3.1. Spatio-temporal reconstruction of time-averaged velocity fields

The comparison of the time-averaged flow fields between the reconstructed results and the PIV reference is presented in figure 4. It can be seen that the $\times$ 4, $\times$ 8 and $\times$ 16 SR cases are able to accurately reconstruct the distribution and magnitude of both $U^*$ and $V^*$ components, successfully capturing the jet spreading and velocity decay along the streamwise direction. This is because time averaging substantially suppresses the small-scale coherent vortical structures, reducing instantaneous fluctuation effects and enhancing the large-scale flow features, which allows even the $\times$ 16 case to recover the main jet characteristics effectively. However, as the SR factor increases, the reconstruction accuracy gradually decreases, particularly in regions close to the plasma actuator ( $x^* \lt 5$ in figure 4 b). The corresponding Reynolds stress fields $-\overline {uv}$ are shown in figure 4(c). The $\times$ 4 and $\times$ 8 cases successfully reconstruct the primary stress distribution, including the strong dipole structures near the plasma actuator and the extended high-shear regions farther downstream. These accurately captured stress fields reflect the models’ ability to reconstruct essential turbulent momentum transport, which is critical for jet spreading and cross-stream mixing processes. In contrast, although the $\times$ 16 case captures the general shape of the stress field, it exhibits more pronounced discrepancies in both spatial continuity and stress intensity. This can be attributed to its limited capability to recover fluctuating velocity components.

Figure 4.

Comparison of time-averaged flow fields between the reconstructed results and the PIV reference: (a) streamwise velocity $U^*$ , (b) vertical velocity $V^*$ and (c) Reynolds stress $-\overline {uv}$ . The white lines in the upper-left panel illustrate the jet width $w_{\!j}$ .

The comparison of the time-averaged maximum streamwise velocity $U_m$ along the jet direction between the reconstructed fields and the PIV reference is presented in figure 5(a). All SR cases closely follow the PIV reference, accurately capturing the rapid acceleration in the near field ( $x^* \lt 10$ ), the gradual decay downstream ( $x^* \gt 15$ ) and finally approaching a plateau. This demonstrates that the model can successfully recover the core jet momentum distribution even at high SR factors. Given that plasma jets are primarily used for active flow control (Corke, Enloe & Wilkinson Reference Corke, Enloe and Wilkinson2010; Wang et al. Reference Wang, Choi, Feng, Jukes and Whalley2013), the plasma-induced forcing plays a critical role in determining control effectiveness. Therefore, the plasma thrust per unit width $M_x$ (Jukes & Choi Reference Jukes and Choi2009), defined as

(3.1) \begin{equation} M_x = \int \rho U^{*2} \mathrm{d}y^*, \end{equation}

where $\rho$ is the air density, is analysed along the streamwise direction, as shown in figure 5(b). It can be seen that the proposed model accurately reconstructs the jet forcing at all SR factors, demonstrating its potential for evaluating plasma jet control performance. Furthermore, the jet entrainment coefficient $\mu$ (Hussein, Capp & George Reference Hussein, Capp and George1994), defined as

(3.2) \begin{equation} \mu = \dfrac {\int _{-\infty }^{\infty } \dfrac {U^*}{U_m} \dfrac {y^*}{x^*} \mathrm{d} \dfrac {y^*}{x^*} }{ \sqrt {\int _{-\infty }^{\infty } \dfrac {U^{*2}}{U_m^2} \dfrac {y^*}{x^*} \mathrm{d} \dfrac {y^*}{x^*}} }, \end{equation}

is computed and presented in figure 5(c). Similarly, all SR cases successfully capture the evolution of $\mu$ , confirming the superior capability of the proposed spatio-temporal SR model in accurately reconstructing time-averaged jet flow fields.

Figure 5.

Quantitative comparison of time-averaged jets reconstructed by different SR cases and the PIV reference: (a) streamwise variation of the maximum velocity $U_m$ , (b) plasma thrust per unit width $M_x$ and (c) jet entrainment coefficient $\mu$ .

Figure 6.

Comparison of the vertical spreading of the plasma jet: (a) streamwise evolution of the jet width $w_{\!j}$ and (b) normalised streamwise velocity profiles at $x^* = 5$ .

The vertical spreading behaviour of the plasma jet is further evaluated and compared in figure 6. The jet width $w_{\!j}$ is defined at the location where $U^* = 0.5 U_m(x^*)$ , as illustrated in figure 4. Due to the symmetric arrangement of the plasma actuators on both sides of the flat plate, a step-like separation region forms near the plasma actuators (figure 4 a). Therefore, the jet width is considered only for $x^* \gt 2$ to avoid initial separation effects. It is evident that all SR cases accurately capture both the initial jet contraction induced by plate effects and the downstream jet spreading driven by enhanced vertical momentum transfer due to vortex breakdown. The reconstructed jet widths are in excellent agreement with the PIV reference data in the entire range. Moreover, for symmetric jets, the vertical velocity profiles are expected to exhibit self-similar behaviour (Hussein et al. Reference Hussein, Capp and George1994). In this analysis, the velocity is normalised by the local maximum velocity $U_m$ and the $y$ -coordinate is non-dimensionalised using both the centreline position $y_m$ and the jet width $w_{\!j}$ . Theoretically, the peak streamwise velocity should appear at the centreline ( $y^* = 0$ ). However, slight random lateral displacements of the peak are observed in plasma jets, a phenomenon also reported in studies of plasma synthetic jets (Zong & Kotsonis Reference Zong and Kotsonis2018). Figure 6(b) shows the normalised streamwise velocity profiles at $x^* = 5$ . Profiles at other streamwise positions show similar trends and are thus not presented here for clarity. The $\times$ 4 and $\times$ 8 SR cases produce velocity profiles nearly identical to the PIV reference, demonstrating a Gaussian-like distribution. In contrast, the $\times$ 16 case maintains good agreement only within $-1 \lt (y^*-y_m)/w_{\!j} \lt 1$ , while larger reconstruction errors are observed outside this core region, especially near the edges of the PIV measurement domain, where the velocities are much lower compared with the jet core.

