The Zhukovskii problem of the flow around a sheet pile

The solution of the Zhukovskii problem of the flow around a sheet pile is given using the principles of two-dimensional steady-state seepage in the case when, accompanying the motion of the seeping water, there is a layer of saline ground waters at a certain depth under the sheet pile and this layer is located above an impermeable thickness of rock salt. The mixed boundary-value problem of the theory of analytic functions which arises is solved using Polubarinova-Kochina's method, which is based on the application of the analytical theory of linear differential equations and, also, the method, developed by us, of the conformal mappings of circular polygons in polar meshes, which are extremely typical for the velocity hodograph domains of such flows. While reflecting the specific details and individual properties of such flows, the solution constructed below turns out to be expressed in closed form in terms of elementary functions and, consequently, it is the simplest and most convenient solution. In addition, it is the most general solution for the class of problems being considered. The well known results Zhukovskii, Vedernikov and others are obtained from it as special and limiting çases A detailed hydrodynamic analysis and the specific features of the seepage process being considered, as well as the effects of all the physical parameters of the model on the pattern of the phenomenon, are presented using this solution and by numerical calculations.     

Mathematical modeling of seepage flows in coastal pressure water-bearing strata

We study mathematical models of certain two-dimensional steady-state seepage flows of sweet groundwater. We assume that the flows go out of a rectangular pressure water-bearing stratum and enter a sea with saline water, which has a water sheet above its surface. On the base of these models we propose algorithms, calculating the intrusion of sea saline water into shoaling sweet water strata in cases, when flows of groundwater enter the sea sidewise (the scheme of P. Ya. Polubarinova-Kochina and M. K. Mikhailov) or from below (the scheme of J. Bear and G. Dagan). Using exact analytic dependencies established with the help of the Polubarinova-Kochina method and numerical computations, we study the influence of all physical parameters of the models onto the type of the intrusion and its degree. In a particular case, with no fresh water stratum over the sea surface, we compare the results for both inflow schemes and consider the intrusion specificity, depending on the initial location of fluids interface. We study the limit cases of flows. We prove that a certain particular choice of values of unknown parameters of the conformal mapping, which enter in a solution, leads to the well-known results of P. Ya. Polubarinova-Kochina for the classical filtration problem in a rectangular dam.     

On the use of equations of the Fuchs class to calculate seepage from channels and irrigation ditches

Several schemes for seepage flows from the channels and ditches of irrigation systems through a layer of soil underlaid by a highly permeable artesian water-bearing table or an impermeable foundation are considered within the framework of the theory of the plane steady seepage of an incompressible liquid oblying to Darcy's law. Mixed multiparameter boundary value problems of the theory of analytic functions are formulated for their investigation, which are solved using Polubarinova–Kochina's method and integration of differential equations of the Fuchs class that are characteristic in problems of subterranean hydromechanics. On the basis of these models, algorithms are developed for calculating the dimensions of the saturation zone in cases when, in the seepage of water from channels and irrigation ditches, there is a need to estimate the combined effect on the pattern of motion of such important factors as the backwater from the underlying artesian water table or confining bed, the soil capillarity and the evaporation of ground waters from the free surface. The results of the calculations for all the flow schemes are compared for identical seepage characteristics.     

Simulation of seepage flows from channels

Plane steady-state seepage in a homogeneous isotropic ground from channels through a layer of soil with an underlying highly permeable pressurized water-bearing layer when the ground possesses capillarity and there is evaporation from the free surface is considered in a hydrodynamic formulation. To investigate it, a mixed multiparametric boundary-value problem of the theory of analytical functions is formulated, which is solved using the Polubarinova-Kochina method and conformal mapping of the regions of a special form, typical of problems of underground hydromechanics. On the basis of this model, an algorithm for calculating the capillary spreading of water and seepage flow is developed in situations when the ground capillarity is taken into account in the seepage of water from channels, as well as evaporation from the free surface of the ground waters, and also backwater of the underlying highly permeable stratum. Using the exact analytical relations obtained and numerical calculations, a hydrodynamic analysis of the structure and characteristic features of the simulated process, and also of the effect of all the physical parameters of the system on the seepage characteristics, is carried out. Limit and special cases, related to the absence of one or two of the three factors, characterizing the simulated process are considered: the ground capillarity, evaporation from the free surface, and also backwater of the underlying highly permeable water-bearing layer. The results of the calculations are compared with similar seepage characteristics with a similar scheme, but in which the flow region is underlaid by an impenetrable base.     

