1,315 research outputs found
Quantization of the First-Order Two-Dimensional Einstein-Hilbert Action
A canonical analysis of the first-order two-dimensional Einstein-Hilbert
action has shown it to have no physical degrees of freedom and to possess an
unusual gauge symmetry with a symmetric field acting as a gauge
function. Some consequences of this symmetry are explored. The action is
quantized and it is shown that all loop diagrams beyond one-loop order vanish.
Furthermore, explicit calculation of the one-loop two-point function shows that
it too vanishes, with the contribution of the ghost loop cancelling that of the
``graviton'' loop
On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
The relationship between Koopman mode decomposition, resolvent mode
decomposition and exact invariant solutions of the Navier-Stokes equations is
clarified. The correspondence rests upon the invariance of the system operators
under symmetry operations such as spatial translation. The usual interpretation
of the Koopman operator is generalised to permit combinations of such
operations, in addition to translation in time. This invariance is related to
the spectrum of a spatio-temporal Koopman operator, which has a travelling wave
interpretation. The relationship leads to a generalisation of dynamic mode
decomposition, in which symmetry operations are applied to restrict the dynamic
modes to span a subspace subject to those symmetries. The resolvent is
interpreted as the mapping between the Koopman modes of the Reynolds stress
divergence and the velocity field. It is shown that the singular vectors of the
resolvent (the resolvent modes) are the optimal basis in which to express the
velocity field Koopman modes where the latter are not a priori known
A critical layer model for turbulent pipe flow
A model-based description of the scaling and radial location of turbulent
fluctuations in turbulent pipe flow is presented and used to illuminate the
scaling behaviour of the very large scale motions. The model is derived by
treating the nonlinearity in the perturbation equation (involving the Reynolds
stress) as an unknown forcing, yielding a linear relationship between the
velocity field response and this nonlinearity. We do not assume small
perturbations. We examine propagating modes, permitting comparison of our
results to experimental data, and identify the steady component of the velocity
field that varies only in the wall-normal direction as the turbulent mean
profile. The "optimal" forcing shape, that gives the largest velocity response,
is assumed to lead to modes that will be dominant and hence observed in
turbulent pipe flow.
An investigation of the most amplified velocity response at a given
wavenumber-frequency combination reveals critical layer-like behaviour
reminiscent of the neutrally stable solutions of the Orr-Sommerfeld equation in
linearly unstable flow. Two distinct regions in the flow where the influence of
viscosity becomes important can be identified, namely a wall layer that scales
with and a critical layer, where the propagation velocity is equal
to the local mean velocity, that scales with in pipe flow. This
framework appears to be consistent with several scaling results in wall
turbulence and reveals a mechanism by which the effects of viscosity can extend
well beyond the immediate vicinity of the wall.Comment: Submitted to the Journal of Fluid Mechanics and currently under
revie
On the design of optimal compliant walls for turbulence control
This paper employs the theoretical framework developed by Luhar et al. (J.
Fluid Mech., 768, 415-441) to consider the design of compliant walls for
turbulent skin friction reduction. Specifically, the effects of simple
spring-damper walls are contrasted with the effects of more complex walls
incorporating tension, stiffness and anisotropy. In addition, varying mass
ratios are tested to provide insight into differences between aerodynamic and
hydrodynamic applications. Despite the differing physical responses, all the
walls tested exhibit some important common features. First, the effect of the
walls (positive or negative) is greatest at conditions close to resonance, with
sharp transitions in performance across the resonant frequency or phase speed.
