Recent cnls items

Moments and Probability Density Functions in Turbulent Boundary Layers

Thu, 27 Aug 2015 00:00:00 +0000

Moments and Probability Density Functions in Turbulent Boundary Layers

The KOSL Scaling, Invariant Measure and PDF of Turbulence

Sat, 30 Nov 2013 00:00:00 +0000

The KOSL Scaling, Invariant Measure and PDF of Turbulence

The Kolmogorov-Obukhov-She-Leveque Scaling in Turbulence

Sat, 30 Nov 2013 00:00:00 +0000

We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.

The Stochastic Theory of Fluvial Landsurfaces

Thu, 4 Apr 2013 00:00:00 +0000

A stochastic theory of fluvial landsurfaces is developed for transport-limited erosion, using well-established models for the water and sediment fluxes. The mathematical models and analysis is developed showing that landsurface evolution is described by Markovian stochastic processes. The landsurfaces are described by non-deterministic stochastic processes, characterized by a statistical quantity, the variogram, that exibits characteristic scalings. Thus the landsurfaces are shown to be SOC (Self-organized-critical) systems, or systems of color, possessing both an initial transient state and a stationary state. The theory reproduces established numerical results and measurements from DEMs (digital elevation models).

The Kolmogorov-Obukhov Statistical Theory of Turbulence

Mon, 15 Oct 2012 00:00:00 +0000

In 1941 Kolmogorov and Obukhov proposed that there exists a statistical theory of turbulence that should allow the computation of all the statistical quantities that can be computed and measured in turbulent systems. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier-Stokes equation. The additive noise in the stochastic Navier-Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient.We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov {'}62 scaling hypothesis with the She-Leveque intermittency corrections. Then we compute the invariant measure of turbulence writing the stochastic Navier-Stokes equation...

Ordered, Disordered and Partially Synchronized Schools of Fish

Fri, 10 Aug 2012 00:00:00 +0000

We study how an ODE description of schools of fish by Birnir (2007) changes in the presence of a random acceleration. The model can be reduced to one ODE for the direction of the velocity of a generic fish and another one for its speed. These equations contain the mean speed v and a Kuramoto order parameter for the phases of the fish velocities, r. We show that their stationary solutions consist of an incoherent unstable solution with r=v=0 and a globally stable solution with r=1 and a constant v > 0. In the latter solution, all fishes move uniformly in the same direction with v and the direction of motion determined by the initial configuration of the school.In the second part, the directional headings of the particles are perturbed, in two distinct ways, and the speeds accelerated in order to obtain two distinct classes of non-stationary, complex solutions. We show that the system has similar behavior as the unperturbed one, and derive the resulting constant value of the...

A Dynamic Energy Budget (DEB) model for the energy usage and reproduction of the Icelandic capelin (Mallotus villosus)

Fri, 8 Oct 2010 00:00:00 +0000

We apply a Dynamic Energy Budget (DEB) model to the Icelandic capelin (Mallotus villosus) and introduce a new state variable to capture the roe production of individual fish. Species-specific coefficients are found for the capelin such as the shape-coefficient and the Arrhenius temperature. We show how to link the DEB model to measurable quantities such as weight, length, fat, and roe content. We use data on measured three year old female capelin from the 1999-2000 season from the Marine Research Institute of Iceland (MRI) and Matis, an Icelandic Food and Biotech R&D. We then find plausible parameter values for the DEB model by fitting the output of the model to these data. We obtain good fits between theory and observations, and the DEB model successfully reproduces weight, length, fat percentage and roe percentage of capelin. We discuss the effect of maturity on the spawning route of capelin, and describe how we intend to incorporate these results with an interacting...

Periodicity, Chaos and Localization in a Burridge-Knopoff Model of an Earthquake with Dieterich-Ruina Friction

Sat, 11 Sep 2010 00:00:00 +0000

We investigate the emergent dynamics when the nonlinear Dieterich-Ruina rate and state friction law is attached to a Burridge-Knopoff spring-block model. We derive both the discrete equations and the continuum formulation governing the system in this framework. The discrete system (ODEs) exhibits both periodic and chaotic motion, where the system’s transition to chaos is size-dependent, i.e. how many blocks are considered. From the discrete model we derive the nonlinear elastic wave equation by taking the continuum limit. This results in a nonlinear partial differential equation (PDE) and we find that both temporal and spatial chaos ensues when the same parameter is increased. This critical parameter value needed for the onset of chaos in the continuous model is much smaller than the value needed in the case of a single block and we discuss the implications this has on dynamic modeling of earthquake rupture with this specific friction law. Most importantly, these results suggest...

