Seiches in a channel with a sharp variation in the bottom relief

A mathematical model of seiches is developed for the case of sharp bottom elevation or depression. An effective high-precision numerical and analytical method is applied to determine the natural frequencies and shapes of the lower modes of oscillations. New important hydrodynamic effects of the bottom relief are revealed. The main features of standing waves in a narrow vessel in the presence of sharp bottom elevation are confirmed by laboratory experiments.     

临界产量Dupuit公式的讨论及一种新的方法

临界产量的确定对底水油藏的高效开发起着重要作用. 然而,临界产量Dupuit 公式反映临界产量随着打开程度的降低而单调增加,存在不合理的地方. 本文与一些学者的研究表明,Dupuit 公式推导过程暗含的“当临界状态时水锥正好处于井底” 的假设不合理,水锥不能稳定在井底位置. 在不作水锥到达井底的假设,假设油井上部为径向流,井底周围为半球形流动的情况下,推导出了一种新的底水临界产量方程. 算例与工程实例表明,该方法比较符合实际,能够求得临界水锥高度和对打开程度进行优化.      

The influence of a sloping bottom endwall on the linear stability in the thermally driven baroclinic annulus with a free surface

We present results of a linear stability analysis of non-axisymmetric thermally driven flows in the classical model of the rotating cylindrical gap of fluid with a horizontal temperature gradient [inner (outer) sidewall cool (warm)] and a sloping bottom endwall configuration where fluid depth increases with radius. For comparison, results of a flat-bottomed endwall case study are also discussed. In both cases, the model setup has a free top surface. The analysis is carried out numerically using a Fourier–Legendre spectral element method (in azimuth and in the meridional plane, respectively) well suited to handle the axisymmetry of the fluid container. We find significant differences between the neutral stability curve for the sloping and the flat-bottomed endwall configuration. In case of a sloping bottom endwall, the wave flow regime is extended to lower rotation rates, that is, the transition curve is shifted systematically to lower Taylor numbers. Moreover, in the sloping bottom endwall case, a sharp reversal of the instability curve is found in its upper part, that is, at large temperature differences, whereas the instability line becomes almost horizontal in the flat-bottomed endwall case. The linear onset of instability is then almost independent of the rotation rate.     

The role of friction on sharp indentation

Frictional effects on sharp indentation of strain hardening solids are examined in this paper. The results of finite element simulations in a wide range of solids allow us to derive two simplified equations, accounting for the influence of the friction coefficient on hardness. Comparisons between the simulations and instrumented micro-indentation experiments are undertaken to ensure the validity of the former to metallic materials. Quantitative estimates of the role of friction on the development of pileup and sinking-in around the contact boundary are also given in the paper. These results provide a physical insight into the plastic flow features of distinctly different solids brought into contact with sharp indenters. Overall, the investigation shows that the amount of pileup can be used to set the range of validity of the two hardness equations indicated above. Friction has the largest influence on the contact response of solids exhibiting considerable piling-up effects (whose parameter , see text for details), whereas materials developing moderate pileup or sinking-in are less sensitive to friction. Finally, a methodology is devised to assess the influence of the friction coefficient on mechanical properties extracted through indentation experiments.     

Oil low to a lateral well in the presence of bottom water

Exact solutions are obtained for a number of two-dimensional problems of steady-state fluid flow to a lateral hole in a reservoir with a quiescent bottom fluid of higher density or with a fluid of lower density at the reservoir top __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 114–126, September–October, 2008.     

Exact static and dynamic critical loads of a sinusoidal arch under a point force at the midpoint

This paper derives the exact static and dynamic critical loads for a pinned sinusoidal arch under a concentrated force at the midpoint. For quasi-static loading, the exact critical load can be derived analytically when the rise parameter h is greater than 4.81. In the case when the concentrated force is applied suddenly, exact dynamic critical load can be formulated when the equilibrium configurations exist.     

Near-Critical Gas–Liquid Flow in Porous Media: Monovariant Model, Analytical Solutions and Convective Mass Exchange Effects

We examine a class of hydrocarbon reservoirs whose thermodynamic state remains close to the critical point during the all period of reservoir exploitation. Such a situation is typical for the so-called gas–condensate systems, in which the liquid phase is formed from gas when pressure decreases. Due to proximity to critical point, the mixture contains many components which are neutral with respect to the phase state. This determines a low thermodynamic degree of freedom of the system. As the results, the mathematical flow model allows a significant reduction in the number of conservation equations, whatever the number of chemical components. In the vicinity of a well, the system may be reduced to one transport equation for saturation. This nonlinear model yields exact analytical solutions when the flow is self-similar. In more general case of flow, we develop partially linearized solutions which are shown to be sufficiently exact. The spectrum of examined cases covers the flow in a medium with a sharp heterogeneity and a sharp variation in the flow rate. A significant relative gas flow past liquid gives rise to a convective mass exchange phenomenon which appears highly different from that observed in static. In the case of a medium discontinuity, the convective mass exchange gives rise to a phenomenon of condensate saturation billow formation. A sharp variation in the flow rate leads to a hysteretic behavior of the saturation field.     

