Oscillations of a gas in an open-ended tube near resonance

The pressure as function of time was measured near resonance in different axial locations of an open-ended tube. Flow visualisation showed that transition to turbulence was not influenced by the strong disturbance of the open end, except in a region near the open end which had a length of about three particle displacements. The pressure readings were decomposed into the first, second and third harmonic and compared with two different theories. In one case, the linearized theory for the oscillating flow in a tube was fitted to the boundary conditions, the obvious one at the piston and a model at the open end. In the second case, the nonlinear theory of Chester 1] was used. Both theories assume a relation between pressure and velocity at the open end that contains two free constants. The constants were determined by comparing the amplitude of the first and the second harmonic ofone pressure measurement with the theoretical predictions. Once the constants are fixed, the pressurep(ωt, x/L) is completely determined. For weak nonlinear effects, the pressure is essentially determined by one constantα(=k ₂) and the second constantβ(=k ₁) loses its significance. For the range of parameters given there isα=0.825±0.015. A very good approximation of the pressure near resonance can therefore be calculated with the following simple boundary condition at the open end $$p_E = \frac{{4\alpha }}{{3\pi }}\rho \hat u_E u_E = 0.350 \rho \hat u_E u_E .$$ Both theories predict a resonance frequency slightly above the experimental one. Changing Levine and Schwingers 2], end correction from 0.6133R to 1R eliminates the discrepancy for all tube lengths. For the first harmonic the variation of the amplitude and the phase of the pressure signal withω andx is very well predicted by both theories. The nonlinear theory describes also the small second and third harmonics fairly well while the linear theory predicts only the correct order of magnitude of these higher harmonics. The constantα that determines the energy loss at the open end shows an apparent increase if the boundary layer on the tube wall becomes turbulent. This occurs for \(A = 2\hat u/\sqrt {v\omega } \geqq 550\) to 750 which is close to the value observed in a tube with a closed end.