Erosion and Optimal Transport

Sun, 7 Feb 2010 00:00:00 +0000

We consider the theory of erosion and investigate connections to the theory of optimal transport. The mathematical theory of erosion is based on two coupled nonlinear partial differential equations. The first one describing the water flow can be considered to be an averaged Navier-Stokes equation, and the second one describing the sediment flow is a very degenerate nonlinear parabolic equation. In the first half of this work, we prove existence, uniqueness, and regularity properties of weak solutions to the second model equation describing the sediment flow. This forms the basis to define an optimal transport problem for the movement of sediment; the second half of this work is devoted to this optimal transport problem for the sediment. We solve the optimal transport problem. Furthermore, we demonstrate that the optimal transport problem distinguishes a particular class of solutions to the model equation. The movement of sediment according to the solution of the optimal...

Approximation of the Invariant Measure for the Stochastic Navier-Stokes

Fri, 27 Mar 2009 00:00:00 +0000

Kolmogorov's statistical theory of turbulence is based on the existence of the invariant measure of the Navier-Stokes flow. Recently the existence of the invariant measure was established in the three-dimensional case. It was established earlier by the author for uni-directional flow and for rivers. We discuss how one can try to go about approximating the invariant measure in three dimensions.

The Existence and Uniqueness of Turbulent Solutions of the Stochastic Navier-Stokes Equation

Mon, 3 Nov 2008 00:00:00 +0000

The existence and uniqueness of solutions of the Navier-Stokes equation driven with additive noise in three dimensions is proven, in the presence of a strong uni-directional mean flow with some rotation. The physical relevance of this solution and its relation to the classical solution, whose existence and uniqueness is also proven, is explained. The existence of a unique invariant measure is established and the properties of this measure are described. The invariant measure is used to prove Kolmogorov's scaling in 3-dimensional turbulence including the celebrated -5/3 power law for the decay of the power spectrum of a turbulent 3-dimensional flow.

Modeling and Simulations of the Spawning Migration of Pelagic Fish

Sat, 6 Sep 2008 00:00:00 +0000

We model the spawning migration of the Icelandic capelin stock using an interacting particle model with added environmental fields. Without artificial forcing terms or a homing instinct, we qualitatively reproduce several observed spawning migrations using available temperature data and approximated currents. The simulations include orders of magnitude more particles than many similar models, affecting the global behavior of the system. Without environmental fields, we analyze how various parameters scale with the number of particles. In particular we present scaling behavior between the size of the time step, radii of the sensory zones and the number of particles in the system. We then discuss incorporating environmental data into the model.

Meanderings of fluid streams on acrylic surfaces, driven by external noise

Mon, 16 Jun 2008 00:00:00 +0000

A stream of fluid flowing down a partially wetting inclined plane usually meanders, unless the volume flow rate is maintained at a highly constant value. However, fluctuations in the flow rate are inevitable in naturally occurring flows. Previous studies have conjectured that for some surfaces the meandering of a stream is an inherent instability. In this paper we show that on an acrylic plate we can eliminate the meandering by reducing perturbations entering the flow.By re-introducing controlled fluctuations, we show that they are indeed responsible for the onset of the meandering. We derive a theoretical model for the stream shape from first principles, which includes stream dynamics and forcing by external noise. While the deviation h(x) from a straight linear stream h(x) = 0 shows considerable variability as a function of downstream distance x, when an ensemble average is computed, averaging power spectrum S(k) as a function of wavenumber k for several different times...

The Dynamics of Myxobacteria Life Cycle

Mon, 16 Jun 2008 00:00:00 +0000

We develop the off-lattice model to simulate the life cycle of Myxococcus xanthus. When the food is abundant, they grow as swarms that spread away from the colony. In this stage, their movement and coordination are determined by their A-motility and S-motility engines. However, when they are in starvation, C-signaling between cells takes place and changes their cell-cell coordination. This allows them to form an aggregate which eventually develops into a fruiting body. Cells inside the fruiting body differentiate into round nonmotile spores which are resistant to adverse condition. In this paper, the Dynamic Energy Budget model is used as a trigger mechanism for cell growth and cell division, and then for switching from the swarming stage to the stage of fruiting body formation. Moreover, the logistic equation is implemented to count the number of C-signal molecules on each cell surface, which is then used as a switch for transitions between the stages of fruiting body formation.

Parallel Modeling of Fish Interactions

Sun, 10 Feb 2008 00:00:00 +0000

This paper summarizes work on a parallel algorithm for an interacting particle model, derived from the model by Czirok, Vicsek, et. al. [13] [3] [14] [4] [5]. Our model is particularly geared toward simulating the behavior of sh in large shoals. In this paper, the back- ground and motivation for the problem are given, as well as an introduction to the mathematical model. A discussion of implementing this model in MATLAB and C++ follows. The parallel implementation is discussed with challenges particular to this mathematical model and how the authors addressed these challenges. Both static and dynamic load balancing were performed and are discussed. Finally, a performance analysis follows, using a performance metric to compare the MATLAB, C++, and parallelized code.

