## 1. Introduction

The impinging jet configuration has received many considerations from researchers due to its widespread use in many industrial applications. For example, the impinging jets can be used for cooling of hot surfaces such as turbine blade (Saddington et al. [

1], Hadžiabdić and Hanjalić [

2]), as rocket engine or vertical and short take off and landing aircraft (Krothapalli et al. [

3], Cabrita et al. [

4], Saddington et al. [

5,

6], Wilke and Sesterhenn [

7]), and as torque generator in turbomachinery system.

The flow characteristics of impinging jets, despite the geometrical simplicity, are very complex and have posed challenges to numerical simulations, particularly for turbulence modelling (Hadžiabdić and Hanjalić [

2]). The jet flow configurations discussed in this paper are under-expanded, subsonic and supersonic free jets, i.e., jets exhaust in a quiet medium, and jets impinging on a solid surface, that are commonly found in aerospace engineering applications and turbomachinery systems. An under-expanded jet may occur when the jet nozzle releases a fluid at a pressure greater than the ambient pressure. A comprehensive review of free under-expanded jets can be seen in Franquet et al. [

8]. The flow configuration of supersonic impinging jet has rich and complex flow structures (Henderson [

9]) that are originated from the compressibility and turbulent flows (Weightman et al. [

10]). Besides, the supersonic impinging jet is a highly resonant flowfied that is governed by a well-known aeroacoustics feedback loop (Weightman et al. [

10], Uzun et al. [

11], Akamine et al. [

12]). The feedback loop initiates as instability waves in the jet shear layer, then grow into large-scale vortices as they travel downstream (Brown and Roshko [

13], Tam and Ahuja [

14], Henderson and Powell [

15]). The impact of the large-scale vortices on the solid surface produces pressure fluctuations and acoustic waves propagating upstream (Krothapalli et al. [

3], Henderson [

16], Henderson et al. [

17], Sinibaldi et al. [

18], Gojon et al. [

19]). These upstream propagated waves reach the nozzle outlet and generate the instability waves, closing the feedback loop (Henderson and Powell [

15], Ho and Nosseir [

20], Nosseir and Ho [

21]).

In the current communication, we experimentally study the under-expanded turbulent impinging jets featuring in many important applications of turbomachinery and aerospace engineering. Particularly, the test model is a convergence-divergence nozzle, which is typically installed in turbomachinery systems of many power plants in United States. When in operation, the nozzle will exhaust gas (air and/or steam) to the turbine blades and induce rotations for the turbomachinery shaft. It should be noted that at this stage, the flow configuration presented in this paper is unheated, under-expanded air jets impinging on a flat surface that can be considered as a simplified version of the practical system. Nevertheless, it is still a valuable benchmark to gain a proper understanding of the flow mixing between the subsonic and supersonic jet flows and an ambient surrounding fluid with and without the presence of the impinging surface.

Researchers at Texas A&M University have conducted isothermal velocity measurements of subsonic and supersonic under-expanded free jets and impinging jets. The general purpose of these tests is to perform high-spatial resolution measurements of the velocity fields for different flow configurations of under-expanded jets. The experimental activities provide an experimental database of velocity measurements suitable for validating system-level codes and developing the computational fluid dynamics (CFD) models that are currently considered for subsonic and supersonic jet flows.

The objective of our study is to investigate the flow characteristics of under-expanded turbulent jets impinging on a solid surface with various nozzle pressure ratios (NPRs) ranging from 2 to

$2.77$, and different values of nozzle-to-plate gaps. These spatial distances are

$e=10$ (mm), 20 (mm), and 30 (mm), that are typical in the practical turbomachinery systems. The resulted non-dimensional gap spacings, defined as