PIV observation of instantaneous velocity structure of lock release gravity currents in the slumping phase

The experiments of lock release gravity currents were performed in a rectangular channel with salt water and fresh water. The spreading law of the salt water is validated by using a digital video to record the progress of the current. Detailed instantaneous velocity structure of lock release gravity currents in the slumping phase was studied experimentally with particle image velocimetry (PIV). The time variation of the spatial distribution of velocity and vorticity is obtained, which shows some qualitative characters of the current as well as the effects of the bottom boundary layer and upper mixing layer.     

Linear and nonlinear instabilities of a granular bed: Determination of the scales of ripples and dunes in rivers

Granular media are frequently found in nature and in industry and their transport by a fluid flow is of great importance to human activities. One case of particular interest is the transport of sand in open-channel and river flows. In many instances, the shear stresses exerted by the fluid flow are bounded to certain limits and some grains are entrained as bed-load: a mobile layer which stays in contact with the fixed part of the granular bed. Under these conditions, an initially flat granular bed may be unstable, generating ripples and dunes such as those observed on the bed of rivers. In free-surface water flows, dunes are bedforms that scale with the flow depth, while ripples do not scale with it. This article presents a model for the formation of ripples and dunes based on the proposition that ripples are primary linear instabilities and that dunes are secondary instabilities formed from the competition between the coalescence of ripples and free surface effects. Although simple, the model is able to explain the growth of ripples, their saturation (not explained in previous models) and the evolution from ripples to dunes, presenting a complete picture for the formation of dunes.     

Dynamics of water evaporation fronts

The evolution and shapes of water evaporation fronts caused by long-wave instability of vertical flows with a phase transition in extended two-dimensional horizontal porous domains are analyzed numerically. The plane surface of the phase transition loses stability when the wave number becomes infinite or zero. In the latter case, the transition to instability is accompanied with reversible bifurcations in a subcritical neighborhood of the instability threshold and by the formation of secondary (not necessarily horizontal homogeneous) flows. An example of motion in a porous medium is considered concerning the instability of a water layer lying above a mixture of air and vapor filling a porous layer under isothermal conditions in the presence of capillary forces acting on the phase transition interface.     

Solutions for Initial-Boundary-Value Problems Representing Gravity Currents Arising from Variable Inflow

In this article, we report on theoretical and numerical studies of models for suddenly initiated variable inflow gravity currents in rectangular geometry. These gravity currents enter a lighter, deep ambient fluid at rest at a time‐dependent rate from behind a partially opened lock gate and their subsequent dynamics is modeled in the buoyancy‐inertia regime using ½‐layer shallow water theory. The resistance to flow that is exerted by the ambient fluid on the gravity current is accounted for by a front condition which involves a non‐dimensional parameter that can be chosen in accordance with experimental observations. Flow filament theory is used to arrive at expressions for the variable inflow velocity under the assumptions of an inviscid and incompressible fluid moving through an opening of fixed area which is suddenly opened under a lock gate at one end of a large rectangular tank. The fluid in the lock is subjected to a (possibly) time varying pressure applied uniformly over its surface and the finite movement of the free surface is accounted for. Finding this time‐dependent inflow velocity, which will then serve as a boundary condition for the solution of the shallow‐water equations, involves solving forced non‐linear ordinary differential equations and the form of this velocity equation and its attendant solutions will, in general, rule out finding self‐similar solutions for the shallow‐water equations. The existence of self‐similar solutions requires that the gravity currents have volumes proportional to t ^α , where α≥ 0 and t is the time elapsed from initiation of the flow. This condition requires a point source of fluid with very special properties for which both the area of the gap and the inflow velocity must vary in a related and prescribed time‐dependent manner in order to preserve self‐similarity. These specialized self‐similar solutions are employed here as a check on our numerical approach. In the more natural cases that are treated here in which fluids flow through an opening of fixed dimensions in a container an extra dimensional parameter is introduced thereby ruling out self‐similarity of the solutions for the shallow‐water equations so that the previous analytical approaches to the variable inflow problem, involving the use of phase‐plane analysis, will be inapplicable. The models developed and analyzed here are expected to provide a first step in the study of situations in which a storage container is suddenly ruptured allowing a heavy fluid to debouch at a variable rate through a fixed opening over level terrain. They also can be adapted to the study of other situations where variable inflow gravity currents arise such as, for example, flows of fresh water from spring run‐off into lakes and fjords, flows from volcanoes and magma chambers, discharges from locks and flash floods.     

