### The aero-acoustic Hartmann effect: hundred years of research and the current state of the matter

**Published:**12.09.2018

**Authors:** Bocharova O.V., Lebedev M.G.

**Published in issue: **#9(81)/2018

**DOI: **10.18698/2308-6033-2018-9-1803

**Category:** Mechanics | **Chapter:** Mechanics of Liquid, Gas, and Plasma

At the beginning of the last century (1916–1919) Yul. Hartmann discovered an aero-acoustic effect, later named after him. This effect consists in the fact that when a hollow tube is placed in a supersonic jet flowing into the atmosphere at excess or insufficient pressure, the interaction of the jet stream with the obstacle can occur in a nonstationary (self-oscillating) mode and be accompanied by powerful acoustic radiation into the environment. During the last century various researchers repeatedly studied this effect using numerical methods among others, and it has been studying until now. The reason for such interest in this phenomenon is its numerous technical applications. From a purely scientific point of view, the problem is interesting because it is determined by a large number of parameters (at least 10), and not all domains of this multidimensional space of determining parameters have been studied.

This article describes a broad parametric study of the problem in question carried out with the aim of obtaining sufficiently general laws governing the phenomenon under study. Numerical calculations were performed in the formulation of the inviscid gas model (the Euler equation) by the Godunov method. The calculation results were compared with the results of experiments obtained by many authors. As a rule, there was good agreement of the data. The physical picture of the phenomenon is analyzed based on the results of calculations of more than 200 variants. Some areas of the mentioned space of determining parameters not having been previously studied either in physical or in numerical experiments are touched upon. A conclusion is drawn about the various mechanisms of self-oscillation excitation for shallow and deep cavities. The processing of the results for deep cavities (experimental and numerical results obtained by the author and other researches,) allowed making conclusion that there is a universal (with an accuracy of about 10%) dependence of the dimensionless vibration frequency (the Strouhal number) on the depth of the cavity. The experimental result that the switching from the low-frequency oscillation mode to the high-frequency mode occurs when the thickness of the resonator walls changes is confirmed. The process of aerothermoacoustic heating in a Hartmann resonator is considered

**References**

**[1]**Mach E., Salcher P. Optische Untersuchung der Luftstrahlen. Sitzungberichte der kais. Akad. Wiss., math.-naturw. Classe, 1889, Bd. XCVIII, Abth. II, S. 1303–1309.

**[2]**Mach L. Optische Untersuchung der Luftstrahlen. Sitzungberichte der kais. Akad. Wiss., math.-naturw. Classe, 1897, Bd. CVI, Abth. II, S. 1025–1074.

**[3]**Prandtl L. Über die stationären Wellen in einem Gasstrahl. Phys. Zeit., 1904, Bd. 5 (19), S. 599–602.

**[4]**Sarpotdar S., Raman G., Cain A.B. Powered Resonance Tubes: Resonance Characteristics and Actuation Signal Directivity. Exp. Fluids, 2005, vol. 39 (6), pp. 1084–1095.

**[5]**Hartmann J. Om en ny method til frembringelse af lydvinginger. Dan. Mat. Fys. Medd., 1919, vol. 1 (13), pp. 1–39.

**[6]**Hartmann J. A New Method for the Generation of Sound Waves. Phys. Rev., 1922, vol. 20 (6), pp. 719–727.

**[7]**Sprenger H. Über thermische Effeckte in Resonanzrorhen. Mitteilungen aus dem Institut für Aerodynamik. Zürich, 1954, Nr. 21, S. 18–35.

**[8]**Raman G., Srinivasan K. The Powered Resonance Tube: From Hartmann’s Discovery to Current Active Flow Control Applications. Progr. Aerospace Sci., 2009, vol. 45 (4), pp. 97–123.

**[9]**Kastner J., Samimy M. Development and Characterization of Hartmann Tube Fluidic Actuators for High-Speed Flow Control. AIAA J., 2012, vol. 40 (10), pp. 1926–1934.

**[10]**Cirnu C., Stefan A., Balan G. Sonomicrobiology of Raw Water at the Treatment by Air-Jet Generators. J. Engineering Studies Research, 2012, vol. 18 (2), pp. 31–38.

**[11]**Bouch D.J., Cutler A.D. Investigation of a Hartmann — Sprenger Tube for Passive Heating of Scramjet Injectant Gases. AIAA, 2003, p. 1275.

**[12]**Arefyev K.Yu., Voronetsky A.V., Ilchenko M.A. Fizika goreniya i vzryva — Combustion, Explosion, and Shock Waves, 2013, no. 6, pp. 41–46.

**[13]**Narayan S., Bholanath B., Sundararajan T., Srinivasan K. Journal of Aeroacoustics, 2013, vol. 121, no. 5–6, pp. 557–578.

**[14]**Brocher E., Maresca C., Bournay M. Fluid Dynamics of the Resonant Tube. J. Fluid Mech., 1970, vol. 43 (2), pp. 369–384.

**[15]**Brocher E., Maresca C. Échange de masse dans un tube de Hartmann — Sprenger. J. Méchanique, 1973, vol. 12 (3), pp. 355–374.

