Roughness Noise Research Articles

High order Anderson parabolic model driven by rough noise in space

In this paper, we concern the fourth parabolic model on [Formula: see text] driven by a multiplicative Gaussian noise which behaves like fractional Brownian motion in time and space with Hurst index [Formula: see text] and [Formula: see text], respectively. The existence and uniqueness of mild solution in Skorohod sense are proved, and the weak intermittency is obtained by estimating [Formula: see text]th ([Formula: see text]) moment of the solution. Moreover, the Hölder continuity can be obtained for the time and space variable.

Ultrasound image restoration through orthogonal and oblique subspace projection based on Lanczos decomposition

We introduced an efficient technique to suppress speckle noise in medical ultrasound images while maintaining low computational complexity. The medical ultrasound image is divided into overlapping subimages, and Lanczos decomposition is then applied to the average Hermitian covariance matrix of all subimages. The resultant orthonormal vectors are used for filtering speckle noise through orthogonal and oblique projections, i.e., by projecting noisy signal onto the signal subspace. After sorting the orthonormal vectors, an orthogonal projection matrix is formed by selecting the first K vectors contributing to the signal, whereas an oblique projection matrix is formed by selecting the first K vectors contributing to the signal and the last K vectors contributing to the noise. The procedure of Lanczos is also followed with singular value decomposition (SVD). Schemes are applied to real ultrasound images and two types of speckle noise simulations: fine and rough speckle noise. Numerical and visual results depict that proposed technique has outperformed various popular benchmark schemes, i.e., Frost, Lee, probabilistic nonlocal means, geometric nonlinear diffusion filter, guided speckle reducing bilateral filter, and SVD, while maintaining a competitive computational complexity. Lanczos-based scheme showed a relatively lagging performance in terms of resolution, which was always outshone by a leading performance in terms of key measures such as signal-to-noise ratio (SNR), peak signal-to-noise ratio (PSNR), feature similarity, and mean structural similarity. Orthogonal and oblique projections were found to perform the same except when speckle noise is fine, where insignificant differences were found in terms of SNR, PSNR, and resolution. Lanczos-based scheme tends to offer a better estimation of orthonormal vectors than SVD, consequently a better speckle noise suppression. Furthermore, it offers an efficient tuning parameter per block/subimage size to treat various speckle noise patterns.

An alternative permeable topology design space for trailing-edge noise attenuation

This study focuses upon a new permeable topology design concept as an alternative to porous metal foams, for turbulent boundary layer trailing-edge (TBL-TE) noise attenuation. The present permeable topology has unconventional characteristics with respect to the metal foams: a combination of low flow resistivity r and high form drag coefficient C. The unconventional characteristics are realized by a Kevlar-covered 3D-printed perforated structure. An experimental study featuring a NACA 0018 airfoil model with a Kevlar-covered 3D-printed TE insert at chord-based Reynolds numbers up to [Formula: see text] is carried out. The airfoil with this TE insert gives a broadband TBL-TE noise reduction up to approximately 5 dB, compared to a solid TE. This reduction varies only slightly with airfoil loading (lower than 1 dB variation), in contrast to the porous metal foams (up to 3 dB variation). When comparing the variation of noise attenuation given by all the permeable materials considered, the variation is found to decrease with the increasing C. This is because C specifies the permeable material's ability to withstand the increasing pressure difference, which causes cross flow that might interfere with the noise attenuation mechanism. Additionally, the drag coefficients as well as the roughness noise of the airfoil equipped with the present TE insert are also significantly lower than those of the metal-foam TE, and are mostly negligible compared to the fully solid airfoil. Based on the findings, design guidelines for permeable TE are proposed: the permeable material shall have a combination of a low flow resistivity and a high form drag coefficient as well as a negligible surface roughness.

Spatial averages for the parabolic Anderson model driven by rough noise

In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0 \in (1/2,1)$, $H_1\in (0,1/2)$ and $H_0 + H_1 > 3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.

Comparing the stochastic nonlinear wave and heat equations: a case study

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order $\alpha > 0$) of a space-time white noise. In particular, we show that the well-posedness theory breaks at $\alpha = \frac 12$ for SNLW and at $\alpha = 1$ for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for $0 < \alpha < \frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of $\alpha $. On the other hand, when $\alpha \geq \frac 12$, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for $\alpha \ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $\alpha = 1$. These examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values ($\alpha = \frac 34$ in the wave case and $\alpha = 2$ in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

Random attractors for 2D stochastic micropolar fluid flows on unbounded domains

<p style='text-indent:20px;'>The asymptotic behavior of a model for 2D incompressible stochastic micropolar fluid flows with rough noise on a Poincaré domain is investigated. First, the existence and uniqueness of solutions to an evolution equation arising from the underlying stochastic micropolar fluid model is established via the Galerkin method and energy method. Then the existence of a random attractor is studied by using the theory of random dynamical systems for which the noise is dealt with by appropriate reproducing kernel Hilbert space.

