2.1. Measurement techniques

The wall pressure fluctuations are measured using a pair of miniaturized pressure transducers (Kulite-Mic190M) able to acquire both fluctuating and mean pressure fields and installed in a series of holes manufactured into the plate pressure side. The sensitive diameter is of approximately 3.3 mm, the size of the sensitive surface being small enough not to affect the measured spectra in the range of frequency of interest. The pressure transducers have a sensing diameter that fits the wall pressure taps and a frequency response up to 100 kHz, the unused holes were covered with tape to avoid any vortex shedding effect and cavity resonance. Data were acquired by setting the cutoff filter of the Kulite signal conditioner at 70 kHz and the sampling frequency at 200 kHz, accordingly to the Nyquist–Shannon theorem. Measurements were performed exploring different streamwise positions from $x/D=1$ up to $x/D=10$ and a schematic view of the transducers positioning is also given in figure 1.

Pointwise velocity measurements were performed using a Pitot probe with the scope of highlighting the modification of the jet flow field due to the presence of the plate. The Pitot tube has been installed on a Thorlabs 2 axis traverse system and moved in an axial direction by step of one nozzle exhaust diameter up to $x/D=10$, whilst in the radial direction the entire jet plume has been spanned by moving the probe by step of $1/6D$.

In order to provide an overall qualitative picture of the flow behaviour at the different flow regimes and different jet-plate distances, flow visualizations were performed by the application of the BOS technique.

The principles of such a non-intrusive technique, providing an estimation of the density gradient integrated along the optical path, have been extensively described in the past (Richard & Raffel Reference Richard and Raffel2001; Gojani, Kamishi & Obayashi Reference Gojani, Kamishi and Obayashi2013; Raffel Reference Raffel2015), and its applicability in the study of free jets (Clem, Zaman & Fagan Reference Clem, Zaman and Fagan2012; Clem, Brown & Fagan Reference Clem, Brown and Fagan2013) and jet-plate configurations has been assessed in the literature (De Paola et al. Reference De Paola, Di Marco, Meloni and Camussi2019; Camussi et al. Reference Camussi, Ahmad, Meloni, de Paola and Di Marco2022).

The first step of this procedure is the acquisition of a reference image generated by recording a background pattern observed through the air at rest. Thereafter, an additional exposure through the flow under investigation leads to a local displacement of the background pattern. The resulting images of both exposures can now be evaluated by correlation methods. Existing algorithms, developed and optimized for particle image velocimetry (PIV), can be used to determine the displacement of patterns at multiple locations throughout the image. In the present study, a structured background pattern was created in Matlab ambient by generating a $2000\times 2000$ size matrix of random numbers whose elements are normally distributed and then printed out as a binary image of white dots. The size of the recorded dots is kept at approximately 2–3 pixels as required by the technique in order to obtain a reliable solution. The CCD camera used for the measurements is a LaVision SX 4M with a resolution of $2360\times 1776$ pixels and equipped with a Nikon lens characterised by a focal length of 50 mm. The recording frequency was set to 10 Hz. The pattern was uniformly back illuminated by a white LED screen. To determine the image displacements a PIV cross-correlation algorithm was applied using the Davis software provided by LaVision. The interrogation window was set at a constant size corresponding to $16\times 16$ pixels with a 50 % overlap. Tests were performed by placing the camera and the back-illuminated pattern at the same distance from the nozzle axis as shown in figure 2. For each case, 50 images were acquired and ensemble averaged providing a mean pixel displacement on a grey scale that represents the dimension proportional to mean density gradient fields integrated over the line of sight. It has been checked that the related standard deviation ensured sufficient statistical accuracy for the estimation of the mean values.

Figure 2.

Flow visualization experimental set-up.

3.1. Velocity measurements

The mean velocity profiles in the radial and axial directions obtained by the Pitot measurements are reported in figure 3, comparing the isolated case with two different positions of the plate. The local mean velocity is reported in terms of the Mach number and is normalised by the Mach measured at the nozzle exhaust. The curves reported refer to the design conditions but similar results are obtained for the other NPRs and are not reported here for brevity.

Figure 3.

Normalised velocity profiles for various jet-plate configurations. (a) Axial evolution at $y/D=0$. (b) Radial profiles at $x/D=8$.

