Coherent Vortical Structures Research Articles

Coherent turbulent flow structure under plunging breaker on movable grain bed simulated by 3D DEM-MPS method

ABSTRACT Breaking waves transport momentum, gas, and sediment vertically. Recent particle image velocimetry (PIV) measurements have elucidated the coherent vortex and turbulence structures under breaking waves. However, clarifying three-dimensional fluid motion under breaking waves with PIV remains challenging. Although particle methods have performed wave-breaking simulations, they have never been validated through velocity distributions for breaking waves above movable sediment beds. This study simulated the sediment transport process under plunging waves using a 3D Lagrangian fluid-particle mixture model. The investigation focuses on the first wave plunging for a fundamental investigation. Our simulation shows the instantaneous velocity distributions at different wave phases, which agree with the PIV measurement. A physically-based turbulence extraction method is introduced: a numerical filter based on the coherence between the water elevation and horizontal velocity. Visualized three-dimensional vortex and turbulence structures are consistent with previous studies. The transport equation of the kinetic turbulence energy investigates the turbulence budget near the bed surface and emphasizes the negative contribution of the fluid-particle interaction. Our findings show that a contribution of the pressure gradient to offshore sediment transport. The pressure gradient is difficult to correlate to the water surface gradient and is derived from the large vortex induced by the plunging wave.

Turbulent structure and circulation inside different planar contractions

We study the turbulent circulation and coherent vortical structures for flow through three separate two-dimensional planar 4:1 contractions, where the turbulence is subjected to different rates of mean strain. We use four high-speed video cameras for Lagrangian particle tracking velocimetry, to measure time-resolved velocity fields in volumes inside the contractions. We identify the coherent vortical structures and quantify their alignment with the mean strain. The intermediate strain contraction shows the strongest alignment. We also compute circulations around square loops in three perpendicular planes, to form a “circulation vector,” whose instantaneous alignment mirrors that of the coherent structures, with strong inhomogeneities between circulation components at the exit of the contractions. The circulation probability density functions show extended exponential tails for the smallest loops, with more Gaussian shapes in the inertial range.

Large eddy simulation of shock wave/boundary layer interactions in a transonic compressor cascade

In this paper, large eddy simulation (LES) was performed to investigate the shock wave/boundary layer interaction (SBLI) phenomenon in transonic compressor cascades with a chord Reynolds number of 2.12 × 106. A comprehensive analysis was conducted on both the SBLI structures inherent to the transonic compressor cascade and the coherent vortex structures within the boundary layer. The underlying mechanisms of the shock-induced boundary layer transition and the shock low-frequency unsteadiness in the transonic compressor cascade were elucidated through spectral and dynamic mode decomposition (DMD) analysis. The results revealed that boundary layer separation induced by the SBLI cannot reattach, leading to the formation of large-scale coherent vortex structures. Spectral analysis revealed that the shock-induced boundary layer transition in the transonic compressor cascade was dominated by inviscid Kelvin–Helmholtz (K–H) and secondary instability mechanisms, characterized by a dimensionless Strouhal number of 0.06. Additionally, pressure signals showed the variations in sub-frequency from the separated shear layer to the main flow. The oscillation amplitude of the shock foot was significantly greater than that of the shock main body, and the oscillation frequency of the shock foot was consistent with the sub-frequency. The oscillation frequency of the shock main body coincided with that of the compression ramp and flat plate configurations. Finally, DMD modal analysis indicated that high-frequency modes were correlated with turbulent fluctuations in the boundary layer, while medium- and low-frequency modes corresponded to shedding motion in the separated shear layer and low-frequency motion of the shock. This work promotes the understanding of the complex flow mechanisms of SBLI in the transonic compressor cascades.

