Abstract
Abstract
Highlights
Sound propagation in uniformly lined straight ducts with uniform mean flow is well established by its analytically exact description in duct modes (Rienstra 2003b, 2016a; Rienstra & Hirschberg 2004)
First the mean flow is assumed (Bouthier 1972) to be slowly varying everywhere so the mean flow equations are rewritten in the slow coordinate X = εx, rescaled and simplified to leading order in ε, the small parameter that measures the slenderness of the duct variations
If we denote the mean flow by V = Uex + Vey, the impermeable duct wall yields the mean flow boundary condition (V · n) = 0, or
Summary
Sound propagation in uniformly lined straight ducts with uniform mean flow is well established by its analytically exact description in duct modes (Rienstra 2003b, 2016a; Rienstra & Hirschberg 2004). First the mean flow is assumed (Bouthier 1972) to be slowly varying everywhere (no entrance effects) so the mean flow equations are rewritten in the slow coordinate X = εx, rescaled and simplified to leading order in ε, the small parameter that measures the slenderness of the duct variations These equations are solved analytically if possible (apart from an algebraic equation, this is usually the case for potential flow: Rienstra 1999; Peake & Cooper 2001; Rienstra 2003a; Brambley & Peake 2008), or otherwise numerically (often for rotational flow; Cooper & Peake 2001; Lloyd & Peake 2013). Apart from some minor notational differences (like Ω for Ω/C, Ψ for M, κ for μ) and algebraic corrections, the main differences are the slowly varying lined walls, the found complete and incomplete adiabatic invariants and the use of a numerical solution of the central Pridmore-Brown equation, giving access to a very wide range of parameters
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