Modelling the non-linear interaction of wind and tide: Its influence on current profiles

A single-point model in the vertical is used to examine the coupling between tidal currents and wind-driven flows in shallow near-coastal regions. Calculations using both a linear slip and a no-slip condition at the sea bed clearly show that coupling between tidal and wind-driven currents cannot occur in a linear model with a time-independent eddy viscosity. However with a physically more realistic time-varying viscosity related to the flow field, coupling does occur, the magnitude of this non-linear interaction depending upon the change in eddy viscosity over a tidal cycle and the intensity of shear in the vertical. A point model in the vertical with flow induced by an oscillatory pressure gradient and an additional constant wind stress is used to examine the influence of viscosity parametrization and water depth upon this coupling. The solution in the vertical is accomplished using both a functional approach and a finite difference method. Some conclusions as to the relative merits of these approaches, particularly the use of a transformed grid in the case of high-shear surface and bed boundary layers, are made in the paper.     

Parameter estimation for a three‐dimensional numerical barotropic tidal model with adjoint method

The parameters of a three‐dimensional (3‐D) barotropic tidal model are estimated using the adjoint method. The mode splitting technique is employed in both forward and adjoint models. In the external mode, the alternating direction implicit method is used to discretize the two‐dimensional depth‐averaged equations and a semi‐implicit scheme is used for the 3‐D internal mode computations. In this model the bottom friction is expressed in terms of bottom velocity which is different from the previous works. Besides, the bottom friction coefficients (BFCs) are supposed to be spatially varying, i.e. the BFC at some grid points are selected as the independent BFC, while the BFC at the other grid points can be obtained through linear interpolation with these independent BFCs. On the basis of the simulation of M₂ tide in the Bohai and North Yellow Seas (BNYS), twin experiments are carried out to invert the prescribed distributions of model parameters. The parameters inverted are the Fourier coefficients of open boundary conditions (OBCs), the BFC and the vertical eddy viscosity profiles. In these twin experiments, the real topography of BNYS is installed. The ‘observations’ are produced by the tidal model and recorded at the position of TOPEX/Poseidon altimeter data, tidal gauge data and current data. The experiments discuss the influence of initial guesses, model errors and data number. The inversion has obtained satisfactory results and the prescribed distributions have been successfully inverted. The results indicate that the inversion of BFC is more sensitive to data error than that of OBC and the vertical eddy viscosity profiles. Copyright © 2007 John Wiley & Sons, Ltd.     

Turbulence modeling for the axially rotating pipe from the viewpoint of analytical closures

A single-point model eddy viscosity model of rotation effects on the turbulent flow in an axially rotating pipe is developed based on two-point closure theories. Rotation is known to impede energy transfer in turbulence; this fact is reflected in the present model through a reduced eddy viscosity, leading to laminarization of the mean velocity profile and return to a laminar friction law in the rapid rotation limit. This model is compared with other proposals including linear redistribution effects through the rapid pressure-strain correlation, Richardson number modification of the eddy viscosity in a model of non-rotating turbulence, and the reduction of turbulence through the suppression of near-wall production mechanisms. PACS 47.27.Eq, 47.32.-y     

A dynamic framework for functional parameterisations of the eddy viscosity coefficient in two-dimensional turbulence

This paper puts forth a dynamic framework for investigating the subgrid scale physics of decaying two-dimensional turbulence utilising a modular approach with eddy viscosities in various functional forms. The derivation of the low-pass spatially filtered implementation of the Navier–Stokes equations is given by using the vorticity-streamfunction formulation. Two different implementations of the viscosity kernels based on the representation of the eddy viscosity terms are proposed and tested by solving a canonical two-dimensional decaying turbulence problem. It is seen that the implementation of the eddy viscosity formulation plays a distinct role in the dissipative behaviour of the different viscosity kernels. Among eddy viscosity kernels tested, we found that the Leith eddy viscosity formulation yields superior results with higher correlation coefficients.     