3.2. Spatio-temporal reconstruction of instantaneous velocity fields

The model-reconstructed instantaneous streamwise and vertical velocities over 10 pulses are compared with the reference PIV results in figure 7, at different vertical locations with $y^* = 0$ and 4. At $x^* = 5$ , where the flow is dominated by the shear layer flow at $y^* = 0$ and periodic coherent vortices at $y^* = 4$ , all reconstructed curves closely match the PIV reference data, accurately capturing the primary periodic oscillations and amplitude variations. Only minor deviations are observed in the $\times$ 16 case for the $V^*$ -component at $y^* = 0$ and the $U^*$ -component at $y^* = 4$ , primarily manifested as an underestimation of local peaks, reflecting the difficulty in reconstructing fine-scale structures from highly downsampled inputs. As $x^*$ increases to 20 and 35, corresponding to the transition and breakdown regions, the flow exhibits more irregular temporal fluctuations due to vortex deformation and increased turbulence. Despite the presence of small-scale fluctuations, the overall temporal trends of both velocity components remain well reconstructed. Noticeable discrepancies still appear in the $\times$ 16 case, which can be attributed to the limited information in the highly downsampled inputs, making it challenging to accurately recover small-scale unsteady features. Overall, the two-component instantaneous velocities are successfully reconstructed within both large-scale and small-scale vortex regions. However, the reconstruction accuracy naturally decreases as the downsampling factor increases.

Figure 7.

Comparison of reconstructed and PIV-measured HR instantaneous velocities over 10 pulses at different vertical locations for (a) $y^* = 0$ and (b) $y^* = 4$ . The left and right columns correspond to the streamwise velocity $U^*$ and the vertical velocity $V^*$ , respectively. The sampling location in the flow field is indicated at the top.

Figure 8 compares the instantaneous streamwise and vertical velocity peaks obtained from the model SR reconstructions and the PIV reference data, where $\hat {U}$ and $\hat {V}$ denote the reconstructed velocities. The probability density of different velocity magnitudes is indicated by the colour scale. For the $\times$ 4 case (figure 8 a), most peak velocities and the high-density region align closely along the diagonal line, indicating excellent agreement between the reconstructed and PIV reference velocities for both components. In the $\times$ 8 case (figure 8 b), the peak velocities remain concentrated near the diagonal, though a slight upward deviation is observed at higher velocity magnitudes, suggesting a minor underestimation of peak values by the model. For the $\times$ 16 case (figure 8 c), the velocities exhibit a more noticeable spread from the diagonal, although the overall trend is still well captured. This increased dispersion indicates reduced accuracy in reconstructing fine-scale velocity peaks, consistent with the temporal velocity variations observed in figure 7.

Figure 8.

Comparison of the instantaneous streamwise and vertical velocity peaks obtained from the model SR reconstructions and the PIV reference data: (a) $\times$ 4, (b) $\times$ 8 and (c) $\times$ 16. The colour scale indicates the probability density of different velocity magnitudes.

The mean squared error (MSE) $\epsilon$ between the reconstructed velocities and the PIV reference data is calculated to assess reconstruction accuracy, defined as

(3.3) \begin{equation} \epsilon = \frac {1}{N} \sum _{i=1}^{N} \big(\hat {U}_i - U^*_i\big)^2, \end{equation}

where $N$ represents the total number of grid points used for averaging. Figure 9 shows the MSE results for the streamwise and vertical velocities under different SR factors ( $\times$ 4, $\times$ 8 and $\times$ 16). Here, $n=1$ and $n=5$ correspond to spatial SR cases with available LR spatial inputs, whereas $n=2$ to $n=4$ represent purely temporal SR cases without spatial input information. For each sequence index $n$ , $\epsilon$ values are averaged over all periodic samples sharing the same sequence position. A lower SR factor yields the smallest $\epsilon$ , indicating more accurate reconstruction from the LR flow fields. As the SR factor increases, $\epsilon$ progressively rises, becoming particularly pronounced in the $\times$ 16 case, suggesting substantial degradation in reconstruction accuracy due to the severely limited information content in highly downsampled inputs. The spatial SR cases ( $n=1$ and $n=5$ ) consistently exhibit lower errors compared with the purely temporal SR cases ( $n=2$ to $n=4$ ). The reconstruction error increases with longer temporal distances from the samples with LR spatial input, reaching a maximum at $n=3$ , which is consistent with the findings of Fukami et al. (Reference Fukami, Fukagata and Taira2021a ). This is attributed to reduced temporal correlation, which leads to increasing difficulty in accurately reconstructing flow structures without spatial information. Both velocity components show similar $\epsilon$ trends; however, the vertical velocity component exhibits higher errors. This may be due to its overall smaller magnitude compared with the streamwise component, as shown in figure 8.

