Numerical simulations of the edge tone

Numerical simulations of the edge tone I. Vaik, G. Paál Department of Hydrodynamic Systems, Budapest University of Technology and Economics, P.O. Box 91., 1521 Budapest, Hungary, {vaik, paal}@vizgep.bme.hu High precision unsteady numerical simulations have been carried out on the well-known low Mach number edge tone configuration playing a central role in many wind instruments. Detailed parametric investigations have been performed (velocity, inlet profile, jet-edge distance). Several interesting physical phenomena known from experimental investigations were reproduced. The results agree well with measurements. Possibilities to calculate sound emission are discussed. 1 Introduction The edge tone has been the subject of intensive research for over one and a half centuries [1]. The configuration consists of a plane jet, and a wedgeshaped object placed roughly opposite to the jet exit, traditionally called the edge (Figure 1.). Despite its simplicity, the configuration displays a remarkably complex behaviour. The jet oscillates around the edge with a stable frequency and under certain circumstances emits an audible tone. The configuration is thought to be the central element of some wind instruments. Figure 1. Basic configuration of the edge tone This frequency depends on two primary factors, the jetedge distance, h, and the mean exit velocity v. The system behaviour is also influenced by a number of secondary factors, such as the exit velocity profile, the shape of the nozzle, the shape and the transversal position of the edge, etc. The basic relationship between the frequency f and the above-mentioned parameters is: v f = C (1) 3 2 h where C is a constant [2, 3]. The formation of the oscillation is attributed to a feedback loop [1]: the oscillating jet creates a dipole type sound source on the wedge which initiates disturbances at the jet exit. The disturbances grow streamwise along the jet, providing the oscillating jet motion for the formation of the dipole. If either of the above parameters is changed continuously, the edge tone exhibits various stages or hydrodynamic modes meaning that at certain values the frequency suddenly jumps to another value and that the qualitative appearance of the flow also changes. In some parameter regions two modes can coexist which can also be interpreted as a hysteresis phenomenon. The existence of stages is due to the fact that between the jet exit and the edge a certain phase relationship must be maintained. Within a parameter region this can be achieved by adjusting the frequency by keeping the wavelength approximately constant, over a certain value, however, the wavelength must be adjusted to a completely different value. The phase relationship is summarized in the following equation: h = λ ( N + ε ) (2) where λ is the wavelength of the disturbance, N is a whole number corresponding to the stage number, ε is a small number indicating that the effective resonance length of the edge tone system is somewhat longer than h, or in other words the effective attacking point of the pressure-induced force on the edge is ελ distance away from the tip. There is no agreement in the literature about the value of ε, it may also depend on the details of the configuration and on stage number. The wavelength can be determined from the frequency of oscillation and the phase speed of the disturbance. The goal of this paper was to reproduce certain features of the edge tone system known from experimental work [4, 5, 7] and from theoretical considerations on the basis of detailed and accurate direct numerical simulations. There is one paper known to the authors with a similar aim [6], nevertheless the present work covers a wider range of parameters and allows broader conclusions. After a satisfactory level of confidence in the methodology is gained, the simulation can be used as a tool to perform further parametric studies and to get new physical insights into the fluid mechanics of the edge tone. 2 Simulation details The simulations were performed using the ANSYS CFX 5.7 commercial CFD software (CFD = Computational Fluid Dynamics). The geometry and the mesh was prepared using the ANSYS-ICEM CFD 5.1. 635