Discrete and continuous models of the dynamics of pelagic fish: application to the capelin

Thu, 24 Jan 2008 00:00:00 +0000

In this paper, we study simulations of the schooling and swarming behavior of a mathematical model for the motion of pelagic fish. We use a derivative of a discrete model of interacting particles originated by Vicsek, Czir´ok et al. [6] [5] [23] [24]. Recently, a system of ODEs was derived from this model [2], and using these ODEs, we find transitory and long-term behavior of the discrete system. In particular, we numerically find stationary, migratory, and circling behavior in both the discrete and the ODE model and two types of swarming behavior in the discrete model. The migratory solutions are numerically stable and the circling solutions are metastable. We find a stable circulating ring solution of the discrete system where the fish travel in opposite directions within an annulus. We also find the origin of noise-driven swarming when repulsion and attraction are absent and the fish interact solely via orientation.

Turbulent Rivers

Mon, 21 Jan 2008 00:00:00 +0000

The existence of solutions describing the turbulent flow in rivers is proven. The existence of an associated invariant measure describing the statistical properties of this one dimensional turbulence is established. The turbulent solutions are not smooth but H\"older continuous with exponent $3/4$. The scaling of the solutions' second structure (or width) function gives rise to Hack's law \cite{H57}; stating that the length of the main river, in mature river basins, scales with the area of the basin $l \sim A^{h}$, $h = 0.568$ being Hack's exponent.

Turbulence of a Unidirectional Flow

Thu, 22 Nov 2007 00:00:00 +0000

Recent advances in the theory of turbulent solutions of the Navier-Stokes equations are discussed and the existence of their associated invariant measures. The statistical theory given by the invariant measures is described and associated with historically-known scaling laws. These are Hack's law in one dimension, the Bachelor-Kraichnan law in two dimensions and the Kolmogorov's scaling law in three dimensions. Applications to problems in turbulence are discussed and applications to Reynolds Averaged Navier Stokes (RANS) and Large Eddy Simulation (LES) models in computational turbulence.

A Model for Aperiodicity in Earthquakes

Sat, 18 Aug 2007 00:00:00 +0000

Conditions under which a single oscillator model coupled with Dieterich-Ruina's rate and state dependent friction exhibits chaotic dynamics is studied. Properties of spring-block models are discussed. The parameter values of the system are explored and the corresponding numerical solutions presented. Bifurcation analysis is performed to determine the bifurcations and stability of stationary solutions and we find that the system undergoes a Hopf bifurcation to a periodic orbit. This periodic orbit then undergoes a period doubling cascade into a strange attractor, recognized as broadband noise in the power spectrum. The implications for earthquakes are discussed.

Shocks in the Evolution of an Eroding Channel

Sun, 4 Feb 2007 00:00:00 +0000

Analysis of an evolution model for a river channel shows how three types of shocks determine the profile of the channel. This model shows that in a young river channel, evolution is driven by white noise magnifying into a bore followed by a hydraulic jump. This mechanism produces a convex profile typical of young landscapes. A small knick-point then develops at the bottom of the unstable convex profile. This knick-point is magnified and colored into a diffusive shock which travels upslope, digging into the convex profile until the profile becomes concave, typical of mature landscapes.

An ODE Model of the Motion of Pelagic Fish

Sun, 4 Feb 2007 00:00:00 +0000

A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czir\'ok et al. 1995, describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are dissussed and simulation of the ODEs presented.

Derivation of the Viscous Moore-Greitzer Equation for Aeroengine Flow

Mon, 22 Jan 2007 00:00:00 +0000

The viscous Moore-Greitzer equation modeling the airflow through the compression system in turbomachines, such as a jet engine, is derived using a scaled Navier-Stokes equation. The method utilizes a separation of scales argument, based on the different spatial scales in the engine and the different time scales in the flow. The pitch and size of the rotor-stator pair of blades provides a small parameter, which is the size of the local cell. The motion of the stator and rotor blades in the compressor produces a very turbulent flow on a fast time scale. The leading order equation, for the fast-time and local scale, describes this turbulent flow. The next order equations, produce an axi-symmetric swirl and a flow-pattern analogous to Rayleigh-B´enard convection rolls in Rayleigh-B´enard convection. On a much larger spatial scale and a slower time scale, there exist modulations of the flow including instabilities called surge and stall. A higher order equation, in the small parameter,...