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Register a new userAcoustic scattering by cascades with complex boundary conditions: compliance, porosity and impedance
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We present a solution for the scattered field caused by an incident wave interacting with an infinite cascade of blades with complex boundary conditions. This extends previous studies by allowing the blades to be compliant, porous or satisfying a generalised impedance condition. Beginning with the convected wave equation, we employ Fourier transforms to obtain an integral equation amenable to the Wiener--Hopf method. This Wiener--Hopf system is solved using a method that avoids the factorisation of matrix functions. The Fourier transform is inverted to obtain an expression for the acoustic potential function that is valid throughout the entire domain. We observe that the principal effect of complex boundary conditions is to perturb the zeros of the Wiener--Hopf kernel, which correspond to the duct modes in the inter-blade region. We focus efforts on understanding the role of porosity, and present a range of results on sound transmission and generation. The behaviour of the duct modes is discussed in detail in order to explain the physical mechanisms behind the associated noise reductions. In particular, we show that cut-on duct modes do not exist for arbitrary porosity coefficients. Conversely, the acoustic modes are unchanged by modifications to the boundary conditions. Consequently, we observe that even modest values of porosity can result in reductions in the sound power level of $5$ dB for the first mode and $20$ dB for the second mode. The solution is essentially analytic (the only numerical requirements are matrix inversion and root finding) and is therefore extremely rapid to compute.
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We study time-harmonic scattering in $mathbb{R}^n$ ($n=2,3$) by a planar screen (a crack in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset $Gamma$ of the hyperplane $mathbb{R}^{n-1}times {0}$, on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, $Gamma$ can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such $Gamma$ we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when $partialGamma$ is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother prefractal screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake.
We report on a comprehensive experimental-computational study of a simple T-shaped mixer for Reynolds numbers up to $4000$. In the experiments, we determine the mixing time by applying the Villermaux--Dushman characterization to a water-water mixture. In the numerical simulations, we resolve down to the smallest (Kolmogorov) flow scales in space and time. Excellent agreement is obtained between the experimentally measured mixing time and numerically computed intensity of segregation, especially in the turbulent regime, which validates both approaches. We confirm that the mixing time is mainly determined by the specific power input, as assumed in most mixing-models. However, we show that by suitably manipulating the inflow conditions, the power input necessary to achieve a given mixing time can be reduced by a factor of six. Our study enables detailed investigations of the influence of hydrodynamics on chemical reactions and precipitation processes, as well as the detailed testing of turbulence and micromixing models.
Human impedance parameters play an integral role in the dynamics of strength amplification exoskeletons. Many methods are used to estimate the stiffness of human muscles, but few are used to improve the performance of strength amplification controllers for these devices. We propose a compliance shaping amplification controller incorporating an accurate online human stiffness estimation from surface electromyography (sEMG) sensors and stretch sensors connected to the forearm and upper arm of the human. These sensor values along with exoskeleton position and velocity are used to train a random forest regression model that accurately predicts a persons stiffness despite varying movement, relaxation, and muscle co-contraction. Our models accuracy is verified using experimental test data and the model is implemented into the compliance shaping controller. Ultimately we show that the online estimation of stiffness can improve the bandwidth and amplification of the controller while remaining robustly stable.
We report an experimental study aiming to clarify the role of boundary conditions (BC) in high Rayleigh number $10^8 < {rm{Ra}} < 3 times 10^{12}$ turbulent thermal convection of cryogenic helium gas. We switch between BC closer to constant heat flux (CF) and constant temperature (CT) applied to the highly conducting bottom plate of the aspect ratio one cylindrical cell 30 cm in size, leading to dramatic changes in the temperature probability density function and in power spectral density of the temperature fluctuations measured at the bottom plate, while the dynamic thermal behaviour of the top plate and bulk convective flow remain unaffected. Within our experimental accuracy, we find no appreciable changes in Reynolds number Re(Ra) scaling, in the dimensionless heat transfer efficiency expressed via Nusselt number Nu(Ra) scaling, nor in the rate of direction reversals of large scale circulation.
A series of direct numerical simulations of Rayleigh-Benard convection, the flow in a fluid layer heated from below and cooled from above, were conducted to investigate the effect of mixed insulating and conducting boundary conditions on convective flows. Rayleigh numbers between $text{Ra}=10^7$ and $text{Ra}=10^9$ were considered, for Prandtl numbers $text{Pr}=1$ and $text{Pr}=10$. The bottom plate was divided into patterns of conducting and insulating stripes. The size ratio between these stripes was fixed to unity and the total number of stripes was varied. Global quantities such as the heat transport and average bulk temperature and local quantities such as the temperature just below the insulating boundary wall were investigated. For the case with the top boundary divided into two halves, one conducting and one insulating, the heat transfer was found to be approximately two thirds of the fully conducting case. Increasing the pattern frequency increased the heat transfer which asymptotically approached the fully conducting case, even if only half of the surface is conducting. Fourier analysis of the temperature field revealed that the imprinted pattern of the plates is diffused in the thermal boundary layers, and cannot be detected in the bulk. With conducting-insulating patterns on both plates, the trends previously described were similar, however, the half-and-half division led to a heat transfer of about a half of the fully conducting case instead of two-thirds. The effect of the ratio of conducting and insulating areas was also analyzed, and it was found that even for systems with a top plate with only $25%$ conducting surface, heat-transport of $60%$ of the fully conducting case can be seen. Changing the 1D stripe pattern to 2D checkerboard tessellations does not result in a significantly different response of the system.
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