Eddy Viscosity Profile Research Articles

Truncation-error-based Vertical Grids For ShallowWater Models

A new method to optimize vertical nodal placement in 3D shallow water models is developed and validated. The method is based on equalizing along the water column the truncation errors for the vertical diffusion term in the momentum equations. Truncation errors are estimated a priori, based on simplifying assumptions on the eddy viscosity profile. Grids are then generated iteratively by moving the nodes to where the largest errors are expected. A validation of the overall approach for ID and 3D barotropic tidal simulations shows that grids generated with the new method compare favorably with nonuniform grids previously proposed.

A three‐dimensional model of diurnal and semidiurnal tides on the European shelf

A three‐dimensional hydrodynamic model of the continental shelf is used to examine the sensitivity of computed diurnal (K1 and O1) and semidiurnal (M2, S2, and N2) tidal elevations and currents to changes in eddy viscosity profile, friction coefficient, and water depth. The influence of the tide‐generating potential is also considered. The model uses a standard finite difference grid in the horizontal, with a functional approach in the vertical, thereby giving a continuous tidal current profile from sea surface to seabed. The magnitude of the vertical eddy viscosity depends upon the flow field. Comparisons are made between model‐computed profiles and harmonic analyses of 278 current time series, at a range of locations in varying water depths, with particular emphasis on locations where there are current measurements at a number of points in the vertical. A detailed examination of these comparisons shows that frictional effects in the near‐bed region are particularly important in determining near‐bed shear and that an accurate specification of water depth is critical for tidal current magnitude. The spatial distribution of computed current ellipses shows that the semidiurnal currents are strongest in shallow water, with the diurnal currents exhibiting an intensification in the shelf edge region. The inclusion of the equilibrium tide is particularly important for the correct determination of the magnitude of the K1 tidal current as well as K1 elevations in the North Sea. The detailed comparisons show that the magnitude of the tidal currents is correctly determined to within ±5 cm s−1 at 220 locations, with the model accurately (to within ±2 cm s−1) reproducing the vertical variation (the most sensitive test of a three‐dimensional model) of the tidal current at most locations.

Eigenfunctions in the Galerkin method of tidal current modelling

Empirical orthogonal function (EOF) analysis of acoustic doppler current profiler data is used to provide a physical justification for the use of eigenfunctions derived from simple vertical eddy viscosity profiles in the Galerkin method of three-dimensional, tidal current modelling in a homogenous sea region. A good fit of EOF modes to eigenfunctions derived from a simple parameterized vertical eddy viscosity profile is obtained. An empirical determination of the most appropriate expression for the temporal dependence of such vertical eddy viscosity profiles is also given.

Analysis of unsteady forced convection in turbulent duct flow

The results of a theoretical analysis and experimental study of unsteady forced convection with turbulent flow in the thermal entrance region of a circular duct, subjected to a periodically varying inlet temperature, are presented. The related unsteady energy equation for the thermal entrance region is analytically solved under a general boundary condition of the fifth-kind, accounting for the effects of both external convection and wall heat capacitance. Together with empirical models for expressing both eddy viscosity and turbulent velocity profiles, the generalized integral transform technique is used to provide hybrid analytical-numerical solutions for the thermal response of the fluid with a prescribed accuracy. An experimental setup was built and used to validate the mathematical modeling wherein the thermal response of the fluid is obtained in terms of amplitudes and decay indices with respect to the inlet condition for fully developed turbulent flow, due to a sinusoidal variation of the inlet temperature. Satisfactory agreement prevailed between the theoretically and experimentally determined heat transfer characteristics for a succession of axial positions, thereby establishing the theoretical model and numerical technique. To further enhance the model's practical applicability to engineering problems, the analytical technique is extended in obtaining results on the effects of the modified Biot number, fluid-towall thermal capacitance ratio, and the Reynolds number, on the thermal response of the fluid within the temperature field, which are presented in tabular forms.

A transport approach to the convolution method for numerical modelling of linearized 3D circulation in shallow seas

AbstractA new method for solving the linearized equations of motion is presented in this paper, which is the implementation of an outstanding idea suggested by Welander: a transport approach to the convolution method. The present work focuses on the case of constant eddy viscosity and constant density but can be easily extended to the case of arbitrary but time‐invariant eddy viscosity or density structure. As two of the three equations of motion are solved analytically and the main numerical ‘do‐loop’ only updates the sea level and the transport, the method features succinctness and fast convergence.The method is tested in Heaps' basin and the results are compared with Heaps' results for the transient state and with analytical solutions for the steady state. The comparison yields satisfactory agreement. The computational advantage of the method compared with Heaps' spectral method and Jelesnianski's bottom stress method is analysed and illustrated with examples.Attention is also paid to the recent efforts made in the spectral method to accelerate the convergence of the velocity profile. This study suggests that an efficient way to accelerate the convergence is to extract both the windinduced surface Ekman spiral and the pressure‐induced bottom Ekman spiral as a prespecified part of the profile.The present work also provides a direct way to find the eigenfunctions for arbitrary eddy viscosity profile. In addition, mode‐trucated errors are analysed and tabulated as functions of mode number and the ratio of the Ekman depth to the water depth, which allows a determination of a proper mode number given an error tolerance.

