Absolute instability in shock-containing jets

We present an analysis of the linear stability characteristics of shock-containing jets. The flow is linearised around a spatially periodic mean, which acts as a surrogate for a mean flow with a shock-cell structure, leading to a set of partial differential equations with periodic coefficients in space. Disturbances are written using the Floquet ansatz and Fourier modes in the streamwise direction, leading to an eigenvalue problem for the Floquet exponent. The characteristics of the solution are directly compared with the locally parallel case, and some of the features are similar. The inclusion of periodicity induces minor changes in the growth rate and phase velocity of the relevant modes for small shock amplitudes. On the other hand, the eigenfunctions are now subject to modulation related to the periodicity of the flow. Analysis of the spatio-temporal growth rates led to the identification of a saddle point between the Kelvin–Helmholtz mode and the guided jet mode, characterising an absolute instability mechanism. Frequencies and mode shapes related to the saddle points for two conditions (associated with axisymmetric and helical modes) are compared with screech frequencies and the most energetic coherent structures of screeching jets, resulting in a good agreement for both. The analysis shows that a periodic shock-cell structure has an impulse response that grows upstream, leading to oscillator behaviour. The results suggest that screech can occur in the absence of a nozzle, and that the upstream reflection condition is not essential for screech frequency selection. Connections to previous models are also discussed.

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Sound generated by instability wave/shock-cell interaction in supersonic jets

Journal of Fluid Mechanics ◽

10.1017/s0022112007007306 ◽

2007 ◽

Vol 587 ◽

pp. 173-215 ◽

Cited By ~ 21

Author(s):

PRASUN K. RAY ◽

SANJIVA K. LELE

Keyword(s):

Cell Structure ◽

Cell Interaction ◽

Linear Instability ◽

Supersonic Jets ◽

Instability Wave ◽

Destructive Interference ◽

Source Terms ◽

Mean Flow ◽

Shock Cell ◽

Instability Waves

Broadband shock-associated noise is an important component of the overall noise generated by modern airplanes. In this study, sound generated by the weakly nonlinear interaction between linear instability waves and the shock-cell structure in supersonic jets is investigated numerically in order to gain insight into the broadband shock-noise problem. The model formulation decomposes the overall flow into a mean flow, linear instability waves, the shock-cell structure and shock-noise. The mean flow is obtained by solving RANSequations with a k-ε model. Locally parallel stability equations are solved for the shock structure, and linear parabolized stability equations are solved for the instability waves. Then, source terms representing the instability wave/shock-cell interaction are assembled and the inhomogeneous linearized Euler equations are solved for the shock-noise.Three cases are considered, a cold under-expanded Mj = 1.22 jet, a hot under-expanded Mj = 1.22 jet, and a cold over-expanded Mj = 1.36 jet.Shock-noise computations are used to identify and understand significant trends in peak sound amplitudes and radiation angles. The peak sound radiation angles are explained well with the Mach wave model of Tam & Tanna J. Sound Vib. Vol. 81, 1982, p. 337). The observed reduction of peak sound amplitudes with frequency correlates well with the corresponding reduction of instability wave growth with frequency. However, in order to account for variation of sound amplitude for different azimuthal modes, the radial structure of the instability waves must be considered in additionto streamwise growth. The effect of heating on the Mj = 1.22 jet is shown to enhance the sound radiated due to the axisymmetric instability waves while the other modesare relatively unaffected. Solutions to a Lilley–Goldstein equation show that soundgenerated by ‘thermodynamic’ source terms is small relative to sound from ‘momentum’ sources though heating does increase the relative importance of the thermodynamic source. Furthermore, heating preferentially amplifies sound associated with the axisymmetric modes owing to constructive interference between sound from the momentumand thermodynamic sources. However, higher modes show destructive interference between these two sources and are relatively unaffected by heating.

