# Energy-Dynamics Resulting in Turbulent and Acoustic Phenomena in an Underexpanded Jet

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Physical Problem and Methodology

#### 2.1. Flowfield Parameters

#### 2.2. Numerical Technique for the Navier-Stokes Solver

#### 2.3. Validation

#### 2.4. Energy-Based Decomposition: Momentum Potential Theory

#### 2.4.1. Theoretical Considerations in MPT

#### 2.4.2. Numerical Implementation of MPT

- The mean solenoidal field, $\overline{\mathbf{B}}$ in Equation (3), is calculated as the average of instantaneous $\rho \mathit{u}$ from the LES.
- The source term for the total scalar potential in Equation (4), $\partial {\rho}^{\prime}/\partial t$, is obtained from the LES data, by obtaining the time-derivative of instantaneous density. This Poisson equation is then solved to obtain the total scalar potential, ${\psi}^{\prime}$.
- The source term for the acoustic scalar potential in Equation (5), $(1/{c}^{2})(\partial {p}^{\prime}/\partial t)$, is also calculated from the LES data, by obtaining the time-derivative of instantaneous pressure. Solution of this Poisson equation yields the acoustic scalar potential, ${\psi}_{A}^{\prime}$.
- The thermal scalar potential is obtained using the relation, ${\psi}_{T}^{\prime}={\psi}^{\prime}-{\psi}_{A}^{\prime}$.
- Finally, the fluctuating solenoidal component is obtained as, ${\mathbf{B}}^{\prime}=\rho \mathbf{u}-\overline{\mathbf{B}}+\nabla {\psi}^{\prime}$.

