Wavenumber Pairs Research Articles

Control of linear instabilities in two-dimensional airfoil boundary layers

The present study deals with the control of linear, modal and non-modal growth of perturbations in a trapezoidal wing boundary layer. The asymmetric NACA 2515 airfoil is chosen for the analysis. The boundary layer velocity profiles, at different locations on the airfoil, are obtained using the Falkner–Skan equations. The non-modal linear stability theory is used to obtain the amplification of the initial perturbations, for which the state-space approach is adopted. The control is brought about using a Linear Quadratic Regulator (LQR) controller by the means of addition of fluid momentum in the wall-normal direction from the wing surface. Maximum transient energy growth is calculated across a range of wavenumber pairs, and control is applied to the pair with the highest energy growth. In this numerical analysis, we show that the transient growth is higher towards the trailing edge because of the adverse pressure gradient of the flow. Upon control actuation, the reduction in the transient energy growth of perturbations is achieved up to a maximum of 54.6%.

Linear and nonlinear receptivity mechanisms in boundary layers subject to free-stream turbulence

Large-eddy simulations of a flat-plate boundary layer, without a leading edge, subject to multiple levels of incoming free-stream turbulence are considered in the present work. Within an input–output model, where nonlinear terms of the incompressible Navier–Stokes equations are treated as an external forcing, we manage to separate inputs related to perturbations coming through the intake of the numerical domain, whose evolution represents a linear mechanism, and the volumetric nonlinear forcing due to triadic interactions. With these, we perform the full reconstruction of the statistics of the flow, as measured in the simulations, to quantify pairs of wavenumbers and frequencies more affected by either linear or nonlinear receptivity mechanisms. Inside the boundary layer, different wavenumbers at near-zero frequency reveal streaky structures. Those that are amplified predominantly via linear interactions with the incoming vorticity occur upstream and display transient growth, while those generated by the nonlinear forcing are the most energetic and appear in more downstream positions. The latter feature vortices growing proportionally to the laminar boundary layer thickness, along with a velocity profile that agrees with the optimal amplification obtained by linear transient growth theory. The numerical approach presented is general and could potentially be extended to any simulation for which receptivity to incoming perturbations needs to be assessed.

Biorthogonal decomposition of the disturbance flow field generated by particle impingement on a hypersonic boundary layer

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

Dispersion in lattices with patterns of hardening and softening stiffness nonlinearity

Recent focus has been given to analytically predicting amplitude-dependent dispersion in periodic structures with weak stiffness nonlinearity. These dispersion relationships may inform devices such as amplitude-dependent filters, waveguides, and diodes. However, attention is generally restricted to either spatially uniform nonlinearity or mutually exclusive distributions of hardening or softening stiffness. This study investigates dispersion in lattices with spatial modulations in stiffness nonlinearity. Special attention is given to modulations consisting of both hardening and softening nonlinear terms. A multiple scales perturbation analysis reveals that patterns of hardening and softening stiffness enable both lifting and lowering of a passband’s frequencies. In such cases, passbands contain discrete frequency and wavenumber pairs that prevent amplitude-dependent dispersion shifting. Numerical integration of the lattice equations of motion are carried out to confirm the analytically predicted dispersion behavior. A design of experiment is proposed in which strength and sign of nonlinearity can be tuned with the initial angle of additively manufactured grounding springs. [Work supported by the Office of Naval Research.]

Nonlinear input/output analysis: application to boundary layer transition

Abstract

Determination of Plate Corrosion Dimension Using Nd:YAG Pulsed Laser-generated Wavefield and Experimental Dispersion Curves.

Corrosion detection using a pulsed laser scanning system can be performed via ultrasonic wave propagation imaging. This method outputs illustrations of the wave field within the host structure; thus, it can depict wave–corrosion area interactions. Additionally, post-processing can be performed to enhance the visualization of corroded areas. The wavefield energy computed using RMS (Root Mean Square) is a validated post-processing tool capable of displaying the location and area of corrosion-damaged regions. Nonetheless, to characterize corrosion, it is necessary to determine its depth. The measurement of depth in conjunction with that of the corroded area via the RMS distribution enables the determination of all dimensions of corrosion damage. Thereafter, the flaw severity can be evaluated. This study employed a wavefield within a plate on which corrosion was developed artificially to generate frequency–wavenumber dispersion curves. The curves were compared with their counterparts from a corrosion-free plate. Alternatively, they could be compared with dispersion curves drawn using the depth and material properties of a pristine plate via a computer program. Frequency–wavenumber pairs were extracted from the dispersion curves produced using the portion of the wavefield within the corroded area. These were inserted into the Rayleigh–Lamb equation, from which depths were calculated and averaged.