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RANS and Large Eddy Simulation of Internal Combustion Engine Flows—A Comparative Study

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.4026165 ◽

2014 ◽

Vol 136 (5) ◽

Cited By ~ 22

Author(s):

Xiaofeng Yang ◽

Saurabh Gupta ◽

Tang-Wei Kuo ◽

Venkatesh Gopalakrishnan

Keyword(s):

Large Eddy Simulation ◽

High Speed ◽

Combustion Engine ◽

Flow Analysis ◽

Partial Geometry ◽

Computational Domain ◽

Mean Flow ◽

Eddy Simulation ◽

Large Eddy ◽

Rans Simulations

A comparative cold flow analysis between Reynolds-averaged Navier–Stokes (RANS) and large eddy simulation (LES) cycle-averaged velocity and turbulence predictions is carried out for a single cylinder engine with a transparent combustion chamber (TCC) under motored conditions using high-speed particle image velocimetry (PIV) measurements as the reference data. Simulations are done using a commercial computationally fluid dynamics (CFD) code CONVERGE with the implementation of standard k-ε and RNG k-ε turbulent models for RANS and a one-equation eddy viscosity model for LES. The following aspects are analyzed in this study: The effects of computational domain geometry (with or without intake and exhaust plenums) on mean flow and turbulence predictions for both LES and RANS simulations. And comparison of LES versus RANS simulations in terms of their capability to predict mean flow and turbulence. Both RANS and LES full and partial geometry simulations are able to capture the overall mean flow trends qualitatively; but the intake jet structure, velocity magnitudes, turbulence magnitudes, and its distribution are more accurately predicted by LES full geometry simulations. The guideline therefore for CFD engineers is that RANS partial geometry simulations (computationally least expensive) with a RNG k-ε turbulent model and one cycle or more are good enough for capturing overall qualitative flow trends for the engineering applications. However, if one is interested in getting reasonably accurate estimates of velocity magnitudes, flow structures, turbulence magnitudes, and its distribution, they must resort to LES simulations. Furthermore, to get the most accurate turbulence distributions, one must consider running LES full geometry simulations.

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Stability and Sensitivity Analysis of Hydrodynamic Instabilities in Industrial Swirled Injection Systems

Volume 4A: Combustion, Fuels and Emissions ◽

10.1115/gt2017-63649 ◽

2017 ◽

Author(s):

Thomas L. Kaiser ◽

Thierry Poinsot ◽

Kilian Oberleithner

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Linear Stability Analysis ◽

Vortex Core ◽

Large Eddy Simulations ◽

Mode Shapes ◽

Mean Flow ◽

Precessing Vortex Core ◽

Large Eddy ◽

Good Agreement

The hydrodynamic instability in an industrial, two-staged, counter-rotative, swirled injector of highly complex geometry is under investigation. Large eddy simulations show that the complicated and strongly nonparallel flow field in the injector is superimposed by a strong precessing vortex core. Mean flow fields of large eddy simulations, validated by experimental particle image velocimetry measurements are used as input for both local and global linear stability analysis. It is shown that the origin of the instability is located at the exit plane of the primary injector. Mode shapes of both global and local linear stability analysis are compared to a dynamic mode decomposition based on large eddy simulation snapshots, showing good agreement. The estimated frequencies for the instability are in good agreement with both the experiment and the simulation. Furthermore, the adjoint mode shapes retrieved by the global approach are used to find the best location for periodic forcing in order to control the precessing vortex core.

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On mean flow universality of turbulent wall flows. II. Asymptotic flow analysis

Journal of Turbulence ◽

10.1080/14685248.2019.1593425 ◽

2019 ◽

Vol 20 (2) ◽

pp. 174-193 ◽

Cited By ~ 4

Author(s):

Stefan Heinz

Keyword(s):

Flow Analysis ◽

Mean Flow ◽

Wall Flows ◽

Turbulent Wall

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Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1995.0061 ◽

1995 ◽

Vol 352 (1698) ◽

pp. 125-168 ◽

Cited By ~ 8

Keyword(s):

Hopf Bifurcation ◽

Periodic Pattern ◽

Invariant Equilibrium ◽

Periodic Patterns ◽

Planar Lattice ◽

Diffusive Convection ◽

Spatially Periodic ◽

Spatio Temporal ◽

Boussinesq Fluid ◽

Time Periodic

In Rayleigh-Bénard convection, the spatially uniform motionless state of a fluid loses stability as the Rayleigh number is increased beyond a critical value. In the simplest case of convection in a pure Boussinesq fluid, the instability is a symmetry-breaking steady-state bifurcation that leads to the formation of spatially periodic patterns. However, in many double-diffusive convection systems the heat-conduction solution actually loses stability via Hopf bifurcation. These hydrodynamic systems provide motivation for the present study of spatiotemporally periodic pattern formation in Euclidean equivariant systems. We call such patterns planforms . We classify, according to spatio-temporal symmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a group-invariant equilibrium. Instead of focusing on plan- forms periodic with respect to a specified planar lattice, as has been done in previous investigations, we consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular differential equation.

