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Janne Pesonen - One of the best experts on this subject based on the ideXlab platform.
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eckart frame vibration rotation hamiltonians Contravariant metric tensor
Journal of Chemical Physics, 2014Co-Authors: Janne PesonenAbstract:Eckart frame is a unique embedding in the theory of molecular vibrations and rotations. It is defined by the condition that the Coriolis coupling of the reference structure of the molecule is zero for every choice of the shape coordinates. It is far from trivial to set up Eckart kinetic energy operators (KEOs), when the shape of the molecule is described by curvilinear coordinates. In order to obtain the KEO, one needs to set up the corresponding Contravariant metric tensor. Here, I derive explicitly the Eckart frame rotational measuring vectors. Their inner products with themselves give the rotational elements, and their inner products with the vibrational measuring vectors (which, in the absence of constraints, are the mass-weighted gradients of the shape coordinates) give the Coriolis elements of the Contravariant metric tensor. The vibrational elements are given as the inner products of the vibrational measuring vectors with themselves, and these elements do not depend on the choice of the body-frame....
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constrained molecular vibration rotation hamiltonians Contravariant metric tensor
Journal of Chemical Physics, 2013Co-Authors: Janne PesonenAbstract:Here, I present a practical recipe for obtaining Contravariant vibration-rotation metric tensors, and thus the kinetic energy operators, when some degrees of freedom are constrained rigidly. An element of the Contravariant metric tensor is obtained as a sum of dot products of Contravariant measuring vectors, which are obtained from their unconstrained counterparts by adding a frozen mode correction. The present method applies in principle for any choice of shape coordinates and a body-frame for which the Contravariant measuring vectors can be evaluated. In contrast to the existing methods, the present method does not involve evaluation of covariant metric tensors, matrix inversions, chain rules of derivation, or numerical differentiation. It is applied in the sequel paper [L. Partanen, J. Pesonen, E. Sjoholm, and L. Halonen, J. Chem. Phys. 139, 144311 (2013)] to study the effects of several different approximations to the kinetic energy operator, when the two large-amplitude OH-torsional motions in H2SO4 ...
Thomas R. Bewley - One of the best experts on this subject based on the ideXlab platform.
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on the Contravariant form of the navier stokes equations in time dependent curvilinear coordinate systems
Journal of Computational Physics, 2004Co-Authors: Thomas R. BewleyAbstract:The Contravariant form of the Navier-Stokes equations in a fixed curvilinear coordinate system is well known. However, when the curvilinear coordinate system is time-varying, such as when a body-fitted grid is used to compute the flow over a compliant surface, considerable care is needed to handle the momentum term correctly. The present paper derives the complete Contravariant form of the Navier-Stokes equations in a time-dependent curvilinear coordinate system from the intrinsic derivative of Contravariant vectors in a moving frame. The result is verified via direct transformation. These complete equations are then applied to compute incompressible flow in a 2D channel with prescribed boundary motion, and the significant effect of some terms which are sometimes either overlooked or assumed to be negligible in such a derivation is quantified.
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On the Contravariant form of the Navier-Stokes equations in time-dependent curvilinear coordinate systems
Journal of Computational Physics, 2004Co-Authors: Haoxiang Luo, Thomas R. BewleyAbstract:The Contravariant form of the Navier-Stokes equations in a fixed curvilinear coordinate system is well known. However, when the curvilinear coordinate system is time-varying, such as when a body-fitted grid is used to compute the flow over a compliant surface, considerable care is needed to handle the momentum term correctly. The present paper derives the complete Contravariant form of the Navier-Stokes equations in a time-dependent curvilinear coordinate system from the intrinsic derivative of Contravariant vectors in a moving frame. The result is verified via direct transformation. These complete equations are then applied to compute incompressible flow in a 2D channel with prescribed boundary motion, and the significant effect of some terms which are sometimes either overlooked or assumed to be negligible in such a derivation is quantified. © 2004 Elsevier Inc. All rights reserved.
Haoxiang Luo - One of the best experts on this subject based on the ideXlab platform.
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On the Contravariant form of the Navier-Stokes equations in time-dependent curvilinear coordinate systems
Journal of Computational Physics, 2004Co-Authors: Haoxiang Luo, Thomas R. BewleyAbstract:The Contravariant form of the Navier-Stokes equations in a fixed curvilinear coordinate system is well known. However, when the curvilinear coordinate system is time-varying, such as when a body-fitted grid is used to compute the flow over a compliant surface, considerable care is needed to handle the momentum term correctly. The present paper derives the complete Contravariant form of the Navier-Stokes equations in a time-dependent curvilinear coordinate system from the intrinsic derivative of Contravariant vectors in a moving frame. The result is verified via direct transformation. These complete equations are then applied to compute incompressible flow in a 2D channel with prescribed boundary motion, and the significant effect of some terms which are sometimes either overlooked or assumed to be negligible in such a derivation is quantified. © 2004 Elsevier Inc. All rights reserved.
Francesco Gallerano - One of the best experts on this subject based on the ideXlab platform.