Customization of reciprocal mass matrices via log‐det heuristic

SummaryCustomization of finite elements for low‐dispersion error through grid dispersion analysis requires a symbolic expansion of a determinant of a representative dynamic stiffness matrix. Such an expansion turns out to be a bottleneck for many practical cases with the size of the representative matrix greater than eight or ten even if the modern computer algebra systems are applied. In this contribution, we propose an alternative approach for low‐dispersion customization that avoids explicit determinant expansion. This approach reduces the customization problem to a series of quadratic programming problems and consist of two main steps. First, the customization problem is reformulated as a rank minimization problem for the representative dynamic stiffness matrix evaluated at several discrete pairs of wavenumbers and frequencies. Second, the rank minimization problem is solved approximately via log‐det heuristic. Examples for customization of reciprocal mass matrices illustrate capabilities of the proposed approach.

A conjecture on the least stable mode for the energy stability of plane parallel flows

In the energy stability theory, the critical Reynolds number is usually defined as the minimum of the first positive eigenvalue $R_{1}$ of an eigenvalue equation for all wavenumber pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the streamwise and spanwise wavenumbers of the normal mode. We prove that $(\cos \unicode[STIX]{x1D703}\pm 1)R_{1}$ are decreasing functions of $\unicode[STIX]{x1D703}=\arctan (\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC})$ for the parallel flows between no-slip or slip parallel plates with or without variations in temperature. Numerical results inspire us to conjecture that $R_{1}$ is also a decreasing function of $\unicode[STIX]{x1D703}$ for the parallel shear flows under the no-slip boundary condition and without variations in temperature. If the conjecture is correct, the least stable normal modes for the energy stability will be streamwise vortices for these base flows.

Stochastic receptivity analysis of boundary layer flow

We utilize the externally forced linearized Navier-Stokes equations to study the receptivity of pre-transitional boundary layers to persistent sources of stochastic excitation. Stochastic forcing is used to model the effect of free-stream turbulence that enters at various wall-normal locations and the fluctuation dynamics are studied via linearized models that arise from locally parallel and global perspectives. In contrast to the widely used resolvent analysis that quantifies the amplification of deterministic disturbances at a given temporal frequency, our approach examines the steady-state response to stochastic excitation that is uncorrelated in time. In addition to stochastic forcing with identity covariance, we utilize the spatial spectrum of homogeneous isotropic turbulence to model the effect of free-stream turbulence. Even though locally parallel analysis does not account for the effect of the spatially evolving base flow, we demonstrate that it captures the essential mechanisms and the prevailing length-scales in stochastically forced boundary layer flows. On the other hand, global analysis, which accounts for the spatially evolving nature of the boundary layer flow, predicts the amplification of a cascade of streamwise scales throughout the streamwise domain. We show that the flow structures that can be extracted from a modal decomposition of the resulting velocity covariance matrix, can be closely captured by conducting locally parallel analysis at various streamwise locations and over different wall-parallel wavenumber pairs. Our approach does not rely on costly stochastic simulations and it provides insight into mechanisms for perturbation growth including the interaction of the slowly varying base flow with streaks and Tollmien-Schlichting waves.

