. After this phase of the motion, for subcritical values of Fr, i.e. less than 1/π, internal wave interactions with the current resulted in an oscillation of the velocity of its leading edge. For supercritical values, velocity decay was monotonic for the geometries tested. A two-dimensional numerical model incorporating a no-slip bottom boundary condition has been found to agree with the experimental velocity magnitudes to within ±1:5%.

Circulation and boundary layers in differentially heated rotating stratified fluid

Recent laboratory experiments with rotating stratified water in a cylinder have revealed many of the predictions of linearized, analytic theory. Earlier measurements of the velocity field generated in a cylinder by top heating compared well with theory. Large stratification clearly suppressed Ekman pumping so that the interior velocity field (primarily azimuthal) responded by satisfying no-slip top and bottom boundary conditions without the need for Ekman layers . This interior flow also occupied a boundary layer of greater thickness than the Ekman layer under some conditions. Theory and experiments have now been conducted for sidewall heating. As before, experiment and theory agree well over some parameter ranges. But for some parameters, the flow is unstable. The exact nature of the instability remains poorly understood. The size of one combination of both vertical and horizontal boundary layers is governed by the Rossby radius of deformation multiplied by the square root of the Prandtl number . Sidewall boundary layers and their scales will be reviewed with the present results in mind.

Three-Dimensional Boundary-Fitted Circulation Model

A spherical coordinate, three-dimensional, nonorthogonal, boundary-fitted circulation model (contravariant formulation) for application to estuarine, coastal sea, and continental shelf waters is presented. The model employs a split mode technique where the equations are decomposed into exterior and interior modes. The exterior mode (vertically averaged) described in an earlier paper (Muin and Spaulding 1996) is solved using a semiimplicit solution technique. The interior mode (vertical structure) is solved explicitly, except for the vertical diffusion terms that are solved implicitly. The temporally and spatially varying eddy viscosity and diffusivity are determined from a turbulent kinetic energy equation and an empirically specified length scale. A series of tests are presented to evaluate model performance where analytical solutions or other numerical solutions are available for comparison. The model's ability to predict the point vertical structure of tidal flow is tested against analytic solutions employing (1) constant viscosity; and (2) an eddy viscosity varying linearly with depth with a no-slip bottom boundary condition. The ability of the model to simulate three-dimensional tidal flow was tested against an exact solution for an annular section channel with quadratically varying bathymetry. The model was also tested against analytic solutions for steady residual flow generated by density gradient, wind, and river flow in a channel. The model predicted turbulent energy distributions generated from a bottom boundary were compared to those from a previous numerical study by Davies and Jones (1990). No-slip and bottom stress formulations at the sea bed, and their effect on the vertical structure of the flow are analyzed. The model was used to predict the salinity distribution in a simple rectangular channel identical to the Rotterdam Waterway. The computational method is very economical, stable, and accurate with the CFL stability condition up to 100.

A Bottom Boundary Layer-resolving Three-Dimensional Tidal Model: A Sensitivity Study of Eddy Viscosity Formulation

Abstract A three-dimensional hydrodynamic model of the Irish Sea that resolves the bottom boundary layer, including the logarithmic layer, is developed. A finite-difference grid is employed in the horizontal, with the current profile through the vertical being determined from an expansion of functions satisfying a no-slip bottom boundary condition. By this means a continuous current profile in the vertical, resolving the bottom boundary layer, is determined. The sensitivity of computed tidal current profiles and tidal elevations to a range of eddy viscosity parameterizations and profiles is examined, with a view to determining appropriate formulations to use in three-dimensional hydrodynamic models with no-slip bottom boundary conditions. The importance of near-bed eddy viscosity in determining the dissipation of tidal energy (in a model with a no-slip condition) and hence tidal elevations, is considered together with its influence upon current profiles. Computed and observed tidal elevations and currents...

On extracting current profiles from vertically integrated numerical models

A transformation method is presented by which current profiles (of tidal or wind-induced origin) can be extracted at any horizontal position and moment in time from a vertically integrated, two-dimensional, hydrodynamic numerical model. An arbitrary vertical variation of eddy viscosity can be included in the method, which can incorporate a no-slip bottom boundary condition. The technique assumes that the sea is homogeneous. The method is used to improve the representation of bottom stress within the two-dimensional model, whereby the bottom stress is no longer related simply to the depth-mean current as in the “conventional” two-dimensional, vertically integrated model. Idealized calculations for a range of eddy viscosity profiles, show that elevations, current profiles, and time series of current extracted from this “enhanced” two-dimensional numerical model are in good agreement with currents obtained from a full three-dimensional model.

On the determination of vertical structure functions for time-dependent flow problems

A method of solving the viscous three-dimensional hydrodynamic equations is developed using an expansion of vertical structure functions (modes). Both slip and no-slip bottom boundary conditions can be incorporated in the method. Vertical structure functions for arbitrary vertical variations of viscosity are computed using either the Runge-Kutta-Merson method or an expansion in terms of spline functions. The accuracy and relative merits of these two methods is compared and contrasted. The influence of vertical eddy viscosity upon modal structure, and profiles of wind drift currents is examined for both homogeneous and stratified sea regions. The excitation and subsequent decay of the modes in response to a suddenly applied wind stress is considered, as is the influence of depth and viscosity upon the rate of convergence of expansions of vertical structure functions.

On determining the profile of steady wind-induced currents

A technique is developed to solve the steady-state hydrodynamic equations which describe wind-induced currents at a point. By using the Galerkin approach, with an expansion of complex coefficients and real functions, a continuous current profile can be computed from sea surface to sea bed for arbitrary vertical variations of viscosity. The method can incorporate slip or no-slip bottom boundary conditions, and gradients of sea surface elevation. The technique is used to investigate the influence of viscosity profile, bed stress and elevation gradient on surface current magnitude and direction. Calculations suggest that viscosity profile, bed stress, and elevation gradient all affect surface current, offering some explanation for the range of surface current magnitudes and directions reported in the literature.