Improvements for the DDES model with consideration of grid aspect ratio and coherent vortex structures

The delayed detached eddy simulation (DDES) model has achieved great success in simulations of massive separations, but has been found to significantly delay the Kelvin-Helmholtz (K-H) instability in weak separations due to the “grey area” problem. A new DDES model was proposed by considering the grid aspect ratio and local coherent vortex structures, named the DDES with an adaptive grid-scale and adaptive coefficient (DDES-AGAC). The DDES-AGAC model was examined in detail by computing four typical types of turbulent flows: fully attached turbulent boundary layer, massive separation, weakly unstable flow with a fixed separation point, and weak separation over a smooth surface due to pressure gradient. The results confirm the advantages of the new DDES-AGAC model over the standard DDES. It exhibits high efficiency and insensitivity to spanwise or circumferential grid spacing in flow simulations of quasi-three-dimensional flows. For the weakly unsteady separation, it helps to accelerate the rapid switch from Reynolds-averaged Navier-Stokes equations to large eddy simulation, unlock the K-H instability in the shear layer, and hence accurately resolve the turbulent structures, profiles of velocity and turbulent kinetic energy, and spectra in the separation regions. Furthermore, it does not result in a noticeable drag or skin friction reduction in the fully attached flow simulations. The DDES-AGAC model is expected to be widely used in simulations of engineering flows with separations.

Flame structure and dynamics in pool fires of different geometries: Experimental and numerical investigation

Fire accidents might be caused by liquid fuel leakage, which can accumulate in a prescribed area under the action of physical barriers. Therefore, pool fires could be of an arbitrary geometry in less ideal configurations. The current objective is to uncover the underlying physics of flame structure and dynamics in pool fires of different geometries. In this work, pool fires with seven different geometries were examined. Flame base structure and instantaneous flame height in pool fires of different geometries were recorded. Temperature, velocity/vorticity fields and coherent vortical structures were further presented by Large Eddy Simulation (LES). Results show that mean flame height of circular pool fire is the highest, while that of right triangle fires is the lowest. Mean flame height of the other noncircular geometries lies in these two extremes. The presence of vertices and hypotenuses has a dramatic influence on the formation and development of vortical structures. Triangular pool fires have the highest oscillation frequency, followed by quadrangular and circular geometries. To account for the physics of air entrainment and vortex dynamics due to pool fire geometry, pool perimeter is used as a characteristic length scale to propose new scaling relations for describing flame height and oscillation frequency.

A Study of spatiotemporal features of sweeping jets acting on afterbody vortices using low-operation-rate stereo PIV

Although low-operation-rate particle image velocimetry (PIV) provides a good spatial accuracy in measurements at relatively affordable costs, it faces some challenges in capturing unsteady features of oscillatory flow. In this paper, a single sweeping jet actuated to control afterbody vortices from a 30◦ slanted-base cylinder is investigated at a Reynolds number of 87,000. Phase-locked stereo PIV measurements combining triggering reference obtaining and real-time processing via field programmable gate array (FPGA) are leveraged to reveal the unsteady characteristics of the sweeping jet. The examined cases show that the phase-locked method can well identify jet’s spatiotemporal development process in each oscillation cycle. A sinusoidal-like interaction along phases between the jet and the afterbody vortex can be reasonably detected. At each moment, coherent small vortical structures form at the upper and bottom jet/ambient interfaces, which are caused by Kelvin-Helmholtz instability. Since the induced vortex has the same rotation direction as the afterbody vortex on each side, they merge with each other as the jet approaches the vortex, causing an increase in vorticity. Meanwhile, the sweeping jet’s intrusion into the vortex region induces a rise in turbulent kinetic energy in that area, causing turbulence ingestion of the vortex which weakens the velocity gradient. The sweeping behavior of the jet dominates the afterbody vortex to meander as the jet pushes its way underneath the vortex. The findings of this study provide encouraging evidence for future applications of sweeping jets in control of afterbody vortices.

Flow–acoustic resonance mechanism in tandem deep cavities coupled with acoustic eigenmodes in turbulent shear layers