Figure 9.

Mean squared error $\epsilon$ between the model-reconstructed velocities and the PIV reference data under different SR factors ( $\times$ 4, $\times$ 8 and $\times$ 16): (a) streamwise velocity and (b) vertical velocity. Here, $n=1$ and $n=5$ represent the frame with LR spatial inputs, while $n=2$ to $n=4$ indicate purely temporal SR without spatial information.

To further evaluate the model performance in reconstructing velocity gradients, figure 10 compares the vorticity $\omega$ distributions at an arbitrarily selected time instant $\tau ^* = 40.6$ . The HR PIV reference is shown in the top-right corner, together with results obtained from the CNN-based SR model (figure 10 b) and from the proposed model (figure 10 c) at different input downsampling factors. Compared with the PIV reference, the GAN-based model reconstructed vorticity fields successfully capture the primary vortex cores and fine-scale structures, even at the $\times$ 16 downsampling input. This is further confirmed by the absolute error maps between the reconstructed and reference vorticity fields shown in figure 10(d), which indicate acceptable deviations mainly accumulating farther downstream ( $x^*\gt 20$ ). In contrast, the CNN-based reconstructions can only accurately capture the primary vortex structure near $x^* = 5$ for all factors, while substantially missing the secondary vortices around the primary cores and the downstream small-scale dynamics, leading to large discrepancies from the measured PIV fields. This issue of over-smoothing has also been reported by Ozawa et al. (Reference Ozawa, Honda and Nonomura2024) and Nista et al. (Reference Nista, Pitsch, Schumann, Bode, Grenga, MacArt and Attili2024). The GAN-based reconstructions, by comparison, preserve the physical details of the flow remarkably well. In the $\times$ 4 and $\times$ 8 cases, both the primary vortex cores, secondary vortices and small-scale features generated during the vortex evolution are well recovered, in terms of both spatial structure and vorticity intensity. Even in the $\times$ 16 case, although some reduction in accuracy for small-scale vortices is observed, the essential multi-scale vortex structures remain identifiable.

Figure 10.

Comparison of instantaneous vorticity fields $\omega$ at $\tau ^* = 40.6$ : (a) LR input fields at different downsampling factors, (b) HR fields obtained via the CNN-based SR model from Fukami et al. (Reference Fukami, Fukagata and Taira2019), (c) reconstructed HR fields from the proposed model and (d) the absolute error between the proposed model-reconstructed and reference vorticity field. The HR PIV reference is shown in the top-right corner.

The POD is adopted to evaluate the model’s capability in reconstructing large coherent vortical structures. Figure 11 presents the first three POD modes obtained from the reconstructed and PIV reference velocity fields, calculated using the streamwise and vertical fluctuation velocities $u$ and $v$ . Modes 1 and 2, shown in figures 11(a) and 11(b), represent the most energetic and dominant coherent structures in the flow. At all SR factors ( $\times$ 4, $\times$ 8 and $\times$ 16), the spatio-temporal SR flow fields successfully capture the primary large-scale vortical patterns with high fidelity. In the $\times$ 4 and $\times$ 8 cases, these coherent structures closely match the PIV reference in both spatial distribution and intensity. Even for the $\times$ 16 case, the essential large-scale features remain evident, although some attenuation and spatial distortion appear, particularly in the far downstream region ( $x^* \gt 20$ ). Mode 3, presented in figure 11(c), characterises finer-scale and higher-order fluctuation structures. As the SR factor increases, the reconstructed mode 3 increasingly deviates from the PIV reference. While the $\times$ 4 case retains a reasonable level of agreement, the $\times$ 8 and especially the $\times$ 16 cases show noticeable discrepancies, reflecting the model’s limited ability to capture high-order, small-scale flow structures.

Figure 11.

Comparison of the first three POD modes calculated from the model SR reconstructions and the PIV reference data: (a) mode 1, (b) mode 2 and (c) mode 3.

To further quantify the reconstruction performance in mode space, figure 12 compares the POD energy distributions and time coefficients between the model reconstructions and the PIV reference data. Considering the negligible energy content of higher-order modes, figure 12(a) only shows the normalised energy distribution of the first 20 POD modes, where each mode energy is calculated as $\lambda _j / \varSigma \lambda _j$ , with $\lambda _j$ denoting the eigenvalue corresponding to the $j$ th mode. It can be seen that more than 60 % of the total energy is concentrated in the first two modes. The SR reconstructions with factors $\times$ 4 and $\times$ 8 closely follow the PIV reference results, even in the higher-order modes, accurately preserving the dominant coherent structures. In the $\times$ 16 case, the overall energy distribution remains similar to that of the PIV reference; however, slight deviations are observed, mainly due to the loss of small-scale coherent structures. Figure 12(b) compares the temporal coefficients of the first two POD modes, $a_1$ and $a_2$ , which are computed by projecting the velocity field vector $\textbf{U} = (U^*, V^*)$ onto the corresponding POD spatial modes $\boldsymbol{\psi }_j = (\psi _j^U, \psi _j^V)$ (Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993), defined as

