Turbulence Correlation Length-Scale Relationships for the Prediction of Aeroacoustic Response

Abstract

Expressions to describe the correlation length scales of turbulent inflow to an aerodynamic body are derived as functions of the classic integral length scale and anisotropy correction factors. These one-point parameters are significantly easier to determine experimentally than traditional correlation-scale measurement techniques, which involve multiple probes at multiple locations. As such correlation scales are necessary to properly estimate the aeroacoustic response of the body, such a technique could have substantial benefit in a wide variety of applications. The approach is applied to a recent experimental study examining the response of a stator downstream of a propeller that is itself ingesting broadband turbulence. Results suggest that the derived expressions not only accurately represent correlation length scales, but also enable the accurate prediction of the acoustic output of the stator. Nomenclaturevector of quantity i i 1 = quantity i in the freestream direction i 2 = quantity i in the direction normal to airfoil (stator) surface i 3 = quantity i in the spanwise direction of airfoil (stator) surface i ∞ = freestream quantity i * = complex conjugate of quantity i k = wave number L i = geometric length scale M = Mach number q = dynamic pressure R i j = correlation function between turbulent velocity components u i and u j r = separation distance between two points Se = aerodynamic response function (Sears function) U = mean velocity component u = turbulent velocity component x = flow location (position) α c , β c = anisotropy scaling factors β = angle between acoustic source and receiver, measured with respect to dipole axis γ 2 = coherence function δ = Dirac delta function , 1 = classic turbulence integral length scale i | j = turbulence correlation scale of jth component of velocity in ith direction ρ 0 = density of fluid ii = autopower spectrum in the ith direction ii (r) = cross-power spectrum in the ith direction (separated by distance r)p rad = radiated far-field acoustic pressure φ ii (r) = nondimensional cross-spectral density function of turbulence in the ith direction (separated by distance r) ω = gust or event frequency (=2π f )

Broadband Trailing-Edge Noise Predictions—Overview of BANC-III Results

“…The correlation length scale Λ 2 (x 2 ) is defined as function of frequency (in addition to its x 2 -dependency) and of the anisotropy stretching factors following the derivation proposed by Lynch et al 31 4. Herein, the moving-axis spectrum is modeled using the original TNO approach 35 which consists of a Gaussian distribution as…”

Measurement of the unsteady lift of thick airfoils in incompressible turbulent flow

“…If the quantity u 3 rms /L 11 , which is proportional to the turbulent dissipation rate, is independent of the direction of the velocity component, then the f −5/3 range of the spectrum will remain isotropic, even though the low frequency range varies with direction. Another type of anisotropy, in which the length scale is stretched in a particular direction, has been modeled by Lynch et al (2005). This would be appropriate for turbulence passing through a contraction, and it affects the entire frequency range by shifting the peak of the spectrum.…”

Aerodynamic Noise Characterization of a Full-Scale Wind Turbine through High-Frequency Surface Pressure Measurements

“…where the integral length scale Λ characterizes the size of the energy-containing eddies, the coefficients β 1 and β 3 are anisotropic stretching factors in the streamwise and spanwise directions, respectively. Following the approach by Lynch et al [37] and introducing the stretching factors in the derivation, the correlation length L 2 is defined as a frequency dependent quantity:…”