Momentum Potential Theory Research Articles

A wall-resolved large-eddy simulation of deep cavity flow in acoustic resonance

Abstract

A study of noise reduction mechanisms of jets with fluid inserts

The fluid insert technology is an active noise reduction technique that has been shown to successfully reduce supersonic jet noise in model scale nozzles. The goal of this paper is to better understand the noise reduction mechanisms of fluid insert jets using Doak's Momentum Potential Theory and Spectral Proper Orthogonal Decomposition applied to an LES database. It is found that the use of fluid inserts results in a lower convection speed of the wavepacket-like coherent structures in the jet shear layer resulting in a shallower Mach wave radiation angle. It is also shown that the use of fluid inserts results in a reduction of length scales of the turbulent fluctuations in the shear layer, which correlates correctly with the observed noise reductions.

A pressure decomposition framework for aeroacoustic analysis of turbulent jets

Aeroacoustic analyses of jet flows have benefited greatly from a decomposition of turbulent pressure fluctuations into hydrodynamic and acoustic components. This is typically accomplished using signal processing techniques based on phase speeds, coherence properties or spectral analyses. We present an approach, building on the Momentum Potential Theory (MPT) approach of (Doak, 1989), to split pressure fluctuations into their hydrodynamic, acoustic and entropic, collectively designated fluid-thermodynamic (FT), components. Key advantages are that the approach is applicable everywhere in the jet i.e, not restricted to the near-acoustic field, and does not need user-defined thresholds. The effectiveness of the technique is demonstrated by analyzing the flowfields of three simulated jets, to encompass moderate-compressible to supersonic conditions. The statistical properties and wavepacket dynamics of each pressure component, and their relationships with the unsplit pressure are elaborated. The acoustic pressure field has the form of a wavepacket that attenuates downstream and whose modal analysis reveals low-rank behavior. At each Mach number examined, the acoustic pressure also identifies the relative prominence of each of three components: i) waves with upstream propagating energy content (negative group velocity), ii) supersonically traveling radiating downstream waves, and iii) subsonically convected evanescent waves, which follow the convection pattern of hydrodynamic eddies in the turbulent region. With increasing Mach number, the radiating and convected bands of energy move closer to each other. The hydrodynamic pressure also displays a wavepacket structure, but its features are different: it displays large-scale subsonically convected structures even past the core collapse region. Thus, in the turbulent region of the jet, the acoustic pressure displays smaller integral time scales of fluctuations than the hydrodynamic component. The acoustic pressure field, which includes a zero-crossing in its radial profiles, displays larger wavelengths than the hydrodynamic pressure field, correlates better with the near-field pressure signal and captures the radiated component of noise, especially at shallow angles. These properties make it a suitable field for informing pressure-based wavepacket models for jet noise.

Analysis of turbulence in convergent-divergent nozzles with fluid insert noise reduction technology

The noise generated by tactical fighter aircraft engines is harmful to personnel working in the vicinity of the aircraft and can cause annoyance to communities in the vicinity of air bases. The primary noise sources in supersonic jets are related to the supersonic convection of large-scale turbulent structures, which generate intense noise in the jet downstream direction, and the interaction of the turbulence with the jet’s shock cell structure that results in broadband shock-associated noise, which radiates predominantly in the sideline and upstream directions. The use of fluid inserts in the divergent section of variable area exhaust nozzles has been shown experimentally, at small and moderate scale, to reduce noise radiation from both noise sources. To understand the fluid insert noise reduction mechanisms, a Large Eddy Simulation database is developed and analyzed. The focus is on differences between turbulence properties in a baseline and a fluid insert nozzle. The flow database is analyzed using Spectral Proper Orthogonal Decomposition and Doak’s Momentum Potential Theory. The latter theory permits the decomposition of the flow into hydrodynamic, thermal and acoustic components. The changes in the acoustic component are related to the observed changes in the radiated noise.

Interactions between vortical, acoustic and thermal components during hypersonic transition