This study presents the interplay of flow and acoustics within tandem deep cavities, focusing on the resonance mechanism occurring between turbulent shear layers and acoustic eigenmodes. The arrangement inside the tandem deep cavities includes both close and remote configurations. A combined fully coupled and decoupled aeroacoustic simulation strategy was devised. Employing an advanced high-order spectral/hp element method in conjunction with implicit large eddy simulation, the nonlinear compressible Navier–Stokes equations were solved to acquire internal flow–acoustic resonant field. In parallel, the linearized Navier–Stokes equations were tackled to determine coherent shear layer perturbations with external acoustic forcing. Based on acoustic measurements, the mainstream Reynolds number approaches approximately $R{e_{in}} = {O}({10^5})$ , where we identified the presence of frequency lock-in and a resonance range. Aeroacoustic noise sources were examined by implementing spectral proper orthogonal decomposition to decompose the pressure fields into hydrodynamic and acoustic components. As feedback intensified, the flow characteristics by the acoustic forcing effect and the flow-interactive effect were categorized according to the development of concurrent turbulent shear layers. Subsequently, the alternating and synchronous behaviours of concurrent shear layers resonated with the out-of-phase and in-phase acoustic eigenmodes were identified, and the corresponding large-scale counter-rotating vortex pairs and co-rotating vortex structures at the cavity entrances were extracted. The acoustic power generated by the Coriolis force was calculated using Howe's vortex-sound analogy, and the aeroacoustic energy transfer mechanism between large-scale shear layer vortices with acoustic eigenmodes was further explored. Finally, a linear response of coherent perturbations of the concurrent shear layers by external acoustic forcing was established. The amplification of flow in the streamwise direction toward the main duct led to the formation of coherent vortex structures, accompanied by separation bubbles into the main duct.

Very short time Fourier transform for turbulent analysis

Nominally stationary acoustic sensors, e.g., buoy suspended or bottom anchored, are subject to various environmental flows. Direct numerical simulation of hydrodynamic flow around the acoustic sensor housing show turbulence along the housing surface. Time-space analysis of pressure variation along the surface of the housing shows significant acoustic-like surface interaction at low frequencies, even at very low mean flow velocity. In this case turbulence manifests as intermittently generated, irregular, coherent vortex structures that separate from the surface of the housing and advect downstream. These structures drive rapid surface pressure transients that propagate within the housing to the acoustic sensors as noise. Fractional Fourier Analysis provides joint time-frequency methods to extract transient features of turbulence. Use of Hermite-Gauss functions and phase relations are of particular interest. [This research is supported by 6.2 NRL base program sponsored by the Office of Naval Research. Distribution Statement A: Approved for public release. Distribution unlimited.]

Vortex of a Symmetric Jet Structure in a Natural Gas Pipeline via Proper Orthogonal Decomposition

The impact of particle addition jets on the flow field in natural gas pipelines was investigated, and the structural information of the flow field at different flow velocities in a symmetric jet flow was analyzed via numerical simulation. The results of coherent structures in the high-pressure natural gas pipeline reveal vortex structures of varying sizes both upstream and downstream of the jet flow. To determine the spatial distribution of the main vortex structures in the flow field, proper orthogonal decomposition (POD) mode analysis was performed on the unsteady numerical results. Moreover, the detailed spatial characteristics of the coherent vortex structures represented by each mode were obtained. The results indicate that the large-scale vortex structures within the pipeline are balanced and stable, with their energy increasing as the jet flow velocity increases. Additionally, higher-order modes exhibit significant shedding of small-scale vortex structures downstream of the jet flow. In this research, coherent structures present in symmetric particle addition jets are provided, offering theoretical support for future investigations on the distribution of particle image velocimetry (PIV) flowmeters.

PIV measurements of coherent vortices and turbulence production inside acoustic liner cavity with offset slit

This paper reports an experimental investigation into the vortex dynamics purely excited by periodic acoustic waves within a slit–cavity structure, which is a typical unit element of an acoustic liner for aeroengine applications. Particle image velocimetry (PIV) was used to measure the instantaneous, time-averaged and phase-dependent flow dynamics inside the slit–cavity, which is exposed to high-intensity incident acoustic waves. Different with traditional literature mainly focusing on the acoustic characteristics, we primarily analyze the coherent vortex structures and turbulence production mechanism, accompanying with the variable of slit offset ratio being taken into consideration. The results demonstrate that the acoustic waves can excite the hydrodynamic velocities exceeding almost tens of times than the acoustic particle velocity, yielding multi-scale vortex structures appearing within the enclosed volume. Comparatively, in configuration with offset slit, the maximum hydrodynamic velocities were gradually intensified by wall confinement effect. Subsequent proper orthogonal decomposition analysis revealed the coherent vortex structures and demonstrated the offset configuration will increase their energy proportions and enhance their temporal correlations. However, the phase-dependent flow results revealed that the offset slit can oppositely attenuate the kinematics of the coherent vortices, which were represented by the convection trajectories and convection speeds, especially during the initial acceleration stage. Finally, analysis on turbulence production mechanism confirmed the attenuated turbulent kinetic energy by the offset slit, which was attributed to the less generation of shear stresses.

