Damping in turbulent Navier-Stokes finite element model simulations of wind instruments
Rolf Bader · The Journal of the Acoustical Society of America · 2013
The flute has a very inefficient energy use, where <1% of the blowing energy ends in the produced sound caused by the high impedance of the blowing hole suggesting a heavy damping in the system. Such damping is known from turbulent flows with high Reynolds numbers. In a FEM simulation of the Navier-Stokes equation of the flute, the inflow of blowing velocity into the flue shows unrealistic values. An alternative is the reasoning of Kolmogorov about turbulent flow as a cascade of vortices satisfying the scaling law as a steady-slope in a log-log relation of vortices size and damping. This is incorporated in the Navier-Stokes model as Reynolds-Averaged-Navier-Stokes model. In a FEM simulation of the flute using this model, the inflow of energy into the flute is modeled realistically, pointing to damping as the crucial factor in flute sound production. As a second example, a transient FEM model of a flow-structure coupling of a saxophone mouthpiece also shows strong damping of the inflow into the mouthpiece due to strong turbulence in the mouthpiece. Only because of this strong damping the system coupled to a tube results in stable oscillations and therefore in a stable tone production.