Sensitivity of the Inner-Shelf Circulation to the Form of the Eddy Viscosity Profile

Abstract The sensitivity of the inner-shelf circulation to the form of the vertical mixing is examined using a steady, linear, two-dimensional, eddy viscosity model. For both alongshelf wind stress and pressure gradient forcing, the alongshelf circulation over the inner shelf is insensitive to the form of the eddy viscosity profile. However, the cross-shelf circulation is sensitive to the form of the eddy viscosity profile. In particular, the location and width of the cross-shelf divergence in the Ekman transport over the inner shelf, and hence the corresponding upwelling or downwelling, depend on the form of the eddy viscosity profile.

Optimal estimation of eddy viscosity for a quasi‐three‐dimensional numerical tidal and storm surge model

AbstractIt is shown that the eddy viscosity profile in a quasi‐three‐dimensional numerical tidal and storm surge model can be estimated by assimilation of velocity data from one or more current meters located on the same vertical line. The computational model used is a simplified version of the so‐called vertical/horizontal splitting algorithm proposed by Lardner and Cekirge. We have estimated eddy viscosity both as a constant and as a variable parameter. The numerical scheme consists of a two‐level leapfrog method to solve the depth‐averaged equations and a generalized Crank‐Nicolson scheme to compute the vertical profile of the velocity field. The cost functional in the adjoint scheme consists of two terms. The first term is a certain norm of the difference between computed and observed velocity data and the second term measures the total variation in the eddy viscosity function. The latter term is not needed when the data are exact for the model but is necessary to smooth out the instabilities associated with ‘noisy’ data. It is shown that a satisfactory minimization can be accomplished using either the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) quasi‐Newton algorithm or Nash's truncated Newton algorithm. Very effective estimation of eddy viscosity profiles is shown to be achieved even when the amount of data is quite small.

An intercomparison of three-dimensional tidal hydrodynamic models of the Irish Sea

This paper deals with an intercomparison of two three-dimensional hydrodynamic numerical models based upon very different approaches in the vertical. In the horizontal domain, both models use an identical grid, open boundary conditions, water depths and coastal topography and hence results of the comparison can truly be attributed to variations in the vertical. Moreover, since both models use σ-coordinates in the vertical, changes in topography are represented in a similar manner. A range of algebraic eddy viscosity profiles are used in the models, and the sensitivity of tidal current profiles to variations in this parameter are considered in detail. One model uses a functional or modal approach in the vertical and resolves the bottom boundary layer (a no slip condition is applied at the sea bed). The other model uses a finite difference grid in the vertical with a slip condition at the bed. The roughness height is derived from the Chezy coefficient used in depth averaged calculations. The advantages/disadvantages and the correspondence between the modal approach and the finite difference approach for the vertical are considered. The sensitivity of model results to changes in the representation of internal friction and bed friction are considered in detail by comparing model results with observations of M2 tidal elevations and currents. The results show that both types of modelling approaches are capable to reproduce the water levels equally well. This also holds for the currents and the vertical current variation, although the current speeds tend to be overpredicted. The type of flow considered is modelled equally well with application of a slip condition and a no slip condition at the bed. Notwithstanding their conceptual differences, the behaviour of the models is very similar when compared with the measurements: the model results tend to be clustered.

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...

Inverse Adjoint Estimation of Eddy Viscosity for Coastal Flow Models

This study developed an adjoint inverse estimation method for vertical eddy viscosity studies. The method formulated a cost functional that consists of two terms: the first describes the lack of fit between model results and avilable data, and the second measures the variance of eddy viscosity with depth. Weights were used to vary the relative influence of each term. The details of the study are described and the findings are presented and discussed. The method was tested using velocity data at the center of a rectanular sea domain. Wind and tidal forcing are considered as well as eddy viscosity profiles.