Direct numerical simulations on oscillating flow past surface-mounted finite-height circular cylinder

In this work, a surface-mounted circular cylinder with a fixed aspect ratio (ratio of height of the cylinder to its diameter) of 5 is subjected to a non-zero mean oscillating flow with a range of frequencies and amplitudes. Three-dimensional direct numerical simulations are then conducted on this finite-height cylinder. The mass and momentum equations are resolved using the finite volume-based Open Source Field Operation and Manipulation (OpenFOAM). A fixed Reynolds number Re=UoD/ν of 250 is used in this study, which is defined based on mean velocity at the inlet ( Uo ) and cylinder diameter (D). Here ν is the kinematic viscosity of the working fluid. Non-dimensional velocity oscillation amplitude ( A∗=a/Uo ) is varied from 0.1 to 0.3, while the non-dimensional oscillation frequency ( f∗=f/fo ) takes the values of 0.33, 0.5, 1, 2, and 3. Here a and f are the dimensional oscillation amplitude and frequency, respectively and fo is the vortex shedding frequency corresponding to a uniform flow at Re = 250. The three-dimensional vortex structures, presented with the help of iso-Q surfaces, show that the oscillating flow changes the size and shape of the hairpin-shaped vortices. Wake is found to be synchronized with the oscillation frequency at f* = 2 for each value of the A* and results in the maximum lift force on the cylinder. Hilbert Huang transformation analysis of the transverse velocity signals at a specific point in the wake reveals that the wake is more complex and aperiodic in nature for f* values of 0.33, 0.5, and 1, whereas it is periodic for f* = 2 and 3. In order to further disclose the nonlinearity associated with the oscillating flow, the degree of stationarity is discussed corresponding to each value of A* and f*. Dynamic mode decomposition is exploited to obtain information about the coherent vortical structures and their spatial and temporal behavior in the wake with a change in the value of f*. Effects of A* and f* on the dynamic characteristics are also investigated.

К теории спиральной турбулентности немагнитного астрофизического диска. Образование крупномасштабных вихревых структур

A closed system of three-dimensional hydrodynamic equations of the mean motion scale is presented for modeling spiral turbulence in a rotating astrophysical disk. The diffusion equations for the averaged vortex and the integral vortex spirality transport equation are derived. A general concept of the emergence of energy-consuming mezoscale coherent vortex structures in a thermodynamically open subsystem of turbulent chaos, associated with the realization of the inverse cascade of kine-tic energy in mirror-nonsymmetric disk turbulence, is formulated. It is shown that negative viscosity in the rotating disk three-dimensional system is apparently a manifestation of cascade processes in spiral turbulence, when an inverse energy transfer from small vortices to larger ones is realized. It is also shown that a relatively long decay of turbulence in the disk is associated with the absence of reflective symmetry of the anisotropic field of turbulent velocities relative to its equatorial plane. The work is of a review character, made with the aim of improving new models of astrophysical nonmagnetic disks for which the effects of spiral turbulence play a determining role.

Circulation in turbulent flow through a contraction

We study experimentally the statistical properties and evolution of circulation in a turbulent flow passing through a smooth 2-D contraction. The turbulence is generated with active grids to reach R e λ ≃ 220 at the inlet to the 2.5:1 contraction. We employ time-resolved 3-D Lagrangian Particle Tracking technique with the Shake-The-Box algorithm to obtain volumetric velocity fields which we use to calculate the simultaneous circulation in three perpendicular planes. Forming a circulation vector and studying the PDFs of the relative strength of its components, we can quantify how the mean strain enhances and orients coherent vortical structures with the streamwise direction. This is further studied with streamwise space and time correlations of the circulations over a range of loop sizes. The streamwise component of the circulation, over same-size square loops, shows increased integral length, while the other two components are less affected. The circulation around the compressive direction weakens and reaches prominent negative correlation values, suggesting buckling or sharp reorientation of transverse vortices. The PDFs of circulation transit from non-Gaussian to Gaussian behaviour as the loop size is increased from dissipative to large scales.

