Natural convection in shallow water

Starting from the 3D Boussinesq model and taking the limit as the domain thickness tends to zero, we derive rigorously the 2D model for natural convection in shallow water. The model reduces to a degenerated elliptic equation for the pressure, an explicit formula for horizontal components of the velocity and a vertical diffusion for the vertical component. The macroscopic flow is driven by temperature variations as well as the bottom topography.     

Mathematical modeling of the vertical magnetic dipole field in a two-dimensional nonhomogeneous medium

We consider a quasi-three-dimensional problem of remote marine sounding by a high-power stationary source located on land. A transition from the three-dimensional problem to a family of parametric two-dimensional problems is performed. The solution of the remote marine sounding problem is obtained with high accuracy after solving about 20 two-dimensional problems. The integral equations are solved by the modified integral current method, which has proved highly efficient for field computations inside a strongly conducting anomaly. The electric field amplitude is observed to increase with depth. The width of the coastal current channel is estimated by analyzing the vertical magnetic field component.     

Algorithms for vector field generation in mass consistent models

Diagnostic models in meteorology are based on the fulfillment of some time independent physical constraints as, for instance, mass conservation. A successful method to generate an adjusted wind field, based on mass conservation equation, was proposed by Sasaki and leads to the solution of an elliptic problem for the multiplier. Here we study the problem of generating an adjusted wind field from given horizontal initial velocity data, by two ways. The first one is based on orthogonal projection in Hilbert spaces and leads to the same elliptic problem but with natural boundary conditions for the multiplier. We derive from this approach the so called E–algorithm. An innovative alternative proposal is obtained from a second approach where we consider the saddle–point formulation of the problem—avoiding boundary conditions for the multiplier— and solving this problem by iterative conjugate gradient methods. This leads to an algorithm that we call the CG–algorithm, which is inspired from Glowinsk's approach to solve Stokes–like problems in computational fluid dynamics. Finally, the introduction of new boundary conditions for the multiplier in the elliptic problem generates better adjusted fields than those obtained with the original boundary conditions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

完整Coriolis力作用下带有外源强迫的非线性KdV方程

利用摄动方法,从描写既有Coriolis力垂直分量又含有水平分量的位涡方程出发,给出了近赤道非线性Rossby波所满足的具有外源强迫的非线性KdV方程,并利用Jacobi椭圆函数展开法,求解了改进后的非线性KdV方程的行波解及孤立波解.通过分析KdV方程的行波解,指出Coriolis力的水平分量和外源对Rossby波动的影响.     

液气界面张力垂直分量引起的基底弹性变形

Young方程是毛细理论和润湿的重要方程之一．但是,该方程只描述了3个界面张力的水平分量之间的平衡与接触角的关系,而对液气界面张力垂直分量未作任何描述．现在,随着软材料的广泛应用,该垂直分量将引起基底的表面变形,并在微流体系统的制造过程中起到重要作用,这已是该研究领域的共识．综述了关于表面变形这一问题在理论分析,实验研究和数值模拟等方面取得的进展．而且,还讨论了由垂直分量引起的表面变形对液滴润湿和铺展行为、微悬臂梁的弯曲、弹性毛细现象、电弹性毛细现象等的影响．不仅对该问题的历史发展和目前的研究进展进行了简单的综述,并且也针对后续的研究提出了几点建议．     

Radiation instability of a circular cylinder in a uniform flow of a two-layer fluid

A solution of the plane linear problem of the oscillations of a horizontal circular cylinder in a uniform flow of a two-layer unbounded fluid is obtained using the method of multipole expansions. The direction of the flow is perpendicular to the cylinder axis. The whole cylinder Ges in the upper or lower layer. The fluid is assumed to be ideal and incompressible, the flow in each layer being a potential one. All the components of the radiation load (the apparent masses and damping coefficients) are determined and the regions of existence of radiation instability are found, depending on the flow velocity for a cylinder suspended by horizontal and vertical elastic links. By solving the integro-differential equation numerically the relative oscillations of the body under specified initial conditions are found.     

