Single Fourier Mode Research Articles

Laminar Boundary-Layer Response to Spanwise Periodic Forcing by Dielectric-Barrier-Discharge Plasma-Actuator Arrays

An extensive parametric study of the response of a Blasius boundary layer to forcing by spanwise arrays of plasma actuators was performed. The actuator arrays are designed to produce streamwise-oriented vortex pairs, which cause streaks of alternating low and high streamwise velocity. This work is motivated by the closed-loop control of bypass transition. The streaks are measured using hot-wire anemometry over a streamwise distance of approximately 100 boundary-layer thicknesses. The resulting spanwise periodic disturbance velocity is decomposed into spanwise Fourier modes. Typically, at least 90% of the disturbance energy is contained within a mode related to the spanwise spacing of the exposed electrodes and its first three harmonics. Along the streamwise direction, the growth and decay characteristics of the disturbance are considered for various actuator geometries, as well as excitation voltages and frequencies for a freestream velocity of . The total disturbance energy is shown to scale with the actuator power consumption. Changes in the excitation voltage influence the energy distribution between Fourier modes due to the effect of voltage on the plasma extent and plasma-induced velocity, which changes the separation between vortices generated by the actuators. The induced disturbance is found to be most closely represented by a single Fourier mode when the electrode width is approximately one-third of the electrode spacing. For the other electrode width, significant energy appears at higher harmonics of the electrode spacing.

Nonlinear stability of gravitationally unstable, transient, diffusive boundary layers in porous media

AbstractThe linear stability of transient diffusive boundary layers in porous media has been studied extensively for its applications to carbon dioxide sequestration. The onset of nonlinear convection, however, remains understudied because the transient base state invalidates the traditional stability methods that are used for autonomous systems. We demonstrate that the onset time of nonlinear convection, $t=t_{\mathit{on}}$, can be determined from an expansion that is two orders of magnitude faster than a direct numerical simulation. Using the expansion, we explore the sensitivity of $t_{\mathit{on}}$ to the initial perturbation magnitude and wavelength, as well as the initial time at which a perturbation is initiated. We find that there is an optimal initial time and wavelength that minimize $t_{\mathit{on}}$, and we obtain analytical relationships for these parameters in terms of aquifer properties and initial perturbation magnitude. This importance of the initial perturbation time and magnitude is often overlooked in previous studies. To investigate perturbation evolution at late-times, $t>t_{\mathit{on}}$, we perform direct numerical simulations that reveal two unique features of transient diffusive boundary layers. First, when a boundary layer is perturbed with a single horizontal Fourier mode, nonlinear mechanisms generate a zero-wavenumber response whose magnitude eventually surpasses that of the fundamental mode. Second, when a boundary layer is simultaneously perturbed with many Fourier modes, the late-time perturbation magnitude is concentrated in the zero-wavenumber mode, and there is no clearly dominant, non-zero, wavenumber. These unique results are further interpreted by comparison with direct numerical simulations of Rayleigh–Bénard convection.

Surface-diffusion-driven decay of high-aspect-ratio gratings: Existence of morphologically related classes

We present numerical and theoretical results concerning the technologically important process of evolution of high-aspect-ratio profiles due to surface diffusion under thermal treatment. We show how a broad class of initial gratings adopt, after a short transient stage, a typical shape that can be accurately described as a curve whose curvature has only two single Fourier modes as a function of the arc-length parameter. Moreover, we introduce a set of evolution equations for the relevant parameters that accounts very accurately for both morphological and kinetic aspects of the transformation processes for these curves in a wide region in parameter space. Regarding the decay of rectangular gratings, our numerical results show the existence of geometrically related classes that asymptotically approach to the same trajectory in parameter space. Gratings belonging to the same class pass through the same sequence of morphologies before reaching the final equilibrium state.

Short-time behavior of advecting-diffusing scalar fields in Stokes flows

This article addresses the short-term decay of advecting-diffusing scalar fields in Stokes flows. The analysis is developed in two main subparts. In the first part, we present an analytic approach for a class of simple flow systems expressed mathematically by the one-dimensional advection-diffusion equation w(y)∂(ξ)φ=ε∂(y)(2)φ+iV(y)φ-ε'φ, where ξ is either time or axial coordinate and iV(y) an imaginary potential. This class of systems encompasses both open- and closed-flow models and corresponds to the dynamics of a single Fourier mode in parallel flows. We derive an analytic expression for the short-time (short-length) decay of φ, and show that this decay is characterized by a universal behavior that depends solely on the singularity of the ratio of the transverse-to-axial velocity components V(eff)(y)=V(y)/w(y), corresponding to the effective potential in the imaginary potential formulation. If V(eff)(y) is smooth, then ||φ||(L(2))(ξ)=exp(-ε'ξ-bξ(3)), where b>0 is a constant. Conversely, if the effective potential is singular, then ||φ||(L(2))(ξ)=1-aξ(ν) with a>0. The exponent ν attains the value 5/3 at the very early stages of the process, while for intermediate stages its value is 3/5. By summing over all of the Fourier modes, a stretched exponential decay is obtained in the presence of nonimpulsive initial conditions, while impulsive conditions give rise to an early-stage power-law behavior. In the second part, we consider generic, chaotic, and nonchaotic autonomous Stokes flows, providing a kinematic interpretation of the results found in the first part. The kinematic approach grounded on the warped-time transformation complements the analytical theory developed in the first part.