Numerical problems in simulating tidal flows with a frictional-velocity-dependent eddy viscosity and the influence of stratification

The paper deals with the accurate determination of tidal current profiles in both homogeneous and stratified regions when a no-slip condition is used at the seabed with a flow-dependent eddy viscosity related to the depth-mean current or the bed frictional velocity. Calculations show that it is essential to accurately resolve the high-shear region which occurs at the bed and across the pycnocline/thermocline in the case of stratified flow. A computationally accurate and economic method of resolving these regions is demonstrated using the Galerkin method with a set of basis functions designed to accurately reproduce the high-shear layers which occur in these regions. With a flow-independent eddy viscosity a stability analysis can be readily performed and an unconditionally stable algorithm developed. However, with a flow-dependent viscosity, in particular a viscosity computed from the frictional velocity, a non-linear numerical instability can occur. A method of maintaining numerical stability in this case is also described. The importance of near-bed resolution to the computed value of the frictional velocity is demonstrated and its influence on the total tidal velocity profile is illustrated by a number of idealized calculations using various eddy viscosity formulations. The influence of stratification on the computed tidal profiles is shown in the latter part of the paper.     

On the evolution of a vortical motion in a turbulent fluid

A model turbulence field consisting of essentially discrete but interacting eddies is presented. The development of a single eddy is described by considering its interaction with the remainder of the turbulence field. Vortex line stretching is found to produce initially an increase in the mean square vorticity of the eddy. Consideration of the decay rate of vorticity suggests that the rate of dissipation of turbulent energy per unit mass depends weakly upon the Reynolds number of the turbulence.     

Turbulence and cavitation models for time-dependent turbulent cavitating flows

Cavitation typically occurs when the fluid pressure is lower than the vapor pressure at a local thermodynamic state,and the flow is frequently unsteady and turbulent.To assess the state-of-the-art of computational capabilities for unsteady cavitating flows,different cavitation and turbulence model combinations are conducted.The selected cavitation models include several widely-used models including one based on phenomenological argument and the other utilizing interface dynamics.The kε turbulence model with additional implementation of the filter function and density correction function are considered to reduce the eddy viscosity according to the computed turbulence length scale and local fluid density respectively.We have also blended these alternative cavitation and turbulence treatments,to illustrate that the eddy viscosity near the closure region can significantly influence the capture of detached cavity.From the experimental validations regarding the force analysis,frequency,and the cavity visualization,no single model combination performs best in all aspects.Furthermore,the implications of parameters contained in different cavitation models are investigated.The phase change process is more pronounced around the detached cavity,which is better illus-trated by the interfacial dynamics model.Our study provides insight to aid further modeling development.     

可压缩多介质粘性流体和湍流的大涡模拟

在可压缩多介质粘性流体动力学高精度计算方法MVPPM(multi-viscous-fluid piecewise parabolicmethod)基础上,引入Smagorinsky和Vreman亚格子湍流模型,采用大涡数值模拟方法求解可压缩粘性流体NS(Navier-Stokes)方程,给出适用于可压缩多介质流体界面不稳定性发展演化至湍流阶段的计算方法和二维计算程序MVFT(multi-viscosity-fluid and turbulence)。在2种亚格子湍流模型下计算了LANL(Los Ala-mos National Laboratory)激波管单气柱RM不稳定性实验,分析了气柱的形状、流场速度以及涡的特征,通过与LANL实验和计算结果的比较可知,Vreman模型略优于Smagorinsky模型,MVFT方法和计算程序可用于对界面不稳定性发展演化至湍流阶段的数值模拟。     

Some Results Based on Near-wall Turbulence

In this paper some results based on the near-wall mean characteristics of a bounded turbulent flow are presented. In the study empirical polynomials and experimental data for an attached wall-bounded flow are used with the objective of studying the time-scales similarities in the very-near-wall region. As a result of this analysis a new parameter to characterize the high to low-Reynolds turbulence transition in the context of turbulence models is proposed. A relation for the Reynolds stress in the buffer region is also proposed, which allows a mean velocity profile through the buffer region to be obtained. This mean velocity profile joins the logarithmic ones at the beginning of the inertial sub-layer and fits appropriately to experimental data. Another result derived from the previous analysis is an expression for the eddy viscosity through the very-near-wall region. Comparison of this expression with those relations used by four known low-Reynolds models reveals that it has a very good performance.     

Turbulence time scales in mixing box experiments

Laboratory experiments are performed to examine eddy time scales in turbulence generated by an oscillating grid in homogeneous fluid using tanks with varying depth and fixed high aspect ratio horizontal cross-section. For high ratios of depth to width, a mean circulation develops in the form of a pair of counter-rotating vortices. In a new technique, pearlescent dye is employed to measure eddy time scales and to determine their power law scaling dependence on the distance from the oscillating grid. Two scaling regimes are observed, one near the source of turbulence and one at intermediate depths. At intermediate depths, the scaling exponent is found to increase if the total fluid depth is larger. We discuss the significance of these results on the use of the law-of-the-wall scaling in the upper oceanic mixed layer.     