完整Coriolis力作用下带有外源强迫的非线性ZK方程

在正压流体中,从包含完整Coriolis参数的准地转位涡方程出发,在弱非线性长波近似下,采用多时空尺度和摄动方法,推导出大气非线性Rossby波振幅演变满足带有外源强迫的二维Zakharov-Kuznetsov(ZK)方程.然后利用Jacobi椭圆函数展开法,求解了ZK方程的椭圆正弦波解和孤立波解.分析结果表明,Coriolis参数的水平分量将影响二维Rossby波传播的频率特征,而外源不仅对二维Rossby波动的传播的频率有影响,对振幅也产生一个调制作用.     

Wind identification along a flight trajectory,part 3: 2D-dynamic approach

This paper deals with the identification of the wind profile along a flight trajectory by means of a two-dimensional dynamic approach. In this approach, the wind velocity components are computed as the difference between the inertial velocity components and the airspeed components. The airspeed profile as well as the nominal thrust, drag, and lift profiles are obtained from the available DFDR measurements. The actual values of the thrust, drag, and lift are assumed to be proportional to the respective nominal values via multiplicative parameters, called the thrust, drag, and lift factors. The thrust, drag, and lift factors plus the inertial velocity components at impact are determined by matching the flight trajectory computed from DFDR data with the flight trajectory available from ATCR data. This leads to a least-square problem which is solved analytically under the additional requirement of closeness of the multiplicative factors to unity. Application of the 2D-dynamic approach to the case of Flight Delta 191 shows that, with reference to the last 180 sec before impact, the values of the multiplicative factors were 1.09, 0.84, and 0.89; this implies that the actual values of the thrust, drag, and lift were 9% above, 16% below, and 11% below their respective nominal values. For the last 60 sec before impact, the aircraft was subject to severe windshear, characterized by a horizontal wind velocity difference of 123 fps and a vertical wind velocity difference of 80 fps. The 2D-dynamic approach is applicable to the analysis of windshear accidents in take-off or landing, especially for the case of older-generation, shorter-range aircraft which do not carry the extensive instrumentation of newer-generation, longer-range aircraft. The same methodology can be extended to the investigation of aircraft accidents originating from causes other than windshear (e.g., icing, incorrect flap position, engine malfunction), above all if its precision is further increased by combining the 2D-dynamic approach and the 2D-kinematic approach.     

Control of an aircraft landing in windshear

The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis.The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) motion and lateral motion. For each subsystem, an, auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time derivative. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite.The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the optimal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential game problems is employed in connection with the complete nonlinear system of dynamical equations.The aircraft flight through the wind downburst zone is simulated, and three different downburst models are used. The aircraft trajectories obtained via the minimax control are essentially better than those obtained by traditional autopilot methods.     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

广义缓坡方程

运用表面波Hamilton方法和缓坡逼近假定,分析缓变三维流场和非平整海底对波浪传播的影响,推导出广义缓坡方程。海底地形由两个分量组成:慢变分量,其水平长度尺度大于表面波的波长;快变分量,其振幅与表面波的波长相比为一小量,但是其频率与表面波频率相当。该广义缓坡方程具有广泛的适用范围,以下著名的缓坡方程成为它的特例:经典的Berkhoff缓坡方程;包含环境流效应的Kirby缓坡方程;描述波状海底特征的Dingemans缓坡方程。另外,同时也得到了描述环境流场和快变海底效应的广义浅水方程。     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

Evolution Equations for Long,Nonlinear Internal Waves in Stratified Shear Flows

Evolution equations for long, nonlinear internal waves are derived when the basic stratified shear flow has a slow temporal and spatial variation as well as the usual dependence on the vertical coordinate. When the horizontal waveguide has a limited vertical extent the evolution equation is a variable coefficient Korteweg-deVries equation, while in the deep fluid case the evolution equation is a variable coefficient Benjamin-Davis-Ono equation. Explicit expressions are obtained for the coefficients of these equations.     

