113 research outputs found
Variational integrators for degenerate Lagrangians, with application to point vortices
We develop discrete mechanics and variational integrators
for a class of degenerate Lagrangian systems,
and apply these integrators to a system of
point vortices. Excellent numerical behavior is observed.
A longer term goal is to use these integration
methods in the context of control of mechanical
systems, such as coordinated groups of underwater
vehicles. In fact, numerical evidence given
in related problems, such as those in [2] shows that
in the presence of external forces, these methods
give superior predictions of energy behavior
On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities
Numerical simulations are used to investigate the resonant instabilities in two-dimensional flow past an open cavity. The compressible Navier–Stokes equations are solved directly (no turbulence model) for cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field, which is shown to be in good visual agreement with schlieren photographs from experiments at several different Mach numbers. The results show a transition from a shear-layer mode, primarily for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear-layer mode is characterized well by the acoustic feedback process described by Rossiter (1964), and disturbances in the shear layer compare well with predictions based on linear stability analysis of the Kelvin–Helmholtz mode. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number independent of Mach number. The wake mode oscillation is similar in many ways to that reported by Gharib & Roshko (1987) for incompressible flow with a laminar upstream boundary layer. Transition to wake mode occurs as the length and/or depth of the cavity becomes large compared to the upstream boundary-layer thickness, or as the Mach and/or Reynolds numbers are raised. Under these conditions, it is shown that the Kelvin–Helmholtz instability grows to sufficient strength that a strong recirculating flow is induced in the cavity. The resulting mean flow is similar to wake profiles that are absolutely unstable, and absolute instability may provide an explanation of the hydrodynamic feedback mechanism that leads to wake mode. Predictive criteria for the onset of shear-layer oscillations (from steady flow) and for the transition to wake mode are developed based on linear theory for amplification rates in the shear layer, and a simple model for the acoustic efficiency of edge scattering
Uncertainty Quantification for Airfoil Icing using Polynomial Chaos Expansions
The formation and accretion of ice on the leading edge of a wing can be
detrimental to airplane performance. Complicating this reality is the fact that
even a small amount of uncertainty in the shape of the accreted ice may result
in a large amount of uncertainty in aerodynamic performance metrics (e.g.,
stall angle of attack). The main focus of this work concerns using the
techniques of Polynomial Chaos Expansions (PCE) to quantify icing uncertainty
much more quickly than traditional methods (e.g., Monte Carlo). First, we
present a brief survey of the literature concerning the physics of wing icing,
with the intention of giving a certain amount of intuition for the physical
process. Next, we give a brief overview of the background theory of PCE.
Finally, we compare the results of Monte Carlo simulations to PCE-based
uncertainty quantification for several different airfoil icing scenarios. The
results are in good agreement and confirm that PCE methods are much more
efficient for the canonical airfoil icing uncertainty quantification problem
than Monte Carlo methods.Comment: Submitted and under review for the AIAA Journal of Aircraft and 2015
AIAA Conferenc
A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
The Koopman operator is a linear but infinite dimensional operator that
governs the evolution of scalar observables defined on the state space of an
autonomous dynamical system, and is a powerful tool for the analysis and
decomposition of nonlinear dynamical systems. In this manuscript, we present a
data driven method for approximating the leading eigenvalues, eigenfunctions,
and modes of the Koopman operator. The method requires a data set of snapshot
pairs and a dictionary of scalar observables, but does not require explicit
governing equations or interaction with a "black box" integrator. We will show
that this approach is, in effect, an extension of Dynamic Mode Decomposition
(DMD), which has been used to approximate the Koopman eigenvalues and modes.
Furthermore, if the data provided to the method are generated by a Markov
process instead of a deterministic dynamical system, the algorithm approximates
the eigenfunctions of the Kolmogorov backward equation, which could be
considered as the "stochastic Koopman operator" [1]. Finally, four illustrative
examples are presented: two that highlight the quantitative performance of the
method when presented with either deterministic or stochastic data, and two
that show potential applications of the Koopman eigenfunctions
Identifying Finite-Time Coherent Sets from Limited Quantities of Lagrangian Data
A data-driven procedure for identifying the dominant transport barriers in a
time-varying flow from limited quantities of Lagrangian data is presented. Our
approach partitions state space into pairs of coherent sets, which are sets of
initial conditions chosen to minimize the number of trajectories that "leak"
from one set to the other under the influence of a stochastic flow field during
a pre-specified interval in time. In practice, this partition is computed by
posing an optimization problem, which once solved, yields a pair of functions
whose signs determine set membership. From prior experience with synthetic,
"data rich" test problems and conceptually related methods based on
approximations of the Perron-Frobenius operator, we observe that the functions
of interest typically appear to be smooth. As a result, given a fixed amount of
data our approach, which can use sets of globally supported basis functions,
has the potential to more accurately approximate the desired functions than
other functions tailored to use compactly supported indicator functions. This
difference enables our approach to produce effective approximations of pairs of
coherent sets in problems with relatively limited quantities of Lagrangian
data, which is usually the case with real geophysical data. We apply this
method to three examples of increasing complexity: the first is the double
gyre, the second is the Bickley Jet, and the third is data from numerically
simulated drifters in the Sulu Sea.Comment: 14 pages, 7 figure
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