Precision of Vegetation Height Estimation Using the Dual-Baseline PolInSAR System and RVoG Model With Temporal Decorrelation

Estimating vegetation height from polarimetric interferometric synthetic aperture radar (PolInSAR) data using the Random Volume over Ground model has motivated several studies. Most of these propose estimators and apply them to real data to demonstrate their potential. In previous publications on the single-baseline system, we proposed a complementary approach, which consisted in analyzing the precision of estimations of vegetation height that can be expected depending on the considered model and on the available a priori knowledge. In this paper, we develop such an analysis for the case of a dual-baseline (DB) system. We consider the DB configuration with a PolInSAR set obtained with three PolSAR acquisitions, the extinction coefficient of the volume is assumed unknown, and the level of temporal decorrelation is assumed to be unknown. The observed high sensitivity of the vegetation height Cramer–Rao bound (CRB) with respect to the system parameters and the vegetation characteristics shows that the system optimization cannot guarantee 1-m precision for all vegetation heights, even for large estimation windows with $N=2000$ pixels. Nevertheless, an operating regime exists for which the vegetation height estimation precision is around 1 m for $N=200$ pixels. This regime is obtained for a pair of wavenumbers (0.06 and 0.25 m−1), for vegetation height ranging [20, 50] m, and for polarimetric contrast between the ground and the volume larger than 0.3. Furthermore, we investigate the performance of a maximum-likelihood estimator and compare this to the precision given by the CRB. For the examples considered, with $N=200$ pixels, we observed convergence issues of the estimator when the polarimetric contrast is smaller or equal to 0.3.

Frequency–wavenumber spectral analysis of spatio-temporal flows

We propose a fully spatio-temporal approach for identifying spatially varying modes of oscillation in fluid dynamics simulation output by means of multitaper frequency–wavenumber spectral analysis. One-dimensional spectrum estimation has proven to be a valuable tool in the analysis of turbulence data applied spatially to determine the rate of energy transport between spatial scales, or temporally to determine frequencies of oscillatory flows. It also allows for the quantitative comparison of flow characteristics between two scenarios using a standard basis. It has the limitation, however, that it neglects coupling between spatial and temporal structures. Two-dimensional frequency–wavenumber spectral analysis allows one to decompose waveforms into standing or travelling variety. The extended higher-dimensional multitaper method proposed here is shown to have improved statistical properties over conventional non-parametric spectral estimators, and is accompanied by confidence intervals which estimate their uncertainty. Multitaper frequency–wavenumber analysis is applied to a canonical benchmark problem, namely, a direct numerical simulation of von Kármán vortex shedding off a square wall-mounted cylinder with two inflow scenarios with matching momentum-thickness Reynolds numbers $Re_{\unicode[STIX]{x1D703}}\approx 1000$ at the obstacle. Frequency–wavenumber analysis of a two-dimensional section of these data reveals that although both the laminar and turbulent inflow scenarios show a turbulent $-5/3$ cascade in wavenumber ($\unicode[STIX]{x1D708}$) and frequency ($f$), the flow characteristics differ in that there is a significantly more prominent discrete harmonic oscillation near $(f,\unicode[STIX]{x1D708})=(0.2,0.21)$ in wavenumber and frequency in the laminar inflow scenario than the turbulent scenario. This frequency–wavenumber pair corresponds to a travelling wave with velocity near one near the centre path of the vortex street.

Linear estimation of coherent structures in wall-bounded turbulence at Reτ = 2000

The estimation problem for a fully-developed turbulent channel flow at Reτ = 2000 is considered. Specifically, a Kalman filter is designed using a Navier–Stokes-based linear model. The estimator uses time-resolved velocity measurements at a single wall-normal location (provided by DNS) to estimate the time-resolved velocity field at other wall-normal locations. The estimator is able to reproduce the largest scales with reasonable accuracy for a range of wavenumber pairs, measurement locations and estimation locations. Importantly, the linear model is also able to predict with reasonable accuracy the performance that will be achieved by the estimator when applied to the DNS. A more practical estimation scheme using the shear stress at the wall as measurement is also considered. The estimator is still able to estimate the largest scales with reasonable accuracy, although the estimator’s performance is reduced.

