The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Graham J. Nathan - One of the best experts on this subject based on the ideXlab platform.
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the influence of Nozzle Exit geometric profile on statistical properties of a turbulent plane jet
Experimental Thermal and Fluid Science, 2007Co-Authors: Jianchun Mi, Graham J. NathanAbstract:The paper reports an investigation of the influence of geometric profile of a long slot Nozzle on the statistical properties of a plane jet discharging into a large space. The Nozzle-Exit profile was varied by changing orifice-plates with different Exit radii (r) over the range of 0 < r/h < 3.60, where h is the slot-height. The present measurements were made at a slot-height based Reynolds number (Reh) of 1.80 · 104 and a slot aspect ratio (span/height) of 72. The results obtained show that both the initial flow and the downstream flow are dependent upon the ratio r/h. A ‘‘top-hat’’ mean Exit velocity profile is closely approximated when r/h approaches 3.60. The decay and spread rates of the jet’s mean velocity decrease asymptotically as r/h is increased, with the differences becoming small as r/h approaches 3.60. A decrease in r/h results in a higher formation rate of the primary vortices in the near-field. The far-field values of the centerline turbulence intensity are higher for smaller r/h, and display asymptotic-like convergence as r/h approaches 3.60. Overall, the effect of r/h on the mean and turbulence fields decreases as r/h increases.
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The influence of Nozzle-Exit geometric profile on statistical properties of a turbulent plane jet
Experimental Thermal and Fluid Science, 2007Co-Authors: Ravinesh C. Deo, Graham J. NathanAbstract:© 2007 Published by Elsevier Inc.The paper reports an investigation of the influence of geometric profile of a long slot Nozzle on the statistical properties of a plane jet discharging into a large space. The Nozzle-Exit profile was varied by changing orifice-plates with different Exit radii (r) over the range of 0 < r/h < 3.60, where h is the slot-height. The present measurements were made at a slot-height based Reynolds number (Reh) of 1.80 × 104 and a slot aspect ratio (span/height) of 72. The results obtained show that both the initial flow and the downstream flow are dependent upon the ratio r/h. A “top-hat” mean Exit velocity profile is closely approximated when r/h approaches 3.60. The decay and spread rates of the jet’s mean velocity decrease asymptotically as r/h is increased, with the differences becoming small as r/h approaches 3.60. A decrease in r/h results in a higher formation rate of the primary vortices in the near-field. The far-field values of the centerline turbulence intensity are higher for smaller r/h, and display asymptotic-like convergence as r/h approaches 3.60. Overall, the effect of r/h on the mean and turbulence fields decreases as r/h increases.Ravinesh C. Deo, Jianchun Mi and Graham J. Nathanhttp://www.elsevier.com/wps/find/journaldescription.cws_home/505737/description#descriptio
Ethirajan Rathakrishnan - One of the best experts on this subject based on the ideXlab platform.
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Impact of tab location relative to the Nozzle Exit on the shock structure of a supersonic jet
Physics of Fluids, 2019Co-Authors: P. Arun Kumar, Manideep Aileni, Ethirajan RathakrishnanAbstract:This paper presents an experimental study of a Mach 2.0 jet manipulated using rectangular tabs to understand the mixing enhancement at the overexpanded and perfectly expanded state of the jet. This paper also compares the mixing effectiveness of the tabs in comparison with the fluidic injection reported in our previous work [Kumar et al., “Empirical scaling analysis of supersonic jet control using steady fluidic injection,” Phys. Fluids 31(5), 056107 (2018)]. Tabs used in this investigation were rectangular strips of aspect ratio, AR, 2 (AR = length of the tab/width of the tab) and are positioned at 0De, 0.25De, 0.5De, and 0.95De (De is the Nozzle Exit diameter) downstream of the Nozzle Exit. Pitot pressure measurements were carried out along the jet centerline and in the radial directions to examine the supersonic core length (Lc*) and jet spread, respectively. The jet stream has been visualized using the shadowgraph technique in the orthogonal planes of the manipulated jet. The mixing capability of the manipulated jet quantified based on the reduction in supersonic core length ΔLc* strongly depends on the control technique and its location along the downstream direction. Three types of flow categories are identified, i.e., the “jet bifurcation,” “complex and strong shock-cell structure,” and “weak shock structure,” which depend on the tab location (xt*) and account for the jet mixing. The present study reveals that the tabs should be positioned downstream of the first shock crossover point which results in shorter core length and, hence, higher jet mixing. A conceptual model of the flow structure under control is proposed.This paper presents an experimental study of a Mach 2.0 jet manipulated using rectangular tabs to understand the mixing enhancement at the overexpanded and perfectly expanded state of the jet. This paper also compares the mixing effectiveness of the tabs in comparison with the fluidic injection reported in our previous work [Kumar et al., “Empirical scaling analysis of supersonic jet control using steady fluidic injection,” Phys. Fluids 31(5), 056107 (2018)]. Tabs used in this investigation were rectangular strips of aspect ratio, AR, 2 (AR = length of the tab/width of the tab) and are positioned at 0De, 0.25De, 0.5De, and 0.95De (De is the Nozzle Exit diameter) downstream of the Nozzle Exit. Pitot pressure measurements were carried out along the jet centerline and in the radial directions to examine the supersonic core length (Lc*) and jet spread, respectively. The jet stream has been visualized using the shadowgraph technique in the orthogonal planes of the manipulated jet. The mixing capability of the ...