The mesh was block-structured in order to reduce numerical dissipation. The flow was assumed to be two-dimensional and since the software is threedimensional, a layer of one cell thickness was considered. The simulation geometry can be seen in Figure 2. and the mesh and the block boundaries in the vicinity of the edge tip in Figure 3. As seen in Figure 2. the jet exit is pushed inside the calculation domain in order to avoid problems with the jet entrainment. Both the spatial and the temporal discretisation was of second order accuracy. δ = 1 mm Figure2.CFD Geometry h = 10δ = 10mm α = 30 V1 = V2 = 75.5 mm H1 = 12.5 mm H2 = 75 mm Figure 3. Detail of the mesh with block boundaries The following boundary conditions were used: on the top and bottom plate symmetry, at the jet exit uniform velocity, at the edge wall no slip, at the nozzle wall (length = H1) free slip. At the upstream boundary (V1 and V2) a very low inlet velocity is prescribed (typically 1% of the inlet velocity) in order to stabilize the flow field behind the nozzle without influencing the flow field of interest. All the rest of the boundaries were given a so-called opening B. C. meaning a prescribed static pressure without prescribing flow direction. Contrary to most of the literature cases a top hat velocity profile was used at the jet exit because a higher level of instability and a more pronounced vortex formation was expected. In a later study the influence of the parabolic inlet velocity profile will also be examined. The flow was simulated with one fixed geometry at various Reynolds numbers. In a later paper simulations will be carried out at a fixed Reynolds number with varying jet exit-edge distance. The Reynolds number is based on the jet exit velocity v and the nozzle width δ. The fluid used was air at 25C with a density of 1.185 kg/m 3 and a dynamic viscosity of 1.831e-005. The flow was assumed to be laminar and incompressible. The Reynolds number was systematically changed by changing the jet exit velocity. Pressure and velocity histories were written out in several points of the flow field and by means of FFT the frequencies were determined (using MATLAB 13). No significant differences were found in the frequencies whether velocity or pressure histories were used and whether this or that point was used. Very careful parameter studies have been performed in order to determine the optimum time step, spatial resolution and residual level. Without going into many details at the end it was decided to use a mesh with 36300 elements, the residual was 10-5. The necessary time step τ was about 1/45 of the expected period for the lowest Reynolds number simulated (Re=200). It was proven that the temporal discretisation error is kept constant if τ 2 f 3 is constant. This means that within one stage when the frequency is proportional to the outlet velocity, the time step had to be reduced faster than linear to keep constant accuracy, while increasing the Reynolds number linearly. When the second stage appeared with a significantly higher frequency then the simulation had to be repeated using the time step corresponding to the smaller period. The initial condition was usually the result of a stationary simulation, although later tests showed that other initial conditions led to the same result. Whatever the initial condition, the temporal history of the physical quantities always had a transient part before settling into a quasi-steady oscillation. The temporal length T of the simulation was determined by the need for an appropriate frequency resolution of the spectrum ( f = 1/T). It was decided that a frequency resolution of roughly 1-2% of the expected frequency is to be reached, for example at a fundamental frequency of 200 Hz, a resolution of 2 Hz was used. Using a PC with a 3 GHz CPU at higher Reynolds numbers one simulation took about one week. 3 Results and discussion The flow was simulated for a fixed geometry of h/δ = 10 with varying exit velocity. In Figure 4a and b the frequency and the Strouhal number (St = fδ/v) are presented, respectively, as a function of the Reynolds number (Re = vδ/ν). It can be seen, that for the range of 636

Reynolds numbers considered, Powell s [1] theory and experimental data are reproduced nicely. The frequency increases linearly within one stage with the Reynolds number but with different slope for the different stages. The second stage appears at Re = 250 and the two stages coexist in the whole investigated range. The agreement of the Strouhal numbers is excellent with those of Powell. This is somewhat surprising since Powell s exit velocity profile was parabolic, and ours was top hat. Ségoufin et al. [7], on the other hand, measured significant differences in the Strouhal number by changing the outlet velocity profile. Further investigations are needed to resolve this apparent contradiction by performing simulations with parabolic profile. The start of edge tone activity and also the appearance of the second stage agrees also well with Powell. A difference is, on the other hand that in our case the first stage keeps existing in the whole range, whereas in Powell s case it ceases slightly below Re = 300. this way and there are no jumps but the two stages coexist all the time. A corresponding detail of the pressure time history is shown in Figure 5c. Figure 5a. Time history of the pressure in a point on the edge surface for Re = 250 Figure 5b. Spectrum belonging to Figure 5a Figure 4. Frequency (a, upper figure) and Strouhal number (b, lower figure) as a function of the Reynolds number for the first two stages. Figure 5a shows a detail of the pressure history of the case Re = 250. It is interesting to observe that the oscillation simply switches from one frequency to the other one. These jumps happen during the simulation several times between the stages, apparently randomly. In Figure 5b the corresponding spectrum can be seen with the two peaks. At higher Reynolds numbers, however the two stages cannot be separated temporally Figure 5c. Time history of the pressure in a point on the edge surface for Re = 600 The next question to clarify was the phase velocity of the jet disturbance along the jet. To this end a piece of the time signal of both components of the velocity, and of the pressure was monitored in many points along a 637

