A new method of numerical calculation for stagnation point flows

Zusammenfassung Bei der Berechnung von zweidimensionalen inkompressiblen Potentialströmungen um abgerundete und symmetrische Körper erhält man einen Annäherungswert für den Bereich nahe dem Staupunkt, indem man annimmt, dass die Strömungsart dieselbe ist wie die um einen Kreiszylinder.Mit der Eingliederung dieses Verfahrens in Thom's Squaring Verfahren kann man dann die Berechnung leicht mit Hilfe eines elektronischen Computers durchführen. Dieses Verfahren könnte nützlich sein für eine Annäherungsauswertung ohne unangemessene Verfeinerung des Netzes im Strombereich.     

Quenching of reaction by cellular flows

We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenarios for the initial data that is compactly supported and the size of the support is large enough. If the flow cells are large compared to the reaction length scale, propagating fronts will always form. For small cell size, any finitely supported initial data will be quenched by a sufficiently strong flow. We estimate that the flow amplitude required to quench the initial data of support L₀ is The essence of the problem is the question about the decay of the L^∞-norm of a solution to the advection-diffusion equation, and the relation between this rate of decay and the properties of the Hamiltonian system generated by the two-dimensional incompressible fluid flow. Received: September 2004 Revision: September 2005 Accepted: September 2005     

A novel algorithm based on parameterization method for calculation of curvature of the free surface flows

In this paper, a new approach based on parameterization method is presented for calculation of curvature on the free surface flows. In some phenomena such as droplet and bubble, surface tension is prominent. Therefore in these cases, accurate estimation of the curvature is vital. Volume of fluid (VOF) is a surface capturing method for free surface modeling. In this method, free surface curvature is calculated based on gradient of scalar transport parameter which is regarded as original method in this paper. However, calculation of curvature for a circle and other known geometries based on this method is not accurate. For instance, in practice curvature of a circle in interface cells is constant, while this method predicts different curvatures for it. In this research a novel algorithm based on parameterization method for improvement of the curvature calculation is presented. To show the application of parameterization method, two methods are employed. In the first approach denoted by, three line method, a curve is fitted to the free surface so that the distance between curve and linear interface approximation is minimized. In the second approach namely four point method, a curve is fitted to intersect points with grid lines for central and two neighboring cells. These approaches are treated as calculus of variation problems. Then, using the parameterization method, these cases are converted into the sequences of time-varying nonlinear programming problems. With some treatments a conventional equivalent model is obtained. It is finally proved that the solution of these sequences in the models tends to the solution of the calculus of variation problems. For verification of the presented methods, curvature of some geometrical shapes such as circle, elliptic and sinusoidal profile is calculated and compared with original method used in VOF process and analytical solutions. Finally, as a more practical problem, spurious currents are studied. The results showed that more accurate curve prediction is obtained by these approaches than the original method in VOF approach.     

Exterior axisymmetric solution for a rotating charged body in the relativistic theory of gravitation

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 2, pp. 305–313, February, 1989.     

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows: Examples of pulsating flows with instabilities

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows is described. The scheme is explicit, conservative, and second-order accurate in space and time. The numerical results obtained for pulsating flows and contact discontinuity instabilities are discussed. The mechanisms of flow pulsation and instability generation are described.     

Analytic solutions of unstable interfaces for all density ratios in axisymmetric flows

An interface between two fluids subject to an external force is hydrodynamically unstable. We extend the Layzer-type potential flow model for unstable interfaces to the system of finite density ratio in axially symmetric geometry and derive analytic solutions for growth rates of unstable interfaces over all times. The analytic expressions for bubble growth rates at finite times are given for arbitrary Atwood number. Predictions of the analytic solutions for growth rates are in excellent agreements with numerical results.     

Uniqueness of exterior axisymmetric solution for a rotating charged body in the relativistic theory of gravitation

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 3, pp. 394–403, June, 1989.     

Note on the uniqueness of subsonic Euler flows in an axisymmetric nozzle

In this paper, we consider the uniqueness of globally subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and satisfies no-flow boundary conditions on the nozzle walls. We will show that for given mass flux and Bernoulli’s function in the upstream, the subsonic flow is unique in the class of all axisymmetric solutions, which possess the asymptotic behaviors at the far fields. This result extends the uniqueness of solutions in the previous paper Du and Duan (2011) [1].     

