The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction

Abstract

The Kutta condition is applied to aerofoils with sharp trailing edges to allow for viscous effects to be considered within a simplified system of equations that are inviscid. This paper discusses in detail the inclusion of an unsteady Kutta condition at a sharp trailing edge during gust-aerofoil interaction, and illustrates how the analytic solution for the far-field noise generated by this interaction changes if the unsteady Kutta condition is neglected, or more precisely, if the unsteady pressure is permitted to be singular at the trailing edge. The analytic solution, both with and without the unsteady Kutta condition, is compared with numerical results that have no imposed unsteady Kutta condition. Importantly the results agree well only when the unsteady Kutta condition is neglected in the analytic solution. This paper highlights where the far-field acoustics are most affected by neglecting the unsteady Kutta condition for a variety of singularities that can occur in the unsteady pressure at the trailing edge, and shows that results permitting different behaviour in the unsteady surface pressure at the trailing edge could give significantly different far-field noise predictions, even though the surface pressure elsewhere along the aerofoil surface agrees with benchmark solutions.

Acoustic scattering by cascades with complex boundary conditions: compliance, porosity and impedance

“…In a similar way to § A.1, we now apply the typical Wiener-Hopf argument. We enforce the unsteady Kutta condition (Ayton, Gill & Peake 2016) which restricts the pressure at the trailing edge to be finite. Consequently, and using the results of appendix B, the left side of (A 26) decays as |γ | → ∞ in L + and the left side of (A 26) tends to an unknown constant as |γ | → ∞ in L − .…”

Downstream Perforations for the Reduction of Turbulence-Aerofoil Interaction Noise: Part II - Theoretical Investigation

“…We additionally require continuity of pressure, as the surface cannot support a pressure jump. We allow a vortex sheet to be shed from the upstream edge of the slot in order to regularise the pressure at this edge and so satisfy an unsteady Kutta condition [9,10]. This is naturally imposed in the course of the Wiener-Hopf procedure as discussed later, and noted in previous works [10].…”

“…In our model of a finite length slot, a vortex sheet is allowed to be shed from junction 2 in order for an unsteady Kutta condition to be satisfied there [9,10]. This is incorporated within an aeroacoustic scattering problem for a otherwise rigid semi-infinite, infinitesimally thin plate in uniform flow.…”

Analytic solutions for flows through cascades

“…In most aeroacoustic applications, physical considerations imply that p = q = 0 so that the entire function is a constant. Further physical arguments generally yield that the entire function vanishes (Ayton et al, 2016).…”