Sensitivity of plume dynamics to the parameterization of vertical mixing

A three‐dimensional primitive equation, baroclinic numerical model incorporating a range of turbulence closure schemes is used to investigate the effects of vertical diffusion of momentum and density upon the spread of a freshwater plume, with particular reference to the Ebro plume. Initial calculations show that there are some differences in the horizontal spread and vertical mixing of the plume when diffusion coefficients are computed from a two‐equation turbulence energy model compared with a one‐equation model. To understand results from the turbulence energy models, the sensitivity of the plume dynamics to variations in the coefficient of vertical eddy viscosity and diffusivity is also considered, with increases in these parameters having a significant effect upon the cross‐shore and along‐shore spread of the plume. Also, increasing these parameters changes the plume characteristics from supercritical to subcritical and reduces the occurrence of meandering and baroclinic instability along the plume's off‐shore edge. However, differences in the southerly spread (the direction of Kelvin wave propagation in the model) of the plume (although not its northerly spread) produced by changes in diffusion coefficients are small compared with the influence of changes in the bottom slope, upon the along‐shore southerly spread of the plume, which moves in the direction of Kelvin wave propagation in the near coastal region. Results from the series of calculations are used as a guide in experimental design, with reference to a planned experiment in the Ebro region involving a coastal HF Radar deployment, as well as off‐shore measurements. Calculations suggest that surface current measurements from a coastal HF Radar, together with a detailed survey of the density field associated with the plume, may be an appropriate, although indirect, means of determining suitable mixing coefficients to use in plume discharge problems. Detailed measurements of water depth variation will also be required. Copyright © 1999 John Wiley & Sons, Ltd.     

Effective eddy viscosities in implicit modeling of decaying high Reynolds number turbulence with and without rotation

We assess the applicability of the numerical dissipation as an implicit turbulence model. The nonoscillatory finite volume numerical scheme MPDATA developed for simulations of geophysical flows is employed as an example of a scheme with an implicit turbulence model. A series of low resolution simulations of decaying homogeneous turbulence with and without Coriolis forces in the limit of zero molecular viscosity are performed. To assess the implicit model the long-time evolution of turbulence in the simulations is investigated and the numerical velocity fields are analyzed to determine the effective spectral eddy viscosity that is attributed to the numerical discretization. The detailed qualitative and quantitative comparisons are made between the numerical eddy viscosity and the theoretical results as well as the intrinsic eddy viscosity computed exactly from the velocity fields by introducing an artificial wave number cutoff. We find that the numerical dissipation depends on the time step and exhibits contradictory dependence on rotation: it is overestimated for rapid rotation cases and is underestimated for nonrotating cases. These results indicate that the numerical dissipation may fail to represent the effects of the physical subgrid scale processes unless the parameters of a numerical scheme are carefully chosen.     

A divergence-conforming hybridized discontinuous Galerkin method for the incompressible Reynolds-averaged Navier-Stokes equations

We present a hybridized discontinuous Galerkin (HDG) method for the incompressible Reynolds-averaged Navier-Stokes equations coupled with the Spalart-Allmaras one-equation turbulence model. The method extends upon an HDG method recently introduced by Rhebergen and Wells for the incompressible Navier-Stokes equations. With a special choice of velocity and pressure spaces for both element and trace degrees of freedom (DOFs), the method returns pointwise divergence-free mean velocity fields and properly balances momentum and energy. We further examine the use of different polynomial degrees and meshes to see how the order of the scalar eddy viscosity affects the convergence of the mean velocity and pressure fields, specifically for the method of manufactured solutions. As is standard with HDG methods, static condensation can be employed to remove the element DOFs and thus dramatically reduce the global number of DOFs. Numerical results illustrate the effectiveness of the proposed methodology.     

