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Kutta Method (kutta + method)
Selected AbstractsA computational stream function method for two-dimensional incompressible viscous flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2005Marcelo H. Kobayashi Abstract This work concerns the development of a numerical method based on the stream function formulation of the Navier,Stokes equations to simulate two-dimensional,plane or axisymmetric,viscous flows. The main features of the proposed method are: the use of the high order finite-difference compact method for the discretization of the stream function equation, the implicit pseudo-transient Newton,Krylov-multigrid matrix free method for the stationary stream function equation and the fourth order Runge,Kutta method for the integration of non-stationary flows. Copyright © 2005 John Wiley & Sons, Ltd. [source] Stability and accuracy of the iterative differential quadrature methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2003Stefania Tomasiello Abstract In this paper the stability and accuracy of an iterative method based on differential quadrature rules will be discussed. The method has already been proposed by the author in a previous work, where its good performance has been shown. Nevertheless, discussion about stability and accuracy remained open. An answer to this question will be provided in this paper, where the conditional stability of the method will be pointed out, in addition to an examination of the possible errors which arise under certain conditions. The discussion will be preceded by an overview of the method and its foundations, i.e. the differential quadrature rules, and followed by a numerical case which shows how the method behaves when applied to reduce continuous systems to two-degree-of-freedom systems in the non-linear range. In particular, here the case of oscillators coupled in non-linear terms will be treated. The numerical results, used to draw Poincaré maps, will be compared with those obtained by using the Runge,Kutta method with a high precision goal. Copyright © 2003 John Wiley & Sons, Ltd. [source] Numerical simulation of free-surface flow using the level-set method with global mass correctionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010Yali Zhang Abstract A new numerical method that couples the incompressible Navier,Stokes equations with the global mass correction level-set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier,Stokes equations with the two-step projection method on a staggered Cartesian grid. The free-surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third-order essentially non-oscillatory schemes and a five stage Runge,Kutta method, to accomplish advection and re-distancing of the level-set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS-VOF method. The simulations reveal some interesting free-surface phenomena such as the free-surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd. [source] Fifth-order Hermitian schemes for computational linear aeroacousticsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2007Article first published online: 17 APR 200, G. Capdeville Abstract We develop a class of fifth-order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth-order accuracy. A one-dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy,Kovaleskaya (the ,,-P5 scheme'); the second one relies on a semi-discrete form and evolves in time the discrete unknowns by using a five-stage Runge,Kutta method (the ,RGK-P5 scheme'). Although the RGK-P5 scheme shares the same local spatial interpolator with the ,-P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi-dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210:133,170; J. Comput. Phys. 2006; 217:530,562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ,reconstruction stage', we define a fifth-order local spatial interpolator based on an upwind stencil. In the ,decomposition stage', we decompose the time derivatives into simple wave contributions. In the ,evolution stage', we use these fluctuations to update either by a Cauchy,Kovaleskaya procedure or by a five-stage Runge,Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ,evolution stage', two fifth-order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK-P5 appears as the best compromise between simplicity and accuracy, while the ,-P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd. [source] Semi-analytical method for departure point determinationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005Nick Martin Abstract A new method for departure point determination on Cartesian grids, the semi-analytical upwind path line tracing (SUT) method, is presented and compared to two typical departure point determination methods used in semi-Lagrangian advection schemes, the Euler method and the four-step Runge,Kutta method. Rigorous comparisons of the three methods were conducted for a severely curving hypothetical flow field and for advective transport in the rotation of a Gaussian concentration hill. The SUT method was shown to have equivalent accuracy to the Runge,Kutta method but with significantly improved computational efficiency. Depending on the case being simulated, the SUT method provides either far greater or equivalent computational efficiency and more certain accuracy than the Euler method. Copyright © 2004 John Wiley & Sons, Ltd. [source] A Lagrangian,Eulerian model of particle dispersion in a turbulent plane mixing layerINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2002L.A. Oliveira Abstract A Lagrangian,Eulerian model for the dispersion of solid particles in a two-dimensional, incompressible, turbulent flow is reported and validated. Prediction of the continuous phase is done by solving an Eulerian model using a control-volume finite element method (CVFEM). A Lagrangian model is also applied, using a Runge,Kutta method to obtain the particle trajectories. The effect of fluid turbulence upon particle dispersion is taken into consideration through a simple stochastic approach. Validation tests are performed by comparing predictions for both phases in a particle-laden, plane mixing layer airflow with corresponding measurements formerly reported by other authors. Even though some limitations are detected in the calculation of particle dispersion, on the whole the validation results are rather successful. Copyright © 2002 John Wiley & Sons, Ltd. [source] Analysis of super compact finite difference method and application to simulation of vortex,shock interactionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2001Fu Dexun Abstract Turbulence and aeroacoustic noise high-order accurate schemes are required, and preferred, for solving complex flow fields with multi-scale structures. In this paper a super compact finite difference method (SCFDM) is presented, the accuracy is analysed and the method is compared with a sixth-order traditional and compact finite difference approximation. The comparison shows that the sixth-order accurate super compact method has higher resolving efficiency. The sixth-order super compact method, with a three-stage Runge,Kutta method for approximation of the compressible Navier,Stokes equations, is used to solve the complex flow structures induced by vortex,shock interactions. The basic nature of the near-field sound generated by interaction is studied. Copyright © 2001 John Wiley & Sons, Ltd. [source] Dynamic Predictive Model for Growth of Salmonella Enteritidis in Egg YolkJOURNAL OF FOOD SCIENCE, Issue 7 2007V. Gumudavelli ABSTRACT:,Salmonella Enteritidis (SE) contamination of poultry eggs is a major human health concern worldwide. The risk of SE from shell eggs can be significantly reduced through rapid cooling of eggs after they are laid and their storage under safe temperature conditions. Predictive models for the growth of SE in egg yolk under varying ambient temperature conditions (dynamic) were developed. The growth of SE in egg yolk under several isothermal conditions (10, 15, 20, 25, 30, 35, 37, 39, 41, and 43 °C) was determined. The Baranyi model, a primary model, was fitted with growth data for each temperature and corresponding maximum specific growth rates were estimated. Root mean squared error (RMSE) values were less than 0.44 log10 CFU/g and pseudo- R2 values were greater than 0.98 for the primary model fitting. For developing the secondary model, the estimated maximum specific growth rates were then modeled as a function of temperature using the modified Ratkowsky's equation. The RMSE and pseudo- R2 were 0.05/h and 0.99, respectively. A dynamic model was developed by integrating the primary and secondary models and solving it numerically using the 4th-order Runge,Kutta method to predict the growth of SE in egg yolk under varying temperature conditions. The integrated dynamic model was then validated with 4 temperature profiles (varying) such as linear heating, exponential heating, exponential cooling, and sinusoidal temperatures. The predicted values agreed well with the observed growth data with RMSE values less than 0.29 log10 CFU/g. The developed dynamic model can predict the growth SE in egg yolk under varying temperature profiles. [source] Optimal control for linear system using genetic programmingOPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 1 2009A. Vincent Antony Kumar Abstract In this paper, optimal control for a linear system with quadratic performance is obtained using genetic programming (GP). The goal is to find the optimal control with reduced calculus effort using non-traditional methods. The obtained GP solution is compared with the traditional Runge,Kutta method. To obtain optimal control, the solution of matrix Riccati differential equation is computed based on grammatical evolution. The accuracy of the solution of the GP approach to the problem is qualitatively better than traditional methods. An illustrative numerical example is presented for the proposed method. Copyright © 2008 John Wiley & Sons, Ltd. [source] | ||||||||||