(3.4) \begin{equation} a_j(\tau ^*) = \iint \textbf{U}(x^*, y^*, \tau ^*) \boldsymbol{\cdot }\boldsymbol{\psi }_j(x^*, y^*) \mathrm{d}x^* \mathrm{d}y^*, \end{equation}

where $\psi _j^U$ and $\psi _j^V$ are the $j$ th POD mode components in the streamwise and vertical directions, respectively. For clarity, only one-fifth of the results are shown. The nearly circular shape of the $a_1$ – $a_2$ trajectory indicates the periodic evolution of the dominant coherent structures. The $\times$ 4 and $\times$ 8 SR reconstructions show excellent agreement with the PIV reference, exhibiting almost overlapping distributions. The $\times$ 16 case also captures the main flow features but shows noticeable deviations, reflecting higher reconstruction errors and reduced phase accuracy at higher downsampling factors. Similar results are observed for the higher-order temporal coefficients of the POD modes, as shown in figure 12(c) for $a_3$ and $a_4$ , and in figure 12(d) for $a_5$ and $a_6$ . The $\times$ 4 and $\times$ 8 SR reconstructions accurately capture these higher-order, small-scale structures, with temporal coefficients that almost overlap with those of the reference field. In contrast, the $\times$ 16 case exhibits noticeable reconstruction errors and reduced fidelity in capturing the finer-scale dynamics, highlighting its limitations in reconstructing high-order flow features.

Figure 12.

Comparison of POD results between the PIV reference data and model reconstructions at different SR factors: (a) normalised energy distribution of the first 20 POD modes, (b) phase-space trajectories of the first two temporal coefficients $a_1$ and $a_2$ , (c) phase-space trajectories of the third and fourth temporal coefficients $a_3$ and $a_4$ and (d) phase-space trajectories of the fifth and sixth temporal coefficients $a_5$ and $a_6$ .

For the reconstruction of instantaneous velocity fields, turbulent channel flow is examined in addition to the plasma-induced flow to evaluate the robustness and applicability of the proposed model in different flow topologies. The reconstructed results at various SR factors, compared with the reference DNS, are presented in Appendix A. Although the $\times$ 16 cases exhibit noticeable discrepancies, the results still highlight the potential of the proposed model for applications in a wide range of fluid dynamics.

3.3. Spatio-temporal reconstruction of phase-averaged velocity fields

To compare the evolution of phase-averaged vortical structures, the fast Fourier transform is applied to extract the characteristic frequencies of the flow field before defining its phase information. The total global power spectrum is calculated from the fluctuation velocities $u$ and $v$ and defined as

(3.5) \begin{equation} P_{\textit{total}} = \sqrt { \left [ 2 \left | \frac {1}{N_t} \sum _{n=0}^{N_t-1} u_n e^{-i 2\pi f_k t_n} \right | \right ]^2 + \left [ 2 \left | \frac {1}{N_t} \sum _{n=0}^{N_t-1} v_n e^{-i 2\pi f_k t_n} \right | \right ]^2 }, \end{equation}

where $f_k$ represents the frequency corresponding to the $k$ th mode. The power spectra $P_{\textit{total}}$ obtained from the model reconstructions and the PIV reference data are shown in figure 13(a). The spectra reveal distinct peaks at frequencies that are integer multiples of the plasma modulation frequency $f_m$ , indicating the absence of vortex pairing and confirming that the primary vortex shedding dynamics remains strictly synchronised with the actuation. The reconstructed flow fields accurately capture the spectral characteristics of the reference data, as evidenced by the nearly overlapping energy spectra for $f/f_m \lt 2$ . This confirms that the large-scale periodic dynamics of the flow fields is reliably recovered at different SR factors. However, as $f/f_m$ increases beyond 2, the power spectra of the reconstructed fields become consistently lower than that of the PIV reference. This deviation at higher $f/f_m$ arises because high frequencies correspond to small-scale structures, which are partially lost due to limited input resolution and the model’s restricted ability to fully recover fine-scale dynamics, resulting in overall lower spectral amplitudes. Despite this amplitude discrepancy at higher frequencies, the characteristic peak frequencies are still accurately captured.

Figure 13.

(a) Power spectral $P_{\textit{total}}$ comparison between the reconstructed flow fields and PIV reference data, (b) definition of the phase using the temporal coefficients of the first two POD modes ( $a_1$ , $a_2$ ), where the fitted circle is shown as a solid red line and the divided phase angles $\phi$ are indicated by dashed lines.

Figure 13(b) illustrates the phase definition using the temporal coefficients of the first two POD modes ( $a_1$ , $a_2$ ). Based on this nearly circular distribution, the vortex phase is defined using a fitted circle obtained by a least-squares method to determine the circle centre and radius (Oudheusden et al. Reference Oudheusden, Scarano, Hinsberg and Watt2005; Pan, Wang & Wang Reference Pan, Wang and Wang2013). Given that the characteristic frequency of the vortex flow matches the plasma modulation frequency $f_m$ , as shown in figure 13(a), the phase circle is divided counterclockwise into 48 equal sections, corresponding to the number of PIV samples per pulse ( $f_s/f_m$ ), as indicated in figure 13(b). Each instantaneous flow field within an angular range of $\pm 3.75^{\circ }$ around a given phase angle $\phi$ is considered to represent the same phase. For example, the phase-averaged flow field at $\phi = 0^{\circ }$ is obtained by averaging all instantaneous fields whose phase angles fall within $\phi = 0^{\circ } \pm 3.75^{\circ }$ .

