Dynamic adaptation for parabolic equations

A dynamic adaptation method is presented that is based on the idea of using an arbitrary time-dependent system of coordinates that moves at a velocity determined by the unknown solution. Using some model problems as examples, the generation of grids that adapt to the solution is considered for parabolic equations. Among these problems are the nonlinear heat transfer problem concerning the formation of stationary and moving temperature fronts and the convection-diffusion problems described by the nonlinear Burgers and Buckley-Leverette equations. A detailed analysis of differential approximations and numerical results shows that the idea of using an arbitrary time-dependent system of coordinates for adapted grid generation in combination with the principle of quasi-stationarity makes the dynamic adaptation method universal, effective, and algorithmically simple. The universality is achieved due to the use of an arbitrary time-dependent system of coordinates that moves at a velocity determined by the unknown solution. This universal approach makes it possible to generate adapted grids for time-dependent problems of mathematical physics with various mathematical features. Among these features are large gradients, propagation of weak and strong discontinuities in nonlinear transport and heat transfer problems, and moving contact and free boundaries in fluid dynamics. The efficiency is determined by automatically fitting the velocity of the moving nodes to the dynamics of the solution. The close relationship between the adaptation mechanism and the structure of the parabolic equations allows one to automatically control the nodes’ motion so that their trajectories do not intersect. This mechanism can be applied to all parabolic equations in contrast to the hyperbolic equations, which do not include repulsive components. The simplicity of the algorithm is achieved due to the general approach to the adaptive grid generation, which is independent of the form and type of the differential equations.     

Relative equilibrium and collapse configurations of four point vortices

Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.      

A combined finite element and Marker and cell method for solving Navier-Stokes equations

Summary A new method is developed to approximate Navier-Stokes equations for incompressible fluids on polygonal domains. This method is based on a simple finite element approximation adapted to the Marker and Cell technique. Its main originality lies in the fact that the components of the velocity are approximated on distinct interlaced networks. The error analysis shows that this method is of order one.     

An immersed interface method for the Navier-Stokes equations on irregular domains

An augmented method based on a Cartesian grid is proposed for the incompressible Navier Stokes equations on an irregular domain. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be used in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results an ill-conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improve the condition number of the system for the augmented variable significantly. Using the immersed interface method, we achieve second order accuracy for the velocity. Numerical results and comparisons are given to validate the new method. Some interesting new numerical experiments results are also presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wavelets and Optical Flow Motion Estimation

This article describes the implementation of a simple wavelet-based optical-flow motion estimator dedicated to continuous motions such as fluid flows. The wavelet representation of the unknown velocity field is considered. This scale-space representation, associated to a simple gradient-based optimization algorithm, sets up a well-defined multiresolution framework for the optical flow estimation. Moreover, a very simple closure mechanism, approaching locally the solution by high-order polynomials is provided by truncating the wavelet basis at fine scales. Accuracy and efficiency of the proposed method are evaluated on image sequences of turbulent fluid flows.     

Enforcing the continuity equation in numerical models of geophysical fluid flows

A method is described for modifying the velocity field in geophysical fluid models so as to enforce the continuity equation. A corrective mass flux is introduced, which derives from a scalar potential. The latter is the solution of a Poisson problem which is formulated in such a way that a suitable norm of the corrective velocity be minimum. It is seen that a generalised vertical coordinate may be used. Finally, an elementary, one-dimensional illustration of the functioning of the method suggested is provided.     

Application of a non-asymptotic approach to prediction of the propagation of a flame through a fuel and/or oxidant droplet cloud

The mathematical analysis of laminar premixed spray propagation has generally been based on exploiting the inverse of the large chemical activation energy as an appropriate parameter for asymptotic analysis. In the current work we apply a modification of a recently suggested non-asymptotic approach for gaseous flames which makes use of a different approximation. In it, only the Arrhenius exponential term in the reaction rate expression is approximated using a step function chosen so that the two functions are in proximity in an integral sense. Application of this approach is more amenable and is shown to yield a simple formula for the burning velocity of a flame propagating through a cloud of fuel and/or oxidant droplets, for the fuel rich off-stoichiometric case in which the only reactant present in the chemical reaction term is the deficient oxidant which appears linearly. Results computed with the new analytical solutions are presented and a comparison is made with the predictions using the usual large activation energy approach. In addition, a double spray is considered for the first time in which both liquid oxidant and liquid fuel feature as sprays of droplets in the unburned pre-mixture. Such a situation arises in rocket engines in which two initially separate spray streams mix in a turbulent shear flow so that locally one dimensionally propagating double spray premixed flames are created. The analysis leads to an analytical expression for the laminar burning velocity dependent on the spray- and gas-related parameters. Typical thermal and velocity maps in parametric space are presented.     