The route to dissipation in strongly stratified and rotating flows

AbstractWe investigate the route to dissipation in strongly stratified and rotating systems through high-resolution numerical simulations of the Boussinesq equations (BQs) and the primitive equations (PEs) in a triply periodic domain forced at large scales. By applying geostrophic scaling to the BQs and using the same horizontal length scale in defining the Rossby and the Froude numbers, $\mathit{Ro}$ and $\mathit{Fr}$, we show that the PEs can be obtained from the BQs by taking the limit ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} \rightarrow 0$. When ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} $ is small the difference between the results from the BQ and the PE simulations is shown to be small. For large rotation rates, quasi-geostrophic dynamics are recovered with a forward enstrophy cascade and an inverse energy cascade. As the rotation rate is reduced, a fraction of the energy starts to cascade towards smaller scales, leading to a shallowing of the horizontal spectra from ${ k}_{h}^{- 3} $ to ${ k}_{h}^{- 5/ 3} $ at the small-scale end. The vertical spectra show a similar transition as the horizontal spectra and we find that Charney isotropy is approximately valid also at larger wavenumbers than the transition wavenumber. The high resolutions employed allow us to capture both ranges within the same simulation. At the transition scale, kinetic energy in the rotational and in the horizontally divergent modes attain comparable values. The divergent energy is several orders of magnitude larger than the quasi-geostrophic divergent energy given by the $\Omega $-equation. The amount of energy cascading downscale is mainly controlled by the rotation rate, with a weaker dependence on the stratification. A larger degree of stratification favours a downscale energy cascade. For intermediate degrees of rotation and stratification, a constant energy flux and a constant enstrophy flux coexist within the same range of scales. In this range, the enstrophy flux is a result of triad interactions involving three geostrophic modes, while the energy flux is a result of triad interactions involving at least one ageostrophic mode, with a dominant contribution from interactions involving two ageostrophic and one geostrophic mode. Dividing the ageostrophic motions into two classes depending on the sign of the linear wave frequency, we show that the energy transfer is for the largest part supported by interactions within the same class, ruling out the wave–wave–vortex resonant triad interaction as a mean of the downscale energy transfer. The role of inertia-gravity waves is studied through analyses of time-frequency spectra of single Fourier modes. At large scales, distinct peaks at frequencies predicted for linear waves are observed, whereas at small scales no clear wave activity is observed. Triad interactions show a behaviour which is consistent with turbulent dynamics, with a large exchange of energy in triads with one small and two large comparable wavenumbers. The exchange of energy is mainly between the modes with two comparable wavenumbers.

Limits on semiclassical fluctuations in the primordial universe

We place limits on semiclassical fluctuations that might be present in the primordial perturbation spectrum. These can arise if some signatures ofpre-inflationary features survive the expansion, or could be created by whatever mechanism ends inflation. We study two possible models for such remnant fluctuations, both of which break the isotropy of CMB on large scales. We first consider a semiclassical fluctuation in one Fourier mode of primordial perturbations. The second scenario we analyze is a semiclassical Gaussian bump somewhere in space. These models are tested with the seven-year WMAP data using a Markov Chain Monte Carlo Bayesian analysis, and we place limits on these fluctuations. The upper bound for the amplitude of a fluctuation in a single Fourier mode is a ⩽ 10−4, while for the Gaussian bump a ⩽ 10−3.

Monte Carlo algorithm for buckling search and neutron leakage-corrected calculations

A Monte Carlo algorithm has been developed for taking into account neutron leakage effect specified by a buckling. The algorithm enables one to perform buckling search mode eigenvalue calculations where the buckling is treated as an eigenvalue. For introducing neutron leakage effect, the spatial dependence of neutron flux is approximated by a single Fourier mode. The particle weight is generally a complex number. The imaginary part, however, vanishes and only the real part needs to be treated if symmetry exists with respect to the direction represented by the buckling and the scattering and fission neutron emission are isotropic. The algorithm has been verified by comparing bucklings obtained by buckling search mode eigenvalue calculations with those by the diffusion approximation and the B1 method. The newly developed algorithm can also be applied to leakage-corrected Monte Carlo calculations, generating leakage-corrected neutron spectra and flux distributions. Although this paper has presented only one-group calculations, the algorithm can contribute to the generation of leakage-corrected group constants based on Monte Carlo calculation techniques.

