ON WHISTLING OF PIPES WITH A CORRUGATED PIPE SEGMENT

Transcription

1 ON WHISTLING OF PIPES WITH A CORRUGATED PIPE SEGMENT Oleksii Rudenko, Gunes Nakiboglu and Avraham Hirschberg Department of Applied Physics, Eindhoven University of Technology, Eindhoven 5600 MB, The Netherlands o.rudenko@tue.nl ABSTRACT A generic experiment is conducted in which a corrugated pipe segment is placed between two smooth pipe segments with open terminations. Such a composite pipe system, widely utilized in industry, can produce high amplitude whistling noises. Experiments with pipes of 1 cm diameter and various lengths of the downstream smooth segment were carried out for flow velocities up to 55 m/s at room conditions. We observed a maximum of whistling amplitude at a critical Strouhal number between 0.36 and There is a critical smooth downstream pipe length much larger than the length of the corrugated pipe segment), which is sufficient to prevent pulsations at room conditions for the pipes considered. A simple model is proposed allowing the prediction of a critical Mach number above which the whistling occurs. Preliminary results focus on the worst-case scenario in which the corrugated segment is placed close to one of the pipe terminations with a high reflection coefficient. In this configuration there is a long downstream smooth pipe and a short upstream smooth pipe. INTRODUCTION A corrugated pipe is a tube with a periodically changing diameter. This undulatory shape makes the thinwalled corrugated pipes locally rigid and globally flexible. Various industrial applications benefit from corrugated pipes utilization ranging from vacuum cleaners to offshore natural gas production [1]. A drawback of corrugated pipes is the production of strong whistling sounds. This whistling is an environmental nuisance and associated vibration can lead to a mechanical failure [2]. In this paper we consider a generic experiment in which a corrugated pipe segment is placed between two smooth pipe segments. The pipe has open terminations. Address all correspondence to this author. We report the existence of a critical Mach number, M cr, above which a substantial increase of the whistling amplitude is observed. A simple model is provided explaining this observation. WHISTLING OF CORRUGATE PIPES The sound generation in corrugated pipes has been intensely studied [2 15]. The whistling of a corrugated pipe is the result of a coupling between local vortex shedding at the cavities formed by the corrugations and longitudinal acoustic waves traveling along the pipe. Whistling is a self-sustained oscillation. For relatively short pipes, involving strong acoustical reflections at the pipe ends, the vortex shedding at the upstream edge of the corrugations is triggered by the grazing oscillating velocity u associated with acoustic standing waves along the pipe. The unsteadiness of the flow at each corrugation results into a fluctuating hydrodynamic force in the direction of the pipe axis. This fluctuating force reinforces the acoustic oscillation. At very small acoustic amplitudes, when the regime is linear, the oscillation amplitude can grow exponentially until a non-linear regime is reached resulting in a saturation of the amplitude, and a steady whistling amplitude is established limit cycle) [9, 11]. Alternatively, perturbations decay exponentially in time. The whistling is observed when the travel time of the vortices across the cavity width W of a corrugation matches the oscillation period of the standing wave. This corresponds to a critical Strouhal number, Sr cr = fw/u cp, 1) where f is the oscillation frequency and U cp is the steady flow velocity in the corrugated pipe averaged over the inner cross-section of the pipe. An empirical relation was suggested in Ref. [11] relating the critical Strouhal num-

2 ber with the ratio of W/D cp, i.e. the cavity width W to the minimal corrugated pipe diameter D cp : Sr cr 0.58 W/D cp ) ) The steady whistling amplitude is a result of the balance between the power produced by the sound sources and the power losses. The acoustic power generated by a single corrugation has been studied by means of computer simulations in [11, 13]. Within the linear approximation see e.g. [16]), the acoustic source power averaged over a period of acoustic osculation, P src, scales as ρ u 2 U cp S cp,where ρ is the mean fluid density, u is the amplitude of the local grazing acoustic velocity and S cp = πd 2 cp /4isthe minimal cross-section area of the corrugated pipe. Fig. 1 adopted from the Ref. [13] shows the dependence of the acoustic source power produced by a single cavity on the relative acoustic amplitude, u /U cp.thisis for a fully developed turbulent flow in a corrugated pipe with a geometry similar