Subsonic and supersonic disturbances in the generation of surface waves in a liquid layer adjacent to a high-speed gas-stream

In an effort to shed further light upon the nature of “supersonic” disturbances as distinct from that of ‘subsonic’ disturbances in parallel compressible flows, this paper makes an investigation of the stability characteristics of the surface waves generated in a liquid layer adjacent to a high-speed gas-stream. It turns out that the nature of the surface waves generated in the liquid layer depends markedly upon the type of disturbances present in the high-speed gas-stream. For the case of ‘subsonic’ disturbances it is shown that the energy transfer from the gas stream to the surface waves is contributed predominantly by the Fourier component of the normal gas-pressure force-field in phase with the slope of the wavy surface. For the case of ‘supersonic’ disturbances, this energy transfer is shown to be predominantly due to the component of the pressure-field in phase with the surface-wave displacement and is related to the presence of travelling periodic waves in the gas-stream—this energy transfer is shown to promote always the growth of the surface waves.     

Steady subsonic boundary layer in domains of local surface heating

The perturbed flow in a laminar boundary layer is investigated when there are heating elements on the body surface. The flow is assumed to be laminar and subsonic everywhere. It is shown that such flows can be described within the scope of free interaction theory. A solution of a plane steady-state problem is constructed in the linear approximation. The results of analytical and numerical analysis are presented.     

Investigation of the effect of seepage or evaporation from a free surface by the method of circular polygons

It is proposed to use a technique developed for polygons in polar nets to integrate equations of the Fuchs class that arise when solving a wide range of problems of plane steady seepage flow using the Polubarinova-Kochina method, based on the use of the analytical theory of linear differential equations. It is shown that, for a large class of pentagons in domains where the flows,which are very characteristic of seepage problems when there is infiltration or evaporation from the free surface, have a complex velocity, the solution of the problem of determining the unknown parameters which appear in the conformal mapping can be completed. In this case, the mapping is carried out in closed form in terms of elementary functions and it is simple and convenient for subsequent application. The results obtained are used to solve the problem of seepage from a channel, taking account of the capillarity of the ground when there is evaporation from the free surface. The results of numerical calculations are presented and a hydrodynamic analysis of the effect of the basic physical parameters of the model on the dimensions of the saturation zone is given.     

A nonlinear two phase fluid flow through a porous medium in presence of a well

We study a flow of fresh and salt water in a two dimensional axially symmetric coastal aquifer with a well on the central axis. The flow is governed by a nonlinear Darcy's law. We also show the behaviour of the solution when the out flow of salt water at well goes to 0. Received May 1999     

Energy method as solution for deformation of geosynthetic-reinforced embankment on Pasternak foundation

To more accurately analyze the settlement of geosynthetic-reinforced embankments on soft soil foundation, we simplified the ground surface structures as an Euler–Bernoulli beam, and the geosynthetic-reinforced layer as a Timoshenko beam. The granular fill and the soft soil foundation were both modelled using two-parameter Pasternak foundation models. We used energy method to establish energy balance equations for the system, and then we used the principle of resident potential energy to derive the governing differential equations of the settlement of the ground surface structures and the geosynthetic-reinforced layer. The MATLAB solver bvp4c was used to obtain numerical solutions for the settlement of the ground surface structures and the geosynthetic-reinforced layer on Pasternak foundations. By comparing to test results and existing models, the validity of the proposed solution is verified. This study analyzed the effects of parameters, such as flexural rigidity of the geosynthetic-reinforced layer, shear modulus of the granular fill, thickness of the soft soil foundation, and horizontal shear modulus of the Timoshenko beam, on the settlement of the ground surface structures and the geosynthetic-reinforced layer. The results showed that the model using a Pasternak foundation and a Timoshenko beam is more accurate at predicting settlement than that using a Winkler foundation and an Euler–Bernoulli beam.     

Mathematical simulation of the evolution of a liquid-liquid interface in piecewise inhomogeneous layers of complex geological structure

A mathematical model describing two incompressible liquids displacing one another in piecewise inhomogeneous porous ground layers is constructed. The ground layers are assumed to contain impermeable and semipermeable inclusions and free-liquid cavities. The square root of the conductivity of a layer is modeled by a metaharmonic (harmonic) function of coordinates.     