$e/{D}_{j}$ (

${D}_{j}$ is the jet hydraulic diameter), are

$0.82$,

$1.64$, and

$2.46$, respectively. The flowfield characteristics in the central plane of the test nozzle and near the impingement surface are obtained by using the two-dimensional two-component (2D2C) particle image velocimetry (PIV) technique. The PIV measurements will provide the full-field flow characteristics of the flow mixing between the under-expanded free jets, impinging jets and surrounding areas. The obtained experimental results at high-spatial resolutions can be then used for validation of theoretical and CFD models, and for providing technical supports to the system design. The structure of this paper is as followed. The experimental facility of impinging jet configuration and PIV experimental setup to acquire the velocity are presented in

Section 2. From the obtained PIV velocity vector fields corresponding to various spatial gaps and NPRs, statistical results including the first- and second-order flow statistics, such as mean velocity, root-mean-square fluctuating velocity and Reynolds stress, are discussed in

Section 3. Effects of the impingement surface and NPRs to the flow patterns and comparisons of statistical profiles are discussed in

Section 3.2. Finally, proper orthogonal decomposition (POD) analysis is applied to reveal the statistically dominant flow structures that play important roles to the flow dynamics and acoustic characteristics of impinging jets. Results from the velocity decomposition are discussed in

Section 4, followed by the conclusions in

Section 5.

## 4. Proper Orthogonal Decomposition Analysis to the Free Jet and Impinging Jet Flows

This section describes the proper orthogonal decomposition (POD) analysis of the 2D2C PIV velocity snapshots using to extract the dominant flow structures that play important roles in the flow dynamics and acoustic characteristics of under-expanded free jets and impinging jets. The velocity vector fields were obtained from the flow measurements of free jets and impinging jets for various values of NPRs and nozzle-to-plate distances.

Lumley [

44] introduced POD, i.e., also called or Karhunen-Loéve decomposition, into turbulence flow studies to identify statistically dominant flow features (coherent structures) in acquired experimental data and numerical simulations. For a given flow, the velocity field

$\mathit{u}(\mathit{x})$ is decomposed into a set of spatially orthogonal modes and a set of temporal coefficients, which vary, respectively, only in space and time (Nguyen et al. [

45]). The POD modes extracted from the velocity vector fields yield an optimal representation of the flow field such that, for any given number of modes, the two-norm of the truncation error between the original velocity data and the projection of the original velocity data onto such modes is minimized (Berkooz et al. [

46], Holmes et al. [

47]). In addition, the original velocity fields can be approximated or reconstructed using the few lowest-order POD modes that capture the highest amount of flow kinetic energy and the associated temporal coefficients (Nguyen et al. [

48]). Detailed descriptions of the POD analysis can be reviewed in Berkooz et al. [

46], Holmes et al. [

47], and Sirovich and Kirby [

49]. A brief review of the snapshot POD of the velocity fields is provided here. A POD analysis of a given velocity vector field

$\mathit{u}(\mathit{x},0\le t\le T)$ (

T is a finite time direction) can be described as

where

N is the number of velocity snapshots,

${\zeta}_{k}(t)$ and

$\mathbf{\psi}(\mathit{x})$ are the POD temporal coefficients and POD basis functions, respectively.

$\mathbf{\psi}(\mathit{x})$ are the eigenfunctions of a two-point correlation matrix

$\mathit{R}(\mathit{x},{\mathit{x}}^{\prime})$ defined as

It is common that velocity vector fields obtained by experiments and numerical simulations are discrete, the snapshot POD (Sirovich and Kirby [

49]) is usually used. In the current study, we apply the snapshot POD analysis to the collections of 2D2C PIV velocity vector fields obtained from PIV measurements of free jets and impinging jets corresponding to various values of nozzle-to-plate distances

e and NPRs. First, a correlation matrix is defined as

and the POD temporal coefficients and POD basis functions are computed as

In the above equations, coefficients

${\alpha}_{ki}$ are defined as

where

${\upsilon}_{i}^{k}$ is the

ith element of the eigenvector

${\upsilon}^{k}$ associated to the eigenvalue

${\lambda}_{k}$ of the matrix

$\mathit{C}$. The correlation matrix

$\mathit{C}$ is built from instantaneous velocity snapshots, therefore, a derived eigenvalue

${\lambda}_{k}$ associated with a POD mode

k represents the flow kinetic energy contained by that mode.