Discrete unstable modes of hypersonic laminar boundary layers, obtained from an eigenvalue analysis, provide insight into key transition scenarios. The character of such modes near the leading edge is often identified with the corresponding asymptotic free-stream behaviour of acoustic, vortical or entropic (thermal) content, which we designate fluid-thermodynamic (FT) components. In downstream regions, however, this direct one-to-one correspondence between discrete modes and FT components does not hold, since FT components interact in well-defined ways with the basic state and with each other (even under linear scenarios). In the present work, we perform an FT decomposition of discrete modes using momentum potential theory, to yield a physics-based analysis that complements linear stability theory in the linear regime, and seamlessly extends to the nonlinear domain where direct numerical simulations are appropriate. Linear and nonlinear saturated disturbance effects, different forcing types and wall thermal conditions are considered, with emphasis on phenomena occurring near stability-mode synchronization locations. The results show that, in the linear regime, each discrete mode contains all FT components, whose relative amplitudes vary with streamwise distance. Vortical components are always the largest, followed by thermal and acoustic components. These latter two show distinct fore and aft signatures near mode synchronization. The vortical component displays a series of rope-shaped recirculation-cell patterns across the generalized inflection point. However, both acoustic and thermal components display ‘trapped’ structures. The former contains an alternating monopole array between the wall and the critical layer, while the latter is confined to an undulating region between the wall and a wavy locus straddling the generalized inflection point. Nonlinear saturation in the region of Mack-mode growth further strengthens the rope-shaped structures in the vortical component and higher harmonics appear, whose form and location depend on the specific component. Wall cooling modifies the eigenfunctions such that the acoustic component accounts for more of its composition, consistent with its destabilization. Analysis of energy interactions among the FT components indicates that, even though the vorticity component is the largest, the thermal component induces the most significant source term for the growth of acoustic perturbations, possibly due to the trapped nature of both.

Analysis of nonequilibrium effects characterizes shock wave-boundary layer interactions in hypersonic flow

The study combines direct simulation Monte Carlo calculations, linear global instability analysis and momentum potential theory to characterize nonequilibrium effects.

On the unsteadiness of shock–laminar boundary layer interactions of hypersonic flows over a double cone

Unsteadiness of axisymmetric shock-dominated hypersonic laminar separated flow over a double cone is studied for the first time using a combination of time accurate Direct Simulation Monte Carlo (DSMC) calculations, linear global instability analysis, and momentum potential theory (MPT). Close to steady state linear analysis reveals the spatial structure of the underlying temporally stable global modes. At all Reynolds numbers examined, the amplitude functions demonstrate the strong coupling between the separated flow region at the cone junction with the entire shock system, including pressure and temperature waves generated behind the shock and spatially amplified Kelvin-Helmholtz waves. In addition, as the Reynolds number is increased, temporally damped harmonic shock oscillations and multiple-reflected λ-shock patterns emerge in the eigenfunctions. Application of the MPT (valid for both linear and nonlinear signals) to the highest Reynolds number DSMC results shows that large acoustic and thermal potential variations exist in the vicinity of the separation shock, the λ-shock patterns, and the shear layers. It is further shown that the motion of the bow shock system is highly affected by non-uniformities in the acoustic field. At the highest Reynolds number considered here, the unsteadiness is characterized by Strouhal numbers in the shear layer and bow-shock regions and is found to be in qualitative agreement with earlier experimental and numerical work.

Transfer mechanisms from stochastic turbulence to organized acoustic radiation in a supersonic jet

We examine the three-dimensional energy-exchange mechanisms between stochastic turbulence and an organized radiating wavepacket, and subsequent scattering and transmission of acoustic energy. A validated round-jet large-eddy simulation (LES) is subjected to Kovásznay-type decomposition into hydrodynamic, acoustic and thermal modes, as encapsulated in the generalized framework of the momentum potential theory (MPT). The results are first assimilated with known and postulated aspects of jet-noise dynamics to provide an alternative perspective. Subsequently, new insights are derived on acoustic-energy generation and transport processes. The prolonged axial and azimuthal spatio-temporal coherence of the jittering acoustic mode, which manifests as a wavepacket, and the localization of its energy to the low-order azimuthal modes (m=0,1 and 2) are shown to arise naturally in the decomposed field. The spatial supports of the internal and radiating components of the acoustic-mode are identified with dynamic mode decomposition (DMD). The resulting spectra are observed to be similar to those of the global instability modes of the mean flow. In contrast, the hydrodynamic mode, which is the primary source of energy in this cold jet, has broadband spectral structure and limited spatio-temporal coherence. This knowledge is then leveraged to analyze and quantify sound scattering versus transmission through energy interactions between the relatively chaotic hydrodynamic mode and its stimulation of an ordered response in the acoustic mode. Surface integral analyses reveal the mechanisms by which the omni-directional nature of the turbulent scattering is established. The acoustic mode plays a central role in imparting directionality to the acoustic transmission process, with its axisymmetric component contributing to downstream and shallow angle radiation and higher azimuthal modes contributing increasingly to sideline radiation. Volumetric integral analyses combine these observations to show that the unorganized hydrodynamic mode generates localized noise-producing regions, resulting in a coherent form of sound radiation from the jet.