to that of the corrugated pipes used in our experiments. The source power was computed for the most intense whistling, i.e. at the critical Strouhal number, which is equal to 0.46 for the present corrugated pipe geometry. The horizontal asymptote in the graph for small u /U cp corresponds to the linear regime. Psrc ΡUcp u' 2 Scp u'u cp FIGURE 1: SINGLE CORRUGATION DIMENSION- LESS SOURCE POWER VS. RELATIVE PULSATION AMPLITUDE. ADOPTED FROM REF. [13]. EXPERIMENTAL SET-UP The set-up and the corrugated pipe segment are sketched in Fig. 2. The pipe system consists of a short upstream smooth pipe segment of length L up = 59 mm followed by the corrugated pipe segment of length L cp = 228 mm and a downstream smooth pipe segment. Four different lengths of the downstream smooth pipe segment, L dn, were tested: 59 mm, 509 mm, 988 mm and 2009 mm. The smooth pipe segments are RVS buis UHP 12.7x1.22mm ultron. The inner smooth pipe diameter is D sp = mm, the outer is 12.7 mm. The corrugated pipe segment is BOA stainless steel corrugated pipe, type PNR made of AISI 316. The number of corrugation is N = 53. The pitch wavelength) of the corrugations is p = 4.3 mm corresponding to a cavity width W 3p/ mm [11]. The depth of the cavities is estimated to be 3 mm. The the outer diameter of corrugated pipe is 16.1 mm and the inner one is D cp = mm D cp /W 3.18). The flow through the pipe is driven by a centrifugal ventilator attached to a settling chamber volume 0.5 m x 0.5 m x 1.8 m). A 10 cm thick layer of acoustic absorbing material foam) covers the side-walls of the settling chamber. This avoids acoustic resonances of the settling chamber and approximates free field radiation conditions for the upstream open end inlet) of the pipe flanged pipe termination). At the downstream side of the set-up outlet), the flow leaves the pipe through an un-flanged open pipe termination. A constant temperature hot wire anemometer is placed on the pipe axis inside the pipe 4 mm upstream of the downstream open pipe termination. The hotwire anemometer was a Dantec 90C10 CTA module installed within a Dantec 90N10 frame. The signal was amplified and low-pass filtered through a low-noise preamplifier Stanford Research Systems, Model SR560) and sent to the computer via a National Instrument BNC data acquisition board with a 12-bit resolution at a sampling rate of 10 khz. This calibrated hot-wire provides a measurement of the time dependent velocity ut), which then is split into a time-averaged velocity U cl at the centerline and a fluctuating acoustic) velocity of amplitude u and frequency f. The acoustic velocity is uniform over the pipe crosssection outside the viscous boundary layers of thickness ν/π f ) < 0.07 mm, where ν is the kinematic viscosity). The center-line velocity, U cl, is related to the average velocity U = Q/S sp where Q is the volume flow and S sp = πd 2 sp /4 is the cross-section area of the smooth pipe) by the empirical equation [17]: U U cl / ) Fr, 3) Where the friction factor, Fr, for a smooth pipe is given

3 inlet outlet D sp x L cp L dn D cp L dn L 0 p W FIGURE 2: EXPERIMENTAL SET-UP & CORRUGATED PIPE GEOMETRY. by the formula of Blasius [17]: Fr 0.316Re 0.25, 4) with Re= UD sp /ν is the Reynolds number. For typical Reynolds numbers in our experiments, < Re < 64000, we approximate U U cl /1.19. Due to slight mismatch in the smooth pipe and corrugated pipe diameters, average velocity in corrugated pipe is U cp = US sp /S cp. Thus U cp U cl /a, a = 1.16, 5) i.e. the steady cross-section averaged velocity in the corrugated pipe is about a factor 1.16 lower than the centerline velocity at the end of the downstream smooth pipe segment measured by means of the hot wire. After establishing a stable flow velocity the hot wire signal was recorded during 30 s at a sample rate of 10 khz. The amplitude u of the acoustic velocity fluctuations was determined by carrying a Fast Fourier Transform FFT) of the hot wire signal and by integrating the energy in the dominating peak above 20 Hz) over a bandwidth of 10 Hz. Then u corresponds to an amplitude of a sinusoidal signal with the same energy [10]. In discussing the results, an experiment is denoted by the length of the upstream smooth pipe segment, the length of the corrugated pipe segment and the length of the downstream smooth pipe segment, i.e. Lup, L cp, L dn ). These lengths are rounded and given in millimeters. RESULTS AND DISCUSSION Whistling Frequency In Fig. 3, the Helmholtz number, He, is presented versus the Mach number, M, defined as He Lf c 0 = LSr cr aw M, M U cl c 0, 6) where L = L up + L cp + L dn is the total length of the composite pipe and c m/s is the speed of sound in the air at room temperature. The symbols in Fig. 3, left, are the experimental