Unsteady boundary layer flows induced by a wavy plate

Summary Unsteady boundary layer flows generated in an incompressible, homogeneous, nonrotating viscous fluid bounded by a rigid wavy plate are studied theoretically. The Laplace transform method is employed to obtain exact solutions of the unsteady boundary layer equations in a wavy plate configuration. The structures of the unsteady velocity distribution and the associated boundary layers are determined explicitly and several particular solutions are recovered as special cases of this analysis. The physical interpretation of the mathematical results are examined.     

On similarity solution of an unsteady laminar boundary layer along a flat plate

Summary A unified analysis has been made to obtain all possible similarity solutions of the steady and unsteady, forced flow, inside a boundary layer along a flat plate. Though previously, attempts were made to obtain similarity solutions of a steady boundary layer flow neglecting viscous dissipation term in the energy conservation equation but the treatments were not complete. Here we have taken account of the viscous dissipation term. In the steady case it has been shown that for a similarity solution of both velocity and temperature, there should be a relation between the undisturbed flow outside the boundary layer and the temperature of the plate. It has been shown that the similarity solution exists in the unsteady case if we neglet the viscous dissipation term in the energy equation.     

Some exact solutions of unsteady boundary layer equations-II

Summary Some exact solutions are presented for the unsteady boundary layer flows of a homogeneous, viscous, incompressible fluid bounded by (i) an infinite rigid oscillating flat plate or (ii) two parallel rigid oscillating flat plates. An explicit representation of the velocity fields for both the configurations has been given. The structures of the associated periodic boundary layers are determined with physical interpretations. Several results of interest have been recovered as special cases of this general theory. The Heaviside operational calculus along with the theory of residues of analytic functions is adopted in finding the solutions.     

On the growth of unsteady boundary layers on porous flat plates

Summary An investigation is carried out on the unsteady boundary layer induced in an incompressible, homogeneous, viscous fluid bounded by (i) an infinite horizontal porous plate aty=0, or (ii) two parallel horizontal rigid porous plates aty=0 andy=d. The unsteady motion is generated in the above fluid configurations by moving the plate(s) impulsively in its (their) own plane with a prescribed time-dependent velocity. Solutions for the unsteady velocity field are exactly solved by the Laplace transform treatment combined with the theory of residues. The structures of the associated boundary layers are determined. The effects of suction on the solutions and the boundary layers are investigated in detail, the limiting behaviour of the unsteady solution as timet is examined and Physical interpretations of the mathematical results are given. Finally, the frictional stresses on the plates are stated without detailed calculation.     

Conformal mapping of groundwater flow fields with internal boundaries

An analytical approach is presented for solving problems of steady, two-dimensional groundwater flow with inhomogeneity boundaries. A common approach for such problems is to separate the problem domain into two homogeneous domains, search for solutions in each domain, and then attempt to match conditions, either exactly or approximately, along the inhomogeneity boundary. Here, we use classical solutions to problems with inhomogeneity boundaries with simple geometries, and map conformally the entire domain onto a new one. In this way, existing solutions are used to solve problems with more complex, and more practical, boundary geometries. The approach is general, but subject to some restrictions on the mapping functions that may be used.Using this approach, we develop explicit analytical solutions for two problems of practical interest. The first problem addresses aquifer interaction across a gap in an impermeable separating layer; flow regimes are defined and the interaction is quantified. The second solution represents flow in the vertical plane to a partially clogged stream bed that is partially penetrating the aquifer; the stream bed is modeled as a thin layer of low-permeability silt. Flow regimes for groundwater surface–water interaction are quantified analytically.     

One-dimensional dispersion in unsteady flow in an adsorbing porous medium: An analytical solution

Analytical solutions describing the concentration distribution along one-dimensional unsteady seepage flow through adsorbing saturated finite porous medium have been obtained. An exponential function concentration is enforced at the source of the dispersion, while the change in the concentration is zero at the other boundary. A new time variable has been introduced to solve the unsteady flow problem and the solution is illustrated.     