## 3. Results

#### 3.1. Fluid-Thermodynamic Modal Features

#### 3.2. Relationship of Nearfield Pressure and the Acoustic Mode

#### 3.3. Shock-Cell Dynamics and Their Acoustic Imprint

#### 3.4. Predictive Advantages of the Acoustic Mode

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Suzuki, T. Wave-Packet Representation of Shock-Cell Noise for a Single Round Jet. AIAA J.
**2016**, 54, 3903–3917. [Google Scholar] [CrossRef] - Panda, J.; Seasholtz, R.G. Measurement of shock structure and shock–vortex interaction in underexpanded jets using Rayleigh scattering. Phys. Fluids
**1999**, 11, 3761–3777. [Google Scholar] [CrossRef] - Owston, R.; Magi, V.; Abraham, J. Fuel-Air Mixing Characteristics of DI Hydrogen Jets. SAE Int. J. Eng.
**2009**, 1, 693–712. [Google Scholar] [CrossRef] - Adamson, T.C.; Nicholls, J.A. On the structure of jets from highly underexpanded nozzles into still air. J. Aerosp. Sci.
**1959**, 26, 16–24. [Google Scholar] [CrossRef] - Bonelli, F.; Viggiano, A.; Magi, V. A numerical analysis of hydrogen underexpanded jets under real gas assumption. J. Fluids Eng.
**2013**, 135, 121101. [Google Scholar] [CrossRef] - Avital, G.; Cohen, Y.; Gamss, L.; Kanelbaum, Y.; Macales, J.; Trieman, B.; Yaniv, S.; Lev, M.; Stricker, J.; Sternlieb, A. Experimental and computational study of infrared emission from underexpanded rocket exhaust plumes. J. Thermophys. Heat Transf.
**2001**, 15, 377–383. [Google Scholar] [CrossRef] - Norum, T.D.; Seiner, J.M. Measurements of Mean Static Pressure and Far Field Acoustics of Shock Containing Supersonic Jets; NASA Technical Report; NASA Langley Research Center: Hampton, VA, USA, September 1982.
- André, B.; Castelain, T.; Bailly, C. Experimental exploration of underexpanded supersonic jets. Shock Waves
**2014**, 24, 21–32. [Google Scholar] [CrossRef] - Raman, G. Supersonic jet screech: Half-century from Powell to the present. J Sound Vib.
**1999**, 225, 543–571. [Google Scholar] [CrossRef] - Tam, C.K.W.; Viswanathan, K.; Ahuja, K.K.; Panda, J. The sources of jet noise: Experimental evidence. J. Fluid Mech.
**2008**, 615, 253–292. [Google Scholar] [CrossRef] - Tam, C.K.W. Jet noise generated by large-scale coherent motion. In Aeroacoustics of Flight Vehicles: Theory and Practice. Volume 1: Noise Sources; NASA Langley Research Center, Aeroacoustics of Flight Vehicles: Hampton, VA, USA, 1991; Volume 1. [Google Scholar]
- Tam, C.K.W. Supersonic jet noise. Ann. Rev. Fluid Mech.
**1995**, 27, 17–43. [Google Scholar] [CrossRef] - Powell, A. On the mechanism of choked jet noise. Proc. Phys. Soc. Sect. B
**1953**, 66, 1039. [Google Scholar] [CrossRef] - Harper-Bourne, M. The noise from shock waves in supersonic jets. AGARD-CP-131
**1973**, 11, 1–13. [Google Scholar] - Tam, C.K.W.; Tanna, H.K. Shock associated noise of supersonic jets from convergent-divergent nozzles. J. Sound Vib.
**1982**, 81, 337–358. [Google Scholar] [CrossRef] - Magstadt, A.S.; Berry, M.G.; Berger, Z.P.; Shea, P.R.; Ruscher, C.J.; Gogineni, S.P.; Glauser, M.N. Flow Structures Associated with Turbulent Mixing Noise and Screech Tones in Axisymmetric Jets. Flow Turbul. Combust.
**2017**, 98, 725–750. [Google Scholar] [CrossRef] - Freund, J.B. Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9. J. Fluid Mech.
**2001**, 438, 277–305. [Google Scholar] [CrossRef] - Freund, J.B.; Lele, S.K.; Moin, P. Numerical simulation of a Mach 1.92 turbulent jet and its sound field. AIAA J.
**2000**, 38, 2023–2031. [Google Scholar] [CrossRef] - Bogey, C.; Bailly, C. Computation of a high Reynolds number jet and its radiated noise using large eddy simulation based on explicit filtering. Comput. fluids
**2006**, 35, 1344–1358. [Google Scholar] [CrossRef] - Schulze, J.; Sesterhenn, J. Numerical simulation of supersonic jet-noise. Proc. Appl. Math. Mech.