输送流体管道的固——液耦合动力学研究

根据Ｈａｍｉｌｔｏｎ原理推导输送流体管道固—液耦合振动方程，得到反对称的固—液耦合阻尼矩阵和对称的固—液耦合刚度矩阵；用ＱＲ法计算管道固有频率，给出了管道前４阶固有频率—流速曲线；讨论了流体的流速、压强变化以及固—液耦合阻尼和固—液耦合刚度对管道固有频率的影响；用Ｎｅｗｍａｒｋ法计算不同流速时管道对阶跃载荷的动力响应；发现了各阶固有频率都有随流速的提高而降低、再提高、再降低的周而复始现象·     

A Mixed Analytical/Numerical Method for Velocity and Heat Transfer of Laminar Power-Law Fluids

This paper presents a relatively simple numerical method to investigate the flow and heat transfer of laminar power-law fluids over a semi-infinite plate in the presence of viscous dissipation and anisotropy radiation. On one hand, unlike most classical works, the effects of power-law viscosity on velocity and temperature fields are taken into account when both the dynamic viscosity and the thermal diffusivity vary as a power-law function. On the other hand, boundary layer equations are derived by Taylor expansion, and a mixed analytical/numerical method (a pseudo-similarity method) is proposed to effectively solve the boundary layer equations. This method has been justified by comparing its results with those of the original governing equations obtained by a finite element method. These results agree very well especially when the Reynolds number is large. We also observe that the robustness and accuracy of the algorithm are better when thermal boundary layer is thinner than velocity boundary layer.     

The Design of axisymmetric ducts for incompressible flow with vorticity.

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function and the function as independent variables where for irrotational flow can be recognized as the velocity potential function, for rotational flow ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences on a uniform mesh is employed. The technique described is capable of tackling the so–called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct wall shapes. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dipole and Multipole Flows with Point Vortices and Vortex Sheets

An exact method is presented for obtaining uniformly translating distributions of vorticity in a two-dimensional ideal fluid, or equivalently, stationary distributions in the presence of a uniform background flow. These distributions are generalizations of the well-known vortex dipole and consist of a collection of point vortices and an equal number of bounded vortex sheets. Both the vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a simple rational function in which the point vortex positions and strengths appear as parameters. The vortex sheets lie on heteroclinic streamlines of the flow. Dipoles and multipoles that move parallel to a straight fluid boundary are also obtained. By setting the translation velocity to zero, equilibrium configurations of point vortices and vortex sheets are found.     

A contractive operator solution of an airfoil design inverse problem

This paper deals with a numerical method for an airfoil design which was presented in [3, 5]. This method is intended for design of an airfoil from a given velocity distribution along a mean camber line. The method is based on searching for a fixed point of a contractive operator. This operator combines an inexact inverse operator and equations describing the flow. A subsonic flow is assumed, the flow is described by a system of the Euler equations which is solved by an implicit finite volume method. The Newton method is applied to the solution of the nonlinear system. The resulting system of linear algebraic equations is solved by GMRES method, the Jacobian-free version is described. The inexact inverse operator consists of a middle curve function and a thickness function, both depending on the given velocity distribution. In addition to the velocity distribution the velocity in infinity is given. The angle of attack is determined so that the stagnation point is in a specific position. Successful numerical results are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximum Divert for Planetary Landing Using Convex Optimization

This paper presents a real-time solution method of the maximum divert trajectory optimization problem for planetary landing. In mid-course, the vehicle is to abort and retarget to a landing site as far from the nominal as physically possible. The divert trajectory must satisfy velocity constraints in the range and cross range directions and a total speed constraint. The thrust magnitude is bounded above and below so that once on, the engine cannot be turned off. Because this constraint is not convex, it is relaxed to a convex constraint and lossless convexification is proved. A transformation of variables is introduced in the nonlinear dynamics and an approximation is made so that the problem becomes a second-order cone problem, which can be solved to global optimality in polynomial time whenever a feasible solution exists. A number of examples are solved to illustrate the effectiveness and efficiency of the solution method.     