Solution of conical-subsonic flows by the method of series truncation

Summary The solution of the windward conical-subsonic region over cones at angle of attack in steady, supersonic, inviscid flow is obtained by coordinate expansions about the plane of symmetry of the flow field, using the method of series truncation. Cases of a circular cone at large angles of attack are calculated, and good agreement with experimental results is obtained.

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Résumé La solution de la région conique-subsonique sur le côté exposé au vent des cônes en incidence dans l'écoulement supersonique non-visqueux stationnaire est obtenue par développement des coordonnées à partir du plan de symétrie du champ d'écoulement, utilisant la méthode des troncatures des séries. Des cas d'un cône circulaire à grands angles d'incidence sont calculés et de bon rapports sont obtenus avec des résultats expérimentaux.

     

Suppression of magnetic islands by additional heating of tokamak plasma

Translated from Matematicheskie Modeli i Vychislitel'nye Metody, pp. 174–184, 1987.     

Inverse problem for the reaction diffusion system by optimization method

In this paper, we study an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final measurement. First the given problem is transformed into an optimization problem by using optimal control framework and the existence of the minimizer for the control functional is established. Then we prove the stability estimate for two coefficients with the upper bound given by some Sobolev norms of the final measurement.     

On the numerical solution of axisymmetric domain optimization problems by dual finite element method

Shape optimization of an axisymmetric three-dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.     

The closed form solutions for axisymmetric modeling of thermal stress due to repetitive pulse laser heating

Repetitive pulse laser heating is increasingly being used in the laser processing industry such as laser welding, laser cutting and laser drilling due to its energy accumulation effect. Modeling of thermo-mechanical coupling effect of repetitive pulse laser heating can enhance the understanding of its thermo physical process. In this paper, an axisymmetric modeling of thermal stress for repetitive pulse laser heating solid materials is introduced, and its closed form solutions are obtained based on a semi-analytical method. The accuracy and efficiency of the closed form solutions are verified by comparison with the existing numerical modeling software based on the finite element method. Distributions of thermal stress for different radial locations, axial locations and duty cycles are calculated, and effects of these parameters on temporal profile of the thermal stress distributions are analyzed. In the region of laser irradiation, three components of thermal stress are all shown as compressive stress. And outside the region of laser irradiation, the hoop component is shown as tensile stress. On the material surface, the change of thermal stresses is very sharp as a result of direct absorption of incident laser energy. With the increasing of the depth, the rises and decreases of thermal stresses become smooth gradually. The duty cycle has a significant effect on the profiles of thermal stresses. In the same conditions, the maximum value of thermal stresses gets higher as the duty cycle decreases. Results of this study can provide some theoretical basis for parameter inversion and optimization of repetitive pulse laser heating, as well as some corresponding experimental research.     

Rate-of-convergence bounds of the grid method for the axisymmetric Dirichlet problem in spherical geometry

We consider the spherically symmetric Dirichlet problem for the Poisson equation in a ball. Existence and uniqueness conditions are established for the generalized solution of the problem in weighted Sobolev spaces .For numerical solution of the probelm by the interpolation method, a difference scheme is proposed whose rate of convergence is compatible with the smoothness of the solution of the differential problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 38–42, 1987.     

Reduction of continuous symmetries of chaotic flows by the method of slices

We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a ‘slice’ defined by minimizing the distance to a single generic ‘template’ intersects the group orbit of every point in the full state space. Global symmetry reduction by a single slice is, however, not natural for a chaotic/ turbulent flow; it is better to cover the reduced state space by a set of slices, one for each dynamically prominent unstable pattern. Judiciously chosen, such tessellation eliminates the singular traversals of the inflection hyperplane that comes along with each slice, an artifact of using the templates local group linearization globally. We compute the jump in the reduced state space induced by crossing the inflection hyperplane. As an illustration of the method, we reduce the SO (2) symmetry of the complex Lorenz equations.