After obtaining the phase-averaged velocity fields using the above method, the MSE between the reconstructed phase-averaged fields and the PIV reference data is presented in figure 14. It is observed that the $\times$ 4 and $\times$ 8 SR cases achieve highly accurate reconstructions for both velocity components, with overall small errors. However, an intermittent increase in error appears between $\phi = 150^{\circ }$ and $270^{\circ }$ . This is attributed to the primary vortex undergoing the shedding process due to large velocity gradients ${\mathrm{d} U^*}/{\mathrm{d} y^*}$ between vortices (Dong et al. Reference Dong, Zhang, Ren, Yang, Xiao, Yao and Mao2025). This process introduces complex multi-scale fluctuations into the flow, thereby increasing the reconstruction difficulty. Another notable rise in error is observed for $\phi \gt 350^{\circ }$ , which is associated with the breakdown of the primary vortex and the emergence of small-scale structures. In contrast to the $\times$ 4 and $\times$ 8 cases, the $\times$ 16 SR case exhibits significantly higher errors in the entire phases, particularly during the initial vortex formation stage at $\phi \lt 100^{\circ }$ . This can be attributed to the relatively weak vortices with smaller vortex cores during the early stages, as shown in figure 15(d), making it difficult for the model to accurately recover the vortex characteristics from overly smoothed inputs, as illustrated in figure 10.

Figure 14.

Phase-dependent MSE $\epsilon$ between the reconstructed phase-averaged velocity fields and the PIV reference data at different SR factors: (a) streamwise velocity $U^*$ and (b) vertical velocity $V^*$ .

Figure 15.

Comparison of phase-averaged vortex evolution identified using the $\lambda _2$ criterion at different phase angles $\phi$ : (a) PIV reference data and reconstructed fields with (b) $\times$ 4, (c) $\times$ 8 and (d) $\times$ 16 SR factors.

The evolution of phase-averaged vortical structures is further analysed using the $\lambda _2$ criterion (Jeong & Hussain Reference Jeong and Hussain1995), which effectively identifies coherent vortex cores by highlighting regions of concentrated swirling motion. The phase-averaged $\lambda _2$ fields are shown in figure 15, illustrating the temporal evolution of the vortices. At early phases ( $\phi \lt 60^{\circ }$ ), both the $\times$ 4 and $\times$ 8 SR reconstructions closely match the PIV reference, accurately capturing the shape and position of the primary vortex cores. In contrast, the $\times$ 16 case significantly overestimates the vortex strength and fails to accurately reconstruct the initial weak vortex structures at this stage. As the phase progresses to $\phi = 240^{\circ }$ , corresponding to the vortex growth and early shedding stages, the $\times$ 4 and $\times$ 8 cases continue to reproduce the main vortex structures with high fidelity, preserving the overall coherent patterns. However, slight discrepancies in the size and strength of the primary and shedding vortices begin to appear in the $\times$ 8 case. Interestingly, the $\times$ 16 case appears to improve its reconstruction of the main vortex pattern during this stage, providing a reasonable representation of the vortex structure; however, its accuracy remains lower than that of the $\times$ 4 and $\times$ 8 cases. At later phases ( $\phi \gt 300^{\circ }$ ), representing the vortex breakdown stage, the $\times$ 4 and $\times$ 8 reconstructions still maintain good agreement with the reference, capturing both the primary vortices and the more diffused residual structures. In contrast, the $\times$ 16 case continues to struggle in recovering fine-scale vortex features, with significantly more diffused cores.

Figure 16.

Quantitative comparison of phase-averaged vortices extracted from the PIV reference data and reconstructed fields at different SR factors: (a) streamwise vortex centroid $x_c$ , (b) vertical vortex centroid $y_c$ , (c) vortex core radius $r_c$ and (d) vortex circulation $\varGamma _c$ .

To further assess the model’s ability to reconstruct vortex structures, quantitative vortex statistics are compared. The vortex cores are identified at the minimum $\lambda _2$ value and their centroids ( $x_c$ , $y_c$ ) are shown in figure 16. Overall, all SR cases accurately capture the streamwise evolution of the vortex cores $x_c$ , while maintaining stable vertical positions $y_c$ . The $\times$ 4 and $\times$ 8 reconstructions show excellent agreement with the PIV reference for all phases. In contrast, the $\times$ 16 case displays larger deviations at later phases ( $\phi \gt 300^{\circ }$ ), where sudden position shifts occur due to the more diffused and less coherent reconstructed vortex cores. The vortex radius $r_c$ is estimated by assuming an equivalent circular shape and is calculated as

(3.6) \begin{equation} r_c = \sqrt {\frac {S_c}{\pi }}, \end{equation}

where $S_c$ denotes the area of the vortex core defined by $\lambda _2 \leq -0.01$ , as illustrated in figure 16(c). The evolution of $r_c$ initially increases during vortex growth and slightly decreases during the breakdown stage. The $\times$ 4 and $\times$ 8 cases closely follow the trend of the PIV reference, accurately capturing both the expansion and subsequent decay of the vortex cores. In contrast, the $\times$ 16 case overestimates the radius in the early phases and underestimates it in later phases, consistent with the results in figure 15(d). The vortex circulation $\varGamma _c$ is calculated as

(3.7) \begin{equation} \varGamma _c = \iint _{S_c} \omega \mathrm{d}S, \end{equation}

as shown in figure 16(d). Similar to the radius, the $\times$ 4 and $\times$ 8 reconstructions successfully recover the peak circulation and its decay, showing strong agreement with the reference data. In contrast, the $\times$ 16 case exhibits significant discrepancies, particularly for $\phi \lt 100^{\circ }$ and $\phi \gt 250^{\circ }$ , further indicating challenges in accurately capturing vortex strength during the initial formation and breakdown phases.