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Effect of Cross-Wire Location on the Mixing of Underexpanded Sonic Jets
JOURNAL OF AEROSPACE ENGINEERING, 2007Co-Authors: P Lovaraju, Ethirajan RathakrishnanAbstract:Abstract: This paper presents the results of an experimental investigation carried out to study the effectiveness of a passive control in the form of a cross-wire located at the Nozzle Exit and downstream of the Nozzle Exit for promoting jet mixing. The advantage of locating the cross-wire at downstream of the Nozzle Exit is to keep the flow at the Nozzle Exit undisturbed, to avoid thrust loss. Sonic jets at Nozzle pressure ratios NPR of 3, 5, and 7 were investigated with cross-wire at the Nozzle Exit, at 1De and 2De from the Nozzle Exit. The cross-wire influenced the supersonic core, causing the significant reduction of core length and mixing enhancement. The cross-wire also results in diffusion of shocks for all the cross-wire locations and at all tested NPRs.
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Studies on oblique impinging subsonic and sonic jets
Fluid Dynamics Research, 1996Co-Authors: T.j. Ignatius, Ethirajan RathakrishnanAbstract:This paper describes an experimental investigation on the normal and oblique subsonic and underexpanded jet impingement. The shadowgraph pictures of normal and oblique impingement show that the stand-off shock at the wall remains parallel to the Nozzle Exit plane when the wall is moved away from the Nozzle Exit. The jet field is investigated based on the field characteristics such as spread, wall half-pressure width and wall pressure similarity. The jet spread is strongly influenced by obliqueness of the wall, whereas the wall pressure similarity is independent of obliqueness. The positive pressure zone at the center of impingement planes disappears for a wall at 10 times Nozzle Exit diameter for subsonic impingement, whereas for underexpanded it disappears as early as 4 times the Nozzle Exit diameter.
S. P. Venkateshan - One of the best experts on this subject based on the ideXlab platform.
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Influence of wire mesh at Nozzle Exit on heat transfer from square jets - An experimental investigation
2017Co-Authors: Pullarao Muvvala, Chakravarthy Balaji, S. P. VenkateshanAbstract:The goal of this work is to compare the fluid flow and heat transfer characteristics of square jet in the presence of a wire mesh at the Nozzle Exit. Towards this, in-house experiments are carried out by impinging a square jet on a uniformly heated plate of finite thickness, with air as the cooling medium. Two different sets of square jets are employed in this study namely a jet in the absence of mesh and a jet in the presence of a wire mesh at the Nozzle Exit. Jet centerline mean velocity and turbulence intensity measurements are done with a hot-wire anemometer to support the heat transfer results.The goal of this work is to compare the fluid flow and heat transfer characteristics of square jet in the presence of a wire mesh at the Nozzle Exit. Towards this, in-house experiments are carried out by impinging a square jet on a uniformly heated plate of finite thickness, with air as the cooling medium. Two different sets of square jets are employed in this study namely a jet in the absence of mesh and a jet in the presence of a wire mesh at the Nozzle Exit. Jet centerline mean velocity and turbulence intensity measurements are done with a hot-wire anemometer to support the heat transfer results.