**2008**, 8, 10703–10704. [Google Scholar] [CrossRef] - Bodony, D.J.; Lele, S.K. On using large-eddy simulation for the prediction of noise from cold and heated turbulent jets. Phys. Fluids
**2005**, 17, 085103. [Google Scholar] [CrossRef] - Bonelli, F.; Viggiano, A.; Magi, V. How does a high density ratio affect the near-and intermediate-field of high-Re hydrogen jets? Int. J. Hydrogen Energy
**2016**, 41, 15007–15025. [Google Scholar] [CrossRef] - Gaitonde, D.V.; Samimy, M. Coherent structures in plasma-actuator controlled supersonic jets: Axisymmetric and mixed azimuthal modes. Phys. Fluids
**2011**, 23, 095104. [Google Scholar] [CrossRef] - Nichols, J.; Ham, F.; Lele, S.; Bridges, J. Aeroacoustics of a supersonic rectangular jet: Experiments and LES predictions. In Proceedings of the 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, Tennessee, 9–12 January 2012. [Google Scholar]
- Li, X.; Zhou, R.; Yao, W.; Fan, X. Flow characteristic of highly underexpanded jets from various nozzle geometries. Appl. Therm. Eng.
**2017**, 125, 240–253. [Google Scholar] [CrossRef] - Morris, P.J.; Miller, S.A.E. Prediction of broadband shock-associated noise using Reynolds-averaged Navier-Stokes computational fluid dynamics. AIAA J.
**2010**, 48, 2931–2944. [Google Scholar] [CrossRef] - Suzuki, T.; Lele, S.K. Shock leakage through an unsteady vortex-laden mixing layer: Application to jet screech. J. Fluid Mech.
**2003**, 490, 139–167. [Google Scholar] [CrossRef] - Doak, P.E. Momentum potential theory of energy flux carried by momentum fluctuations. J. Sound Vib.
**1989**, 131, 67–90. [Google Scholar] [CrossRef] - Goldstein, M.E. On Identifying the Sound Sources in a Turbulent Flow; NASA Technical Report; NASA Glenn Research Center: Cleveland, OH, USA, 2008.
- Jordan, P.; Daviller, G.; Comte, P. Doak’s momentum potential theory of energy flux used to study a solenoidal wavepacket. J. Sound Vib.
**2013**, 332, 3924–3936. [Google Scholar] [CrossRef] - Unnikrishnan, S.; Gaitonde, D.V. Acoustic, hydrodynamic and thermal modes in a supersonic cold jet. J. Fluid Mech.
**2016**, 800, 387–432. [Google Scholar] [CrossRef] - Bogey, C.; Bailly, C. An analysis of the correlations between the turbulent flow and the sound pressure fields of subsonic jets. J. Fluid Mech.
**2007**, 583, 71–97. [Google Scholar] [CrossRef] - Panda, J.; Seasholtz, R.G. Experimental investigation of density fluctuations in high-speed jets and correlation with generated noise. J. Fluid Mech.
**2002**, 450, 97–130. [Google Scholar] [CrossRef] - Arroyo, C.P.; Daviller, G.; Puigt, G.; Airiau, C. Shock-cell noise of supersonic under expanded jets. In Proceedings of the 50th 3AF International Conference on Applied Aerodynamics, Toulouse, France, 29 March–1 April 2015. [Google Scholar]
- Sinayoko, S.; Agarwal, A.; Sandberg, R.D. On wavenumber spectra for sound within subsonic jets. arXiv, 2013; arXiv:1311.5358. [Google Scholar]
- Grizzi, S.; Camussi, R.; Di Marco, A. Experimental Investigation of pressure fluctuations in the near field of subsonic jets at different Mach and Reynolds numbers. In Proceedings of the 18th AIAA/CEAS Aeroacoustics Conference, Colorado Springs, CO, USA, 4–6 June 2012. [Google Scholar]
- Krothapalli, A.; Hsia, Y.; Baganoff, D.; Karamcheti, K. The role of screech tones in mixing of an underexpanded rectangular jet. J. Sound Vib.