In our POD calculation, the eigenvalue

$\lambda $ associated with each POD mode is proportional to the kinetic energy contained in that mode. The decomposition yields statistically dominant flow structures in the few lowest-order POD modes. These modes capture most of the flow’s kinetic energy and are typically associated with large-scale structures. Besides, the POD basis functions computed from these velocity snapshots yield an optimal representation of the flow field in the sense that, for any given number of basis functions, the Hilbert norm of the truncation error between the original velocity data and the projection of the original velocity data onto these basis functions is minimized. The present PIV setup allows us to capture several hundred instantaneous velocity fields. A set of few hundreds PIV realizations suffice for a POD analysis to reveal the statistically dominant structures of the flow (Nguyen et al. [

24]).

Figure 10,

Figure 11,

Figure 12 and

Figure 13 illustrate results from the POD analysis of velocity fields obtained from 2D2C PIV measurements of free jets and impinging jets for various nozzle-to-plate distances of

$e=30$ mm, 20 mm, and 10 mm, respectively. The results are also presented for various values of NPRs.

Figure 10a,b,

Figure 11a,b,

Figure 12a,b and

Figure 13a,b show the energy spectra and the cumulative kinetic energy computed from the POD analysis of PIV instantaneous velocity vector fields for

$NP{R}_{1}$,

$NP{R}_{2}$,

$NP{R}_{3}$, and

$NP{R}_{4}$. In addition, the kinetic energy fractions contained in low-order POD modes 1–4, i.e.,

${\mathsf{\Psi}}_{1}$–

${\mathsf{\Psi}}_{4}$, are listed in

Table 2 for different nozzle-to-plate distances

e and for values of

$NP{R}_{1}$,

$NP{R}_{2}$,

$NP{R}_{3}$, and

$NP{R}_{4}$.

In all the POD analysis to the PIV velocity fields of under-expanded free jets and impinging jets, the kinetic energy fractions contained in the first POD modes were found to increase when the NPRs increased. It is noted that the first POD modes are considered approximately equivalent to the time-averaged velocity fields and such observations can be confirmed in the representation of POD Mode 1 displayed in

Figure 10c,

Figure 11c,

Figure 12c and

Figure 13c. For instance, in the POD analysis of impinging jets with

$e=30$ mm (

$e/{D}_{j}=2.46$), the kinetic energy in the first POD modes increased from

$84.5\%$ to

$90.4\%$ when NPRs increased from 2 to

$2.77$. In comparisons among the first POD modes of the velocity decomposition for the free jets and impinging jets with different spatial gaps

e, it is seen that for the same values of NPRs, POD Mode 1 of the under-expanded free jets had the highest energy fractions. Besides, it is found that the POD Mode 1 captured lower levels of the flow kinetic energy when the spatial gaps

e reduced from 30 mm to 10 mm. For all the POD velocity decompositions of under-expanded free jets and impinging jets, the kinetic energy levels of the low-order POD modes 2, 3, and 4 were less than

$3\%$. This observation indicates that the flow fields of the free jets and impinging jets in this study are highly turbulent and the flow kinetic energy is widely distributed over many flow-structures whose scales are smaller than the time-averaged flows.

For the under-expanded free jets, the total flow kinetic energy levels contained in the first 100 low-order POD modes varied between

$96.4\%$ and

$98.3\%$ when NPRs increased from

$NP{R}_{1}=2$ to

$NP{R}_{4}=2.77$. Analogously, for the under-expanded impinging jets, these values ranged from

$94.5\%$ to

$96.1\%$ with

$e=30$ mm, from

$95.4\%$ to

$96.7\%$ with

$e=20$ mm, and from

$94.6\%$ to

$96.17\%$ with

$e=10$ mm when

$NPRs$ increased from 2 to

$2.77$, respectively. Sirovich and Kirby [

49] suggested a

$99\%$ of total flow energy as a cutoff to accurately represent the flowfield, while Palacios et al. [