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Experimental investigation on the effect of wire mesh at the Nozzle Exit on heat transfer from impinging square jets
Experimental Thermal and Fluid Science, 2017Co-Authors: Pullarao Muvvala, Chakravarthy Balaji, S. P. VenkateshanAbstract:Abstract This paper reports the results of an investigation carried out to study the effect of a wire screen mesh installed at the Nozzle Exit on the heat transfer performance of a square jet. Towards this, experiments are carried out to cool an electrically heated flat plate with an impinging air jet issued from a square orifice. Three wire meshes with area based porosity values of 0.35, 0.41 and 0.66 respectively are employed in this study. Heat transfer characteristics are presented as stagnation point and local Nusselt numbers. The heat transfer characteristics of the jets with and without the wire meshes are compared at (i) the same mass flow rate of the jet fluid and (ii) the same power consumption of the jet across the Nozzle. To support the heat transfer results, mean velocity and turbulence intensities are measured along the jet centerline with a hot-wire anemometer. The effect of dimensionless jet-to-plate distance (2–8.5) on the heat transfer rate is studied as well.
Christophe Bogey - One of the best experts on this subject based on the ideXlab platform.
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Effects of Nozzle-Exit boundary-layer profile on the initial shear-layer instability, flow field and noise of subsonic jets
Journal of Fluid Mechanics, 2019Co-Authors: Christophe Bogey, Roberto SabatiniAbstract:The influence of the Nozzle-Exit boundary-layer profile on high-subsonic jets is investigated by performing compressible large-eddy simulations (LES) for three isothermal jets at a Mach number of 0.9 and a diameter-based Reynolds number of $5\times 10^{4}$ , and by conducting linear stability analyses from the mean-flow fields. At the Exit section of a pipe Nozzle, the jets exhibit boundary layers of momentum thickness of approximately 2.8 % of the Nozzle radius and a peak value of turbulence intensity of 6 %. The boundary-layer shape factors, however, vary and are equal to 2.29, 1.96 and 1.71. The LES flow and sound fields differ significantly between the first jet with a laminar mean Exit velocity profile and the two others with transitional profiles. They are close to each other in these two cases, suggesting that similar results would also be obtained for a jet with a turbulent profile. For the two jets with non-laminar profiles, the instability waves in the near-Nozzle region emerge at higher frequencies, the mixing layers spread more slowly and contain weaker low-frequency velocity fluctuations and the noise levels in the acoustic field are lower by 2–3 dB compared to the laminar case. These trends can be explained by the linear stability analyses. For the laminar boundary-layer profile, the initial shear-layer instability waves are most strongly amplified at a momentum-thickness-based Strouhal number $St_{\unicode[STIX]{x1D703}}=0.018$ , which is very similar to the value obtained downstream in the mixing-layer velocity profiles. For the transitional profiles, on the contrary, they predominantly grow at higher Strouhal numbers, around $St_{\unicode[STIX]{x1D703}}=0.026$ and 0.032, respectively. As a consequence, the instability waves rapidly vanish during the boundary-layer/shear-layer transition in the latter cases, but continue to grow over a large distance from the Nozzle in the former case, leading to persistent large-scale coherent structures in the mixing layers for the jet with a laminar Exit velocity profile.
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Effects of Nozzle-Exit boundary-layer profile on the initial shear-layer instability, flow field and noise of subsonic jets
Journal of Fluid Mechanics, 2019Co-Authors: Christophe Bogey, Roberto SabatiniAbstract:The influence of the Nozzle-Exit boundary-layer profile on high-subsonic jets is investigated by performing compressible large-eddy simulations (LES) for three isothermal jets at a Mach number of 0.9 and a diameter-based Reynolds number of 5 × 10 4 , and by conducting linear stability analyses from the mean-flow fields. At the Exit section of a pipe Nozzle, the jets exhibit boundary layers of momentum thickness of approximately 2.8 % of the Nozzle radius and a peak value of turbulence intensity of 6 %. The boundary-layer shape factors, however, vary and are equal to 2.29, 1.96 and 1.71. The LES flow and sound fields differ significantly between the first jet with a laminar mean Exit velocity profile and the two others with transitional profiles. They are close to each other in these two cases, suggesting that similar results would also be obtained for a jet with a turbulent profile. For the two jets with non-laminar profiles, the instability waves in the near-Nozzle region emerge at higher frequencies, the mixing layers spread more slowly and contain weaker low-frequency velocity fluctuations and the noise levels in the acoustic field are lower by 2-3 dB compared to the laminar case. These trends can be explained by the linear stability analyses. For the laminar boundary-layer profile, the initial shear-layer instability waves are most strongly amplified at a momentum-thickness-based Strouhal number St θ = 0.018, which is very similar to the value obtained downstream in the mixing-layer velocity profiles. For the transitional profiles, on the contrary, they predominantly grow at higher Strouhal numbers, around St θ = 0.026 and 0.032, respectively. As a consequence, the instability waves rapidly vanish during the boundary-layer/shear-layer transition in the latter cases, but continue to grow over a large distance from the Nozzle in the former case, leading to persistent large-scale coherent structures in the mixing layers for the jet with a laminar Exit velocity profile.