Waves behind a Step in an Open Channel

The paper presents experimental data on flow in the vicinity of a sudden elevation of channel bottom (step). The range of external flow parameters is considered for the case where the step generates waves. A distinguishing feature of these waves is that they are formed in transition from subcritical to supercritical flow. It is shown that there is a range of external parameters in which the depth at the channel exit, the depth above the step, and the distance from the step to the first wave trough depend exclusively on flow discharge.     

The dynamic behaviour of sharp V-notches under impact loading

The dynamic behaviour of sharp V-notches which are either symmetric or oblique to the longitudinal boundary of a homogeneous elastic and isotropic strip subjected to an impact plane pulse was studied by the method of caustics. The stress pulse impinging on the flanks of the notch reflects and diffracts in different ways depending on the geometry of the notch relative to the coming pulse. For compressive stress pulses a stress concentration at the bottom of the notch does not create a crack propagation phenomenon, whereas for tensile pulses there is a possibility for an incubation, nucleation and eventual propagation of a crack. A complete experimental study of the incubation nucleation and propagation of cracks from the bottoms of notches in thin strips under tensile stress pulses was undertaken, whereas for compressive stress pulses the stress concentration at the bottom of the notch was evaluated. Interesting results were disclosed concerning the reinforcement of pulses by reflection and caging in, the evolution of stress concentration at the notch and the mode of crack propagation inside the plate. Dynamic stress intensity factors were evaluated all over the paths of crack propagation indicating a close intimacy between crack velocity and values of SIFs.     

Circulation flow around airfoils by a steady plane-parallel flow of a heavy liquid of finite depth with a free surface

Steady problems of a circulation flow around bodies by a flow of a heavy liquid bounded by a free surface and a straight bottom are solved. The method of complex boundary elements is used, which is based on the integral Cauchy formula written for a complex-conjugate velocity. Results of numerical calculations of the flow around a circular contour and the Joukowski airfoil are presented. Shapes of the free surface and the most important hydrodynamic characteristics of the process (velocity circulation over the airfoil and the lifting force and its moment relative to the sharp edge of the airfoil) are given. Kemerovo State University, Kemerovo 650043. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 101–110, May–June, 2000.     

Depeletion of the elastic reserve of a sealed and fractured formation with anomalously high formation pressure

Analysis of the exploitation of deposits with anomalously high formation pressures has shown [1] that when the weighted-mean formation pressure drops below a certain critical value (close to the hydrostatic pressure) a rather sharp sudden fall in output, together with a change in the rate of decline of pressure, is observed. The fall in output is attributable to the closing of the joints and the resulting catastrophic deterioration in the permeability of the reservoir [2]. In this paper an attempt is made to develop a joint closing hypothesis, to calculate the motion of the joint closing front from the bottom of the well to the edge of a homogeneous circular formation, and to derive expressions for predicting the fall in the output of the wells and the pressure in the formation. In order to obtain solutions it is assumed that the reservoir depletion regime is quasisteady, so that the results should be regarded as approximate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 73–83, November–December, 1985.     

Experimental Study of the Bottom Structure Effect on the Damping of Standing Surface Waves in a Rectangular Vessel

The damping coefficient is estimated for standing surface waves in a rectangular vessel: (1) with a smooth horizontal rigid bottom, (2) with developed sandy bottom structures, (3) with a profiled rigid bottom, and with thin bottom layers of (4) fine-grained sand and (5) glass spheres. The results obtained are compared with available theoretical models.     

Experimental and Theoretical Assessment of Brittle Fracture in Engineering Components Containing a Sharp V-Notch

A criterion was proposed to predict brittle fracture in engineering components containing sharp V-shaped notches and subjected to mixed mode I/II loading. The criterion, called SV-MTS, was developed based on the maximum tangential stress (MTS) criterion proposed originally for analyzing crack problems. The curves which are obtained from the SV-MTS criterion could be used conveniently to predict the fracture resistance and also the notch bifurcation angle in sharp V-notched components under pure mode II and also mixed mode loading. To evaluate the validity of the proposed criterion, a set of fracture tests were conducted on a new test specimen, called sharp V-notched Brazilian disc (SV-BD), under mixed mode loading conditions. It is shown that the experimental results obtained from PMMA specimens are in very good agreement with the curves of SV-MTS criterion.     