Analysis of inflow conditions on the flow past a wall-mounted square cylinder with OpenFOAM

In this study, the flow mechanisms around wall-mounted structures was numerically investigated using Large Eddy Simulation (LES). A comparison was made between the uniform and non-uniform inflow condition to explore the impact of the inflow turbulence on the flow physics, the dynamic response and the hydrodynamic performance of the flow past the cylinder. Sparsity Promoting Dynamic Model Decomposition (SPDMD) was applied to select the dominant modes in the downstream flow field, and to further investigate the evolution of the temporal-spatial scales, as well as the mutual interaction among the wake, the cylinder and the boundary layer. The present study revealed a strong interference between the velocity fluctuations and the wake past the cylinder, in which leads to rapid energy transfer from large eddies to small eddies. Strong convection effects were also observed in the far wake region, where significant interference occurs during the energy transfer induced by fluctuating velocities and the coherent vortex structures. The model analysis successfully identifies the prominent modes of the wake dynamic characteristics under different inflow conditions. This study expands our understanding on the wakes past wall-mounted structures, particularly in terms of their evolution and instability mechanisms, and provides valuable insights for the design and optimization of future wall-mounted structures in engineering practice.

Numerical Study on Aerodynamic and Aeroacoustic Performances of Bioinspired Wings

Noise control has become one of the key issues to be considered in modern aeronautical machinery design. Many efforts have been devoted to noise reduction of airfoils and wings, including traditional flow control methods. In fact, some animals in wild nature exhibit superior aerodynamic and aeroacoustic performance, providing novel ideas for solving this engineering problem. In this research, bionic technology is used to obtain quiet and efficient wing. Inspired by the owl’s wing, we propose two bionic configurations, one coupled with leading edge waves and trailing edge serrations. The Large Eddy Simulation and the Ffowcs-Williams and Hawkings equation is applied to simulate the aerodynamic and aeroacoustic characteristics of wings at low-Reynolds number flow. Numerical results demonstrate that the bioinspired wings have excellent aerodynamic performances and remarkable lower overall sound pressure level compared to NACA 0016 which has similar relative thickness. In addition, the unsmooth structure of leading edge waves and trailing edge serrations provide an additional 4.27 dB noise suppression effect, with little impact on aerodynamic characteristics at small angle of attack. The detailed analysis reveals that, due to the special owl-based profile, the flow around two bioinspired wings is mainly turbulent on the upper and lower surfaces, and no laminar separation bubble is detected at the trailing edge. Moreover, the unsmooth structure modifications successfully weaken the scale and scope of coherent vortex structures. These factors contribute to reducing the associated pressure fluctuation, thereby controlling the aeroacoustic noise of wing. Consequently, a coupled bionic wing is presented with the excellent aerodynamic and aeroacoustic characteristics. The conclusions are envisioned to be beneficial to the design of new generation low-noise aeronautical machinery.

Stable Vortex Particle Method Formulation for Meshless Large-Eddy Simulation

A novel formulation of the vortex particle method (VPM) is developed for large-eddy simulation (LES) in a meshless scheme that is numerically stable. A new set of VPM governing equations are derived from the LES-filtered Navier–Stokes equations. The new equations reinforce the conservation of angular momentum by resizing vortex elements subject to vortex stretching. In addition to the VPM reformulation, a new anisotropic dynamic model of subfilter-scale (SFS) vortex stretching is developed. This SFS model is well suited for turbulent flows with coherent vortical structures, where the predominant cascade mechanism is vortex stretching. The mean and fluctuating components of turbulent flow and Reynolds stresses are validated through the simulation of a turbulent round jet. The computational efficiency of the scheme is showcased in the simulation of an aircraft rotor in hover, showing our meshless LES to be 100 times faster than a mesh-based LES with similar fidelity. The implementation of our meshless LES scheme is released as open-source software, called FLOWVPM.