data: black) discs 6,27,6) configuration, blue) squares 6,27,51) configuration and orange) rhombi 6,27,99) configuration. The lines are fits of Eq. 6) to the experimental data at the most intense whistling per mode step or quasi-plateau in the Figure): lower black) gives critical Strouhal number 0.36, middle blue) 0.38 and upper orange) Note, that the Eq. 2) gives Sr cr 0.46, which is 22% larger than the lowest observed critical Strouhal number. This variation in the critical Strouhal number are not understood and deserve further study. The longest pipe configuration 6,23,201), not shown in the Fig. 3, left, was not whistling in the whole range of Mach numbers reachable with the current set-up. In agreement with the previous observations [11, 13], the Helmholtz number, hence the whistling frequency, changes in a step-wise manner illustrating the coupling of the standing wave inside the corrugated pipe) with an appropriate hydrodynamic mode of vortex shading see [11, 13] for more details). In first order approximation, the standing waves correspond to the Helmholtz number changing in steps of 0.5. A stepping close to this is observed in Fig. 3. Whistling Threshold In Fig. 3, the relative whistling amplitude, u 0 /U cl, versus the Mach number is shown. Here u 0 is the acoustic velocity amplitude at x = 0, i.e. at the downstream end. Observe that the increase in length of the downstream smooth pipe segment leads to an expected decrease in the relative amplitude such that for the configuration 6,27,201), not shown in the Fig. 3. The whistling disappears in the flow range considered in our experiments. There are also noticeable sudden variations in whistling amplitudes for high Mach numbers. These variations might be due to acoustic resonances of the room in which the experiment was conducted. Interesting

4 Helmholtz number , 23, 99 Sr cr , 23, 51 Sr cr , 23, 6 Sr cr Mach number Relative whistling amplitude, , 23, 6 Sr cr , 23, 99 Sr cr Mach number 6, 23, 51 Sr cr 0.38 FIGURE 3: LEFT: HELMHOLTZ VS. MACH NUMBER, RIGHT: AMPLITUDE VS. MACH NUMBER. observation is the existence of a critical Mach number, M cr 0.05, below which there is only weak whistling. This is more evident for longer downstream smooth pipe segments. We try to rationalize the existence of the critical Mach number by the following simple model. We suppose that L up L cp L dn, which is very roughly realized in 6,27,99) configuration. For such a configuration one expects a limit behavior in which the upstream open end is strongly reflecting, while the downstream end is almost anechoic. Thus a downstream traveling wave in the long downstream) smooth pipe segment is a first order approximation. In this approximation the radiation losses of the corrugated pipe are P loss 1 2 ρc 0 u dn 2 S sp, 7) where u dn is a typical value of the sound velocity amplitude at x = L dn. In the limit case of vanishingly small oscillatory perturbation, i.e. for u /U cp 1, the linear shear layer instability acts as the sound source. Then a rough estimate for the source power is N P src NAρ u 2 U cp S cp. 8) The single source power P src isshowninfig.1.inthe linear regime we have P src Aρ u 2 U cp S cp with A If the losses overcome the production, the system remains silent. The balance of the losses with production, P src N = P loss, determines the threshold for whistling: NAρ u 2 U cp S cp = 1 2 ρc 0 u dn 2 S sp, 9) Assuming u = u dn,wefind a critical Mach number: M cr = Dsp D cp ) 2 a 2A 1 N. 10) This is the Mach number below which the system remains silent. For the set-up considered N = 53, a = 1.16, A = 0.29, D sp /D cp 1.01) we find: M cr 0.04, 11) which agrees surprisingly well with the experimentally observed value. It is important to stress that we made a very crude approximation assuming that the grazing oscillatory velocity amplitude, u, is uniform through the whole corrugated pipe and equal to u dn. In general this is not true. The weakly unstable shear layers amplify the acoustic wave propagating in the corrugated pipe at every corrugation. Hence we expect a non-uniform acoustic velocity along the corrugated pipe segment. Furthermore, a critical Mach number also exists for other pipes configurations including the symmetric one 6,27,6), where a dominating traveling wave in the downstream smooth pipe segment is a poor approximation. Note that considering damping which is independent of the mean) flow velocity will result in the prediction of a critical Mach number. Hence a quasi-steady model of frictional losses would not predict such a Mach number. A better quantitative model is required. Nevertheless, even a very simple model indicates the existence of a critical Mach number, which value surprisingly well agrees with the experiment.