3.4. Generalisation of spatio-temporal super-resolution models

The performance of the GAN-based spatio-temporal SR framework has been thoroughly validated under a specific baseline condition, showing excellent results at downsampling factors of $(\alpha , \beta ) = ((1/4), (1/4))$ , $((1/8), (1/4))$ and $((1/16), (1/4))$ . However, to enable practical application of the SR model for PIV data analysis, its generalisation capability must be further evaluated across multiple conditions with different vortex evolutions. Therefore, two additional test cases are considered. In the first case, referred to as case A, the plasma actuator is activated at a modulation frequency of $f_m = 8$ Hz. Similar to the baseline condition, the dominant flow frequency matches the modulation frequency; however, the detailed spatio-temporal evolution of the vortical structures exhibits significant differences. In the second case, designated as case B, the modulation frequency is increased to $f_m = 60$ Hz. This case features a completely different vortex evolution pattern, where a leapfrogging phenomenon is observed. Successive vortex pairs undergo pairing interactions, with a trailing vortex overtaking and eventually surpassing the leading vortex pair.

The comparisons of the spatio-temporal SR velocities $U^*$ and $V^*$ between the directly applied original model and the PIV reference for cases A and B are shown in figures 17(a) and 17(b), respectively. It can be observed that the reconstructions of all SR factors exhibit unacceptable errors, significantly deviating from the reference trends and amplitudes. Although the deviation is relatively smaller in the similar-vortex-dynamic case A, the overall accuracy remains insufficient. This result suggests that directly applying the original model to other flow conditions leads to poor reconstruction performance due to the presence of random and case-specific velocity fluctuations.

Figure 17.

Comparison of SR reconstructed velocities with the PIV reference data for cases A and B: (a) and (b) show results from directly applying the original model to cases A and B, respectively, (c) and (d) show results from the transfer-trained model for cases A and B, respectively.

To address this limitation, transfer learning is adopted to improve the reconstruction accuracy. The objective of transfer learning here is to fine tune the parameters of the generator $G$ to enhance reconstruction accuracy. Specifically, only the parameters in the layers after the convolutional layer following the RRDB in $G$ (as shown in figure 1 b) are updated during the transfer training. In other words, only the structures responsible for SR reconstruction are fine tuned. The remaining layers, which primarily encode general spatial features, are kept frozen and do not participate in the training process. It is important to note that the discriminator $D$ is not involved in the transfer learning. Its structure and parameters remain identical to those in the original model and are used solely to evaluate the authenticity of the outputs generated by $G$ . The dataset used for cases A and B in the transfer learning is summarised in table 2. For the SESRGAN transfer learning, the number of training samples $N_s$ is reduced by more than half, while for TESRGAN, $N_s$ is reduced by nearly two thirds. Additionally, the number of epochs is decreased to 200 and 100 for SESRGAN and TESRGAN training, respectively, which significantly reduces the training time and computational cost.

Table 2.

Summary of dataset configurations for transfer learning of cases A and B.

Figures 17(c) and 17(d) present the SR reconstruction results of $U^*$ and $V^*$ after applying transfer learning. It is evident that the reconstruction accuracy is significantly improved compared with the original model. For case A, the instantaneous velocity fields at all SR factors are successfully recovered, showing excellent agreement with the PIV reference in both components. In contrast, for case B, while the $\times$ 4 and $\times$ 8 SR reconstructions achieve good accuracy, the $\times$ 16 case still exhibits noticeable errors. These discrepancies primarily manifest as overestimations in the amplitude of both velocity components.

Figure 18.

Comparison of MSE $\epsilon$ between the original model and transfer-trained model for (a) case A and (b) case B, showing results of the $U^*$ component at different SR factors.

The reconstruction MSE errors of the original and transfer-trained models for cases A and B are compared in figure 18, where only the $U^*$ component errors are presented due to the similar trends observed in $V^*$ . It is evident that transfer learning substantially reduces $\epsilon$ at all SR factors. For example, in the $\times$ 4 case, $\epsilon$ is reduced by more than four times compared with the original model. As the SR factor increases, the reduction in $\epsilon$ also becomes more pronounced, highlighting the effectiveness of transfer learning in improving reconstruction accuracy under challenging downsampling conditions. Overall, $\epsilon$ in case B is consistently higher than that in case A, which can be attributed to the fundamentally different vortex evolution dynamics in case B compared with the baseline condition. Interestingly, the $\epsilon$ values of case A after transfer learning are even lower than those of the original model applied to the baseline condition ( $f_m = 20$ Hz), as shown in figure 9(a). A detailed comparison between the transfer-trained model reconstructions and the reference PIV results is provided in Appendix B. The results indicate that the GAN-based model with transfer learning achieves excellent generalisation for recovering fluctuation parts in flows with similar dynamics, supporting substantial flow data enhancement. However, for case B with a different vortex dynamics, accurate reconstruction is only achieved at $\times$ 4 and $\times$ 8 SR factors. Therefore, it is expected that the combination of GAN-based approaches and transfer learning can reliably achieve at least $\times$ 8 SR for flows with varying vortex dynamics.