In figure 3(a) a relevant difference in the axial evolution of the mean velocity is seen in installed configurations compared with the isolated case at large $x/D$. This is hypothesized as an effect of a local acceleration ascribed to the restricted development of the jet, due to the presence of the plate. This effect was expected considering previous results in the subsonic flow regime (Meloni et al. Reference Meloni, Di Marco, Mancinelli and Camussi2019; Proenca, Lawrence & Self Reference Proenca, Lawrence and Self2019). The radial installed velocity profiles reported in figure 3(b) show a higher jet velocity in the lower shear layer (i.e. close to the plate surface) compared with the isolated case, while installed and isolated jet mean velocity curves collapse well at positive $y/D$ (opposite to the plate). This is observed more intensely at $H/D=0.75$ because of the earlier impact of the jet shear layer over the plate surface. To sum up, as for the subsonic case, the plate restricts the amount of flow entrainment causing a Coanda effect (Proenca et al. Reference Proenca, Lawrence and Self2019).

3.2. Flow visualization

Flow visualizations obtained with the BOS technique provide an overall estimation of the influence of the plate on the shock cell train. Figure 4 illustrates the results obtained in over-expanded (figure 4a,c) and under-expanded (figure 4b,d) conditions. The figures show the trend of the axial component of the displacement vector, which better reveals the shock cell characteristics.

Figure 4.

Flow visualizations at different flow velocities and different plate locations. Results are shown for (a) $H/D=2$ and ${\rm NPR}=2.1$, (b) $H/D=2$ and ${\rm NPR}=2.6$, (c) $H/D=0.75$ and ${\rm NPR}=2.1$, (d) $H/D=0.75$ and ${\rm NPR}=2.6$.

According to the literature (Ben-Dor Reference Ben-Dor2007; Arun Kumar & Rajesh Reference Arun Kumar and Rajesh2017), the cell pattern is significantly modified when moving from over-expanded to under-expanded conditions. As the NPR is raised, the shock cell train exhibits an increasingly pronounced shape and intensity. Also the length of the shock cells changes and increases when the pressure ratio is incremented (André, Castelain & Bailly Reference André, Castelain and Bailly2013). As expected, this length variation is strongly related to the structure of the shock wave reflection. In the over-expanded regime the reflected shock wave does not produce an equal deflection, and the subsequent shock occurs in a closer position than in the under-expanded case (Hadjadj, Kudryavtsev & Ivanov Reference Hadjadj, Kudryavtsev and Ivanov2004; Matsuo et al. Reference Matsuo, Setoguchi, Nagao, Alam and Kim2011). Figure 5 shows the pixel displacement trends along the jet axis ($y/D=0$) normalised with respect to its mean value in the case of over-expansion (figure 5a) and under-expansion (figure 5b), respectively. For all the NPR values, the effect of the plate is not significantly relevant. More specifically, immediately in the downstream of the jet, the shock cell length only slightly differs between the two configurations while, for increasing $x/D$, the separation between one cell and another remains mostly identical, as illustrated in figure 4 and highlighted in figure 5. It is possible to notice an acceleration of the flow due to the presence of the plate that results in a small delay of the shock cell at the nozzle exit for small $H/D$, which tends to disappear further along the axis. The proximity of the surface increases the intensity of the shocks, leading, in the over-expanded case, to a larger number of shock cells (figure 5a) visible at large $x/D$.

Figure 5.

Axial evolution of the pixel displacement at different flow velocities and different plate locations; (a) ${\rm NPR}=2.1$, (b) ${\rm NPR}=2.6$.

5.1. Training set and validation set

The data collected by means of the experimental campaign must be converted into an $N$-dimensional space suitable to the construction of the model. Specifically, for a given $n$ tuple of the radial distance of the plate $H/D$, axial distance $x/D$ and NPR, the database provides two spectra related to the SPL and the coherence $\gamma$. With the aim at modelling simultaneously spectra and coherence values for a given frequency, the following mixed-integer approach has been adopted:

(5.1)\begin{equation} \begin{bmatrix} \text{SPL}_i\\ \gamma_i \end{bmatrix} = \boldsymbol{f}\left({H/D},{x/D},\text{NPR},i_\omega\right). \end{equation}