Estimating large-scale structures in wall turbulence using linear models

A dynamical systems approach is used to devise a linear estimation tool for channel flow at a friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=1000$. The estimator uses time-resolved velocity measurements at a single wall-normal location to estimate the velocity field at other wall-normal locations (the data coming from direct numerical simulations). The estimation tool builds on the work of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) by using a Navier–Stokes-based linear model and treating any nonlinear terms as unknown forcings to an otherwise linear system. In this way nonlinearities are not ignored, but instead treated as an unknown model input. It is shown that, while the linear estimator qualitatively reproduces large-scale flow features, it tends to overpredict the amplitude of velocity fluctuations – particularly for structures that are long in the streamwise direction and thin in the spanwise direction. An alternative linear model is therefore formed in which a simple eddy viscosity is used to model the influence of the small-scale turbulent fluctuations on the large scales of interest. This modification improves the estimator performance significantly. Importantly, as well as improving the performance of the estimator, the linear model with eddy viscosity is also able to predict with reasonable accuracy the range of wavenumber pairs and the range of wall-normal heights over which the estimator will perform well.

A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization

We combine resolvent-mode decomposition with techniques from convex optimization to optimally approximate velocity spectra in a turbulent channel. The velocity is expressed as a weighted sum of resolvent modes that are dynamically significant, non-empirical, and scalable with Reynolds number. To optimally represent direct numerical simulations (DNS) data at friction Reynolds number 2003, we determine the weights of resolvent modes as the solution of a convex optimization problem. Using only 12 modes per wall-parallel wavenumber pair and temporal frequency, we obtain close agreement with DNS-spectra, reducing the wall-normal and temporal resolutions used in the simulation by three orders of magnitude.

Chaotic transport across two-dimensional jet streams

A dynamically consistent model of a jet stream modulated by Rossby waves is used to analyze the onset of chaotic transport of a passive tracer across the jet (cross-jet transport). For odd Rossby wavenumbers, a technique is developed to find a central invariant curve whose breakup signifies the onset of cross-jet transport. Amplitude and resonance mechanisms of breakup of this curve are investigated. For even-odd pairs of wavenumbers, an alternative technique is proposed for detecting cross-jet transport, based on an overlap of northern and southern stochastic layers. In this case, the Rossby-wave amplitudes required for global crossjet transport to occur are larger than those in the case of odd wavenumbers. The predicted results should be tested in laboratory experiments simulating jet streams modulated by Rossby waves.

Structure and dynamics of low Reynolds number turbulent pipe flow

Using large-scale numerical calculations, we explore the proper orthogonal decomposition of low Reynolds number turbulent pipe flow, using both the translational invariant (Fourier) method and the method of snapshots. Each method has benefits and drawbacks, making the 'best' choice dependent on the purpose of the analysis. Owing to its construction, the Fourier method includes all the flow fields that are translational invariants of the simulated flow fields. Thus, the Fourier method converges to an estimate of the dimension of the chaotic attractor in less total simulation time than the method of snapshots. The converse is that for a given simulation, the method of snapshots yields a basis set that is more optimal because it does not include all of the translational invariants that were not a part of the simulation. Using the Fourier method yields smooth structures with definable subclasses based upon Fourier wavenumber pairs, and results in a new dynamical systems insight into turbulent pipe flow. These subclasses include a set of modes that propagate with a nearly constant phase speed, act together as a wave packet and transfer energy from streamwise rolls. It is these interactions that are responsible for bursting events and Reynolds stress generation. These structures and dynamics are similar to those found in turbulent channel flow. A comparison of structures and dynamics in turbulent pipe and channel flows is reported to emphasize the similarities and differences.

Minimizing transient energy growth in plane Poiseuille flow

The feedback control of laminar plane Poiseuille flow is considered. In common with many flows, the dynamics of plane Poiseuille flow is very non-normal. Consequently, small perturbations grow rapidly with a large transient that may trigger non-linearities and lead to turbulence, even though such perturbations would, in a linear flow, eventually decay. This sensitivity can be measured using the maximum transient energy growth. The linearized flow equations are discretized using spectral methods and then considered at one wave-number pair in order to obtain a model of the flow dynamics in a form suitable for advanced control design. State feedback controllers that minimize an upper bound on the maximum transient energy growth are obtained by the repeated solution of a set of linear matrix inequalities. The controllers are tested using a full Navier—Stokes solver, and the transient energy response magnitudes are significantly reduced compared with the uncontrolled case.