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Identification of the effects of the Nozzle-Exit boundary-layer thickness and its corresponding Reynolds number in initially highly disturbed subsonic jets
Physics of Fluids, 2013Co-Authors: Christophe Bogey, Olivier MarsdenAbstract:The influence of the Nozzle-Exit boundary-layer thickness in isothermal round jets at a Mach number of 0.9 and at diameter Reynolds numbers Re D ≃ 5 × 104 is investigated using large-eddy simulations. The originality of this work is that, contrary to previous studies on the topic, the jets are initially highly disturbed, and that the effects of the boundary-layer thickness are explored jointly on the Exit turbulence, the shear-layer and jet flow characteristics, and the acoustic field. The jets originate from a pipe of radius r 0, and exhibit, at the Exit, peak disturbance levels of 9% of the jet velocity, and mean velocity profiles similar to laminar boundary-layer profiles of thickness δ0 = 0.09r 0, 0.15r 0, 0.25r 0, or 0.42r 0, yielding 99% velocity thicknesses between 0.07r 0 and 0.34r 0 and momentum thicknesses δθ(0) between 0.012r 0 and 0.05r 0. Two sets of computations are reported to distinguish, for the first time to the best of our knowledge, between the effects of the ratio δ0/r 0 and of the Reynolds number Reθ based on δθ(0). First, four jets with a fixed diameter, hence at a constant Reynolds number Re D = 5 × 104 giving Reθ = 304, 486, 782, and 1288 depending on δ0, are considered. In this case, due to the increase in Reθ, thickening the initial shear layers mainly results in a weaker mixing-layer development with lower spreading rates and turbulence intensities, and reduced sound levels at all emission angles. Second, four jets at Reynolds numbers Re D between 1.8 × 104 and 8.3 × 104, varying so as to obtain Reθ ≃ 480 in all simulations, are examined. Here, increasing δ0/r 0 has a limited impact on the mixing-layer key features, but clearly leads to a shorter potential core, a more rapid velocity decay, and higher fluctuations on the jet axis, and stronger noise in the downstream direction. Similar trends can be expected for high-Reynolds-number jets in which viscosity plays a negligible role.
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Effects of moderate Reynolds numbers on subsonic round jets with highly disturbed Nozzle-Exit boundary layers
Physics of Fluids, 2012Co-Authors: Christophe Bogey, Olivier Marsden, Christophe BaillyAbstract:The effects of moderate Reynolds numbers on the flow and acoustic fields of initially highly disturbed isothermal round jets at Mach number M = 0.9 and diameter-based Reynolds numbers Re D between 2.5 × 104 and 2 × 105 are investigated using large-eddy simulation under carefully controlled conditions. To the best of our knowledge, this is the first comprehensive study of its kind. The jets originate at z = 0 from a pipe Nozzle of radius r 0, in which a tripping procedure is applied to the boundary layers. At the Nozzle Exit, laminar-like mean velocity profiles of thickness δ ≃ 0.15r 0 and momentum thickness δθ ≃ 0.018r 0, yielding Reynolds numbers Reθ varying from 256 to 1856 depending on Re D , and peak turbulence intensities around 9% of the jet velocity, are thus obtained. As the Reynolds number increases, the mixing layers develop more slowly, with smaller integral length scales and lower levels of velocity fluctuations. The axial profiles of turbulence intensities become smoother, showing a clear overshoot around z = 2r 0 at Re D = 2.5 × 104, but a monotonical growth at Re D = 2 × 105. Velocity spectra downstream of the Nozzle Exit also broaden with Re D , as expected. Large-scale components usually observed in turbulent boundary layers and shear layers, characterized by Strouhal numbers Stθ ≃ 0.013 around z = r 0 and by azimuthal spacings λθ ≃ δ, remain dominant, although the contribution of fine-scale structures with λθ ⩽ δ/2 strengthens. Moreover, with rising Re D , the jet potential core lengthens slightly, but the flow properties do not change significantly farther downstream. Finally, lower sound pressure levels are generated, with a decrease of about 2 dB over the range of Re D considered.