NUMERICAL SIMULATION OF SEDIMENT TRANSPORT IN DYNAMIC FLOW CONDITIONS

1 INTRODUCTION Rivers as a source of life can at the same time impose devastating conditions on the environment. It is , therefore, imperative to analyse and predict river behaviour for different given conditions and engineering activities. Therefore, the use of simulation tools in this field has become a necessity. Many computational tools for simulation of sediment transport in rivers are now available that can be used for prediction and design under different flow conditions. However, …     

Flow to an eccentric well in a leaky circular aquifer with varied lateral replenishment

Summary An analytical solution is obtained for the flow to an eccentric well in a leaky circular aquifer with lateral replenishment, both for steady and unsteady cases. The flows for external boundary conditions of constant head and zero flux, which were treated previously, follow in the limit from a more general boundary condition. Graphs are developed to show the influence of vertical leakage and lateral replenishment on the relationship between drawdown at the well and eccentricity.Other symbols are defined in the text as they occur.     

On unsteady motion of a rotating fluid bounded by flat porous plates

Summary In this paper, an investigation is made of the unsteady flow generated in a viscous, incompressible and homogeneous fluid bounded by (i) an infinite horizontal porous plate atz=0, or (ii) two infinite horizontal porous plates atz=0 andz=D. The fluid together with the plate(s) is in a state of solid body rotation with a constant angular velocity about the z-axis normal to the plate(s), and additionally, the plate(s) performs non-torsional elliptic harmonic oscillations in its (their) own plane(s). A uniform suction or injection is introduced in the configurations through the porous plate(s) and its influence on the unsteady flow and the associated boundary layers is examined. The unsteady flow field as well as the associated boundary layers is obtained explicitly. In contrast to the unsteady rotating flow without suction, solutions of the present problem with suction exhibit no resonant phenomena. It is shown that the suction is responsible for making the boundary layers thinner and for the elimination of the resonant phenomena. It is confirmed that the velocity field and the associated multiple boundary layers are significantly modified by suction. Physical significances of the mathematical results are discussed. Several limiting cases of interest are recovered from this analysis. The initial value problem for both the configurations is exactly solved by the Heaviside operational calculus combined with the theory of residues.     

Unsteady hydromagnetic flow in an annulus with porous Walls when the inner boundary starts moving from rest

Summary The problem of unsteady hydromagnetic flow in a porous annulus when the inner boundary starts moving from rest with a constant longitudinal velocity is considered. For the case of equal Reynolds number and magnetic Reynolds number Laplace transforms of the solutions for the axial components of velocity and magnetic field are obtained in terms of modified Bessel functions. The inversions are effected for the cases of very large and small Reynolds number when the pressure gradient vanishes.     

A nonlinear,resistive boundary layer in rotating hydromagnetic flow

A new nonlinear boundary layer in rotating hydromagnetic flows is presented. The purpose of this layer is to provide a smooth transition between the magnetic field at a rigid, electrically insulating boundary and that far from the boundary. The boundary layer problem is solved in a cylindrical geometry, assuming that the variables obey the hydromagnetic analog of the von Karman similarity. The primary momentum balance within this boundary layer is between pressure, Coriolis and hydromagnetic forces; the viscous and nonlinear inertia forces are unimportant. Diffusion of the magnetic field is balanced by advection where the advecting fluid flow is induced by the action of the hydromagnetic body forces within the boundary layer. The structure of the layer depends upon a single parameter which measures the strength of the normal magnetic field. Steady solutions are presented and their uniqueness and temporal stability are analyzed. The relevance of this layer to the core of the Earth is discussed. It is estimated that this boundary layer would be at least as thick as one tenth of the core radius if it exists within the core.     