On a non-linear 2D fractional wave equation

Nous nous interessons au modele d’equation des ondes stochastique non-lineaire suivant: \begin{equation*}\begin{cases}\partial^{2}_{t}u-\Delta u=u^{2}+\dot{B},\quad t\in[0,T],x\in\mathbb{R}^{2},\\u(0,\cdot)=\phi_{0},\qquad \partial_{t}u(0,\cdot)=\phi_{1},\end{cases}\end{equation*} ou $\phi_{0},\phi_{1}$ sont des conditions initiales deterministes dans un espace de Sobolev approprie et $\dot{B}$ represente un bruit fractionnaire espace-temps. Dans cette situation bi-dimensionnelle, notre strategie est basee sur un developpement de l’equation a l’ordre trois, qui, combine a une procedure de renormalisation de type Wick, nous permet d’etendre les resultats de Deya (2019) a des bruits plus rugueux. Nous mettons egalement en avant les limites de cette strategie particuliere en presence de processus tres irreguliers.

Intermittency for the parabolic Anderson model of Skorohod type driven by a rough noise

In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter $H \in (0, 1/2)$. By using the Feynman-Kac representation for the $L^p(\Omega)$ moments of the solution, we find the upper and lower bounds for the moments.

Bioinspired aerofoil adaptations: the next steps for theoretical models.

The extended introduction in this paper reviews the theoretical modelling of leading- and trailing-edge noise, various bioinspired aerofoil adaptations to both the leading and trailing edges of blades, and how these adaptations aid in the reduction of aerofoil-turbulence interaction noise. Attention is given to the agreement between current theoretical predictions and experimental measurements, in particular, for turbulent interactions at the trailing edge of an aerofoil. Where there is a poor agreement between theoretical models and experimental data the features neglected from the theoretical models are discussed. Notably, it is known that theoretical predictions for porous trailing-edge adaptations do not agree well with experimental measurements. Previous works propose the reason for this: theoretical models do not account for surface roughness due to the porous material and thus omit a key noise source. The remainder of this paper, therefore, presents an analytical model, based upon the acoustic analogy, to predict the far-field noise due to a rough surface at the trailing edge of an aerofoil. Unlike previous roughness noise models which focus on roughness over an infinite wall, the model presented here includes diffraction by a sharp edge. The new results are seen to be in better agreement with experimental data than previous models which neglect diffraction by an edge. This new model could then be used to improve theoretical predictions for far-field noise generated by turbulent interactions with a (rough) porous trailing edge. This article is part of the theme issue 'Frontiers of aeroacoustics research: theory, computation and experiment'.

Strong convergence of bi-spatial random attractors for parabolic equations on thin domains with rough noise

This article concerns bi-spatial random dynamics for the stochastic reaction-diffusion equation on a thin domain, where the noise is described by a general stochastic process instead of the usual Wiener process. A bi-spatial attractor is obtained when the non-initial state space is the p-times Lebesgue space, meanwhile, measurability of the attractor in the Banach space is proved by using measurability of both cocycle and absorbing set. Finally, the $p$-norm convergence of attractors is obtained when the thin domain collapses onto a lower dimensional domain. The method of symbolical truncation is applied to provide some uniformly asymptotic estimates.

Two-Step Root-MUSIC for Direction of Arrival Estimation without EVD/SVD Computation

Most popular techniques for super-resolution direction of arrival (DOA) estimation rely on an eigen-decomposition (EVD) or a singular value decomposition (SVD) computation to determine the signal/noise subspace, which is computationally expensive for real-time applications. A two-step root multiple signal classification (TS-root-MUSIC) algorithm is proposed to avoid the complex EVD/SVD computation using a uniform linear array (ULA) based on a mild assumption that the number of signals is less than half that of sensors. The ULA is divided into two subarrays, and three noise-free cross-correlation matrices are constructed using data collected by the two subarrays. A low-complexity linear operation is derived to obtain a rough noise subspace for a first-step DOA estimate. The performance is further enhanced in the second step by using the first-step result to renew the previous estimated noise subspace with a slightly increased complexity. The new technique can provide close root mean square error (RMSE) performance to root-MUSIC with reduced computational burden, which are verified by numerical simulations.

Rating the perception of jet noise crackle

Crackle is a perceptual aspect of noise caused by impulsive acoustic shocks and observed in supersonic jets, including those from military aircraft and rockets. Overall and long-term spectral noise metrics do not account for the unique perception of crackle. Listening tests were designed to better understand perception of crackle and examine its relationship to physical noise metrics, such as skewness of the first time derivative of the pressure waveform, hereafter derivative skewness. It is hypothesized that as derivative skewness increases, the perception of crackle tends to increase. Two listening tests were conducted with 30 subjects to examine their perception of crackle. In the first test, subjects compared and ordered crackle-containing sounds. In the second test, category scaling was employed with subjects rating the crackle content with category labels: (1) smooth noise with no crackle, (2) rough noise with no crackle, (3) sporadic crackle, (4) continuous crackle, and (5) intense crackle. There is a high correlation between perception of crackle and derivative skewness. This five-point classification scheme appears to be an effective tool to measure crackle perception. These insights will help inform community noise models, allowing them to incorporate annoyance due to jet crackle. [Data courtesy of F-35 JPO.]