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on the spectra of Nozzle Exit velocity disturbances in initially nominally turbulent transitional jets
Physics of Fluids, 2011Co-Authors: Christophe Bogey, Olivier Marsden, Christophe BaillyAbstract:In a recent paper by C. Bogey, O. Marsden, and C. Bailly [“Large-eddy simulation of the flow and acoustic fields of a Reynolds number 105 subsonic jet with tripped Exit boundary layers,” Phys. Fluids 23(3), 035104 (2011)], simulation results were presented for round jets with tripped boundary layers, displaying Nozzle-Exit conditions typical of initially nominally turbulent, transitional jets, namely laminar mean velocity profiles and high fluctuation intensities. The velocity spectra evaluated just downstream of the Nozzle Exit are re-examined here with respect to literature data. They agree qualitatively very well with spectra obtained in a fully turbulent pipe flow using direct numerical simulation. The wave numbers dominating in the azimuthal direction are also consistent with measurements of spanwise energy distribution in fully turbulent boundary layers. The initial turbulent structures in the jets, therefore, appear to be organized similarly to those in fully developed wall-bounded flows.
Christophe Bailly - One of the best experts on this subject based on the ideXlab platform.
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Effects of moderate Reynolds numbers on subsonic round jets with highly disturbed Nozzle-Exit boundary layers
Physics of Fluids, 2012Co-Authors: Christophe Bogey, Olivier Marsden, Christophe BaillyAbstract:The effects of moderate Reynolds numbers on the flow and acoustic fields of initially highly disturbed isothermal round jets at Mach number M = 0.9 and diameter-based Reynolds numbers Re D between 2.5 × 104 and 2 × 105 are investigated using large-eddy simulation under carefully controlled conditions. To the best of our knowledge, this is the first comprehensive study of its kind. The jets originate at z = 0 from a pipe Nozzle of radius r 0, in which a tripping procedure is applied to the boundary layers. At the Nozzle Exit, laminar-like mean velocity profiles of thickness δ ≃ 0.15r 0 and momentum thickness δθ ≃ 0.018r 0, yielding Reynolds numbers Reθ varying from 256 to 1856 depending on Re D , and peak turbulence intensities around 9% of the jet velocity, are thus obtained. As the Reynolds number increases, the mixing layers develop more slowly, with smaller integral length scales and lower levels of velocity fluctuations. The axial profiles of turbulence intensities become smoother, showing a clear overshoot around z = 2r 0 at Re D = 2.5 × 104, but a monotonical growth at Re D = 2 × 105. Velocity spectra downstream of the Nozzle Exit also broaden with Re D , as expected. Large-scale components usually observed in turbulent boundary layers and shear layers, characterized by Strouhal numbers Stθ ≃ 0.013 around z = r 0 and by azimuthal spacings λθ ≃ δ, remain dominant, although the contribution of fine-scale structures with λθ ⩽ δ/2 strengthens. Moreover, with rising Re D , the jet potential core lengthens slightly, but the flow properties do not change significantly farther downstream. Finally, lower sound pressure levels are generated, with a decrease of about 2 dB over the range of Re D considered.