Modeling nitrogen–carbon cycling and oxygen consumption in bottom sediments

A model framework is presented for simulating nitrogen and carbon cycling at the sediment–water interface, and predicting oxygen consumption by oxidation reactions inside the sediments. Based on conservation of mass and invoking simplifying assumptions, a coupled system of diffusive–reactive partial differential equations is formulated for two-layer conceptual model of aerobic–anaerobic sediments. Oxidation reactions are modeled as first-order rate processes and nitrate is assumed to be consumed entirely in the anoxic portion of the sediments. The sediments are delineated into a thin oxygenated surface layer whose thickness is equal to the oxygen penetration depth, and a lower, but much thicker anoxic layer. The sediments are separated from the overlying water column by a relatively thin boundary layer through which mass transfer is diffusion controlled. Transient solutions are derived using the method of Laplace transform and Green’s function, which relate pore-water concentrations of the constituents to their concentrations in the bulk water and to the flux of decomposable settling organic matter. Steady-state pore-water concentrations are also obtained including expressions for the extent of methane saturation zone and methane gas flux. A relationship relating the sediment oxygen demand (SOD) to bulk water oxygen is derived using the two-film concept, which in combination with the depth-integrated solutions forms the basis for predicting the extent of oxygen penetration in the sediment. Iterative procedure and simplification thereof are proposed to estimate the extent of methane saturation zone and thickness of the aerobic layer as functions of time. Sensitivity of steady-state solutions to key parameters illustrates sediment processes interactions and synergistic effects. Simulations indicate that for a relatively thin diffusive boundary layer, d, oxygen uptake is limited by biochemical processes inside the sediments, whereas for a thick boundary layer oxygen transfer through the diffusive boundary layer is limiting. The results show an almost linear relationship between steady-state sediment oxygen demand and bulk water oxygen. For small d methane and nitrogen fluxes are sediment controlled, whereas for large d they are controlled by diffusional transfer through the boundary layer. It is shown that the two-layer model solution converges to the one-layer model (anaerobic layer) solution as the thickness of the oxygenated layer approaches zero, and that the transient solutions approach asymptotically their corresponding steady-state solutions.     

Dynamic response of a flexible plate on saturated soil layer

An analytical approach is developed to study the dynamic response of a flexible plate on single-layered saturated soil. The analysis is based on Biot's two-phased theory of poroelasticity and also on the classical thin-plate theory. First, the governing differential equations for saturated soil are solved by the use of Hankel transform. The general solutions of the skeleton displacements, stresses, and pore pressures, derived in the transformed domain, are subsequently incorporated into the imposed boundary conditions, which leads to a set of dual integral equations describing the corresponding mixed boundary value problem. These governing integral equations are finally reduced to the Fredholm integral equations of the second kind and solved by standard numerical procedures. The accuracy of the present solution is validated via comparisons with existing solutions for an ideal elastic half-space. Furthermore, some numerical results are presented to show the influences of the layer depth, the plate flexibility, and the soil porosity on the dynamic compliances.     

In-plane dynamic Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic(TI)half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf(1985),and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.     

中国太湖梅梁湾湖流及散度场：水—气耦合模式

Research is conducted on the distribution features of flow and divergence fields under the stress of ununiform and unsteady wind at Meiliang Bay on Lake Taihu by using a 3D atmospheric boundary layer model and a 2D hydrodynamic model for the area of the lake. Some meaningful results were achieved.     

On the application of the boundary layer approximation for the simulation of density stratified flows in aquifers