On the accuracy of finite difference and modal methods for computing tidal and wind wave current profiles

This paper deals with the comparative accuracy of using finite difference grids or a modal representation through the vertical in modelling tidally or wind wave induced current profiles. A point model is used in the vertical, with a no-slip condition at the sea bed. In the finite difference approach the high-shear bottom layer is resolved using either a regular grid on a logarithmic or log-linear transformed co-ordinate or an irregular grid, varying in such a manner as to retain second-order accuracy. The accuracy of these various grid schemes is considered in detail. The relative merits of using either the Crank-Nicolson or Dufort-Frankel time integration methods are considered; in the case of a fine grid in a high-viscosity region, some numerical problems are found with the Dufort-Frankel method. An alternative approach to using a finite difference grid in the vertical, namely a modal (spectral) method, is described. The form of the modes is such that they can accurately resolve the high-shear bottom boundary layer. Calculations show that the thickness of the bottom boundary layer in relation to the total water depth is important in determining the choice of grid transform and rates of convergence of solutions using finite difference or modal methods. However, for the majority of problems the modal solution is numerically attractive owing to its computational efficiency and the ease with which solution algorithms based upon it can be coded in vectorizable form suitable for the new generation of vector computers. The influence of viscosity profile, its time variation and water depth upon tidally induced or wave induced currents is considered. Calculations suggest that near-bed measurements of tidal flow in shallow water together with associated modelling would enable appropriate formulations of eddy viscosity to be determined. Similar measurements, though using a laboratory flume, would be appropriate for wind wave problems.     

On the eddy viscosity model of periodic turbulent shear flows

Physical argument shows that eddy viscosity is essentially different from molecular viscosity. By direct numerical simulation, it was shown that for periodic turbulent flows, there is phase difference between Reynolds stress and rate of strain. This finding posed great challenge to turbulence modeling, because most turbulence modeling, which use the idea of eddy viscosity, do not take this effect into account. The project supported by the National Natural Science Foundation of China (19732005) and Liu Hui Center for Applied Mathematics of Nankai & Tianjin University     

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

It 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.     

Turbulence modulation for gas - particle flow in vertical tube and horizontal channel using four-way Eulerian-Lagrangian approach

Turbulence modulation of gas-solid flow in vertical tube and horizontal channel in dilute and moderately dense suspensions is investigated numerically using a four way Eulerian-Lagrangian approach. Low Reynolds number k-l model is used for analyzing the fluid phase motion. A new model is presented based on a source-term formulation, which can predict fluid phase turbulence augmentation due to the presence of large particles and damping of turbulence due to small particles in the core of the channel and tube. Particle-particle and particle-wall collisions are simulated based on a deterministic approach, and coupling terms representing the fluid-particle interactions are also taken into account. The predicted fluid mean velocity and turbulence intensity profiles are in good agreement with the available experimental data. Additional numerical simulation results for variation of the eddy viscosity, turbulence production and dissipation are also presented for different values of loading ratios.     

Memory effect in a turbulent boundary-layer flow due to a relatively strong axial variation of the mean-velocity gradient

Measurements have been made of the distributions of mean velocity, turbulence intensities and turbulence shear-stress in a turbulent boundary-layer downstream of a hemi-spherical cap attached onto the plane rigid wall. The eddy-viscosity, when computed in the classical way according to Boussinesq's concept from the lateral gradient of the mean velocity and the turbulence shear-stress, showed a very strong non-uniform lateral distribution, also across the outer region of the boundary-layer. Moreover, the non-dimensional values of the eddy-viscosity, using the wall-friction velocity and the boundary-layer thickness as the velocity scale and length scale respectively, were higher than those for the boundary-layer when not disturbed by the wake of the spherical cap. However, when account is taken of an axial memory effect of the stream-wise variation of the lateral gradient of the mean-velocity, the values of the non-dimensional eddy viscosity are close to those for the undisturbed boundary-layer.     

Extension of the local domain-free discretization method to large eddy simulation of turbulent flows

In this work, an immersed boundary method, called the local domain-free discretization (DFD) method, is extended to large eddy simulation (LES) of turbulent flows. The discrete form of partial differential equations at an interior node may involve some nodes outside the solution domain. The flow variables at these exterior dependent nodes are evaluated via linear extrapolation along the direction normal to the wall. To alleviate the requirement of mesh resolution in the near-wall region, a wall model based on the turbulence boundary layer equations is introduced. The wall shear stress yielded by the wall model and the no-penetration condition are enforced at the immersed boundary to evaluate the velocity components at an exterior dependent node. For turbulence closure, a dynamic subgrid scale (SGS) model is adopted and the Lagrangian averaging procedure is used to compute the model coefficient. The SGS eddy viscosity at an exterior dependent node is set to be equal to that at the outer layer. To maintain the mass conservation near the immersed boundary, a mass source/sink term is added into the continuity equation. Numerical experiments on relatively coarse meshes with stationary or moving solid boundaries have been conducted to verify the ability of the present LES-DFD method. The predicted results agree well with the published experimental or numerical data.