The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.

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.

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.

During the grant period from January 1,2002 to December 31,2002 work was carried out on three projects to extend the range of applicability of advanced turbulence models. First, a new transition-sensitized turbulence model was tested and refined. Second, the influence of compressibility on the pressure-strain rate correlation was studied. Third, the relationship between time-filtered large eddy simulation (TLES) and Reynolds-averaged Navier Stokes (RANS) modeling was investigated leading to submission of the article. The transition-sensitized turbulence model encompasses the early-stage transition and turbulent flow regimes describing the evolution of the ensemble mean disturbance energy and dissipation rate. It is founded on a consistent mathematical description of the laminar regime with its linear disturbances and the fully turbulent regime with its stochastic fluctuations. The unified description is provided by the ensemble viewpoint.

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 stress vanishes, in which case the Navier–Stokes equations are recovered from the temporally filtered equations. Alternately, in the limit Δ→∞, the residual stress is equivalent to the long-time averaged stress, and the Reynolds-averaged Navier–Stokes equations are recovered from the temporally filtered equations. The predicted behavior at the asymptotic limits of filter width is further validated by numerical simulations of the temporally filtered forced, viscous Burger’s equation. Finally, finite filter widths are also considered, and both a priori and a posteriori analyses of temporal similarity and temporal approximate deconvolution models of the residual stress are conducted for the model problem.

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.

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.

In order to expand the predictive capability of single-point turbulence closure models too 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 distort,ion theory (RDT). The relationship between RDT and linear stability theory is exploit,c d 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 coefficient,s in the ensemble-averaged disturbance equations can he determined.

A classical path integral (CPI) provides a functional integral representation of the kernel which propagates phase space density distributions. In this paper a new formulation of the CPI is developed in which time and energy are promoted to dynamical variables. The reparametrization invariance, inherent in this formalism, is handled by means of the Batalin–Fradkin–Vilkovisky method. The path integral action possesses a set of ISp(2) symmetries connected with reparametrization invariance and an additional set of ISp(2) symmetries connected with the symplectic geometry of the extended phase space. Supersymmetry is also present in the CPI action. This formulation of the CPI allows us to study the dependence on energy of the dynamical evolution of Hamiltonian systems. It naturally incorporates the constraints onto the energy surface.

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.

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.

In this paper we generalize previous work done on the path-integral approach to classical mechanics and its symmetries. We study in particular the case that the components of the symplectic two-form omega(ab), expressed in arbitrary coordinates, are allowed to depend on the phase-space coordinates. This lifts the restriction that the path integral and its symmetry generators be expressed only in terms of canonical coordinates. We show, in particular, that an extra term must be added to the anti-Becchi-Rouet-Stora (anti-BRS) charge in order to preserve the ISp(2) symmetry which reflects the geometry of phase space. The cohomology of this new anti-BRS operator is found to be isomorphic to the de Rham cohomology of phase space. The modification of the anti-BRS charge leads to a modification of one of the supersymmetry generators associated with the classical Hamiltonian. Despite this change in the form of the generators, the classical Kubo-Martin-Schwinger conditions can still be derived from this supersymmetry. We also prove that the requirement of supersymmetric invariance of the states results in a new set of equations that, despite their new form, are still satisfied by the Gibbs states on a general phase-space manifold.

It is shown that a relativistic wave equation with a scalar coupling correctly describes the Dirac oscillator in the sense that in the nonrelativistic limit this results in the Schrödinger equation for the harmonic oscillator with an extra σ⋅r̂ term. It is demonstrated that this equation has built-in supersymmetry, and that it guarantees the stability of the Dirac vacuum.

In this paper we develop the notion of adiabatic holonomy in classical fermionic field theory and apply it to chiral gauge theory. In chiral gauge theory the classical adiabatic holonomy leads to a deformation of the Poisson algebra of translation generators in the space of gauge fields and to an additional term in the Poisson brackets among gauge-transformation generators. We study all this in detail in a (1+1)-dimensional model.

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.

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.