Stratification of the density in groundwater flow stems from the contact between water which contains minerals in low concentration with water containing a high concentration of minerals. The flow in such a flow field should be simulated by solving simultaneously the equations of continuity, motion and solute transport, because solute concentration affects the dynamics of the flow. Such an approach is generally associated with complicated calculations and numerical schemes subject to problems of convergence and stability, as the basic equations are highly nonlinear.This study applies the phenomenological boundary layer approximation, and suggests a reference to three different zones in the flow field: (a) fresh water zone, (b) transition zone, and (c) mineralized water zone. In zones (a) and (c) it is assumed that the potential flow theory can be applied. In zone (b) the flow is nonpotential but the basic similarity conditions typical to boundary layers exist.The approach suggested in this study simplifies the mathematical models that should be used for the flow field simulation. This approach is especially attractive in cases where the Dupuit approximation is applicable. In such cases very often analytical solutions can be obtained for unidirectional flows. In cases that are too complicated for representation by analytical solutions, the method can be used for the creation of simplified numerical schemes.Various examples in this study demonstrate the application of the method for various field problems associated with steady state as well as unsteady state conditions.The simplicity of the method makes it useful for variety of problems. It can be used even by small institutions and small consulting firms, who have usually access to minicomputers and microprocessors.     

大气边界层湍流的动力非平稳性的验证

首次用验证时间序列中是否存在动力非平稳性的一种简单图示方法——space time index法来分析大气边界层湍流的动力平稳性特征.本文以取自淮河流域和威斯康星森林下垫面条件下的三维高精度风速和温度、湿度湍流脉动资料对大气边界层湍流的平稳性特征进行了分析.结果表明space time index方法能有效地检验大气边界层湍流信号中是否存在动力平稳性.另外,均匀下垫面条件（水稻田）及复杂下垫面条件（森林）下的大气边界层湍流信号中几乎都存在动力非平稳性,大气湍流动力学非平稳性可能是边界层湍流信号相当普遍具有的一种特性.大气边界层湍流中的间歇性和相干结构使得其非平稳性图形的特征不同于一般时间序列非平稳性图形的“V”型特征;森林下垫面条件下的湍流信号比相对均匀下垫面（水稻田）下的湍流信号更有组织性,相干结构更强.     

Path planning in uncertain flow fields using ensemble method

An ensemble-based approach is developed to conduct optimal path planning in unsteady ocean currents under uncertainty. We focus our attention on two-dimensional steady and unsteady uncertain flows, and adopt a sampling methodology that is well suited to operational forecasts, where an ensemble of deterministic predictions is used to model and quantify uncertainty. In an operational setting, much about dynamics, topography, and forcing of the ocean environment is uncertain. To address this uncertainty, the flow field is parametrized using a finite number of independent canonical random variables with known densities, and the ensemble is generated by sampling these variables. For each of the resulting realizations of the uncertain current field, we predict the path that minimizes the travel time by solving a boundary value problem (BVP), based on the Pontryagin maximum principle. A family of backward-in-time trajectories starting at the end position is used to generate suitable initial values for the BVP solver. This allows us to examine and analyze the performance of the sampling strategy and to develop insight into extensions dealing with general circulation ocean models. In particular, the ensemble method enables us to perform a statistical analysis of travel times and consequently develop a path planning approach that accounts for these statistics. The proposed methodology is tested for a number of scenarios. We first validate our algorithms by reproducing simple canonical solutions, and then demonstrate our approach in more complex flow fields, including idealized, steady and unsteady double-gyre flows.     

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.     

Broadband Plasma Waves In The MagnetopauseAnd Polar Cap Boundary Layers

Boundary layers occurring in the magnetosphere can support a wide spectrumof plasma waves spanning a frequency range of a few mHz to tens of kHz andbeyond. This review describes the main characteristics of the broadband plasma waves observed in the Earth's low-latitude magnetopause boundary layer (LLBL), in the polar cap boundary layer (PCBL), and the possible generation mechanisms. The broadband waves at the low-latitude boundary layer are sufficiently intense to cause the diffusion of the magnetosheath plasma across the closed magnetospheric field lines at a rate rapid enough to populate and maintain the boundary layer itself. The rapid pitch angle scattering of energetic particles via cyclotron resonant interactionswith the waves can provide sufficient precipitation energy flux to the ionosphere to create the dayside aurora. In general, the broadband plasma waves may play an important part in the processes of local heating/acceleration of the boundary layer plasma.