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On mean flow universality of turbulent wall flows. I. High Reynolds number flow analysis

Journal of Turbulence ◽

10.1080/14685248.2019.1566736 ◽

2018 ◽

Vol 19 (11-12) ◽

pp. 929-958 ◽

Cited By ~ 5

Author(s):

Stefan Heinz

Keyword(s):

Reynolds Number ◽

High Reynolds Number ◽

Flow Analysis ◽

Mean Flow ◽

Reynolds Number Flow ◽

Wall Flows ◽

Number Flow ◽

High Reynolds ◽

Turbulent Wall

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Acoustic modes in jet and wake stability

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.148 ◽

2019 ◽

Vol 867 ◽

pp. 804-834 ◽

Cited By ~ 1

Author(s):

Eduardo Martini ◽

André V. G. Cavalieri ◽

Peter Jordan

Keyword(s):

Dispersion Relation ◽

Acoustic Waves ◽

Reverse Flow ◽

Saddle Points ◽

Vortex Sheet ◽

Necessary Condition ◽

Absolute Instability ◽

Acoustic Modes ◽

Long Time ◽

New Perspective

Motivated by recent studies that have revealed the existence of trapped acoustic waves in subsonic jets (Towne et al., J. Fluid Mech., vol. 825, 2017, pp. 1113–1152), we undertake a more general exploration of the physics associated with acoustic modes in jets and wakes, using a double vortex-sheet model. These acoustic modes are associated with eigenvalues of the vortex-sheet dispersion relation; they are discrete modes, guided by the vortex sheet; they may be either propagative or evanescent; and under certain conditions they behave in the manner of acoustic-duct modes. By analysing these modes we show how jets and wakes may both behave as waveguides under certain conditions, emulating ducts with soft or hard walls, with the vortex-sheet impedance providing effective ‘wall’ conditions. We consider, in particular, the role that upstream-travelling acoustic modes play in the dispersion-relation saddle points that underpin the onset of absolute instability. The analysis illustrates how departure from duct-like behaviour is a necessary condition for absolute instability, and this provides a new perspective on the stabilising and destabilising effects of reverse flow, temperature ratio and compressibility; it also clarifies the differing symmetries of jet (symmetric) and wake (antisymmetric) instabilities. An energy balance, based on the vortex-sheet impedance, is used to determine stability conditions for the acoustic modes: these may become unstable in supersonic flow due to an energy influx through the shear layers. Finally, we construct the impulse response of flows with zero and finite shear-layer thickness. This allows us to show how the long-time wavepacket behaviour is indeed determined by interaction between Kelvin–Helmholtz and acoustic modes.

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Studies on high speed jets from nozzles with internal grooves

The Aeronautical Journal ◽

10.1017/s000192400000498x ◽

2004 ◽

Vol 108 (1079) ◽

pp. 43-50 ◽

Cited By ~ 7

Author(s):

S. Elangovan ◽

E. Rathakrishnan

Keyword(s):

High Speed ◽

Cell Structure ◽

Pressure Profile ◽

Equivalent Diameter ◽

Streamwise Vortices ◽

Shock Cell ◽

Underexpanded Jets ◽

Pitot Pressure ◽

Mean Pressure ◽

The Mean

Experiments were carried out on jets issuing from circular nozzles with grooved exits and the results compared with those of the plain nozzle. The plain nozzle had an exit diameter of 10mm. Because of the introduction of semi-circular grooves at the exit, the effective or equivalent diameter of the grooved nozzles was 10·44mm. The groove lengths were varied as 3, 5 and 8mm. The nozzles were operated at fully expanded sonic and underexpanded exit conditions. The corresponding fully expanded Mach numbers were 1·0 and 1·41. The shock cell structure of the underexpanded jets from grooved nozzles appeared to be weaker than that of the plain nozzle, as indicated by lesser amplitudes of the cyclic variation of the Pitot pressure. The iso-Mach contours indicate that the jet spread along the grooved plane is significantly higher than that along the ungrooved plane. Off-centre peaks were observed in the mean pressure profile of underexpanded jets from grooved nozzles. They were probably due to the streamwise vortices shed from the grooves.

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