3.5. Efficiency and limitation of GAN-based spatio-temporal super-resolution

To evaluate the efficiency improvements enabled by the GAN-based spatio-temporal SR model for accelerating PIV raw particle image-to-velocity field analyses, the training times of both the original and transfer-trained SESRGAN and TESRGAN models are critical. Figure 19 illustrates the evolution of the generator loss $\mathcal{L}_{G}$ over training epochs. For the original SESRGAN models (figure 19 a), higher SR factors require more epochs for $G$ to sufficiently optimise parameters and generate high-fidelity HR velocity fields, with convergence generally achieved after approximately 300 epochs. After convergence, $\mathcal{L}_{G}$ stabilises, indicating robust training. During transfer learning for cases A and B, the pre-trained original model serves as a strong initialisation, significantly accelerating convergence. Case A (with a vortex dynamics similar to the baseline) converges faster than case B (with a distinct vortex dynamics). Nevertheless, all transfer-trained SESRGAN models achieve stable convergence within 200 epochs. For TESRGAN training (figure 19 b), since the temporal downsampling factor $\beta$ remains fixed at (1/4), all original models converge in approximately 100 epochs, while the transfer-trained TESRGAN models achieve convergence even faster, within approximately 50 epochs.

Figure 19.

Evolution of generator loss $\mathcal{L}_{G}$ during training: (a) SESRGAN models and (b) TESRGAN models.

Based on § 3.4, it is demonstrated that the proposed model can maintain good generalisation capability at an $\times$ 8 SR factor even under different vortex dynamics, accurately reconstructing spatio-temporal SR flow fields. Therefore, the following analysis on improving PIV analysis efficiency focuses on the $\times$ 8 case. To ensure convergence with different SR factors, the maximum training epochs are set to 800 and 200 for the SESRGAN original and transfer learning models, and 300 and 100 for the TESRGAN original and transfer learning models, respectively. Their corresponding training times per epoch are denoted as $t_{S,O}$ , $t_{S,{\textit{TL}}}$ , $t_{T,O}$ and $t_{T,{\textit{TL}}}$ . The combined SESRGAN and TESRGAN models enable the generation of HR velocity fields from LR inputs, with the reconstruction time per frame denoted as $t_{\textit{GAN},\textit{HR}}$ . Additionally, the processing times of the adaptive PIV algorithm in DynamicStudio 2015a under different minimum IAs are evaluated, including the commonly used $16 \times 16$ (Zong & Kotsonis Reference Zong and Kotsonis2018; Dong et al. Reference Dong, Zhang, Ren, Yang, Xiao, Yao and Mao2025) and their $\times$ 8 downsampled counterparts (128 $\times$ 128), as well as 32 $\times$ 32 (Yilmaz & Rockwell Reference Yilmaz and Rockwell2012; Yu & Bernal Reference Yu and Bernal2017; Mishra & Panigrahi Reference Mishra and Panigrahi2025) and their $\times$ 8 downsampled counterparts (256 $\,\times\,$ 256). The analysis time per frame for HR PIV is denoted as $t_{\textit{HR}}$ and that for LR PIV as $t_{\textit{LR}}$ . Specific timing details are provided in table 3. Thus, the theoretical net saved time when applying the GAN-based spatio-temporal SR model to PIV analysis is estimated as

(3.8) \begin{align} t_{\textit{net}} &= \underbrace {t_{\textit{HR}} \times N_t}_{\text{PIV HR analysis time}} - \underbrace {t_{\textit{LR}} \times N_t}_{\text{PIV LR analysis time}} - \underbrace {\left [N_O \times t_{\textit{HR}} + (N_c - 1) \times N_{\textit{TL}} \times t_{\textit{HR}}\right ]}_{\text{PIV HR analysis time for training}} \nonumber\\&\quad - \underbrace {\left [t_{S,O} + t_{T,O} + (N_c - 1) \times t_{S,\textit{TL}} + (N_c - 1) \times t_{T,\textit{TL}}\right ]}_{\text{Model training time}} \nonumber\\&\quad - \underbrace {\left [N_t - N_O - (N_c - 1) \times N_{\textit{TL}}\right ] \times t_{\textit{GAN},\textit{HR}}}_{\text{GAN-based HR reconstruction time}}. \end{align}

Here, $N_c$ the number of flow conditions, $N_O$ the number of HR PIV frames used to train the original model (1728, as shown in table 1) and $N_{\textit{TL}}$ the number of HR PIV frames used for transfer learning (approximately 860, as shown in table 2). It should be noted that $t_{\textit{net}}$ represents a theoretical estimate. Practical factors, such as additional operating time required when performing both high- and LR analyses on the same condition, as well as delays from cumulative data processing in multiple measurements, are not considered here.

Table 3.

Summary of PIV analysis and model training times.