Modeling of tsunami generation and propagation by a spreading curvilinear seismic faulting in linearized shallow-water wave theory

The processes of tsunami evolution during its generation in search for possible amplification mechanisms resulting from unilateral spreading of the sea floor uplift is investigated. We study the nature of the tsunami build up and propagation during and after realistic curvilinear source models represented by a slowly uplift faulting and a spreading slip-fault model. The models are used to study the tsunami amplitude amplification as a function of the spreading velocity and rise time. Tsunami waveforms within the frame of the linearized shallow-water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space) for the movable source models. We analyzed the normalized peak amplitude as a function of the propagated uplift length, width and the average depth of the ocean along the propagation path.     

冻结壁系统热力学熵模型(Ⅰ)

本文述评了人工冻土研究的热学理论,指出了其重要意义和不足之处。文中对冻结壁进行了系统分析,根据物质层次的不同,将其划分为三个子系统:冻土分散系统、冻土土质系统和冻结壁系统。它们对应于不同的运动形态。冻结壁系统是一个多方多层次的开放性大系统。冻结壁系统的稳定及其控制问题是人工土冻结技术中的关键问题。利用非平衡热力学和耗散结构理论方法,作者论述了冻结壁系统的形成及其稳定问题,剖析了它们的热力学本质,提出了冻结壁系统的热力学熵模型,其结果令人满意。     

Continuum models in problems of the hypersonic flow of a rarefied gas around blunt bodiest

All possible continuum (hydrodynamic) models in the case of two-dimensional problems of supersonic and hypersonic flows around blunt bodies in the two-layer model (a viscous shock layer and shock-wave structure) over the whole range of Reynolds numbers, Re, from low values (free molecular and transitional flow conditions) up to high values (flow conditions with a thin leading shock wave, a boundary layer and an external inviscid flow in the shock layer) are obtained from the Navier-Stokes equations using an asymptotic analysis. In the case of low Reynolds numbers, the shock layer is considered but the structure of the shock wave is ignored. Together with the well-known models (a boundary layer, a viscous shock layer, a thin viscous shock layer, parabolized Navier-Stokes equations (the single-layer model) for high, moderate and low Re numbers, respectively), a new hydrodynamic model, which follows from the Navier-Stokes equations and reduces to the solution of the simplified (“local”) Stokes equations in a shock layer with vanishing inertial and pressure forces and boundary conditions on the unspecified free boundary (the shock wave) is found at Reynolds numbers, and a density ratio, k, up to and immediately after the leading shock wave, which tend to zero subject to the condition that (k/Re)^1/2 → 0. Unlike in all the models which have been mentioned above, the solution of the problem of the flow around a body in this model gives the free molecular limit for the coefficients of friction, heat transfer and pressure. In particular, the Newtonian limit for the drag is thereby rigorously obtained from the Navier-Stokes equations. At the same time, the Knudsen number, which is governed by the thickness of the shock layer, which vanishes in this model, tends to zero, that is, the conditions for a continuum treatment are satisfied. The structure of the shock wave can be determined both using continuum as well as kinetic models after obtaining the solution in the viscous shock layer for the weak physicochemical processes in the shock wave structure itself. Otherwise, the problem of the shock wave structure and the equations of the viscous shock layer must be jointly solved. The equations for all the continuum models are written in Dorodnitsyn--Lees boundary layer variables, which enables one, prior to solving the problem, to obtain an approximate estimate of second-order effects in boundary-layer theory as a function of Re and the parameter k and to represent all the aerodynamic and thermal characteristic; in the form of a single dependence on Re over the whole range of its variation from zero to infinity.

An efficient numerical method of global iterations, previously developed for solving viscous shock-layer equations, can be used to solve problems of supersonic and hypersonic flows around the windward side of blunt bodies using a single hydrodynamic model of a viscous shock layer for all Re numbers, subject to the condition that the limit (k/Re)^1/2 → 0 is satisfied in the case of small Re numbers. An aerodynamic and thermal calculation using different hydrodynamic models, corresponding to different ranges of variation Re (different types of flow) can thereby, in fact, be replaced by a single calculation using one model for the whole of the trajectory for the descent (entry) of space vehicles and natural cosmic bodies (meteoroids) into the atmosphere.     

Potential flow of fluid from an elevated,two-dimensional source

When fluid is pumped from an elevated source it flows downward and then outward once it hits the base. In this paper we consider a simple two dimensional model of flow from a single line source elevated above a horizontal base and consider its downward flow into a spreading layer on the bottom. A hodograph solution and linear solutions are obtained for high flow rates and full nonlinear solutions are obtained over a range of parameter space. It is found that there is a minimum flow rate beneath which no steady solutions exist. Overhanging surfaces are found for a range of parameter values. This flow serves as a model for a two-dimensional water fountain, or approximates a similar flow in a density stratified environment.