5 CONCLUSIONS Experiments have been carried out on pipes consisting of three segments: upstream smooth segment, corrugated pipe segment and downstream smooth segment. The length of the last smooth pipe segment was varied. Strong whistling was observed for all the lengths except the longest one at which the system remained silent. This supports the idea that a long enough smooth pipe placed after the corrugated one will keep the system silent since the losses brought by the smooth pipe will overcome the sources power, which are located in the corrugated pipe. However, a very simple model predicts another behavior: the whistling should occur for very long downstream pipes above a critical Mach number. The recorded dimensionless whistling frequencies Strouhal numbers) appeared to be in a fair agreement with the earlier findings [11]. A very simple order-ofmagnitude estimate predicts surprisingly good the value for the critical Mach number observed in the experiments below which a corrugated pipe segment attached to a long smooth pipe should not whistle. ACKNOWLEDGMENT This work was made possible by the contributions of STW Technologiestichting Project No. STW ). The authors wish to thank A. Holten, J. F. H. Willems, H. B. M. Manders, E. Cocq and F. M. R. van Uittert for their contributions to the development of the experiments. REFERENCES [1] Belfroid, S., Shatto, D., and Peters, R., Flow induced pulsation caused by corrugated tubes. In Proceeding of ASME Pressure Vessels and Piping Division Conference, San Antonio. [2] Ziada, S., and Bühlmann, E., Flow induced vibration in long corrugated pipes. In Intl Conference on Flow-Induced Vibrations. IMechE, UK. [3] Ziada, S., and Bühlmann, E., Multiple sidebranches as tone generators. In Proceedings of International Mechanical Engineering Conference, paper, Vol. 416, pp [4] Cadwell, L. H., Singing corrugated pipes revisited. American Journal of Physics, 62, pp [5] Elliott, J., Corrugated pipe flow. Imperial College Press, ch. 11, pp [6] Debut, V., Antunes, J., and Moreira, M., Experimental study of the flow-excited acoustical lockin in a corrugated pipe. 14 th International Conference on Sound and Vibration. [7] Debut, V., Antunes, J., and Moreira, M., Flow-acoustic interaction in corrugated pipes: time domain simulation of experimental phenomena. 9 th International Conference on Flow-Induced Vibration. [8] Nakiboğlu, G., Belfroid, S. P. C., Tonon, D., Willems, J., and Hirschberg, A., A parametric study on the whistling of multiple side branch system as a model for corrugated pipes. No. PVP , ASME-PVP. [9] Tonon, D., Landry, B., Belfroid, S., Willems, J., Hofmans, G., and Hirschberg, A., Whistling of a pipe system with multiple side branches: Comparison with corrugated pipes. Journal of Sound and Vibration, 3298), pp [10] Nakiboğlu, G., Belfroid, S., Willems, J., and Hirschberg, A., Whistling behavior of periodic systems: Corrugated pipes and multiple side branch system. International Journal of Mechanical Sciences, 5211), pp [11] Nakiboğlu, G., Belfroid, S. P. C., Golliard, J., and Hirschberg, A., On the whistling of corrugated pipes: effect of pipe length and flow profile. Journal of Fluid Mechanics, 672, pp [12] Kristiansen, U. R., Mattei, P. O., Pinhede, C., and Amielh, M., Experimental study of the influence of low frequency flow modulation on the whistling behavior of a corrugated pipe. The Journal of the Acoustical Society of America, 130, pp [13] Nakiboğlu, G., Rudenko, O., and Hirschberg, A., Aeroacoustics of the swinging corrugated tube: Voice of the dragon. The Journal of the Acoustical Society of America, 131, pp [14] Nakiboğlu, G., Manders, H. B. M., and Hirschberg, A. Aeroacoustic power generated by a compact axisymmetric cavity: prediction of self-sustained oscillation and influence of the depth. Journal of Fluid Mechanics, Submitted. [15] Nakiboğlu, G., and Hirschberg, A. Aeroacoustic power generated by multiple compact axisymmetric cavities: Effect of hydrodynamic interference on the sound production. Physics of Fluids, Accepted. [16] Kop ev, V., Mironov, M., and Solntseva, V., Aeroacoustic interaction in a corrugated duct. Acoustical Physics, 54, pp [17] Daugherty, R., Franzini, J., and Finnemore, E., Fluid mechanics with engineering applications: SI metric ed. McGraw-Hill.