On the application of a hydrodynamic model for a limited sea area

The Saudi Arabian Oil Company (Saudi Aramco) plans to install a new Gas/Oil Separation Plant (GOSP) in the northern Safaniya offshore oil field on the north-western side of the Arabian Gulf. Design engineers require analysis of typical three-dimensional velocity fields in the area of interest, and specification of an appropriate location for the cooling water outfall in order to minimize recirculation. In this paper, a three-dimensional hydrodynamic model for a limited sea region based on a modified Vertical/Horizontal Splitting (VHS) method is developed. The VHS model is modified so that at the open boundaries both water level and height are specified. Thus, the effect of the large scale circulation in the rest of the basin is brought in through the open boundary conditions. Extensive model testing and calibration is carried out. An appropriate eddy viscosity profile is selected. The hydrodynamic model is used to analyze water circulation in the Safaniya sea area for typical seasonal wind conditions. The most appropriate locations for the cooling water outfall are indicated.

Turbulence model effects on separated flow about a prolate spheroid

The three-dimensional separated flow about a prolate spheroid at high incidence is numerically investigated using the F3D thin-layer Navier-Stokes code. The effect of different turbulence models on the flowfield solution and the characteristics of the predicted flow are analyzed. The models used in this study are the Baldvvin-Lomax algebraic model, the Baldvvin-Lomax model as modified for crossflow separation by Degani and Schiff, and a modified version of the Johnson-King model with and without the Degani-Schiff crossflow modifications applied. The Johnson-King model is applied to assess the importance of modeling nonequilibrium effects in predicting flow about a slender body at high incidence. The computations are made for steady-state, fully turbulent flow. The results are compared with experimental pressure data and with computational results obtained by Panaras and Steger, using the identical code and grid, but with their own modifications to the Baldwin-Lomax model. In addition, the computed solutions are analyzed using surface flow patterns, helicity density contours, and turbulent eddy-viscosity profiles. The results of this analysis provide insight into the effects of the turbulence models on flow characteristics and demonstrate the effect of the models on the accurate prediction of highly separated and vortical flows about a slender body.

Modelling currents in highly sheared surface and bed boundary layers

A method is presented for accurately determining currents in the high shear surface and bed boundary layers of a homogeneous sea region under conditions of meteorological and tidal forcing. The Galerkin technique is used in the vertical, with a mixed basis set composed of eigenfunctions (modes) of the eddy viscosity profile and two additional functions to account for shear in the two boundary layers. Calculations of wind induced currents in a closed rectangular basin, and oscillatory pressure driven flow at a point clearly demonstrate that this new approach is significantly more accurate than the “classical” eigenfunction method. For wind induced flow, in which the surface stress is specified, there is no restriction on the form of the additional function used at the surface. However in the case of the additional function at the seabed, analysis and calculations show that there is a stability condition which must be satisfied.

Solution of the 3D linear hydrodynamic equations using an enhanced eigenfunction approach

AbstractA method is presented for solving the 3D hydrodynamic equations in homogeneous sea regions using the Galerkin approach in the vertical with a mixed basis set. The basis set is composed of eigenfunctions of the eddy viscosity profile and a fixed function through the vertical, the amplitude of which is related to the externally applied surface wind stress. By this means the high‐shear near‐surface layer, which has previously been difficult to resolve using eigenfunction expansions, is accurately represented in the solution.The computational advantages of this approach compared with other basis functions, in terms of computer time and memory, and the ease of implementation on parallel processors with vector facilities are briefly discussed.The accuracy of the method and the choice of the additional function is demonstrated for the problem of wind‐induced currents in a rectangular sea region.Calculations clearly show that for wind‐induced currents this new approach is significantly more accurate than the ‘classical’ eigenfunction method. Also, the new method retains the advantages of the eigenfunction approach, namely insight into the mechanisms involved and ease of implementation on vector‐parallel computers, together with minimization of computer time and memory.

Transient effects in wave-current boundary layer flow

Abstract Previous studies of combined wave and current bottom boundary layer flow have concentrated on the final converged state of the flow following the addition of waves to a current. While this final state is of primary interest to modellers and engineers, it pre-supposes that such a state is actually attained in reality, and this may not always be the case. In addition, it overlooks the interesting and complicated transient effects which occur as a wave-current flow evolves from one state to another. The present study concentrates attention on the transient effects predicted by a “one-equation” turbulence closure model. Results of case studies are presented in which waves are superimposed co-linearly on a current (“forward problem”), and are then removed from the converged wave-current flow (“backward problem”). Two formulations of the “forward” and “backward” problems are discussed. In the first the steady component of the pressure gradient driving the mean flow is held constant throughout, and in the second the steady component of the mass flux is held constant. In each case the detailed evolution of the profiles of mean velocity, turbulent energy, mixing length, eddy viscosity and shear stress are discussed. More generally, the question of the convergence timescale of a combined wave-current flow is considered, and a convergence criterion is proposed.