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numerical integration of the Contravariant integral form of the navier stokes equations in time dependent curvilinear coordinate systems for three dimensional free surface flows
Continuum Mechanics and Thermodynamics, 2019Co-Authors: Giovanni Cannata, Chiara Petrelli, Luca Barsi, Francesco GalleranoAbstract:We propose a three-dimensional non-hydrostatic shock-capturing numerical model for the simulation of wave propagation, transformation and breaking, which is based on an original integral formulation of the Contravariant Navier–Stokes equations, devoid of Christoffel symbols, in general time-dependent curvilinear coordinates. A coordinate transformation maps the time-varying irregular physical domain that reproduces the complex geometries of coastal regions to a fixed uniform computational one. The advancing of the solution is performed by a second-order accurate strong stability preserving Runge–Kutta fractional-step method in which, at every stage of the method, a predictor velocity field is obtained by the shock-capturing scheme and a corrector velocity field is added to the previous one, to produce a non-hydrostatic divergence-free velocity field and update the water depth. The corrector velocity field is obtained by numerically solving a Poisson equation, expressed in integral Contravariant form, by a multigrid technique which uses a four-colour Zebra Gauss–Seidel line-by-line method as smoother. Several test cases are used to verify the dispersion and shock-capturing properties of the proposed model in time-dependent curvilinear grids.
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Numerical integration of the Contravariant integral form of the Navier–Stokes equations in time-dependent curvilinear coordinate systems for three-dimensional free surface flows
Continuum Mechanics and Thermodynamics, 2018Co-Authors: Giovanni Cannata, Chiara Petrelli, Luca Barsi, Francesco GalleranoAbstract:We propose a three-dimensional non-hydrostatic shock-capturing numerical model for the simulation of wave propagation, transformation and breaking, which is based on an original integral formulation of the Contravariant Navier–Stokes equations, devoid of Christoffel symbols, in general time-dependent curvilinear coordinates. A coordinate transformation maps the time-varying irregular physical domain that reproduces the complex geometries of coastal regions to a fixed uniform computational one. The advancing of the solution is performed by a second-order accurate strong stability preserving Runge–Kutta fractional-step method in which, at every stage of the method, a predictor velocity field is obtained by the shock-capturing scheme and a corrector velocity field is added to the previous one, to produce a non-hydrostatic divergence-free velocity field and update the water depth. The corrector velocity field is obtained by numerically solving a Poisson equation, expressed in integral Contravariant form, by a multigrid technique which uses a four-colour Zebra Gauss–Seidel line-by-line method as smoother. Several test cases are used to verify the dispersion and shock-capturing properties of the proposed model in time-dependent curvilinear grids.
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numerical simulation of wave transformation breaking and runup by a Contravariant fully non linear boussinesq equations model
Journal of Hydrodynamics, 2016Co-Authors: Francesco Gallerano, Giovanni Cannata, Francesco LasaponaraAbstract:In this paper we propose a new model based on a Contravariant integral form of the fully non-linear Boussinesq equations (FNBE) in order to simulate wave transformation phenomena, wave breaking, runup and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned Contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. A high-order upwind weighted essentially non-oscillatory (WENO) finite volume scheme that involves an exact Riemann solver is implemented. The wave breaking is represented by discontinuities of the weak solution of the integral form of the non-linear shallow water equations (NSWE). On the basis of the shock-capturing high order WENO scheme a new procedure, for the computation of the structure of the solution of a Riemann problem associated with a wet/dry front, is proposed in order to simulate the run up hydrodynamics in swash zone. The capacity of the proposed model to correctly represent wave propagation, wave breaking, run up and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. The proposed model is applied to a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite San Mauro Cilento (Italy).
G Cannata - One of the best experts on this subject based on the ideXlab platform.
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central weno scheme for the integral form of Contravariant shallow water equations
International Journal for Numerical Methods in Fluids, 2011Co-Authors: F Gallerano, G CannataAbstract:A new Central Weighted Essentially Non-Oscillatory scheme for the solution of the shallow-water equations expressed in Contravariant formulation is presented. One of the most efficient methodologies belonging to Central WENO family involves: reconstructions of cell-averaged values of flow variables, reconstruction of point-values of flow variables, advancing from time level tn to time level tn+1 of the cell-averaged values. The extension of the above-mentioned methodology into the Contravariant environment implies that the Contravariant shallow-water equations must be expressed in an integral form. An element of novelty presented in this paper regards the definition of a formal integral expression of the shallow-water equations in a Contravariant formulation, in which the Christoffel symbols are avoided. The WENO reconstructions are performed by a two-dimensional interpolating procedure taking into account the curved coordinate lines. The proposed scheme ensures the satisfaction of the exact C-property. Several two-dimensional test cases are used to verify the good resolution for smooth and discontinuous solutions. Copyright © 2010 John Wiley & Sons, Ltd.
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Central WENO scheme for the integral form of Contravariant shallow‐water equations
International Journal for Numerical Methods in Fluids, 2010Co-Authors: F Gallerano, G CannataAbstract:A new Central Weighted Essentially Non-Oscillatory scheme for the solution of the shallow-water equations expressed in Contravariant formulation is presented. One of the most efficient methodologies belonging to Central WENO family involves: reconstructions of cell-averaged values of flow variables, reconstruction of point-values of flow variables, advancing from time level tn to time level tn+1 of the cell-averaged values. The extension of the above-mentioned methodology into the Contravariant environment implies that the Contravariant shallow-water equations must be expressed in an integral form. An element of novelty presented in this paper regards the definition of a formal integral expression of the shallow-water equations in a Contravariant formulation, in which the Christoffel symbols are avoided. The WENO reconstructions are performed by a two-dimensional interpolating procedure taking into account the curved coordinate lines. The proposed scheme ensures the satisfaction of the exact C-property. Several two-dimensional test cases are used to verify the good resolution for smooth and discontinuous solutions. Copyright © 2010 John Wiley & Sons, Ltd.