**[16]**Mørch K.A. A Theory of the Mode of Operation of the Hartmann Air Jet Generation. J. Fluid Mech., 1964, vol. 20 (1), pp. 141–159.

**[17]**Kawahashi M., Suzuki M. Generative Mechanism of Air Column Oscillation in a Hartmann — Sprenger Tube Excited by an Air Issuing from a Convergent Nozzle. Z. Angew. Math. Phys., 1979, vol. 30(5), pp. 797–810.

**[18]**Naberezhnova G.V. Trudy TsAGI — Proceedings of Central Aerohydrodynamic Institute, 1978, no. 1899, pp. 31–42.

**[19]**Naberezhnova G.V., Nesterov Yu.N. Uchenye zapiski TsAGI — TsAGI Science Journal, 1983, vol. 14, no. 5, pp. 58–64.

**[20]**Bocharova O.V., Lebedev M.G. Matematicheskoe modelirovanie — Mathematical Models and Computer Simulations, 2007, vol. 19, no. 8, pp. 31–36.

**[21]**Lebedev M.G., Bocharova O.V. Self-Oscillatory Regimes of the Sonic Jet/Flat Plate Interaction: Theoretical Predictions vs. Experimental Data. West-East High Speed Flow Field Conference, 19–22 noyabrya 2007, Moscow, Russia. Moscow, TsAGI Publ., 2007, pp. 190–191.

**[22]**Bocharova O.V. Uchenye zapiski TsAGI — TsAGI Science Journal, 2010, vol. 16, no. 2, pp. 59–64.

**[23]**Bocharova O.V., Lebedev M.G. Khimicheskaya fizika — Russian Journal of Physical Chemistry B, 2011, vol. 30, no. 7, pp. 40–47.

**[24]**Bocharova O.V. Matematicheskoe modelirovanie — Mathematical Models and Computer Simulations, 2013, vol. 25, no. 9, pp. 75–84.

**[25]**Kraiko A.N., Pyankov K.S. Izvestiya RAN. Mekhanika zhidkosti i gaza — Fluid Dynamics, 2006, no. 5, pp. 41–54.

**[26]**Love E.S., Grisby C.E., Lee L.P., Woodling M.J. Experimental and Theoretical Studies of Axisymmetric Free Jets. NASA Technical Report, R-6, 1959.

**[27]**Henderson B., Bridges J., Wernet M. An Experimental Study of the Oscillatory Flow Structure of Tone-Producing Supersonic Impinging Jets. J. Fluid Mech., 2005, vol. 342, pp. 115–117.

**[28]**Isaev S.A., Lipnitsky Yu.M., Baranov P.A., Panasenko A.V., Usachov A.E. Inzhenerno-Fizicheskiy Zhurnal —– Journal of Engineering Physics and Thermophysics, 2012, vol. 85, no. 6, pp. 1253–1267.

**[29]**Bocharova O.V., Lebedev M.G. Prikladnaya matematika i mekhanika — Journal of Applied Mathematics and Mechanics, 2016, no. 51, pp. 24–44.

**[30]**Godunov S.K., Zabrodin A.V., Ivanov M.Ya., Kraiko A.N., Prokopov G.P. Chislennoe reshenie mnogomernykh zadach gazovoy dinamiki [Numerical solving multidimensional problems of gas dynamics]. Moscow, Nauka Publ., 1976, 400 p.

**[31]**Lebedev M.G., Sitnik V.V. Prikladnaya matematika i informatika — Journal of Applied Mathematics and Informatics, 2005, no. 20, pp. 40–57.

**[32]**Bendat J.S., Piersol A.G. Measurement and Analysis of Random Data. New York, Wiley Publ., 1966 [In Russ.: Bendat J.S., Piersol A.G. Izmerenie i analiz sluchaynykh protsessov. Moscow, Mir Publ., 1974, 540 p.].

**[33]**Sobieraj G.B., Szumowski A.P. Experimental Investigation of an Underexpanded Jet from a Convergent Nozzle Impinging on a Cavity. J. Sound Vibration, 1991, vol. 149 (3), pp. 375–396.

**[34]**Sarohia V., Back L.H. Experimental Investigation of Flow and Heating in a Resonance Tube. J. Fluid Mech., 1979, vol. 94, pp. 649–672.

**[35]**Murugappan S., Gutmark E. Parametric Study of the Hartmann — Sprenger Tube. Exp. Fluids, 2005, vol. 38 (6), pp. 813–823.

**[36]**Vinoth B.R., Throvagunta P., Rathakrishnan E. Effect of Tube Lip Thickness on the Performance of Hartmann — Sprenger Tubes. Proc. Inst. Mech. Eng., Part G, J. Aerospace Engineering, 2011, vol. 226 (1), pp. 74–87.

**[37]**Narayanan S., Bhave P., Srinivasan K., Ramamurthi K., Sundararajan T. Spectra and Directivity of a Hartmann Whistle. J. Sound Vibration, 2009, vol. 321 (3), pp. 875–892.

**[38]**Dumnov G.E., Telenin G.F. Mekhanika zhidkosti i gaza — Fluid Dynamics, 1978, no. 3, pp. 177–180.