On the importance of time varying eddy viscosity in generating higher tidal harmonics

By using a spectral/modal expansion in the vertical, a method is developed for determining tidal current profiles subject to either a slip or no‐slip bottom boundary condition. An arbitrary vertical profile of eddy viscosity can be used in the calculation, the magnitude of which can be related to the flow field. The nature of the modal solution is such that the high shear bottom boundary layer can be accurately resolved using a few modes (of order five), compared with the order of 45 grid boxes on a logarithmic transformation which would be required in a finite difference approach. Calculations show the importance of a time‐varying eddy viscosity in generating higher harmonics when a no‐slip condition is applied at the seabed. By relating the current one meter above the bed to the bed stress, a value of friction coefficient C100 in reasonable agreement with observations was obtained from these no slip calculations. However the value of friction coefficient was strongly dependent upon the height above the bed, at which the current was calculated, with friction coefficient increasing rapidly in the near‐bed region. Current profiles on the continental shelf determined using the modal method developed in the paper were in good agreement with observations, with computed profiles in the Celtic Sea giving the observed midwater maximum.

On extracting tidal current profiles from vertically integrated two‐dimensional hydrodynamic models

A numerically accurate and computationally efficient method is developed for extracting tidal current profiles from vertically integrated two‐dimensional models. The method can incorporate an arbitrary profile of eddy viscosity and determines tidal current profiles satisfying a no slip condition at the sea bed. Comparisons with observations and current profiles computed with full three‐dimensional models, confirm the accuracy of the method. Bottom boundary layer thicknesses determined from these current profiles clearly show regions of the European continental shelf where the bottom boundary layer extends to the sea surface. The boundary between these regions and other areas where the bottom layer is less than the water depth corresponds quite closely to the known position of tidal fronts.

A hydrodynamic model for wind-driven and tidal circulation in the Arabian Gulf

The vertical/horizontal splitting (VHS) model is extended and adapted to simulate tidal and wind-driven circulation in the Arabian Gulf. The extensions include the use of finite differences for the depth-averaged equations, the use of the depth following coordinate in the vertical, and relating the bottom friction to the bottom velocity. The VHS model is adapted to the Arabian Gulf. Two implementations are presented. The first, HYDRO1, simulates wind-driven and/or tidal circulation in the Arabian Gulf. A two-block grid system in cartesian coordinates is used with a finer grid on the western coast of the gulf. The second, HYDRO2, simulates wind-driven and/or tidal circulation in any user-defined area in the Arabian Gulf. HYDRO2 enables the user to study in detail a specific region of interest in the gulf. Both implementations carry out three-dimensional computation when the conventional two-dimensional hypotheses are locally inadequate and when specified by the user. Elsewhere, two-dimensional computation is employed. Implementation and testing details including specification of appropriate eddy viscosity profile, wind drag coefficient, and Chezy coefficient are discussed. A case study is presented to illustrate the model implementation and capabilities.

The relaxation of a turbulent boundary layer in an adverse pressure gradient

The relaxation of a reattached turbulent boundary layer downstream of a wall fence has been investigated. The boundary layer has an adverse pressure gradient imposed upon it which is adjusted in an attempt to bring the boundary layer into equilibrium. This is done by adjusting the pressure gradient so as to bring the Clauser parameter (G) down to a value of about 11.4 and then maintain it constant. In the region from the reattachment point to 2 or 3 reattachment lengths downstream, the boundary layer recovers from the initial major effects of reattachment. Farther downstream (where G is constant) the pressure-gradient parameter changes very slowly and profiles of non-dimensionalized eddy viscosity appear self-similar. However, pressure gradient and eddy viscosity are both roughly twice as large as expected on the basis of previous studies of equilibrium turbulent boundary layers. It is not known whether equilibrium has been achieved in this downstream region. This is another illustration of the great sensitivity of boundary-layer structure to perturbations.

Comparison of 2-D and 3-D models of the steady wind-driven circulation in shallow waters

The classical theory of the steady wind-driven circulation in shallow waters described by the linear three-dimensional equations is generalized to include arbitrary variations of the vertical eddy viscosity. The stream function equation is compared to the corresponding equation of a two-dimensional model. The influence, on the solution, of various eddy viscosity profiles and/or bottom boundary conditions is discussed. From the comparison of the equations, a parameterization of the bottom stress in terms of transport and wind stress is derived such that the two-dimensional model can yield results identical to those of the three-dimensional model.