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

Underexpanded jets exhibit interactions between turbulent shear layers and shock-cell trains that yield complex phenomena that are absent in the more commonly studied perfectly expanded jets. We quantitatively analyze these mechanisms by considering the interplay between hydrodynamic (turbulence) and acoustic modes, using a validated large-eddy simulation. Using momentum potential theory (MPT) to achieve energy segregation, the following observations are made. The sharp gradients in fluctuations introduced by the shock-cell structure are captured mostly in the hydrodynamic mode, whose amplitude is an order of magnitude larger than the acoustic mode. The acoustic mode has a relatively smoother distribution, exhibiting a compact wavepacket form. Proper orthogonal decomposition (POD) identifies the third-to-sixth cells as the most dynamic structures. The imprint of shock cells is discernible in the nearfield of the acoustic mode, primarily along the sideline direction. Energy interactions that feed the acoustic mode remain compact in nature, facilitating a simple propagation technique for farfield noise prediction. The farfield sound spectra show peak directivity at 30 ∘ to the downstream axis. The POD modes of the acoustic component also identify two main energetic components in the wavepacket: one representative of the periodic internal structure and the other of intermittent downstream lobes. The latter component occurs at exactly the same frequency as, and displays high correlation with, the farfield peak noise spectra, making the acoustic mode a better predictor of the dynamics than velocity fluctuations.

Acoustic, hydrodynamic and thermal modes in a supersonic cold jet

Large-eddy simulation data for a Mach 1.3 round jet are decomposed into acoustic, hydrodynamic and thermal components using Doak’s momentum potential theory. The decomposed fields are then analysed to examine the properties of each mode and their dynamics based on the transport equation for the total fluctuating enthalpy. The solenoidal fluctuations highlight hydrodynamic components of the jet and capture the shear layer growth and breakdown process. The acoustic mode exhibits a jittering coherent wavepacket structure in the turbulent region and consequent highly directional downstream radiation. The expected radial decay rates,$r^{-6}$for hydrodynamic and$r^{-2}$for acoustic, are recovered and closely follow the universal radiation spectra in the sideline and downstream directions. The scalogram of the acoustic mode in the near-acoustic-field region is consistent with that of the pressure perturbation signal in the acoustic-frequency range, but effectively removes the hydrodynamic and thermal content. The time-resolved and mean behaviour of terms in the total fluctuating enthalpy equation is analysed in detail. A large-scale intermittent event in the near-acoustic field is shown to be associated with an intrusion of vortices from the shear layer into the core of the jet. Acoustic sources are created when the resulting negative fluctuations in the solenoidal component interact with positive fluctuations in the Coriolis acceleration term. The latter are associated with regions of high vorticity on the inner side of the shear layer. In contrast, sinks result from the interaction of solenoidal momentum fluctuations with positive entropy gradients along entrainment streaks.

Doak's momentum potential theory of energy flux used to study a solenoidal wavepacket

Doak's momentum potential theory of energy flux is explored using a model problem constructed to facilitate the introduction of solenoidal perturbations, in a controlled manner, to an otherwise well-understood sound generation problem: that of an irrotational, subsonically convecting wavepacket. The solenoidal wavepacket has, in addition to a downstream radiation lobe similar to its irrotational counterpart, lower level radiation in the sideline direction. Helmholtz decomposition of the linear momentum and subsequent exploration of the various source and flux terms that participate, according to Doak's fluctuation-energy balance, in the generation and transport of total fluctuating enthalpy (TFE) reveal a rich inner structure involving work performed on, and extracted from, the fluid system by means of the irrotational and solenoidal components of the wavepacket. The response of the fluid involves the internal transport and attentuation of trapped TFE, as well as the radiation of a small amount of radiating TFE. The analysis shows how the downstream radiation is associated with a mechanism similar to that of the irrotational wavepacket, while the sideline radiation arises due to the scattering of irrotational wavepacket fluctuations by solenoidal momentum fluctuations. It is postulated that such a mechanism might play a role in the sideline radiation of subsonic jets.

Momentum potential theory of energy flux carried by momentum fluctuations

The momentum potential theory of time-stationary fluctuating flows is briefly reviewed and then extended to include energy flux carried by momentum fluctuations. It is shown that the mean (time-averaged) energy flux can be expressed as a linear superposition of mean, turbulent, acoustic and thermal components. A mean energy flux balance relating turbulent, acoustic and thermal energy fluxes only is obtained. For homentropic flows the Cantrell and Hart expression for acoustic intensity is obtained as a special case. Physical and analog interpretations of the mean energy flux balance are presented for the general case. A formal solution of the equation for mean flow of energy due to momentum fluctuations is presented, leading to a definition of local fluctuating dynamical equilibrium of a time-stationary flow and an identification of sources of far field radiated acoustic power. In the Appendix, the momentum potential theory version of Jenvey's partitioning of the stagnation enthalpy into mean, turbulent, acoustic and thermal parts is presented.