W. D. Thacker

The temporally filtered Navier–Stokes equations: Properties of the residual stress

Recent interest in the development of a unifying framework among direct numerical simulations, large-eddy simulations, and statistically averaged formulations of the Navier–Stokes equations, provides the motivation for the present paper. Toward that goal, the properties of the residual (subgrid-scale) stress of the temporally filtered Navier–Stokes equations are carefully examined. This includes the frame-invariance properties of the filtered equations and the resulting residual stress. Causal time-domain filters, parametrized by a temporal filter width 0<Δ<∞, are considered. For several reasons, the differential forms of such filters are preferred to their corresponding integral forms; among these, storage requirements for differential forms are typically much less than for integral forms and, for some filters, are independent of Δ. The behavior of the residual stress in the limits of both vanishing and infinite filter widths is examined. It is shown analytically that, in the limit Δ→0, the residual stre...

Variational methods via supersymmetric techniques

Abstract In this paper we exploit the Schrodinger factorization technique, in its modern supersymmetric version, to propose a variational method for excited states of one-dimensional systems.

On removing Berry's phase

Abstract In this paper we show that the Berry phase can always be cancelled by a unitary transformation if the space of external parameters is appropriately restricted, but it simply reappears as part of the dynamical phase. Moreover the overall extra geometrical phase measured in experiments is unambiguous and retains its geometrical character.

Analysis of Transition-Sensitized Turbulent Transport Equations

The dynamics of an ensemble of linear disturbances in boundary-layer flows at various Reynolds numbers is studied through an analysis of the transport equations for the mean disturbance kinetic energy and energy dissipation rate. Effects of adverse and favorable pressure-gradients on the disturbance dynamics are also included in the analysis. Unlike the fully turbulent regime where nonlinear phase scrambling of the fluctuations affects the flow field even in proximity to the wall, the early stage transition regime fluctuations studied here are influenced across the boundary layer by the solid boundary. The dominating dynamics in the disturbance kinetic energy and dissipation rate equations are described. These results are then used to formulate transitionsensitized turbulent transport equations, which are solved in a two-step process and applied to zero-pressuregradient flow over a flat plate. Computed results are in good agreement with experimental data.

Modeling the Dynamics of Ensemble-Averaged Linear Disturbances in Homogeneous Shear Flow

In order to expand the predictive capability of single-point turbulence closure models to account for the early-stage transition regime, a methodology for the formulation and calibration of model equations for the ensemble-averaged disturbance kinetic energy and energy dissipation rate is presented. The calibration is based on homogeneous shear flow where disturbances can be described by rapid distortion theory (RDT). The relationship between RDT and linear stability theory is exploited in order to obtain a closed set of modeled equations. The linear disturbance equations are solved directly so that the numerical simulation yields a database from which the closure coefficients in the ensemble-averaged disturbance equations can be determined.

Analyzing mean transport equations of turbulence and linear disturbances in decaying flows

The decay of laminar disturbances and turbulence in mean shear-free flows is studied. In laminar flows, such disturbances are linear superpositions of modes governed by the Orr–Sommerfeld equation. In turbulent flows, disturbances are described through transport equations for representative mean quantities. The link between a description based on a deterministic evolution equation and a probability-based mean transport equation is established. Because an uncertainty in initial conditions exists in the laminar as well as the turbulent regime, a probability distribution must be defined even in the laminar case. Using this probability distribution, it is shown that the exponential decay of the linear modes in the laminar regime can be related to a power law decay of both the (ensemble) mean disturbance kinetic energy and the dissipation rate. The evolution of these mean disturbance quantities is then described by transport equations similar to those for the corresponding turbulent decaying flow.

On the Toda criterion

Abstract Using the path-integral formulation of classical mechanics, we present a further study of the Toda Criterion. This is an approximate criterion to detect local transitions from ordered to stochastic/ergodic motion or viceversa. We analyze the criterion by studying those minima of the classical path-integral weight that are not invariant under a universal supersymmetry present in any classical Hamiltonian system. This analysis relies on a theorem, that we previously proved, which says that systems which are in a phase with this supersymmetry un-broken are also in the ergodic phase, while systems which are in a phase characterized by ordered motion always have, in that phase, the supersymmetry broken. This study confirms that the Toda Criterion is neither a sufficient nor a necessary condition for the transition from ordered to stochastic motion. In the conclusions some ideas are put forward to find a true criterion based on our supersymmetry.

On the removability of Berry's phase

The authors present a thorough analysis of the removability of the Berry phase. They show that for any system they can, by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one. However, this extra shift in the dynamical phase retains its truly geometrical character. The removability is thus only apparent and the geometrical phase can still be measured in interference experiments, as indeed has already been done several times.

A path integral for turbulence in incompressible fluids

In this paper a classical path integral is formulated for incompressible fluids that evolve according to the Navier–Stokes equation. The path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields. We construct a set of ISp(2) charges associated with the geometry of gvol and its Poisson structure, and a pair of supersymmetry charges connected with the Hamiltonian. These charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero. When the effect of dissipation is included, the charges associated with the Poisson structure and the Hamiltonian are no longer conserved. Charges that generate Kolmogorov scaling and Galilean transformations are also constructed. The classical path integral is formulated in terms of vorticity as well.

POD Study of the Coherent Structures Within a Turbulent Spot

In this experimental study, turbulent spots were created in the boundary layer on a flat plate inside a water tunnel using a peristaltic pump. Digital Particle Image Velocimetry (DPIV) obtained velocity vector field plots of turbulent spots and the Proper Orthogonal Decomposition (POD) analysis was used in order to identify and study the coherent structures within turbulent spots. The part of the turbulent spot studied was a 5 x 5 cm region of the trailing edge, since it was impossible to capture the entire spot due to size constraints. This region of the trailing edge was also chosen because it corresponded to the best data obtained from the DPIV system. The POD analysis resulted in eigenvalues, which represent the energy contributed by each coherent structure. The velocity vector fields corresponding to the POD eigenvectors were obtained and plotted in order to visualize each coherent structure. The results revealed the presence of low and high-speed streaks, as well as hairpin vortices within the turbulent spot.