Research on the Method to Decompose Propagation Sound Wave in Annular Duct

There are many benchmark problems which have exact solutions or experiment data to gauge the simulation of aeroacoustic problems. Annular duct acoustic modes are good benchmark problems of this kind. The waves propagated along the duct can be considered as the summation of infinite Fourier modes. The energy relations of a single Fourier mode are deduced from governing equations in this paper. To investigate the properties of acoustic wave propagating along the duct, it’s necessary to decompose acoustic waves in the duct into multiple Fourier modes; the decomposition methods are presented.

A Resolution of the Sommerfeld Paradox

The Sommerfeld paradox roughly says that mathematically Couette linear shear is linearly stable for all Reynolds numbers, but experimentally arbitrarily small perturbations can induce the transition from the linear shear to turbulence when the Reynolds number is large enough. The main idea of our resolution of this paradox is to show that there is a sequence of linearly unstable shears which approaches the linear shear in the kinetic energy norm but not in the enstrophy (vorticity) norm. These oscillatory shears are single Fourier modes in the Fourier series of all the shears. In experiments, such linear instabilities will manifest themselves as transient nonlinear growth leading to the transition from the linear shear to turbulence. Under the Euler dynamics, these oscillatory shears are steady, and cat's eye structures bifurcate from them as traveling waves. The three-dimensional shears $(U(y,z),0,0)$ in a neighborhood of these oscillatory shears are linearly unstable too. Under the Navier–Stokes dynamics, these oscillatory shears are not steady but rather slowly drifting. When these oscillatory shears are viewed as frozen, the corresponding Orr–Sommerfeld operator has unstable eigenvalues which approach the corresponding inviscid eigenvalues when the Reynolds number tends to infinity. All the linear instabilities mentioned above offer a resolution to the Sommerfeld paradox and an initiator for the transition from the linear shear to turbulence.

Axial-symmetry breaking in constrained membranes

We study the equilibrium shapes of a lipid membrane, attached to a fixed circular substrate. We show how the weakening of the boundary conditions is able to break the axial symmetry of the optimal equilibrium configuration. We derive the critical threshold of the symmetry-breaking transition, and obtain the analytical expression of the free-energy minimizers in the quasi-planar approximation. Metastable states turn out to contain contributions only from the axisymmetric mode, and at most one single non-trivial Fourier mode.

Fourier analysis of the parametric resonance in neutrino oscillations

Parametric enhancement of the appearance probability of the neutrino oscillation under the inhomogeneous matter is studied. Fourier expansion of the matter density profile leads to a simple resonance condition and manifests that each Fourier mode modifies the energy spectrum of oscillation probability at around the corresponding energy; below the MSW resonance energy, a large-scale variation modifies the spectrum in high energies while a small-scale one does in low energies. In contrast to the simple parametric resonance, the enhancement of the oscillation probability is itself an slow oscillation as demonstrated by a numerical analysis with a single Fourier mode of the matter density. We derive an analytic solution to the evolution equation on the resonance energy, including the expression of frequency of the slow oscillation.

Ergodicity of ideal Galerkin three-dimensional magnetohydrodynamics and Hall magnetohydrodynamics models

We explore the problem of the ergodicity of magnetohydrodynamics and Hall magnetohydrodynamics in three-dimensional, ideal Galerkin systems that are truncated to a finite number of Fourier modes. We show how single Fourier modes follow the Gibbs ensemble prediction, and how the ergodicity of the phase space is restored for long-time Galerkin solutions. Running time averages and two-time correlation functions show, at long times, a convergence towards zero of time averaged single Fourier modes. This suggests a delayed approach to, rather than a breaking of, ergodicity. Finally, we present some preliminary ideas concerning the origin of the associated time scales.

Construction of circle bifurcations of a two-dimensional spatially periodic flow

The study by Yudovich [V.I. Yudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid, J. Math. Mech. 29 (1965) 587–603] on spatially periodic flows forced by a single Fourier mode proved the existence of two-dimensional spectral spaces and each space gives rise to a bifurcating steady-state solution. The investigation discussed herein provides a structure of secondary steady-state flows. It is constructed explicitly by an expansion that when the Reynolds number increases across each of its critical values, a unique steady-state solution bifurcates from the basic flow along each normal vector of the two-dimensional spectral space. Thus, at a single Reynolds number supercritical value, the bifurcating steady-state solutions arising from the basic solution form a circle.

Nonlinear Dynamics of Inviscid Reduced MHD Plasmas: The Appearance of Quasi-Single-Helicity States

The nonlinear dynamics of an inviscid magnetohydrodynamic system in a low-beta plasma is investigated through a simplified model obtained by a Galerkin projection. We show that a single Fourier mode dominates over the whole energy spectrum, a self-organization mechanism of turbulence being responsible for this nonlinear effect. Our results are in agreement with observations of an analogous phenomenon, say, the emergence of a quasi-single-helicity state both in numerical simulations and real experiments in reversed field pinch laboratory plasmas.