Transient Model for the Hydrodynamic Force in a Hydraulic Capsule Pipeline Transport System

The hydraulic capsule pipeline (HCP) is an eco-friendly and sustainable pipeline transport option. The freight-carrying capsule is driven by hydraulic pipe flow. Fluid drag is generated by the principal dynamic force effect on the capsule, which could influence the capsule’s motion speed. To make the HCP more efficient, a transient model for the hydrodynamic force in an HCP was developed in this study. From a numerical simulation, the coherent vortex structures of fluctuating modes were observed, and the velocity iso-surfaces of the coherent vortex of the wake flow exhibited an annular trend in circumferential connection. Then, the hydrodynamic force was analyzed: the steady component and transient component were resolved, and the general trend in forces in terms of the transient components was that the maximum amplitude of forces reduced with an increase in mode order. Through short-term Fourier transform, the frequency components and their variations in terms of the entire time range could be acquired. The transient model in this study provided a perspective to build the connection between the flow structures and the hydrodynamic force. By the transient model, the transient component of hydrodynamic force can be explained as the fluctuation of coherent vortex structures.

Vorticity cascade and turbulent drag in wall-bounded flows: plane Poiseuille flow

Drag for wall-bounded flows is directly related to the spatial flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution to this wall-normal flux arises from nonlinear advection and stretching of vorticity, interpretable as a cascade. We study this process using numerical simulation data of turbulent channel flow at friction Reynolds number $Re_\tau =1000$ . The net transfer from the wall of spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is ‘down-gradient’ or outward from the wall, where most vorticity concentrates, and the other which is ‘up-gradient’ or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient/down-gradient transport occurs by a mechanism of correlated inflow/outflow and spanwise vortex stretching/contraction that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism, we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra identifying eddies involved in nonlinear vorticity transport in the two opposing directions and (iii) visualizations of coherent vortex structures which contribute to the transport. The ‘D-type’ vortices contributing to down-gradient transport in the log layer are found to be attached, hairpin-type vortices. However, the ‘U-type’ vortices contributing to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.

Air Interactions of Magnetically Driven Plasma Discharges in Crossflow

Motivated by the use of magnetohydrodynamic devices in fluid dynamic flow control research, the current work experimentally studied the actuation mechanism and the resulting aerodynamic interactions of magnetically driven low-current arc-plasma discharges. The investigation was conducted in the context of boundary-layer interactions in low-speed crossflow conditions through the use of coaxial and v-shaped geometries. Time-averaged velocity field measurements showed that the toroidal region of vorticity induced by the coaxial geometry in quiescent air resulted in the formation of a horseshoe vortex in crossflow. Similarly, the v-shaped actuator resulted in the formation of streamwise vortices. The resulting boundary layers had s-shaped streamwise velocity profiles as measured in the wall-normal direction characteristic for the flow downstream of a conventional vortex generator pair, due to the three-dimensional mixing induced by coherent vortex structures. A simplified model based on intermolecular momentum transfer through collisions is proposed to explain the fundamental discharge–air interactions and the induced flowfields. These results inform the applications of various magnetically driven discharges in the field of aerodynamic flow control.

Experimental Investigation of High Frequency Flame Response on Injector Coupling in a Perfectly Premixed Multi-Jet Combustor

Abstract High frequency injector-coupled thermoacoustic instabilities are a major threat to multi-jet combustors in rocket and gas turbine engines. The complex three-dimensional acoustic coupling between the combustion chamber and injector acoustics cause local fluctuations in heat release. In turn, multiple thermoacoustic feedback mechanisms close the thermoacoustic loop and serve as a source of the thermoacoustic instability. Except for the flame deformation and flame displacement mechanism, the underlying feedback mechanisms for high frequency instabilities are to a large extent unknown. The paper at hand gives new insights into the injector-coupled convective driving mechanisms that are present in multi-jet combustors at perfectly premixed conditions. The forced flame response to the first transverse combustor mode is investigated for two distinct injector tube lengths: one with an axial acoustic velocity node and one with a velocity antinode coupling at the injector–combustor interface. Phase locked OH* images reveal convectively transported coherent vortex structures as the main source of the flame response. The origin of the flame response can be linked to the axial acoustic velocity at the injector–combustor interface using numerical simulations. Both configurations show a clear oscillation of the heat release fluctuations in-phase with the acoustic pressure fluctuations. In similarity to time delay models in low frequency thermoacoustics, a wave number model is proposed to estimate the local flame response due to feed flow modulations and validated with the experimental results.