State estimation in wall-bounded flow systems. Part 2. Turbulent flows

This work extends the estimator developed in Part 1 of this study to the problem of estimating a turbulent channel flow at based on a history of noisy measurements on the wall. The key advancement enabling this work is the development and implementation of an efficient technique to extract, from direct numerical simulations, the relevant statistics of an appropriately defined ‘external forcing’ term on the Navier–Stokes equation linearized about the mean turbulent flow profile. This forcing term is designed to account for the unmodelled (nonlinear) terms during the computation of the (linear) Kalman filter feedback gains in Fourier space. Upon inverse transform of the resulting feedback gains computed on an array of wavenumber pairs to physical space, we obtain, as in Part 1, effective and well-resolved feedback convolution kernels for the estimation problem. It is demonstrated that, by applying the feedback so determined, satisfactory correlation between the actual and estimated flow is obtained in the near-wall region. As anticipated, extended Kalman filters (with the nonlinearity of the actual system reintroduced into the estimator model after the feedback gains are determined) outperform standard (linear) Kalman filters on the full system.

Retrieval of semitransparent ice cloud optical thickness from atmospheric infrared sounder (AIRS) measurements

An approach is developed to infer the optical thickness of semitransparent ice clouds (when optical thickness is less than 5) from Atmospheric Infrared Sounder (AIRS) high spectral resolution radiances. A fast cloud radiance model is developed and coupled with an AIRS clear-sky radiative transfer model for simulating AIRS radiances when ice clouds are present. Compared with more accurate calculations based on the discrete ordinates radiative transfer model, the accuracy of the fast cloud radiance model is within 0.5 K (root mean square) in terms of brightness temperature (BT) and runs three orders of magnitude faster. We investigate the sensitivity of AIRS spectral BTs and brightness temperature difference (BTD) values between pairs of wavenumbers to the cloud optical thickness. The spectral BTs for the atmospheric window channels within the region 1070-1135 cm/sup -1/ are sensitive to the ice cloud optical thickness, as is the BTD between 900.562 cm/sup -1/ (located in an atmospheric window) and 1558.692 cm/sup -1/ (located in a strong water vapor absorption band). Similarly, the BTD between a moderate absorption channel (1587.495 cm/sup -1/) and the strong water absorption channel (1558.692 cm/sup -1/) is sensitive to ice cloud optical thickness. Neither of the aforementioned BTDs is sensitive to the effective particle size. Thus, the optical thickness of semitransparent ice clouds can be retrieved reliably. We have developed a spectrum-based approach and a BTD-based method to retrieve the optical thickness of semitransparent ice clouds. The present retrieval methods are applied to a granule of AIRS data. The ice cloud optical thicknesses derived from the AIRS measurements are compared with those retrieved from the Moderate Resolution Imaging Spectroradiometer (MODIS) 1.38and 0.645-/spl mu/m bands. The optical thicknesses inferred from the MODIS measurements are collocated and degraded to the AIRS spatial resolution. Results from the MODIS and AIRS retrievals are in reasonable agreement over a wide range of optical thicknesses.

3D refraction statics in the wavenumber domain

The wavenumber iterative modelling (WIM) method was first introduced to estimate the static corrections for 2D land profiles by performing first‐break inversion in the wavenumber domain. The WIM algorithm presents some useful advantages, of robustness, stability and flexibility. Robustness is obtained by intensive exploitation of all the available data and by application of an automatic function for mispick removal. Stability is the result of an iterative procedure that ensures convergence towards a stable and plausible solution even at the end of the profile where the problem is normally ill‐posed. Finally, flexibility is due to the possibility of solving for multilayered structures and of estimating vertical gradients of the velocity.This work extends the WIM method to three dimensions. The extension is feasible because the three‐dimensional (3D) problem can be decomposed into a number of small independent problems, one for any pair of wavenumbers kx, ky. The extension preserves the above‐mentioned advantages. The parameters of the estimated model are affected differently by noise: the analysis of the input/output noise transfer function demonstrates that the high spatial frequencies of the velocity distributions are the components that are most affected by noise; thus, the algorithm includes a gradual damping of the higher wavenumbers of the velocity parameter. Although the WIM 3D algorithm requires a larger amount of RAM compared with other standard approaches, considerable reduction in CPU run time can be achieved as every wavenumber pair can be treated as an independent linear problem.