In this section, the effects of jet velocity and plate length on the spectral amplitude of an installed jet are addressed. This is performed by finding scaling laws for the far-field spectra for different geometric cases, using only information from the isolated jet. Those scaling laws are deemed to predict the far-field noise independently of the geometric configuration adopted for the plate, reducing the need for testing or computing several cases.

The effect of changing the plate length is shown in Fig. 9. Spectra are obtained for three surface lengths, at fixed radial position *h* = 0.8\({D}_{j}\), for θ = −90◦ and \({M}_{j}\) = 0.9. It is shown that, for longer surfaces, noise increase is higher at low frequencies, with a difference of 9 dB between the curves for *L* = 6.5\({D}_{j}\) and *L* = 2.5\({D}_{j}\), for *h* = 0.8\({D}_{j}\) and *St* = 0.11. For longer plates, the spectral peak also moves towards lower frequencies: for *h* = 0.8\({D}_{j}\) the spectral peak is at *St* = 0.15 and 0.11 for the shortest and longest plates, respectively. This is due to the increase of energy content of large-scale structures in the mixing layer in the downstream direction of the jet (Lawrence et al., 2011). Since these structures generate low-frequency hydrodynamic pressure waves, the scattering effects are also amplified in that frequency range. At low Strouhal numbers, the integral length scales of jet shear layer exhibit near constant behavior (Kerherve ́ & Fitzpatrick, 2011). Based on similarity equation for potential core lengths of jet flow (Witze, 1974), the jet length scale near the trailing edge for jet plate interaction noise prediction is given by:

$${l}_{s-i}=3\left(1-0.16{M}_{j}\right)\sqrt{\frac{L}{{D}_{j}}}{D}_{j}$$

1

where \({M}_{j}\) is jet exit Mach number,\({D}_{j}\) is nozzle diameter.

The far-field noise data in Fig. 9 show higher noise levels and a broader frequency range of amplification when the trailing edge is farther away from the nozzle exit. Based on the respective plate length for each case, the fourth power of the length scale for jet plate interaction noise is used to scale the jet plate interaction.

As shown in Fig. 10, where there is a fairly good agreement between the curves for 0.03 < Helmholtz number < 0.3. For Helmholtz number > 0.3, the curves diverge since this region of the spectra is dominated by noise from jet fine scale turbulent noise sources reflected on the surface. As a result, the sound intensity of jet plate interaction noise with the fourth power of the length scale:

$${I}_{i}\propto {l}_{s-I}^{4}$$

2

The installed far-field noise levels can be scaled with \({M}_{j}^{3}\), based on the respective jet exit velocity for each case. This is shown in Fig. 11, where there is a fairly good agreement between the curves for 0.03 < Helmholtz number < 0.3. As a result,th, sound intensity of jet plate interaction noise with the fourth power of the jet velocity:

$${I}_{i}\propto {M}_{j}^{3}$$

3

As shown in Fig. 9,it is obvious that the Strouhal number of the spectrum peak decreases with an increase of plate length. And the Helmholtz number of the spectrum peak increases with a increase of jet velocity which is shown in Fig. 11. Neither Strouhal number nor Helmholtz number can perfectly reflect the similar characteristics of jet plate interaction noise spectrum. Based on the length scale near the trailing edge for jet plate interaction noise, as shown in Formula (1), an effective Helmholtz number is given as follow:

$${He}_{eff}=\frac{f*{l}_{s-i}}{{c}_{0}+{U}_{j}}$$

4

where *f* is the frequency, \({l}_{s-i}\) is the length scale near the trailing edge for jet plate interaction noise, \({c}_{0}\) is the acoustic velocity and \({U}_{j}\) is the jet exit velocity.

The dimensionless spectrum of jet noise under different experimental conditions is shown in Fig. 12 The horizontal coordinate is the new frequency dimensionless parameter \({He}_{eff}\), and the vertical coordinate is the dimensionless power spectral density (PSD) according to Formula (2) and Formula (3). As shown in Fig. 13, there is a fairly good agreement between the curves in the frequency range 0.025 <\({He}_{eff}\)< 0.1, where jet plate interaction noise is dominant.