Kink-fold onset and development based on the maximum strength theorem

The structure of interest occupies initially a rectangular-shaped domain and is composed of a laminated material with weak interfaces defining the horizontal bedding. It is subjected to a compressive lateral force parallel to the bedding. The postulated fold kinematics relies on the existence of a kink band with two parallel sharp boundaries, or hinges, separating two rigid blocks. One block moves upward, the other sideways, by distances defining the fold amplitude and the horizontal shortening, respectively. Internal work is assumed to be done only along the hinges and the weak interfaces. The orientations of the hinges (hinge dip) and of the beds within the kink band (kink dip) are optimised for every fold amplitude to provide the least upper bound on the compressive force according to the maximum strength theorem.It is shown that the fold onset requires the introduction of a compaction mechanism for the hinges. The kink initiates as a sub-vertical compaction band of finite thickness equal to the bed thickness times the sine of the friction angle over the bedding. During the first phase of the fold development, the kink band rotates with the property that the sum of the kink dip and the hinge dip is always complementary to the friction angle over the bedding. Each bed along the hinges sees the activation of two deformation mechanisms: compaction and opening. The boundary between the regions over each bed where they are activated migrates from the bottom of the bed—pure compaction at the onset—towards a position at 90% of the bed thickness—development dominantly controlled by opening. The second phase of the fold development is marked by a thickening of the kink band with minor evolution of the dips. This two-phase development leads to a sharp decrease of the compressive load from the onset, a minimum as the two dips are approximately equal and then a moderate increase in applied load. In conclusion, it is noted that such combination of postulated fold kinematics and the application of the maximum strength theorem to optimise the structure could be generalised to folding in the presence of ramps, providing a useful tool to comprehend the mechanics of fold-and-thrust belts and of accretionary wedges.     

Shapes of the fastest fish and optimal underwater and floating hulls

A streamlined shape of the best swimmers removes the boundary-layer separation and ensures a laminar flow pattern. The fastest fish have a very sharp convex nose (rostrum), the purpose of which remains unclear. The bodies of revolution similar to their shapes are analyzed in steady underwater and floating motion. The sources and sinks were located on the axis of symmetry and above the water surface to estimate the pressure on the body and the vertical velocities on the water surface. It was shown that the flow patterns on a special shaped body with concave nose has no stagnation points and ensure small values of the water surface elevation. These fact allow diminishing the maximum pressure on the surface and wave drag. Special shapes with the sharp concave nose and negative pressure gradients on their surface could be parts of the low drag underwater and floating hulls.     

The Green’s function of the mild-slope equation:: The case of a monotonic bed profile

In the present work the Green’s function of the mild-slope and the modified mild-slope equations is studied. An effective numerical Fourier inversion scheme has been developed and applied to the construction and study of the source-generated water-wave potential over an uneven bottom profile with different depths at infinity. In this sense, the present work is a prerequisite to the study of the diffraction of water waves by localized bed irregularities superimposed over an uneven bottom. In the case of a monotonic bed profile, the main characteristics of the far-field are: (i) the formation of a shadow zone with an ever expanding width, which is located along the bottom irregularity, and (ii) in each of the two sectors not including the bottom irregularity the asymptotic behavior of the wave field approaches the form of an outgoing cylindrical wave, propagating with an amplitude of order O(R^−1/2), where R is the distance from the source, and wavelength corresponding to the sector-depth at infinity. Moreover, the weak wave system propagating in the shadow zone is of order O(R^−3/2), and along the bottom irregularity consists of the superposition of two outgoing waves with wavelengths corresponding to the two depths at infinity.     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

Numerical simulation of the vertical structure of discontinuous flows

A numerical method to solve the Reynolds‐averaged Navier–Stokes equations with the presence of discontinuities is outlined and discussed. The pressure is decomposed into the sum of a hydrostatic component and a hydrodynamic component. The numerical technique is based upon the classical staggered grids and semi‐implicit finite difference methods applied for quasi‐ and non‐hydrostatic flows. The advection terms in the momentum equations are approximated in order to conserve mass and momentum following the principles recently developed for the numerical simulation of shallow water flows with large gradients. Conservation of these properties is the most important aspect to represent near local discontinuities in the solution, following from sharp bottom gradients or hydraulic jumps. The model is applied to reproduce the flow over a step where a hydraulic jump forms downstream. The hydrostatic pressure assumption fails to represent this type of flow mainly because of the pressure deviation from the hydrostatic values downstream the step. Fairly accurate results are obtained from the numerical model compared with experimental data. Deviation from the data is found to be inherent to the standard k–ε model implemented. Copyright © 2001 John Wiley & Sons, Ltd.     

A stable algorithm for bed friction in three-dimensional shallow sea modal models

The mathematical formulation of a three-dimensional shallow sea model using a modal expansion in the veitical is briefly described. The importance of the time discretization of the vertical diffusion term and bottom friction term is discussed in some detail. Both stability theory and numerical calculations show the importance of time centring or evaluating the modal form of the viscosity term at the higher time step in order to develop a numerically efficient algorithm. Similar analysis and calculations show that in shallow water it is essential to time centre or evaluate bottom friction at the higher time step. In the case of linear bottom friction it is shown that this condition can be readily accomplished. However, using a quadratic friction formulation (a more physically realistic form), this cannot be readily achieved. A new algorithm is presented whereby a stable solution can be obtained even in shallow water using quadratic bottom friction.