line parallel to the jet and δ distance above the jet centerline. When reaching the wedge the line turns parallel to the wedge surface. Using a correlation technique the relative phase of a piece of the signal (5-6 periods) was determined as a function of the axial distance x. The phase reference was the signal in the very first point. One example of such a phase curve is presented in Figure 6. based on the phase of the transversal velocity. It can be seen that the curve is piecewise linear indicating an approximately constant phase velocity. After examining several Reynolds numbers and different variables it turns out that the phase velocity is mostly between 40% and 50% of the exit velocity v. The theoretical result in [8] predicted for disturbance propagation the most unstable frequency for a free 2D jet (without edge) 0.4v, whereas several other authors, for example [7] say that in an edge tone configuration the phase velocity is 0.5v. Since the wavelength λ = u ph /f, this uncertainty leads to an uncertainty in the wavelength: in the cases examined the wavelength for the first stage lies between 0.9h and 1.2h, whereas for the second stage between 0.45h and 0.58h. According to [4] in the first stage λ h. much longer, 7.5h. Since the pressure is non-negligible along almost the whole length of the wedge, the point of force action (x F ) is much more behind the tip in our case than in [4]. Whereas x F =1/4λ = 1/4h was found there, here the force is located at 16-17 mm from the tip for all Reynolds numbers, corresponding to x F = 1.6-1.7h. If the temporal variation of x F is examined, it turns out, that in most of the cycle x F remains in a very narrow range, the only exception being the those phases where the force approaches zero. A further question of interest is the spatial development of the higher harmonics along the oscillating jet. Figure 8. shows this. There is a typical exponential-type rise in the amplitude for all harmonics in the beginning and then the saturation, and a decrease. The last point at the tip of the edge is much higher. Somewhat unusual is the fact that the first overtone is larger than the fundamental along the whole length. This is not so at other Reynolds numbers. Figure 7. RMS pressure distribution along the wedge surface. Re = 200. Figure 6. Relative phase of the transversal velocity signal along the jet axis. Re = 200. Kaykayoglu and Rockwell [4] present measured rms pressure distributions along the edge and on the basis of this they calculated the point of action of the total force of the integrated pressure. They found that after a medium value the pressure reaches a sharp maximum close to the wedge tip and afterwards the pressure decreases roughly as ~x -1/2. They measured the pressure only in a few points. Our rms pressure distribution can be seen in Figure 7. for Re = 200. It is found that the distribution is very similar for all the Reynolds numbers, except that the pressure magnitudes change. The basic features are reproduced with the sharp peak close to the edge tip. Since our spatial resolution is higher, more details can be seen in Figure 7. than in [4]. The distribution can be represented rather with a piecewise linear than with a power function. Another difference is that in [4] the total streamwise length of the wedge was 0.8h, whereas here Figure. 8. Development of the higher harmonics along the wedge. Re = 225 4 Summary A large number of unsteady direct numerical simulations have been performed on the edge tone. The following conclusions can be drawn: the existence of stages and the linear dependence of oscillating frequency has been verified. The 638

numerical values of the Strouhal number agree very well with the literature. Two different ways of coexistence of the two stages have been found: at lower Reynolds numbers they switch back and forth between the pure modes, whereas with higher Reynolds numbers, the two stages exist at the same time. No upper limit has been found for the existence of the first stage so far. The phase velocity of the disturbances has been extracted from the data and found to be in agreement with the literature with some scatter (0.4-0.5v) The pressure distributions along the wedge surface found in [4] have been reproduced well. The point of action of the pressure force is, however much further behind the tip, due to the larger length of our wedge. The harmonics develop first exponentially then after a maximum at 0.8h, decrease. The following steps are planned in the near future: Continue the simulations for a broader range of Reynolds numbers. Examine the influence of the jet exit-edge distance. Examine the influence of the jet velocity profile. Examine the influence of the transversal position of the wedge. Based on the fluid mechanical simulations acoustical simulations will be performed. Financial support for this project was given by the Hungarian Scientific Research Fund. I. Vaik was supported by the System International Foundation for the duration of this work. References [1] A, Powell: On the edgetone, J. Acoust. Soc. Am. Vol. 33. pp395-409 (1961) [2] D. Holger, T. Wilson, G. Beavers: Fluid mechanics of the edgetone J. Acoust. Soc. Am. Vol. 62. pp1116-1128 (1977) [3] D. Crighton: The edgetone feedback cycle. Linear theory for the operating stages, J. Fluid Mech. Vol. 234. pp361-391 (1992) [4] R. Kaykayoglu, D. Rockwell: Unstable jet-edge interaction. Part 1. Instantaneous pressure fields at a single frequency J. Fluid Mech. Vol. 169. pp125-149 (1986) [5] R. Kaykayoglu, D. Rockwell: Unstable jet-edge interaction. Part 2. Multiple frequency pressure fields J. Fluid Mech. Vol. 169. pp151-172 (1986) [6] S. Ohring: Calculations of self-excited impinging jet flow J. Fluid Mech. Vol. 163. pp69-88 (1986) [7] C. Ségoufin, B. Fabre, L. de Lacombe: Experimental investigation of the flue channel geometry influence on edge tone oscillations Acta Acustica united with Acustica Vol. 90. pp966-975 (2004) [8] G. E. Mattingly, W. O. Criminale: Disturbance characteristics in a plane jet Phys. Fluids Vol. 14(11) pp2258-2264 (1971) 639