The Stability of Flows in a Differentially Heated Rotating Fluid System with Rigid Bottom and Free Top

The stability of certain steady flows in a rotating system with rigid bottom and free top surfaces is investigated. The simplest flow states having the essential spatial variations of steady responses of a rotating fluid system to differential heating in the horizontal are studied, that is, those with a constant gradient temperature distribution with both horizontal and vertical components, and the accompanying Coriolis-balanced constant velocity shear (thermal wind). Ekman boundary layers and intermediate boundary layers are encountered in a systematic asymptotic analysis in two small parameters, the Ekman number and an inverse Richardson number. The resulting neutral stability curves indicate the possibility of instabilities above the inviscid stability criterion due to Eady, for some mean flow configurations. The estimate of the critical Taylor number is numerically close to the values obtained in the most nearly applicable experiments.     

Unsteady Flow and its Influence to Aerodynamic and Aeroacoustic of VAWT's

A study is reported of the influence of unsteady flow on the aerodynamics and aeroacoustics of vertical axis wind turbines by numerical simulation. The combination of aerodynamic predictions with a discrete vortex method and aeroacoustic predictions based on Ffowcs Williams-Hawkings equation is used to achieve this goal. The numerical results show that unsteady flow of the turbine has a significant influence on the turbine aerodynamics and can lead to a decrease in generated noise as compared to the conventional horizontal axis wind turbine at the similar aerodynamic performance. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and simulation of the motion of nanoparticles in cylindrical capillaries allowing particle‐to‐wall interactions

The process of transporting nanoparticles at the blood vessels level stumbles upon various physical and physiological obstacles; therefore, a Mathematical modeling will provide a valuable means through which to understand better this complexity. In this paper, we consider the motion of nanoparticles in capillaries having cylindrical shapes (i.e., tubes of finite size). Under the assumption that these particles have spherical shapes, the motion of these particles reduces to the motion of their centers. Under these conditions, we derive the mathematical model, to describe the motion of these centers, from the equilibrium of the gravitational force, the hemodynamic force and the van der Waals interaction forces. We distinguish between the interaction between the particles and the interaction between each particle and the walls of the tube. Assuming that the minimum distance between the particles is large compared with the maximum radius R of the particles and hence neglecting the interactions between the particles, we derive simpler models for each particle taking into account the particles‐to‐wall interactions. At an error of order O(R) or O(R³)(depending if the particles are 'near' or 'very near' to the walls), we show that the horizontal component of each particle's displacement is solution of a nonlinear integral equation that we can solve via the fixed point theory. The vertical components of the displacement are computable in a straightforward manner as soon as the horizontal components are estimated. Finally, we support this theory with several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.     

Notes on a finite differencing scheme for the dispersion equation

Finite difference techniques applied to atmospheric dispersion problems often encounter time step limitations due to the variance in the characteristic length scales (horizontal to vertical) of both the field variables and the computational region. Methods to maximize the integration time step are explored and techniques are described which ensure numerical accuracy and stability of these optimized time step techniques.

To circumvent time step limitations arising from consideration of the vertical diffusion term in the dispersion equation, a column implicitization technique is suggested which, through correction terms added to the differencing equation to compensate for truncation errors, provides an efficient and economical atmospheric dispersion solver which is insensitive to the common time step limitations of explicit schemes when large aspect ratio computational volumes are required. Further, it is shown that a relaxed stability criteria proposed by Leonard and Clancy for explicit differencing of the horizontal terms in the dispersion equation, presents a further saving in computational time provided correction terms to the differencing equation are included to eliminate phase and amplitude errors resulting from the larger time steps employed.     

Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability

Second-order finite-difference methods are developed for the numerical solutions of the eighth-, tenth- and twelfth-order eigenvalue problems arising in the study of the effect of rotation on a horizontal layer of fluid heated from below. Instability setting-in as overstability may be modelled by an eighth-order ordinary differential equation. When a uniform magnetic field also acts across the fluid in the same direction as gravity, instability setting-in as ordinary convection may be modelled by a tenth-order differential equation, while instability setting-in as overstability may be modelled by a twelfth-order differential equation. The numerical methods are developed by making direct replacements of the derivatives in the differential equations and then by computing the eigenvalues, which may incorporate Rayleigh number, horizontal wave speed and a time constant, from the resulting algebraic eigenvalue problem. The eigenvalues are also computed by writing the differential equations as systems of second-order differential equations and then using second- and fourth-order methods to obtain the eigenvalues. Numerical results obtained using the two approaches are compared with estimates appearing in the literature.     