Second, compliant walls are predicted to have a more pronounced effect on
slower-moving structures because such structures generally have larger
wall-pressure signatures. Third, two-dimensional (spanwise constant) structures
are particularly susceptible to further amplification. These features are
consistent with many previous experiments and simulations, suggesting that
mitigating the rise of such two-dimensional structures is essential to
designing performance-improving walls. For instance, it is shown that further
amplification of such large-scale two-dimensional structures explains why the
optimal anisotropic walls identified by Fukagata et al. via DNS (J. Turb., 9,
1-17) only led to drag reduction in very small domains. The above observations
are used to develop design and methodology guidelines for future research on
compliant walls
A foundation for analytical developments in the logarithmic region of turbulent channels
An analytical framework for studying the logarithmic region of turbulent
channels is formulated. We build on recent findings (Moarref et al., J. Fluid
Mech., 734, 2013) that the velocity fluctuations in the logarithmic region can
be decomposed into a weighted sum of geometrically self-similar resolvent
modes. The resolvent modes and the weights represent the linear amplification
mechanisms and the scaling influence of the nonlinear interactions in the
Navier-Stokes equations (NSE), respectively (McKeon & Sharma, J. Fluid Mech.,
658, 2010). Originating from the NSE, this framework provides an analytical
support for Townsend's attached-eddy model. Our main result is that
self-similarity enables order reduction in modeling the logarithmic region by
establishing a quantitative link between the self-similar structures and the
velocity spectra. Specifically, the energy intensities, the Reynolds stresses,
and the energy budget are expressed in terms of the resolvent modes with speeds
corresponding to the top of the logarithmic region. The weights of the triad
modes -the modes that directly interact via the quadratic nonlinearity in the
NSE- are coupled via the interaction coefficients that depend solely on the
resolvent modes (McKeon et al., Phys. Fluids, 25, 2013). We use the hierarchies
of self-similar modes in the logarithmic region to extend the notion of triad
modes to triad hierarchies. It is shown that the interaction coefficients for
the triad modes that belong to a triad hierarchy follow an exponential
function. The combination of these findings can be used to better understand
the dynamics and interaction of flow structures in the logarithmic region. The
compatibility of the proposed model with theoretical and experimental results
is further discussed.Comment: Submitted to J. Fluid Mec
A Massive Renormalizable Abelian Gauge Theory in 2+1 Dimensions
The standard formulation of a massive Abelian vector field in
dimensions involves a Maxwell kinetic term plus a Chern-Simons mass term; in
its place we consider a Chern-Simons kinetic term plus a Stuekelberg mass term.
In this latter model, we still have a massive vector field, but now the
interaction with a charged spinor field is renormalizable (as opposed to super
renormalizable). By choosing an appropriate gauge fixing term, the Stuekelberg
auxiliary scalar field decouples from the vector field. The one-loop spinor
self energy is computed using operator regularization, a technique which
respects the three dimensional character of the antisymmetric tensor
. This method is used to evaluate the vector self
energy to two-loop order; it is found to vanish showing that the beta function
is zero to two-loop order. The canonical structure of the model is examined
using the Dirac constraint formalism.Comment: LaTeX, 17 pages, expanded reference list and discussion of
relationship to previous wor
On coherent structure in wall turbulence
A new theory of coherent structure in wall turbulence is presented. The theory is
the first to predict packets of hairpin vortices and other structure in turbulence,
and their dynamics, based on an analysis of the Navier–Stokes equations, under an
assumption of a turbulent mean profile. The assumption of the turbulent mean acts
as a restriction on the class of possible structures. It is shown that the coherent
structure is a manifestation of essentially low-dimensional flow dynamics, arising from
a critical-layer mechanism. Using the decomposition presented in McKeon & Sharma
(J. Fluid Mech., vol. 658, 2010, pp. 336–382), complex coherent structure is recreated
from minimal superpositions of response modes predicted by the analysis, which take
the form of radially varying travelling waves. The leading modes effectively constitute
a low-dimensional description of the turbulent flow, which is optimal in the sense of
describing the resonant effects around the critical layer and which minimally predicts
all types of structure. The approach is general for the full range of scales. By way
of example, simple combinations of these modes are offered that predict hairpins
and modulated hairpin packets. The example combinations are chosen to represent
observed structure, consistent with the nonlinear triadic interaction for wavenumbers
that is required for self-interaction of structures. The combination of the three leading
response modes at streamwise wavenumbers 6; 1; 7 and spanwise wavenumbers
±6; ±6; ±12, respectively, with phase velocity 2/3, is understood to represent a
turbulence ‘kernel’, which, it is proposed, constitutes a self-exciting process analogous
to the near-wall cycle. Together, these interactions explain how the mode combinations
may self-organize and self-sustain to produce experimentally observed structure. The
phase interaction also leads to insight into skewness and correlation results known in
the literature. It is also shown that the very large-scale motions act to organize hairpin-like
structures such that they co-locate with areas of low streamwise momentum,
by a mechanism of locally altering the shear profile. These energetic streamwise
structures arise naturally from the resolvent analysis, rather than by a summation of
hairpin packets. In addition, these packets are modulated through a ‘beat’ effect. The
relationship between Taylor’s hypothesis and coherence is discussed, and both are
shown to be the consequence of the localization of the response modes around the
critical layer. A pleasing link is made to the classical laminar inviscid theory, whereby
the essential mechanism underlying the hairpin vortex is captured by two obliquely
interacting Kelvin–Stuart (cat’s eye) vortices. Evidence for the theory is presented
based on comparison with observations of structure in turbulent flow reported in the experimental and numerical simulation literature and with exact solutions reported in
the transitional literature
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