Abstract
This work presents analyses of high-speed schlieren images that depict the spatio-temporal structure of near-field sound in uniformly and non-uniformly heated supersonic round jets. The non-uniformly heated jet has a concentrated region of locally lower total temperature flow around the centerline of an ideally expanded jet. Compared to the uniform jet, the non-uniform jet is shown to reduce jet noise by up to 2 ± 0.5 dB in the peak narrowband sound pressure level at polar angles upstream of the peak directivity. Space-time correlations are performed on frequency-filtered time series of fluctuating schlieren image intensities, an analog for the fluctuating near-field density gradients. The effect of path integration is evaluated using synthetic schlieren of the dominant azimuthal jet modes, which are simulated using the azimuthal basis function of the Fourier transform. Hydrodynamic structures are identified at low frequencies and are shown to be modified by the thermal non-uniformity at axial locations in the near- and far-nozzle regions. The mid-frequency range is dominated by convecting Mach waves that are decorrelated in the thermally non-uniform jet in the near- and far-nozzle regions. Correlations of the high frequency content capture the emission of an acoustic beam. Results indicate the perturbations induced by the thermal non-uniformity can persist far into the developing flow field and reduce the length scale of coherent structures in regions far from the nozzle exhaust. This suggests centerline base flow changes can be optimized to reduce the acoustic efficiency of unsteady flow structures present near strong noise-producing areas such as the potential core collapse region.
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Abbreviations
- A :
-
Schlieren intensity, counts
- a :
-
Upper limit of the radial basis function (m)
- c :
-
Speed of sound (m/s)
- D :
-
Nozzle diameter (m)
- F :
-
Static thrust (N)
- f :
-
Frequency (Hz)
- \(f_\mathrm{s}\) :
-
Sampling frequency (Hz)
- \(G_{11}\) :
-
Auto-spectral density, counts\(^2\)/Hz
- \(G_{12}\) :
-
Cross-spectral density, counts\(^2\)/Hz
- \(J_m\) :
-
mth order Bessel function
- k :
-
Wave number (1/m)
- M :
-
Mach number
- \({\dot{m}}\) :
-
Mass flow rate (kg/s)
- N :
-
Record length
- \(N_{\mathrm{rec}}\) :
-
Number of records
- \(N_\mathrm{n}\) :
-
Normalization factor
- NPR:
-
Nozzle pressure ratio, \(p_0/p_{\infty }\)
- n :
-
Refractive index
- p :
-
Static pressure (pa)
- \(p_0\) :
-
Total pressure (pa)
- \(R_{11}\) :
-
Auto-correlation, counts\(^2\)
- \(R_{12}\) :
-
Cross-correlation, counts\(^2\)
- \(R_{nm}\) :
-
Radial part of the basis function
- \(\mathrm{Re}\) :
-
Reynolds number, \(\rho U D/\mu\)
- r :
-
Radius coordinate from jet centerline (m)
- SPL:
-
Sound pressure level
- T :
-
Static temperature (K)
- \(T_0\) :
-
Total temperature, K
- TTR:
-
Total temperature radio, \(T_0/T_{\infty }\)
- t :
-
Time (s)
- U :
-
Mean axial velocity (m/s)
- x :
-
Jet axial coordinate relative to exhaust (m)
- y :
-
Jet vertical coordinate relative to centerline (m)
- z :
-
Jet horizontal coordinate relative to centerline (m)
- \(\gamma\) :
-
Specific heat ratio
- \(\gamma _{12}^2\) :
-
Coherence
- \(\varDelta\) :
-
Difference between the normalized synthetic schlieren and the pressure gradient along the radius.
- \(\epsilon\) :
-
Light ray deflection on the x−y image plane
- \(\zeta\) :
-
Separation in y (m)
- \(\theta _0\) :
-
Polar angle (°)
- \(\kappa\) :
-
Gladstone dale constant (m)\(^3\)/kg
- \(\xi\) :
-
Separation in x (m)
- \(\rho\) :
-
Density (kg/m)\(^3\)
- \(\rho _{12}\) :
-
Normalized cross correlation coefficient
- \(\tau\) :
-
Separation in t (s)
- \(\tau ^*\) :
-
Non-dimensional time lag, \(\tau U/D\)
- \(\varPhi _{nm}\) :
-
Angular component of the basis function
- \(\phi\) :
-
Azimuthal angle, deg
- \(\varPsi _{nm}\) :
-
Basis function
- \(\bar{\omega ^2}\) :
-
Root mean square of the window function
- i :
-
Incident angle
- j :
-
Uniform flow condition
- m :
-
Azimuthal mode number
- n :
-
Eigenvalue
- p :
-
Primary heated flow
- s :
-
Secondary un-heated flow
- t :
-
Transmitted angle
- \(\infty\) :
-
Ambient condition
- \(E{[\cdot ]}\) :
-
Ensemble average
- \({[\cdot ]}^{*}\) :
-
Complex conjugate
- \({[\cdot ]}'\) :
-
Fluctuating component
- \({<\cdot >}\) :
-
Temporal average
- \({\mathcal{F}}{[\cdot ]}\) :
-
Discrete Fourier transform
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Funding
This work was sponsored by Navy Grants N00014-16-1-2444 and N00014-14-1-2836, which are funded by the Office of Naval Research and managed by Steven Martens.
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Daniel, K.A., Mayo, D.E., Lowe, K.T. et al. The density near-field of a non-uniformly heated supersonic jet. Exp Fluids 63, 67 (2022). https://doi.org/10.1007/s00348-022-03413-w
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DOI: https://doi.org/10.1007/s00348-022-03413-w