Boundary-layer pressures and the Corcos model: a development to incorporate low-wavenumber constraints

Abstract

This paper re-examines the theoretical arguments that indicate the structure of the pressure field induced on a flat surface by boundary-layer turbulence at low Mach number. The long-wave elements are shown to be dictated by the acoustics of the flow, and the limit of the acoustic range is the coincidence condition of grazing waves where the spectrum is singular and proportional to the logarithm of the flow scale. The surface spectrum is shown to be proportional to the square of frequency at low-enough frequency and to the square of wavenumber at those low wavenumbers with subsonic phase speed.The similarity model successfully used by Corcos for the main convective elements of the field is used in this paper to model the turbulent sources of pressure, not the pressure itself, so that a Corcos-like description of the pressure spectrum is derived that is consistent with constraints imposed by the governing equations. This results in a fairly compact specification of the pressure spectrum with yet-undetermined constants, which must be derived from experiment. Despite an extensive search of published data on the pressure field, it is concluded that existing information is an inadequate basis for setting those constants and that new free field experiments are needed. Boundary layers formed on gliders or buoyant underwater bodies offer the most promising source of such data.The paper concludes with a study of how large flush-mounted transducers discriminate against the local flow noise field and i t is shown that they do so at a rate of 9 decibels per doubling of transducer diameter. This different conclusion from Corcos’ correct 6 decibel rate for small transducers is entirely due to the low- wavenumber constraints on the spectrum, which are misrepresented in the simple similarity model. This result, which conforms with the constraints imposed by the weak compressibility of the fluid, is the same as that later suggested by Corcos for transducers that are large on the boundary-layer scale.

Assessment of the Contribution of Surface Roughness to Airframe Noise

“…is repeated with different wall pressure spectrum models by Corcos [7], Efimtsov [8], Smol'yakov and Tkachenko [9], and Chase [11,12]. The Ffowcs Williams model [10] is rejected, as suggested by Graham [6], because its divergent behavior in high wave number region does not satisfy the integral requirement:…”

“…(32) for two types of roughness elements, that is, R 4 mm and R 3 mm, respectively. The experimental results § These models are the Corcos model [7], the Efimtsov model [8], the Smol'yakov and Tkachenko model [9], the Ffowcs Williams model [10], the Chase 1 model [11], and the Chase 2 model [12], respectively (see the Appendix for details).…”

“…Finally, the sound pressure level (SPL) of the rough noise spectrum is obtained by comparing P R x; ! with the reference sound pressure level SPL 10log 10 P …”

“…Graham [6] discussed the different models p ; ! presented by Corcos [7], Efimtsov [8], Smol'yakov and Tkachenko [9], Ffowcs Williams [10], and Chase [11,12]. These models can be related to P s ; !…”

Turbulent boundary-layer noise: direct radiation at Mach number 0.5

“…A good discussion of this problem is provided by Ffowcs Williams (1965Williams ( , 1982. The KraichnanPhillips theorem (Howe (1992)), named after the studies of Kraichnan (1956) andPhillips (1956), states that Φ pp (k, ω) must tend toward zero as k 2 for an incompressible flow (Bull (1996)).…”