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on the spectra of Nozzle Exit velocity disturbances in initially nominally turbulent transitional jets
Physics of Fluids, 2011Co-Authors: Christophe Bogey, Olivier Marsden, Christophe BaillyAbstract:In a recent paper by C. Bogey, O. Marsden, and C. Bailly [“Large-eddy simulation of the flow and acoustic fields of a Reynolds number 105 subsonic jet with tripped Exit boundary layers,” Phys. Fluids 23(3), 035104 (2011)], simulation results were presented for round jets with tripped boundary layers, displaying Nozzle-Exit conditions typical of initially nominally turbulent, transitional jets, namely laminar mean velocity profiles and high fluctuation intensities. The velocity spectra evaluated just downstream of the Nozzle Exit are re-examined here with respect to literature data. They agree qualitatively very well with spectra obtained in a fully turbulent pipe flow using direct numerical simulation. The wave numbers dominating in the azimuthal direction are also consistent with measurements of spanwise energy distribution in fully turbulent boundary layers. The initial turbulent structures in the jets, therefore, appear to be organized similarly to those in fully developed wall-bounded flows.
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A computational study of the effects of Nozzle-Exit turbulence level on the flow and acoustic fields of a subsonic jet
17th AIAA CEAS Aeroacoustics Conference (32nd AIAA Aeroacoustics Conference), 2011Co-Authors: Christophe Bogey, Olivier Marsden, Christophe BaillyAbstract:Five isothermal round jets at Mach number M = 0.9 and diameter-based Reynolds number ReD = 10 5 originating from a pipe Nozzle are computed by Large-Eddy Simulations to investigate the effects of initial turbulence on flow development and noise generation. In the pipe, the boundary layers are tripped in order to impose, at the Nozzle Exit, laminar mean velocity profiles of momentum thickness equal to 1.8% of the jet radius, yielding Reynolds number Re� = 900, and peak turbulence intensities of 0, 3, 6, 9 and 12% of the jet velocity. As the Nozzle-Exit turbulence level increases, the shear-layer development is strongly modified. Vortex roll-ups and pairings and, more generally, coherent structures gradually disappear, leading to lower shear-layer spreading rate and rms fluctuating velocities. The jets also develop farther downstream, resulting in longer potential cores. With rising Exit turbulence intensity, the noise levels generated by the present jets at ReD = 10 5 are moreover found to decrease, and to tend asymptotically towards the levels measured for jets at Reynolds numbers higher than 5 × 10 5 , which are likely to be initially turbulent and to emit negligible vortex-pairing noise. These results correspond well to experimental observations available in the literature, usually obtained separately for either mixing layers, jet flow or sound fields.
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influence of Nozzle Exit boundary layer conditions on the flow and acoustic fields of initially laminar jets
Journal of Fluid Mechanics, 2010Co-Authors: Christophe Bogey, Christophe BaillyAbstract:Round jets originating from a pipe Nozzle are computed by large-eddy simulations (LES) to investigate the effects of the Nozzle-Exit conditions on the flow and sound fields of initially laminar jets. The jets are at Mach number 0.9 and Reynolds number 10 5 , and exhibit Exit boundary layers characterized by Blasius velocity profiles, maximum root-mean-square (r.m.s.) axial velocity fluctuations between 0.2 and 1.9 % of the jet velocity, and momentum thicknesses varying from 0.003 to 0.023 times the jet radius. The far-field noise is determined from the LES data on a cylindrical surface by solving the acoustic equations. Jets with a thinner boundary layer develop earlier but at a slower rate, yielding longer potential cores and lower centreline turbulent intensities. Adding random pressure disturbances of low magnitude in the Nozzle also increases the potential core length and reduces peak r.m.s. radial velocity fluctuations in the shear layer. In all the jets, the shear-layer transition is dominated by vortex rolling-ups and pairings, which generate strong additional acoustic components, but also amplify the downstream-dominant low-frequency noise component when the Exit boundary layer is thick. The introduction of inlet noise however results in weaker pairings, thus spectacularly reducing their contributions to the sound field. This high sensitivity to the initial conditions is in good agreement with experimental observations.
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On the importance of specifying appropriate Nozzle-Exit conditions in jet noise prediction
Procedia Engineering, 2010Co-Authors: Christophe Bogey, Christophe BaillyAbstract:Abstract In this paper, the importance of initial conditions on subsonic jet noise is emphasized by showing numerical results obtained by large-eddy simulations for initially laminar round jets at Mach number 0.9 and Reynolds number 105. The near and the far sound pressure fields of the jets are found to significantly vary with the flow parameters, namely the boundary-layer thickness and the turbulence levels, at the Nozzle Exit. With respect to initially turbulent jets, strong additional noise components generated by pairings of coherent vortical structures in the transitional shear layers are in addition observed, in agreement with experiments.