**1986**, 106, 119–143. [Google Scholar] [CrossRef] - Tam, C.K.W. Stochastic model theory of broadband shock associated noise from supersonic jets. J. Sound Vib.
**1987**, 116, 265–302. [Google Scholar] [CrossRef] - Lumley, J.L. The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation; House Nauka: Moscow, USSR, 1967; pp. 166–178. [Google Scholar]
- Lighthill, M.J. On Sound Generated Aerodynamically: I. General Theory. Proc. R. Soc. Lond. A
**1952**, 211, 564–587. [Google Scholar] [CrossRef] - Lighthill, M.J. On Sound Generated Aerodynamically: II. Turbulence as a Source of Sound. Proc. R. Soc. Lond. A
**1954**, 222, 1–32. [Google Scholar] [CrossRef] - Ffowcs Williams, J.E. The noise from turbulence convected at high speed. Philos. Trans. R. Soc. Lond. A
**1963**, 255, 469–503. [Google Scholar] [CrossRef] - Shea, P.R.; Berger, Z.P.; Berry, M.G.; Glauser, M.N.; Gogineni, S. Low-dimensional modeling of a Mach 0.6 axisymmetric jet. In Proceedings of the 52nd Aerospace Sciences Meeting, National Harbor, MA, USA, 13–17 January 2014. [Google Scholar]
- Gaitonde, D.V. Analysis of the near field in a plasma-actuator-controlled supersonic jet. J. Propuls. Power
**2012**, 28, 281–292. [Google Scholar] [CrossRef] - Speth, R.L.; Gaitonde, D.V. Parametric Study of a Mach 1.3 Cold Jet Excited by the Flapping Mode Using Plasma Actuators. Comput. Fluids
**2013**, 84, 16–34. [Google Scholar] [CrossRef] - González, D.R.; Speth, R.L.; Gaitonde, D.V.; Lewis, M.J. Finite-time Lyapunov exponent-based analysis for compressible flows. Chaos Int. J. Nonlinear Sci.
**2016**, 26, 083112. [Google Scholar] [CrossRef] [PubMed] - Steger, J.L. Implicit finite-difference simulation of flow about arbitrary two-dimensional geometries. AiAA J.
**1978**, 16, 679–686. [Google Scholar] [CrossRef] - Vinokur, M. Conservation equations of gasdynamics in curvilinear coordinate systems. J. Comput. Phys.
**1974**, 14, 105–125. [Google Scholar] [CrossRef] - Rizzetta, D.P.; Visbal, M.R. Large-eddy simulation of plasma-based turbulent boundary-layer separation control. AIAA J.
**2010**, 48, 2793–2810. [Google Scholar] [CrossRef] - Roe, P.L. Approximate Riemann Solvers, Parameter Vectors and Difference Schemes. J. Comput. Phys.
**1981**, 43, 357–372. [Google Scholar] [CrossRef] - Van Leer, B. Towards the Ultimate Conservation Difference Scheme V, A Second-Order Sequel to Godunov’s Method. J. Comput. Phys.
**1979**, 32, 101–136. [Google Scholar] [CrossRef] - Pulliam, T.H.; Chaussee, D.S. A Diagonal Form of an Implicit Approximate-Factorization Algorithm. J. Comp. Phys.
**1981**, 39, 347–363. [Google Scholar] [CrossRef] - Beam, R.; Warming, R. An Implicit Factored Scheme for the Compressible Navier-Stokes Equations. AIAA J.
**1978**, 16, 393–402. [Google Scholar] [CrossRef] - Goparaju, K.; Gaitonde, D.V. Large-Eddy Simulation of Plasma-Based Active Control on Imperfectly Expanded Jets. J. Fluids Eng.
**2016**, 138, 071101. [Google Scholar] [CrossRef] - Berger, Z.P. The Effects of Active Flow Control on High-Speed Jet Flow Physics and Noise. Ph.D. Thesis, Syracuse University, New York, NY, USA, 2014. [Google Scholar]
- Powell, A. On Prandtl’s formulas for supersonic jet cell length. Int. J. Aeroacoust.
**2010**, 9, 207–236. [Google Scholar] [CrossRef] - Kovásznay, L.S.G. Turbulence in supersonic flow. J. Aeronaut. Sc.
**1953**, 20, 657–674. [Google Scholar] [CrossRef] - Truesdell, C. Intrinsic Equations of Spatial Gas Flow. ZAMM-J. Appl. Math. Mech. Z. Angew. Math. Mech.
**1960**, 40, 9–14. [Google Scholar] [CrossRef] - Doak, P.E. Fluctuating total enthalpy as a generalized acoustic field. Acoust. Phys.
**1995**, 41, 677–685. [Google Scholar] [CrossRef] - Doak, P.E. Fluctuating total enthalpy as the basic generalized acoustic field. Theor. Comput. Fluid Dyn.
**1998**, 10, 115–133. [Google Scholar] [CrossRef] - Van der Vorst, H.A. