An analytical theory of resonant oscillations of a gas in an open-ended tube is developed. The gas flow in the tube is assumed to be turbulent. A model of gas flow near the open end of the tube is constructed. This model allows a boundary condition that is free of empirical parameters to be obtained. Theoretical results are in reasonable agreement with experimental data obtained by other authors. Thetheory of resonant oscillations of a gas in tubes-resonators is one of the most interesting problems of hydrodynamic acoustics. High-intensity oscillations of a gas are usually excited by a piston which harmonically oscillates at one end of the tube [1-4]. Of particular interest from the practical viewpoint are open-ended tubesresonators. Oscillations in such systems are accompanied by a number of interesting effects: an oscillating jet is formed at the open end of a tube [5], a nonuniform temperature field is established in a tube [6], etc.The qualitative theory of the phenomenon is not yet complete. This is connected with the complexity of the boundary condition at the open end of a tube [7, 8] and poor knowledge of the specifics of an oscillating turbulent flow in a tube [9]. An analytical model of the processes at the open end of the tube has been constructed recently, and, using this model, the boundary condition has been determined [10]. The models of tube turbulence, which were proposed in [11, 12] for the first time, offer a description of experimental results, but have significant drawbacks: (a) they ignore the heat transfer between the tube wall and a gas; (b) they -are based on the assumption of a quasi-stationary regime'of turbulence; (c) they do not consider dispersion in a turbulent medium. In this paper, an attempt is made to construct a model of resonant oscillations of a gas at one end of a tube in a turbulent flow regime that is free of the above drawbacks.Oscillations in a cylindrical tube of length L and radius R, which are excited by a harmonically oscillating piston with displacement amplitude l0 << L, are characterized by the following dimensionless parameters [7-9, 13]:
H=Rvu, sinh=--~--, Mp= , Re~=--. cO topHere V is the amplitude of velocity fluctuations in a velocity loop (for the first resonance, at the open end of the tube), w is the cyclic frequency of oscillations, v is the kinematic viscosity, and cO is the speed of sound in an undisturbed gas. Since 10 ~ L, for oscillations around the basic resonance frequency w0 = ~rcO/(2L) [13] we obtain Mp << 1. In experiment, we usually have H >> 1 (the effect of the acoustic boundary layer on the flow core is small) and sinh ~< 1. For a long tube (L/R >> 1), the condition sinh ~< 1 leads to e << 1, i.e., the problem can be solved by the methods of disturbance theory [13]. The criterion Re~ indicates a turbulence regime: if 10 s ~ Re~ ~< 6-10 s, the regime of weak turbulence occurs [9]. This regime is of interest because the majority of experiments were conducted under these conditions [1][2][3][4].