The formation of energy pass bands from periodically spaced point masses on a ribbed cylindrical shell

An infinite cylindrical shell has bending energy that is able to propagate down the shell at all frequencies with little attenuation other than from material damping and the surrounding medium. In contrast, making the shell finite causes bending resonances to occur only at discrete frequencies. The addition of periodically spaced circumferential stiffeners causes the bending wave dispersion curves to be seen at additional axial modes, which is referred to as Brillouin folding. The stiffeners also cause energy stop bands to occur when the load is a single Fourier mode. In this presentation, periodically spaced point masses in the circumferential direction are added to a ribbed finite cylindrical shell and analyzed via the finite element method. It is seen that Brillouin folding occurs circumferentially. Also, energy pass bands are created in a certain pattern. Analysis of the axisymmetric version of this shell combined with information on the type of aperiodicity yields a prediction of the energy pass band behavior. This provides a qualitative approach toward understanding the motion of nonaxisymmetric shell structures. [Work supported by ONR.]

Kolmogorov flow in three dimensions

A numerical study of the long-time evolution of incompressible Navier-Stokes turbulence forced at a single long-wavelength Fourier mode, i.e., a Kolmogorov flow, has been completed. The boundary conditions are periodic in three dimensions and the forcing is effected by imposing a steady, two-dimensional, sinusoidal shear velocity which is directed along the x-direction and varies along the z-direction. A comparison with experimental data shows agreement with measured cross-correlations of the turbulent velocity components which lie in the mean-flow plane. A statistical analysis reveals that the shear-driven turbulence studied here has significant spectral anisotropy which increases with wave number.

Collision operators for partially linearized particle simulation codes.

Algorithms are presented for energy- and momentum-conserving like-particle Coulomb collisions in partially linearized (\ensuremath{\delta}f) particle simulations. They are developed and implemented in particular for gyrokinetic simulation models of a strongly magnetized plasma. The collision operators include both drag and diffusion terms, are not restricted to a single or few Fourier modes, and approximately conserve both momentum and energy locally in space in a statistical sense. The first algorithm is a many-mode generalization of a test-particle--plus--source algorithm previously proposed. The second is easier to implement and improves upon the first significantly by not requiring many time steps for good conservation. Implementations for the case for ion-ion collisions are given and conservation properties are demonstrated, both directly with non-self-consistent test simulation runs and indirectly with self-consistent runs. The computational cost of particle pushing and solving for fields depends on the relative collisionality and can result in a tripling of the total computational costs if collisions are done at each time step, but typically will be a small fraction of the total simulation cost. It is also shown that binary-collision-based algorithms are unsuitable for partially linearized simulations.

Stability of two-dimensional needle growth.

The stability of two-dimensional well-defined needle growth is considered beyond the perturbative approach, until singularity formation, when surface tension may no longer be neglected. The needle growth is shown to be represented by the development of a single dominant Fourier mode, and the conditions under which all other modes are negligibly small, and not influenced by this dominant one, are derived. The dominant mode divides the Fourier spectrum into two parts with different criteria for stability. It is shown that for modes higher than the dominant one the initial smoothness (i.e., the decay of the Fourier spectrum with wave numbers) should be at least exponential for needle stability, whereas for modes lower than the dominant one the condition for stability is more restrictive in that the initial law of decay should be faster than Gaussian.

Minimal modelling of atmospheric energy cycle and high-frequency variability

In this paper we build a minimal model able to describe the essential features of the atmospheric energy cycle and high-frequency variability. The model is obtained by projecting the quasi-geostrophic, two-level system on a single Fourier mode in the longitudinal directions and two eigenfunctions of a Sturm-Liouville problem in the meridional direction. The dynamical system, consisting of 14 equations, shows statistical properties in agreement with the observations and other low-order models.

Large-eddy simulation of transitional channel flow

A large-eddy simulation (LES) of transition in plane channel flow has been carried out. The LES results have been compared with those of a fine direct numerical simulation (DNS), and with those of a coarse DNS that uses the same mesh as the LES, but no residual stress model. While at the early stages of transition LES and coarse DNS give the same results, the presence of the residual stress model was found to be necessary to predict accurately mean velocity and Reynolds stress profiles during the late stages of transition (after the second spike stage). The evolution of single Fourier modes is also predicted more accurately by the LES than by the coarse DNS. As small scales are generated, the dissipative character of the residual stress causes it to reproduce correctly the energy cascade; as transition progresses, then, and the flow approaches its fully developed turbulent state, the subgrid scales tend towards equilibrium and the model becomes more accurate.