50] discussed that a

$75\%$ of total flow energy could be sufficient for a reasonable representation of the system. Moreno et al. [

51] applied POD analysis to PIV velocity vectors obtained from experimental measurements of supersonic rectangular convergence-divergence jet at a Mach number of

$1.44$. The authors quantified the accuracy of low-order flow reconstruction using the first two POD modes capturing

$90\%$ of the total flow energy. They reported the mean square error of less than

$1.2\%$, thus concluded that the reconstructed flowfield using two modes is a good representation of the flowfield (Moreno et al. [

51]). In the current study, it is found that for the under-expanded free jets, the total flow kinetic energy captured by the first two POD modes are

$87.15\%$,

$89.92\%$,

$92.6\%$, and

$94.95\%$ corresponding to

$NP{R}_{1}=2$,

$NP{R}_{2}=2.2$,

$NP{R}_{3}=2.5$, and

$NP{R}_{4}=2.77$, respectively. These values are comparable to those reported in the study of Moreno et al. [

51].

Figure 10c–f,

Figure 11c–f,

Figure 12c–f and

Figure 13c–f show the in-plane components of the low-order POD modes 1, 2, 3, and 4 extracted from the POD velocity decomposition for the under-expanded free jets and impinging jets with

$e=30$ mm, 20 mm, and 10 mm, respectively. In these figures, results from the POD analysis of velocity fields for various

$NPRs$ are also illustrated for comparisons. It is noted that in the current POD velocity decomposition, the POD spatial functions, i.e., POD modes, were non-dimensional.

It is obviously seen that the first POD modes displayed flow structures that are statistically similar to the time-averaged velocity fields for all the studied cases. For the specific values of the spatial gap e and NPR, lower-order POD modes illustrated the statistically dominant flow structures with larger shapes and sizes compared with those structures depicted by the higher-order POD modes. In addition, for a specific value of e, while the spatial flow structures illustrated by POD mode 1 are similar when NPRs increased, the dominant flow structures revealed by POD modes 2, 3, and 4 were not entirely analogous. One may also find that for results from the POD analysis to velocity fields of $NP{R}_{1}=2$ and $NP{R}_{2}=2.2$, the extracted dominant flow structures are the same, while those extracted from the POD analysis to velocity fields of $NP{R}_{3}=2.5$ and $NP{R}_{4}=2.77$ are quite similar. This indicates the differences in large-scale flow structures and transition of energy-contained eddies when the under-expanded free jet and impinging jet flows have undergone from near sonic to supersonic conditions.

It is noticed that the presence of stand-off shock is captured in the visualization of POD mode 2, especially for the value of

$NP{R}_{4}=2.77$. Additionally, one may find that the POD modes 2, 3, and 4 obtained for lower

$NPR$ values displayed the large-scale flow structures with considerable large shapes and sizes compared to the nozzle diameter

${D}_{j}$ and spatial gaps

e. Furthermore, spatial locations of these structures are found to distribute within the jet core and the inner sides of the shear layers. On the other hand, for higher values of NPRs, the statistically dominant flow structures depicted by the low-order POD modes 2, 3, and 4 were likely the resemblances of stand-off shock near the impingement surface and the jet shear layers. Such observations confirmed the discussions on the characteristics of flow fields and acoustics of the supersonic impinging jets in many of previous studies, for examples studies of Krothapalli et al. [

3], Tam and Ahuja [

14], Henderson and Powell [

15], Henderson [

16], Sinibaldi et al. [

18], Ho and Nosseir [

20], Nosseir and Ho [

21], Wilke and Sesterhenn [

52], and Wilke and Sesterhenn [

7], to name a few. In their experimental and numerical investigations, these authors studied and described the mechanism of acoustic generation as a feedback loop in supersonic flows. For instance, Tam and Ahuja [