Here $i_\omega$ is the frequency index; in this view, SPL and $\gamma$ at the $i$th frequency are the output of the single-frequency multi-point (SFMP) model. The rationale underlying the choice of a SFMP model relies, for example, on the possibility of using the network within an optimal design process, where the objective function must be evaluated thousands of times. Since the network can reproduce single- and two-point statistics with a unique function, providing simultaneously SPL and coherence for a given frequency, the complete set of network parameters (weight matrices and bias vectors) have to be loaded only once by the function evaluator. Such a problem can be translated into a network with four input neurons and two output neurons (in addition to a unknown hidden topology) and $M n_\omega$ points ($M$ being the number of experiments and $n_\omega$ the number of frequencies) are used to build the model. Since a large number of training points could have represented a computational burden, it has been decided to use the bands representation for the spectra. Moreover, to collect only the desired physical phenomena, some bands were cut both at the beginning and at the end of the spectra. Figure 11 shows how the 1/12 octave band approximation is capable of preserving all the relevant physical information related to both the broadband and tonal components, using a relatively small number of bands.

Figure 11.

Comparison between the broadband spectra and its bands representation.

It is also worth noting that, as depicted in figure 11, the tonal screech components location is determined with remarkable accuracy.

The data of the experimental campaign are therefore used to create a database consisting of 7884 elements that define the four variables ($H/D$, $x/D$, NPR and $i_\omega$) with associated two function values (SPL and $\gamma$). The dataset has been randomly shuffled and 5913 elements were used as training set $\mathcal {T}$, whereas 1971 elements were used as validation set $\mathcal {V}$.

5.2. Network architecture and training

The architecture of the model has been found by solving a minimisation problem in the design space defined by the number of layers, the number of neurons per layer and the kind of activation functions. Specifically, the maximum number of hidden layers has been set equal to three, and five activations are used for the tournament (Gaussian, swish, identity, hyperbolic tangent and cardinal sine). Each optimizer iteration consists of training the current network architecture, and is controlled by an exit rule based on the normalised root-mean-squared error (RMSE) on the training set points RMSE$_\mathcal {T}$,

(5.2)\begin{equation} \text{RMSE}_\mathcal{T}= \sqrt{\frac{1}{NM}\sum_{i=1}^M\sum_{j=1}^N\left[ \frac{f_j(\boldsymbol{x}_i)-\hat{f}_j(\boldsymbol{x}_i) }{\max(f_j)-\min(f_j)} \right]^2}, \end{equation}

with $N$ the number of modelled functions, $M$ the size of the training set $\mathcal {T}$, $\hat {f}_j(\boldsymbol {x}_i)$ the predictions and $f_j(\boldsymbol {x}_i)$ the true values (the terms $\max (f_j)$ and $\min (f_j)$ are used to normalise the error value, making it independent of the functions range). In order to avoid burdensome training of underperforming network configurations, suitable early stop criteria, based on the analysis of different performance metrics, are implemented within the optimizer. The validation error RMSE$_\mathcal {V}$ is also analysed in runtime during the training of the current architecture, and is used to stop the training in case of overfitting.

The calculations for the construction of the model were made on a computer equipped with an Intel Xeon CPU E5-2620 v4. Despite being strongly parallelizable, the optimization algorithm used here makes intentional use of a non-parallel implementation to better highlight the relationship between computing cost and desired accuracy.

For the specific problem, the RMSE$_\mathcal {T}$ target accuracy was set equal to $1.25\,\%$ (the convergence of the network response as a function of the target accuracy is presented in Appendix B), and the optimization algorithm reached the convergence value after 147 iterations, as depicted in figure 12.

Figure 12.

The RMSE$_\mathcal {T}$ as a function of the number of iterations.

Figure 12 shows that the optimizer is capable of improving the accuracy (with respect to the initial value) by an order of magnitude in a few iterations, while further improvements are achieved at great computational cost – such behaviour is similar to that of some meta-heuristic optimization algorithms. This is further confirmed by figure 13, which reports the computation time as a function of RMSE$_\mathcal {T}$.

Figure 13.

Computation time as a function of RMSE$_\mathcal {T}$ (Intel Xeon CPU E5-2620 v4).

The analysis of figure 13 shows that an accuracy of 5 % in terms of RMSE$_\mathcal {T}$ is reached in less than 5 min, whereas the optimizer drops below the $1.5\,\%$ threshold in just over an hour. To reach the RMSE$_\mathcal {T}$ required target accuracy (set equal to 1.25 %), it takes approximately 5 days. The configuration of the model obtained by means of the optimization process described above has the characteristics reported in table 2.

Table 2.

The ANN metamodel final configuration.