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput.
**1992**, 13, 631–644. [Google Scholar] [CrossRef] - Seiner, J. Advances in high speed jet aeroacoustics. In Proceedings of the 9th Aeroacoustics Conference, Williamsburg, VA, USA, 10–15 October 1984; p. 2275. [Google Scholar]
- Bodony, D.; Ryu, J.; Ray, P.; Lele, S. Investigating broadband shock-associated noise of axisymmetric jets using large-eddy simulation. In Proceedings of the 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference), Cambridge, MA, USA, 8–10 May 2006. [Google Scholar]
- Hileman, J.; Samimy, M. Turbulence structures and the acoustic far field of a Mach 1.3 jet. AIAA J.
**2001**, 39, 1716–1727. [Google Scholar] [CrossRef] - Kearney-Fischer, M.; Sinha, A.; Samimy, M. Intermittent nature of subsonic jet noise. AIAA J.
**2013**, 51, 1142–1155. [Google Scholar] [CrossRef] - Samimy, M.; Kim, J.H.; Kastner, J.; Adamovich, I.; Utkin, Y. Active control of high-speed and high-Reynolds-number jets using plasma actuators. J. Fluid Mech.
**2007**, 578, 305–330. [Google Scholar] [CrossRef] - Berland, J.; Bogey, C.; Bailly, C. Large eddy simulation of screech tone generation in a planar underexpanded jet. In Proceedings of the 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference), Cambridge, MA, USA, 8–10 May 2006. [Google Scholar]
- Arroyo, C.P.; Daviller, G.; Puigt, G.; Airiau, C.; Moreau, S. Identification of temporal and spatial signatures of broadband shock-associated noise. Shock Waves
**2018**, 1–18. [Google Scholar] [CrossRef] - Howe, M.S.; Ffowcs, J.E. On the noise generated by an imperfectly expanded supersonic jet. Philos. Trans. R. Soc. Lond. A
**1978**, 289, 271–314. [Google Scholar] [CrossRef] - Lo, S.C.; Aikens, K.M.; Blaisdell, G.A.; Lyrintzis, A.S. Numerical investigation of 3-D supersonic jet flows using large-eddy simulation. Int. J. Aeroacoust.
**2012**, 11, 783–812. [Google Scholar] [CrossRef] - Sinha, A.; Rodríguez, D.; Brès, G.A.; Colonius, T. Wavepacket models for supersonic jet noise. J. Fluid Mech.
**2014**, 742, 71–95. [Google Scholar] [CrossRef] - Bailly, C.; André, B.; Castelain, T.; Henry, C.; Bodard, G.; Porta, M. An analysis of shock noise components. AerospaceLab
**2014**, 1–8. [Google Scholar] [CrossRef] - Magstadt, A.S.; Berry, M.G.; Berger, Z.P.; Shea, P.R.; Glauser, M.N.; Ruscher, C.J.; Gogineni, S. An investigation of sonic & supersonic axisymmetric jets: correlations between flow physics and far-field noise. In Proceedings of the 9th International Symposium on Turbulence and Shear Flow Phenomena, Melbourne, Australia, 30 June–1 July 2015. [Google Scholar]
- Savarese, A.; Jordan, P.; Girard, S.; Collin, E.; Porta, M.; Gervais, Y. Experimental study of shock-cell noise in underexpanded supersonic jets. In Proceedings of the 19th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2013–2080, Berlin, Germany, 27–29 May 2013. [Google Scholar]
- Tam, C.K.W. Broadband shock associated noise from supersonic jets measured by a ground observer. AIAA J.
**1992**, 30, 2395–2401. [Google Scholar] [CrossRef] - Sirovich, L. Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Math.
**1987**, 45, 561–571. [Google Scholar] [CrossRef] - Sirovich, L. Turbulence and the dynamics of coherent structures. II. Symmetries and transformations. Q. Appl. Math.
**1987**, 45, 573–582. [Google Scholar] [CrossRef] - Norum, T.D.; Seiner, J.M. Broadband shock noise from supersonic jets. AIAA journal
**1982**, 20, 68–73. [Google Scholar] - Cavalieri, A.V.G.; Jordan, P.; Agarwal, A.; Gervais, Y. Jittering wave-packet models for subsonic jet noise. J. Sound Vib.
**2011**, 330, 4474–4492. [Google Scholar] [CrossRef]