Figure 20 illustrates the estimated net saved time $t_{\textit{net}}$ achievable using the GAN-based $\times$ 8 SR models under different numbers of flow conditions $N_c$ and total PIV frames $N_t$ . Figure 20(a) corresponds to results reconstructed to PIV resolution with IA = $16 \times 16$ while figure 20(b) to IA = $32 \times 32$ . The cyan line indicates $t_{\textit{net}} = 0$ , above which the SR model yields positive time savings and below which it results in negative time savings. The blank region at the bottom denotes where the number of PIV frames is insufficient to support model training, thus making the SR approach inapplicable. It can be observed that achieving significant time benefits from the SR model requires a large $N_t$ and these benefits diminish as $N_c$ increases due to additional time costs introduced by transfer learning for each new condition. Moreover, increasing the flow field resolution demand, for example, by reducing IA from $32 \times 32$ to $16 \times 16$ , nearly doubles the time savings under the same $N_t$ and $N_c$ . For example, the plasma-induced vortices data used in this work in fact involve six conditions with a total of 17 280 PIV frames. Applying the GAN-based SR model to these data results in negative time savings for both IA = $16 \times 16$ and IA = $32 \times 32$ cases, due to the relatively small number of frames per condition. In contrast, in our earlier investigation of vortex dynamics around a pitching flat plate (Dong et al. Reference Dong, Choi, Mao and Wang2022a ), four pitching conditions were measured, each with 32 planar PIV slices, resulting in a total of 1450 000 PIV particle images. HR PIV analysis with IA = $16 \times 16$ would require approximately 12 179 704 s ( $\sim$ 141 days) and with IA = $32 \times 32$ , approximately 7829 810 s ( $\sim$ 91 days), making it impractical to process all frames. If combined with the proposed SR model, the estimated time savings are approximately 5277 874 s ( $\sim$ 61 days) for IA = $16 \times 16$ and 2651 650 s ( $\sim$ 31 days) for IA = $32 \times 32$ . These savings make it feasible to analyse the entire instantaneous flow field dataset, thereby enabling a more comprehensive understanding of detailed vortex evolution mechanisms.

Figure 20.

Estimated net saved time $t_{\textit{net}}$ when applying GAN-based $\times$ 8 SR models for different numbers of flow conditions $N_c$ and total PIV frames $N_t$ : (a) IA = $16 \times 16$ and (b) IA = $32 \times 32$ , where the cyan line indicates the zero-savings boundary ( $t_{\textit{net}} = 0$ ). The hollow star marker denotes the application of the SR model on the plasma-induced velocity field analysis, while the filled star marker indicates the application on the pitching flat plate analysis (Dong et al. Reference Dong, Choi, Mao and Wang2022a ).

Figure 21.

Spatio-temporal autocorrelation functions for different $y^*$ locations at $x^* = 5$ : (a) streamwise velocity autocorrelation $\rho _{uu}$ and (b) vertical velocity autocorrelation $\rho _{vv}$ .

The application of GAN-based SR models for accelerating PIV particle image analysis has been discussed above. In fact, not only the SR factors $\times$ 4, $\times$ 8 and $\times$ 16 presented here have been tested, but higher factors such as $\times$ 32 and $\times$ 64 were also attempted. However, at these higher factors, the model fails to converge and cannot generate physically realistic reconstruction results. Furthermore, we observed that while the model can achieve up to $\times$ 16 spatial SR, it is only able to realise $\times$ 4 temporal SR; attempts at $\times$ 8 and $\times$ 16 temporal SR failed to converge. This limitation is considered to stem from changes in data correlation. To verify this, autocorrelation functions are employed to quantify spatio-temporal data coherence, as illustrated in figure 21. The spatial autocorrelation for the streamwise component is defined as

(3.9) \begin{equation} \begin{aligned} \rho _{uu}(\triangle x^*/\alpha ) = \frac {\langle u(x^*) u(x^* + \triangle x^*/\alpha ) \rangle }{\langle u(x^*) u(x^*) \rangle }, \end{aligned} \end{equation}

and the temporal autocorrelation as

(3.10) \begin{equation} \begin{aligned} \rho _{uu}(\triangle \tau ^*/\beta ) = \frac {\langle u(\tau ^*) u(\tau ^* + \triangle \tau ^*/\beta ) \rangle }{\langle u(\tau ^*) u(\tau ^*) \rangle }, \end{aligned} \end{equation}

where $\triangle x^*$ and $\triangle \tau ^*$ represent the spatial gap and temporal interval of the HR PIV data, respectively. The same definitions apply to $\rho _{vv}$ . It is observed that $\rho _{uu}$ decays more slowly with increasing spatial downsampling factor than with temporal downsampling and maintains higher values even at large downsampling factors. In contrast, $\rho _{vv}$ exhibits a more rapid decay in both spatial and temporal directions, reflecting weaker intrinsic coherence. Considering that the plasma jet flow is dominated by the $U^*$ component, the degradation in $\rho _{uu}$ is identified as the primary factor limiting the achievable SR factor. For temporal SR, training data are constructed by sampling instantaneous velocity fields at the same $x^*$ position but at different $\tau ^*$ instants (see § 2.3). From a Lagrangian perspective, fluids at $\tau ^*$ have already convected downstream by $\tau ^*+\triangle \tau ^*/\beta$ , thus reducing data correlation in the $y^*-\tau ^*$ plane and then hindering model convergence. Therefore, when designing GAN-based SR models, it is crucial to consider the inherent convection and spatio-temporal coherence of the flow, as these factors directly affect data correlation and set the fundamental limits on achievable SR factors.