The Sound of Flow Over Rigid Walls

An overview of the acoustics of boundary layer flows over rough surfaces and surfaces with discontinuities at low Mach number is presented. Roughness noise is dominated by dipole radiation produced by unsteady tangential pressure forces on the uneven surface. Pressure fluctuations may be generated by turbulence of the overriding boundary layer or by the wakes of upstream roughness features, but in either case the sound can be predicted from the wall pressure frequency spectrum and the surface geometry. Small discontinuities, such as steps and gaps, are special cases of isolated two-dimensional roughness. Forward steps are much louder than backward steps because the former generate strong turbulence close to the step. Gap noise is dominated by any exposed forward-step portion of the gap. Rounding can substantially reduce forward step noise, and moderate sweep does not alter the noise-generation mechanism.

Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise

This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the basic convergence. A bi-spatial attractor is obtained when the non-initial space is $p$-times Lebesgue space or Sobolev space. The measurability of the solution operator is proved, which leads to the measurability of the attractor in both state spaces. Finally, the upper semi-continuity of attractors under the $p$-norm is established when the narrow domain degenerates onto a lower dimensional set. Both methods of symbolical truncation and spectral decomposition provide all required uniform estimates in both Lebesgue and Sobolev spaces.

Extension of roughness noise to bluff bodies using the boundary element method

A prediction model of roughness noise generated by bluff body flow at high Reynolds numbers is proposed. Howe's roughness noise theory extended by Liu and Dowling is used, and the boundary layer inputs to the theory have been modified for a bluff body. The scattering due to the bluff body has been accounted for by the boundary element method. The procedure to couple the roughness noise sources to the tailored Green's function is detailed for the case where the boundary element method mesh is orthogonal and aligned with the boundary layer outer velocity. The proposed method has been implemented and compared to experimental results for the particular case of a circular cylinder with large roughness. Two different estimations of the skin friction, which is an input to the roughness noise theory, are considered. One is a zero-pressure gradient model, and the second is based on published experimental data of the skin friction on a rough circular cylinder, but with smaller roughness than was used in the experiments. The zero-pressure gradient skin friction estimate leads to a better prediction of the effect of changes in the area covered by roughness elements. The success of the zero-pressure gradient skin friction estimate is encouraging as the only modifications that need to be made to the boundary layer model to account for a bluff body are the boundary layer outer velocity distribution and the location of separation.

Approximating Stochastic Evolution Equations with Additive White and Rough Noises

In this paper, we analyze Galerkin approximations for stochastic evolution equations driven by an additive Gaussian noise which is temporally white and spatially fractional with Hurst index less than or equal to 1/2. First we regularize the noise by the Wong--Zakai approximation and obtain its optimal order of convergence. Then we apply the Galerkin method to discretize the stochastic evolution equations with regularized noises. Optimal error estimates are obtained for the Galerkin approximations. In particular, our error estimates remove an infinitesimal factor which appears in the error estimates of various numerical methods for stochastic evolution equations in existing literatures.

Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.

Bioinspired Trailing-Edge Noise Control

Strategies for trailing-edge noise control have been inspired by the downy canopy that covers the surface of exposed flight feathers of many owl species. Previous wind-tunnel measurements demonstrate that canopies of similar characteristics can reduce pressure fluctuations on the underlying surface by as much as 30 dB and significantly attenuate roughness noise generated by that surface. In the present work, surface treatments are designed to replicate the effects of the canopy in a form suitable for application to an airfoil. These treatments are installed directly upstream of the trailing edge to modify the boundary-layer turbulence before acoustic scattering by the edge. Over 20 variants of these designs have been tested by performing aeroacoustic wind-tunnel measurements on a tripped DU96-W180 airfoil at chord Reynolds numbers of up to 3 million. Compared with the unmodified airfoil, the treatments provided up to 10 dB of broadband attenuation of trailing-edge noise. The effectiveness of the treatment is not highly dependent on a particular geometry, but there appears to be strong potential for optimization. The surface treatments remain effective over an angle-of-attack range that extends over 9 deg from zero lift. Aerodynamic impact of the treatment appears minimal.

Intermittency for the Hyperbolic Anderson Model with rough noise in space

In this article, we consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index H∈(14,12). Initial data are assumed to be constant. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the p-th moment the solution, for any p≥2. Condition H>14 turns out to be necessary for the existence of solution. Secondly, we show that this solution coincides with the one obtained by the authors in a recent publication, in which the solution is interpreted in the Itô sense. Finally, we prove that the solution of the equation in the Skorohod sense is weakly intermittent.

SPDEs with rough noise in space: Hölder continuity of the solution

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H∈(14,12). The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than H, and Hölder continuous in time of order smaller than γ, where γ=H for the wave equation and γ=H/2 for the heat equation.