**Figure 1.**Computational grid used for the LES. The nozzle is shown on the left using the gray surface. The azimuthal (red), axial (blue) and radial (green) surfaces are also marked to display the grid lines. Every tenth axial and radial point, and every other azimuthal point is marked on these surfaces. The dotted arrow indicate the jet-flow direction.

**Figure 2.**(

**a**) Comparison of mean-streamwise velocity along the centerline of the jet; (

**b**) Comparison of shock cell structure on a vertical plane. Current computational results compared with corresponding experimental values. The experimental values plotted in (

**a**) has an uncertainty of $1.4\%$ [55]. The horizontal dotted arrows indicate the flow direction in (

**b**).

**Figure 3.**Instantaneous snapshot of the jet. The expansion and compression regions in the potential core are highlighted using $\mathit{u}\xb7\nabla p$, in the region $0\le r\le 0.5$ and $0\le x\le 8$. 10 equally spaced contour levels are used within the range 1 and $-1$. The shear layer is shown using magnitude of vorticity, using 11 contour levels between $1.5$ and 5, with blue and red representing the minimum and maximum levels, respectively. The acoustic emissions in the nearfield are visualized using divergence of velocity, using 10 contour levels between $-0.01$ and $0.01$. The jet-flow direction is from left to right.

**Figure 4.**(

**a**) Instantaneous snapshot of the axial component of the hydrodynamic (contours) and acoustic (gray scale) modes; (

**b**) Corresponding three-dimensional form of the acoustic wavepacket, shown using iso-levels of ${A}_{x}^{\prime}$. The jet-flow direction in (

**b**) is along the x-axis, as indicated by the axis-marker. The vertical lines in (

**b**) are marked on the $z=0$ vertical plane and corresponds to $x=2,4,6,8$ and $x=10$ respectively, from left to right. The horizontal lines in (

**b**) are marked on the $z=0$ vertical plane and corresponds to $y=-4,-2,0,2$ and $y=4$ respectively, from bottom to top.

**Figure 6.**Instantaneous snapshots of (

**a**) pressure perturbations; (

**b**) axial; (

**c**) radial and (

**d**) azimuthal components of the acoustic mode. The axial extent of the contours begin from the nozzle-exit station, and the jet-flow direction is from left to right. Frames

**a**,

**b**,

**c**and

**d**share the same abscissa and ordinate.

**Figure 7.**Comparison of PSDs obtained from pressure fluctuations and the acoustic mode at various locations in the computational nearfield, as indicated in each frame. Frames

**a**-

**f**share the same abscissa and ordinate.

**Figure 8.**Spectral variation of acoustic signature along the axial direction, at $r=2$. The contours represent PSD of (

**a**) ${A}_{x}^{\prime}$; (

**b**) ${A}_{r}^{\prime}$; (

**c**) upstream propagating components in ${A}_{x}^{\prime}$ and (

**d**) upstream propagating components in ${A}_{r}^{\prime}$.

**Figure 9.**POD modes of the field $\mathit{u}\xb7\nabla p$, along with the PSD of the corresponding modal time-coefficient. The modes shown are: (

**a**) mode 1; (

**b**) mode 3; (

**c**) mode 5 and (

**d**) mode 7. The horizontal dotted arrow in (

**a**) marks the jet-flow direction. Frames

**a**-

**d**share the same abscissa and ordinate.

**Figure 10.**Mean (

**a**) hydrodynamic; (

**b**) acoustic and (

**c**) thermal sources of TFE; (

**d**) Radial profile of the three source terms along the dotted vertical line at $x=2$, shown in (

**a**). The horizontal dotted arrow in (

**a**) marks the mean-flow direction of the jet. Frames

**a**,

**b**, and

**c**share the same abscissa and ordinate.

**Figure 11.**(

**a**) An instantaneous acoustic field, ${A}_{x}^{\prime}$, extracted from the LES; (

**b**) The predicted ${A}_{x}^{\prime}$ field at the corresponding time-instant, using a homogeneous wave propagator. The wave propagator is forced at $r=1.5$ (marked using a horizontal dotted line). The color and gray-scale contours in (

**a**) simply demarcate the zone of prediction in the wave propagator solution for easy comparison. The jet-flow direction is from left to right.

**Figure 13.**POD modes of the acoustic field, ${A}_{x}^{\prime}$. The modes shown are: (

**a**) mode 1; (

**b**) mode 2; (

**c**) mode 3; (

**d**) mode 4; (

**e**) mode 5 and (

**f**) mode 6. The horizontal dotted arrow in (

**a**) marks the jet-flow direction. Frames

**a**-

**f**share the same abscissa and ordinate.

**Figure 14.**Maximum cross-correlation values between the time coefficients of the first 12 POD modes of the acoustic wavepacket, and the farfield acoustic signal rerecorded at various polar angles at a distance of 75 jet diameters.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sasidharan Nair, U.; Goparaju, K.; Gaitonde, D. Energy-Dynamics Resulting in Turbulent and Acoustic Phenomena in an Underexpanded Jet. *Aerospace* **2018**, *5*, 49.
https://doi.org/10.3390/aerospace5020049

**AMA Style**

Sasidharan Nair U, Goparaju K, Gaitonde D. Energy-Dynamics Resulting in Turbulent and Acoustic Phenomena in an Underexpanded Jet. *Aerospace*. 2018; 5(2):49.
https://doi.org/10.3390/aerospace5020049

**Chicago/Turabian Style**

Sasidharan Nair, Unnikrishnan, Kalyan Goparaju, and Datta Gaitonde. 2018. "Energy-Dynamics Resulting in Turbulent and Acoustic Phenomena in an Underexpanded Jet" *Aerospace* 5, no. 2: 49.
https://doi.org/10.3390/aerospace5020049