Vortex-Shedding Noise – The Reversed Sears’ Problem
Vortex-shedding noise is referred to in this chapter as the generation of sound by the formation of structured vortices, such as a von HArman street, downstream of a blunted airfoil trailing edge. The mechanism differs from aforementioned trailing-edge noise produced by vortical patterns already present upstream in the developing boundary layers (note that yet some authors use the same terminology). As stated by Brooks & Hodgson (1981), the occurrence or not of structured vortex shedding is a matter of compared values of the physical thickness of the trailing edge h and of the boundary layer displacement thickness V Typically the vortex street cannot develop if h/5 < 0.3. Vortex shedding is known to radiate narrow-band noise
around the Strouhal frequency f0 = 0.2 U0/h. In contrast trailing-edge noise is either broadband or tonal depending whether the boundary layers are turbulent or laminar-unstable. Vortex-shedding sound in fan technology can be difficult to recognize because the different values of the characteristic flow speed along the span of a blade result in a broader frequency range.
(a) (b) Figure 11. Model von Karman vortex street in the wake of a thick plate (a) and associated reversed Sears’ problem (b) from Roger et al (2006). Pure vortex-shedding configuration.
Imposing a Kutta condition in the present generic model would make no sense because both experiments and numerical simulations show evidence of lift fluctuations concentrating at the trailing edge with phase opposition between both sides. This symmetry with what happens at a leading edge impinged by upstream disturbances and referred to as Sears’ problem in section 2 suggests that a similar mathematical statement is also relevant. This is why the configuration is interpreted as a reversed Sears’ problem by Roger et al (2006) (note that another modeling approach has been proposed by Blake (1986)). This is simply made by reversing the flow direction in Fig.2-a and assuming that the leading edge becomes the trailing edge and vice versa. The wake oscillation defines an upwash w which is con – vected downstream and has to be canceled farther upstream on the airfoil surface. When applied to the vortex-shedding noise from a thick plate, a refinement must be introduced to account for a convection speed Uc of the upwash lower than the external flow speed U0 (Fig. 11). Again using Schwarzschild’s technique, the dominant first iteration for the induced lift is obtained as
. (a — 1) k
E [—© і y l]
with ©і = ak+p (1+M0), Щ = akf [1—M02 (1 — 1/a)2] V2, akf ~ 0.2nc/h. Since vortex shedding is known to have quite a large spanwise correlation length, between 5 and 7 trailing-edge thicknesses h, and because the observer is generally in the mid-span plane in validation experiments, the solution is derived only for two-dimensional gusts. The result reduces to the main term of parallel-gust impingement onto a leading edge as the convection speed is set equal to U0 and as the Mach number and chord-wise coordinates are given the opposite sign. The mathematical expression is defined up to —to upstream, but the unsteady loads concentrate at the trailing edge and negligible values are expected at the leading edge. Only the loads distributed over the actual plate surface are taken into account for the acoustic calculations.
The normalized radiation integral of a gust at angular frequency ш is derived in a way similar to preceding cases. It is found as