14] and Henderson and Powell [

15] discussed that the instability waves appeared in the jet shear layers supplied the energy for the feedback loop. Generated by acoustic excitation in the vicinity of the nozzle outlet, these waves propagate further downstream and grow in size as the large-scale flow structures that are captured in flow visualizations of Krothapalli et al. [

3], and in our experimental velocity vector fields. Moreover, Krothapalli et al. [

3], Henderson [

16] and Sinibaldi et al. [

18] suggested that when these large-scale vortical structures impinging on the solid wall, they generate coherent pressure fluctuations, yielding acoustic waves at significant strengths. Later, the resulted acoustic waves travel in the upstream direction and eventually reach the nozzle outlet. Such interactions excite the jet shear layers and cause the generation of instability waves, thus close the feedback loop (Krothapalli et al. [

3]). Using direct numerical simulations of subsonic and supersonic impinging jets, Wilke and Sesterhenn [

52] and Wilke and Sesterhenn [

7] have intensively shown that primary vortices are generated in the jet shear layers and initially transported with the flows. The impact of the jet shear layers to the solid surface generates new vortices (secondary vortices) that pair with the primary ones. Wilke and Sesterhenn [

53] also discussed that the formation of primary and secondary vortices is a periodical phenomenon associated with a characteristic frequency. The flow mode can be distorted due to interactions between large-scale flow structures in the jet and shear layers, and the feedback waves resulted from the impingement.

Although the current 2D2C PIV measurements were not able to provide the temporal evolution of the large-scale structures, their existences can be confirmed in

Figure 10,

Figure 11,

Figure 12 and

Figure 13 as the results of velocity decomposition via POD analysis. The results from the POD velocity decomposition have revealed the appearances of statistically large-scale flow structures in the regions of stand-off shock, i.e., in the vicinity of impingement surface, and in the regions of jet shear layers. It is found that the first POD modes representing the mean flow fields of the free jets captured higher kinetic energy levels than those of the jet impinging on the solid surface. However, the flow kinetic energy levels contained in the low-order POD modes, presenting the coherent large-scale structures, increased when the nozzle-to-plate distances decreased. The statistically dominant large-scale structures with considerable sizes and shapes are found within regions between the jet core and shear layers for lower NPRs. However, for higher NPRs, the large-scale structures are resemblances of the stand-off shock near the solid surface and the jet shear layers.

It is noted that in this study, the velocity snapshots were obtained from PIV measurements with a sampling rate of 15 Hz, facilitating the POD analysis to the statistically independent snapshots to form the POD spatial basis functions. The current experimental setup, however, is not able to provide transient behavior of the flowfield and acoustic characteristics of under-expanded supersonic free jets and impinging jets. Even at moderate supersonic speeds, it is still a challenge for current time-resolved PIV systems to obtain large enough numbers of velocity snapshots that could enable low-dimensional analysis (Berry et al. [

54]). An experimental setup combining a PIV system for velocity measurement and microphone system for acoustic measurement could provide details about the flowfield and noise characteristics of supersonic free jets and impinging jets. Experimental measurements of flowfield and acoustics as can be reviewed in numerous studies of Krothapalli et al. [

3], Henderson [

16], Henderson et al. [

17], Sinibaldi et al. [

18], Alvi and Iyer [

43], and Guariglia et al. [

55], to name a few. With the recent improvement in digital imaging cameras, high-speed schlieren is enable to acquire large datasets of time-resolved flowfield information (Berry et al. [

56]), although the measured quantities are scalars derived from density gradients (Berry et al. [

54]). The statistical dominant flow structures and dynamical evolution of large-scale flow structures can be extracted from the time-resolved schlieren images or from the time-resolved velocity fields obtained from large-eddy simulations using POD (Weightman et al. [

10], Berry et al. [

54], Nair et al. [

57], Berry et al. [

58], Weightman et al. [

59]), spectral POD (Karami and Soria [

60]), and dynamic mode decomposition (DMD) (Berry et al. [

56]) techniques.