This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) pro... more This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). The research reported here produced new effective ways to solve multiscale problems in nonlinear fluid dynamics, such as turbulent flow and global ocean circulation. This was accomplished by first developing new methods for averaging over random or rapidly varying
We present two large families ofŠilnikov-type homoclinic orbits in a two modemodel that describes... more We present two large families ofŠilnikov-type homoclinic orbits in a two modemodel that describes second-harmonic generation in a passive optical cavity. These families of homoclinic orbits give rise to chaotic dynamics in the model.
In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent f... more In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggests that the constant α version of the Camassa-Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order α distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant α region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that α decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply α is independent of Reynolds number. Given the agreement with empirical and numerical data, our current work indicates that the Camassa-Holm equations provide a promising theoretical framework from which to understand some turbulent flows.
Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a syst... more Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e., Leray and LANS−α regularisation. The Leray principle introduces a smoothed transport velocity as part of the regularised convective nonlinearity. The LANS−α principle extends the Leray formulation in a natural way in which a filtered Kelvin circulation theorem, incorporating the smoothed transport velocity, is explicitly satisfied. These regularisation principles give rise to implied subgrid closures which will be applied in large eddy simulation of turbulent mixing. Comparison with filtered direct numerical simulation data, and with predictions obtained from popular dynamic eddy-viscosity modelling, shows that these mathematical regularisation models are considerably more accurate, at a lower computational cost. Particularly, the capturing of flow features characteristic of the smaller resolved scales improves significantly. Variations in spatial resolution and Reynolds number establish that the Leray model is more robust but also slightly less accurate than the LANS−α model. The LANS−α model retains more of the small-scale variability in the resolved solution. This requires a corresponding increase in the required spatial resolution. When using second order finite volume discretisation, the potential accuracy of the implied LANS−α model is found to be realized by using a grid spacing that is not larger than the length scale α that appears in the definition of this model.
Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations is found to pro... more Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations is found to provide a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e.,
The time derivative of the n-volume spanned by g~(t),..., g „(t) is given by (d/dt) IIL (t, AO)(~... more The time derivative of the n-volume spanned by g~(t),..., g „(t) is given by (d/dt) IIL (t, AO)(~ A hL (t, A)(„0II= 2 (Lt, A)(o~ A. AL (t, AO)(„I Re T [r [F (t, A) ooP „(t)] j,(i3) where P „(t) denotes the time-dependent projection of L onto the span of g& (t),..., g „(t) and Tr [F (t, AO) oP „(t)] denotes the trace of thefinite-rank operator F (t, AO) oP „(t). From Eq.(9), the sum of the first n global Lyapunov exponents may be expressed as
Abstract. The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary co... more Abstract. The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear stability analysis of a class of exact solutions establishes lower bounds on the dimension of the universal, or global, attractor and the Fourier spanning dimension, defined here as the number of Fourier modes required to span the universal attractor. We use concepts from the theory of inertial manifolds to determine ...
This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) pro... more This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). The research reported here produced new effective ways to solve multiscale problems in nonlinear fluid dynamics, such as turbulent flow and global ocean circulation. This was accomplished by first developing new methods for averaging over random or rapidly varying
We present two large families ofŠilnikov-type homoclinic orbits in a two modemodel that describes... more We present two large families ofŠilnikov-type homoclinic orbits in a two modemodel that describes second-harmonic generation in a passive optical cavity. These families of homoclinic orbits give rise to chaotic dynamics in the model.
In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent f... more In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggests that the constant α version of the Camassa-Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order α distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant α region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that α decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply α is independent of Reynolds number. Given the agreement with empirical and numerical data, our current work indicates that the Camassa-Holm equations provide a promising theoretical framework from which to understand some turbulent flows.
Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a syst... more Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e., Leray and LANS−α regularisation. The Leray principle introduces a smoothed transport velocity as part of the regularised convective nonlinearity. The LANS−α principle extends the Leray formulation in a natural way in which a filtered Kelvin circulation theorem, incorporating the smoothed transport velocity, is explicitly satisfied. These regularisation principles give rise to implied subgrid closures which will be applied in large eddy simulation of turbulent mixing. Comparison with filtered direct numerical simulation data, and with predictions obtained from popular dynamic eddy-viscosity modelling, shows that these mathematical regularisation models are considerably more accurate, at a lower computational cost. Particularly, the capturing of flow features characteristic of the smaller resolved scales improves significantly. Variations in spatial resolution and Reynolds number establish that the Leray model is more robust but also slightly less accurate than the LANS−α model. The LANS−α model retains more of the small-scale variability in the resolved solution. This requires a corresponding increase in the required spatial resolution. When using second order finite volume discretisation, the potential accuracy of the implied LANS−α model is found to be realized by using a grid spacing that is not larger than the length scale α that appears in the definition of this model.
Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations is found to pro... more Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations is found to provide a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e.,
The time derivative of the n-volume spanned by g~(t),..., g „(t) is given by (d/dt) IIL (t, AO)(~... more The time derivative of the n-volume spanned by g~(t),..., g „(t) is given by (d/dt) IIL (t, AO)(~ A hL (t, A)(„0II= 2 (Lt, A)(o~ A. AL (t, AO)(„I Re T [r [F (t, A) ooP „(t)] j,(i3) where P „(t) denotes the time-dependent projection of L onto the span of g& (t),..., g „(t) and Tr [F (t, AO) oP „(t)] denotes the trace of thefinite-rank operator F (t, AO) oP „(t). From Eq.(9), the sum of the first n global Lyapunov exponents may be expressed as
Abstract. The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary co... more Abstract. The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear stability analysis of a class of exact solutions establishes lower bounds on the dimension of the universal, or global, attractor and the Fourier spanning dimension, defined here as the number of Fourier modes required to span the universal attractor. We use concepts from the theory of inertial manifolds to determine ...
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Papers by Darryl Holm