Stochastic process model for solute transport and the associated transport equation

A mathematical model of Lagrangian motions of a particle in turbulent flows is developed on the basis of a stochastic differential equation. The model expresses uncertainties involved in turbulence by standard Brownian motion. Because the model does not guarantee smoothness of the path of the particle, local velocity is newly defined so as to be suitable for observation of a velocity time series at a fixed point. Then, it is shown that the newly defined local velocity is governed by a Gaussian distribution. In addition, an estimation method of the turbulent diffusion coefficient involved in the model is proposed by using the local velocity. The estimation method does not require tracer experiments. In order to assess the validity of the proposed local velocity, velocity measurements with three-dimensional acoustic Doppler velocimeters were conducted in agricultural drainage canals. Also, the turbulent diffusion coefficient was estimated by the derived time series of the observed local velocity. Finally, a transport equation of conservative solute is derived by using the linearity of the Kolmogorov forward equation without using gradient-type lows.     

Numerical solution of the fully non-linear weakly dispersive serre equations for steep gradient flows

We demonstrate a numerical approach for solving the one-dimensional non-linear weakly dispersive Serre equations. By introducing a new conserved quantity the Serre equations can be written in conservation law form, where the velocity is recovered from the conserved quantities at each time step by solving an auxiliary elliptic equation. Numerical techniques for solving equations in conservative law form can then be applied to solve the Serre equations. We demonstrate how this is achieved. The system of conservation equations are solved using the finite volume method and the associated elliptic equation for the velocity is solved using a finite difference method. This robust approach allows us to accurately solve problems with steep gradients in the flow, such as those generated by discontinuities in the initial conditions.The method is shown to be accurate, simple to implement and stable for a range of problems including flows with steep gradients and variable bathymetry.     

Quenching of flames by fluid advection

We consider a simple scalar reaction‐advection‐diffusion equation with ignition‐type nonlinearity and discuss the following question: What kinds of velocity profiles are capable of quenching any given flame, provided the velocity's amplitude is adequately large? Even for shear flows, the answer turns out to be surprisingly subtle. If the velocity profile changes in space so that it is nowhere identically constant, (or if it is identically constant only in a region of small measure) then the flow can quench any initial data. But if the velocity profile is identically constant in a sizable region, then the ensuing flow is incapable of quenching large enough flames, no matter how much larger is the amplitude of this velocity. The constancy region must be wider across than a couple of laminar propagating front‐widths. The proof uses a linear PDE associated to the nonlinear problem and quenching follows when the PDE is hypoelliptic. The techniques used allow the derivation of new, nearly optimal bounds on the speed of traveling wave solutions. © 2000 John Wiley & Sons, Inc.     

Low-order macroelements for two- and three-dimensional Stokes flow

The use of low-order elements for approximating fluid flow is attractive because all the elemental contributions can be quickly and easily obtained. One of the drawbacks is that low-order elements often give rise to spurious pressure modes or incompatible velocity and pressure approximations. In this paper linear velocity and linear pressure elements are described for both two- and three-dimensional flow that always produce stable solutions provided the elements are assembled into simple macroelements following easily used rules. Some examples of this idea are given for Stokes flow and compared with another popular low-order method. © 1993 John Wiley & Sons, Inc.     

Conservative methods of controlling the rotation of a gyrostat

A method for shaping the control of the rotation of a gyrostat consisting of a rigid body, within which there are three rotors rotating about non-coplanar axes rigidly connected to the body, is discussed. The state of the system is defined by the position and angular velocity of rotation of the body, as well as by the angular velocities of the rotors. Control is achieved by torques applied to the rotors. The idea behind the proposed control method is to choose the controlling torques so that the angular velocities of rotation of the rotors are linear functions of the components of the angular velocity vector of the body. The linear dependence thus specified defines a 3 × 3 matrix, that is, a “controlled inertia tensor.” This matrix, which is specified by the parameters of the control selected, does not necessarily have the properties of an inertia tensor. As a result of such a choice of controls, the equations that define the variation of the angular velocity of the body are written in a form similar to Euler's dynamical equations. The system of equations obtained is used to formulate and solve problems of controlling the angular motion of a satellite in a circular orbit. The proposed method for constructing controlling actions enables both the Lagrangian structure of the equations of motion and the fundamental symmetries of the problem to be maintained. Expressions for the torques acting on the rotors and realizing the motion of the required classes are written in explicit form.     

应力协调迭代法在高速冲击动力有限元中的应用

在EPIC[2][3],NONSAP[4]等弹塑性撞击动力有限元程序中,有一个共同的弱点是都采取了静力有限元方法,把位移函数用线性插值表示.单元之间应力是非协调的.因此应用虚功原理的基础不合理.为了克服以上困难,本文引入一个新的方法,即协调应力迭代法.实例表明,这种方法在冲击动力有限元计算中是稳定和精确的,同时具有减小单元刚度的作用.