Wind identification along a flight trajectory,part 2: 2D-kinematic approach

This paper deals with the identification of the wind profile along a flight trajectory by means of a two-dimensional kinematic approach. In this approach, the wind velocity components are computed as the difference between the inertial velocity components and the airspeed components. The airspeed profile is obtained from flight measurements. The inertial velocity profile is obtained by integration of the measured inertial acceleration. The accelerometer biases and the impact values of the inertial velocity components are determined by matching the computed flight trajectory with the measured flight trajectory, available from the digital flight data recorder and air traffic control radar. This leads to a least-square problem, which is solved analytically for both the continuous formulation and the discrete formulation. Key to the precision of the identification process is the proper selection of the integration time. Because the measured data are noise-corrupted, unstable identification occurs if the integration time is too short. On the other hand, if the integration time is too long, the hypothesis of two-dimensional motion (flight trajectory nearly contained in a vertical plane) breaks down. Application of the 2D-kinematic approach to the case of Flight Delta 191 shows that stable identification takes place for integration times in the range τ = 120 to 180 sec before impact. The results of the 2D-kinematic approach are close to those of the 3D-kinematic approach (Ref. 1), particularly in terms of the inertial velocity components at impact (within 1 fps) and the maximum wind velocity differences (within 2 fps). The 2D-kinematic approach is applicable to the analysis of wind-shear accidents in take-off or landing, especially for the case of older-generation, shorter-range aircraft which do not carry the extensive instrumentation of newer-generation, longer-range aircraft.     

Study of storm surge due to Typhoon Linda (1997) in the Gulf of Thailand using a three dimensional ocean model

Numerical integrations using the three dimensional ocean model based on the princeton ocean model (POM) were applied for the study of both sea level elevation and ocean circulation patterns forced by the wind fields during typhoons that moved over the Gulf of Thailand (GoT). The simulation concerned a case of Typhoon Linda which occurred during November 1-4, 1997. Typhoon Linda was one of the worst storms that passed the Gulf of Thailand and hit the southern coastal provinces of Thailand on November 3, 1997. It caused flooding and a strong wind covering large areas of agriculture and fisheries, which destroyed households, utilities and even human lives. The model is the time-dependent, primitive equation, Cartesian coordinates in a horizontal and sigma coordinate in the vertical. The model grid has 37 × 97 orthogonal curvilinear grid points in the horizontal, with variable spacing from 2 km near the head of the GoT to 55 km at the eastern boundary, with 10 sigma levels in the vertical conforming to a realistic bottom topography. Open boundary conditions are determined by using radiation conditions, and the sea surface elevation is prescribed from the archiving, validation and interpretation of satellite oceanographic data (AVISO). The initial condition is determined from the spin up phase of the first model run, which was executed by using wind stress calculated from climatological monthly mean wind, restoring-type surface heat and salt and climatological monthly mean freshwater flux. The model was run in spin up phase until an ocean model reached an equilibrium state under the applied force. A spatially variable wind field taken from the European Centre for Medium-Range Weather Forecasts (ECMWF) is used to compute the wind stress directly from the velocity fluctuations. Comparison of tendency between the sea surface elevations from model and the observed significant wave heights of moored buoys in the Gulf of Thailand under Seawatch project is investigated. The model predicts the sea level elevation up to 68.5 cm at the Cha-Am area located in the north of where the typhoon strands to the shore. Results of sea level elevation show that there is an area of peak set-up in the upper gulf, particularly in the western coast, and the effects of the storm surge are small at the lower gulf. During the entire period of this study, the surge in the gulf was induced by the northeasterly wind blowing over it.