1. Introduction
Fluid mechanicians have developed approaches to measure turbulent flows in a high-fidelity manner. High-fidelity data sets collected from numerical simulations and experiments have been shared in the community (Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, Chen, Szalay and Eyink2008; Towne et al. Reference Towne2023), supporting the development of approaches for fluid flow modelling and control. Such high-fidelity turbulent flow data involve their rich physics such as vortex dynamics, energy cascade processes, and anisotropic structures that evolve across a range of spatiotemporal length scales. While modern machine-learning techniques analyse high-fidelity turbulent flow data, it is challenging to evaluate data-driven models with data sets collected through different approaches since data characteristics such as measurement uncertainties and spatiotemporal length scales covered by each data source are different. Aiming to perform data-driven assessments of turbulent flows across different sources and scales, this paper proposes a nonlinear machine-learning technique that seeks a low-order representation of multi-source turbulent flow data.
To find such a low-rank representation, this study considers the concept of a manifold of turbulent flows. The long-term dynamics of infinite-dimensional turbulence systems often converge to a low-order manifold surface (Graham & Floryan Reference Graham and Floryan2021). This low-rank nature can be used to facilitate understanding and modelling of turbulent flows (Noack et al. Reference Noack, Afanasiev, Morzynski, Tadmor and Thiele2003; Luchtenburg et al. Reference Luchtenburg, Günther, Noack, King and Tadmor2009). Proper orthogonal decomposition (POD; Lumley Reference Lumley, Yaglom and Tatarski1967) has been considered in estimating a low-order manifold from turbulent flow data. However, finding a minimal representation of turbulence with such a linear technique becomes challenging as the variance of data increases because it linearly projects data onto a flat manifold. Increasing the Reynolds number of flows of interest falls under such a case since the difference between the minimum and maximum length scales is enlarged (Alfonsi & Primavera Reference Alfonsi and Primavera2007). In dealing with multi-source turbulent flow data, including sensor readings, numerically simulated data, and experimental measurements, extracting features from data is further demanding due to measurement noise, uncertainties, the difference in degree of freedom and length scales captured by each data source.
In response, we consider a nonlinear autoencoder-based compression (Hinton & Salakhutdinov Reference Hinton and Salakhutdinov2006) to seek a manifold capturing turbulence characteristics from various data sources. This technique achieves greater order reduction of turbulence compared with linear techniques, demonstrated with channel flow (Yousif et al. Reference Yousif, Yu and Lim2022), Kolmogorov turbulence (Racca et al. Reference Racca, Doan and Magri2023) and a flow through an urban environment (Eivazi et al. Reference Eivazi, Le Clainche, Hoyas and Vinuesa2022). In a reduced-order space, there often exists a submanifold forming a geometrical structure that compactly represents the characteristics and patterns underlying the flow data (Graham & Floryan Reference Graham and Floryan2021). Furthermore, such low-order representations can be used for modelling (Constante-Amores & Graham Reference Constante-Amores and Graham2024) and controlling unsteady flows (Fukami et al. Reference Fukami, Nakao and Taira2024b ).
While a nonlinear autoencoder achieves significant order reduction of turbulent flows, the mapping from the input high-dimensional space to the latent subspace can involve various transformations even if their decoding leads to the same flow reconstruction. This study discusses the importance of incorporating prior knowledge of turbulent flow physics in the learning formulation of a nonlinear autoencoder to identify a submanifold that best describes the multi-source turbulent flow data in a low-order manner. As an example, the data sets of the transitional turbulent boundary layer collected by direct numerical simulations and particle image velocimetry are considered. We find that a nonlinear autoencoder with physics embedding provides a manifold that captures structural features of flows across the Reynolds number while distinguishing the data source. With machine-learning-based super-resolution, we further discuss that the present manifold can facilitate analysis, comparison and interpretation of turbulent flow data measured through different approaches.
This paper is organised as follows. The approach is described in § 2. Results of the manifold discovery for the transitional turbulent boundary-layer data sets are presented in § 3. Conclusions are offered in § 4.
2. Approach
This study develops a machine-learning approach for identifying a low-order submanifold that presents geometric structures capturing the essential features of turbulent flows, in hopes of facilitating data-driven analyses performed across different data sources. We consider the numerical and experimental data sets of a transitional turbulent boundary layer made available by the database of Towne et al. (Reference Towne2023). These data sets include time-resolved snapshots of a high-fidelity direct numerical simulation (DNS) across a wide domain in the streamwise direction and planar particle image velocimetry (PIV) measurements at different Reynolds numbers. As there is a variation of noise, measurement uncertainties, resolved length scales and Reynolds number across the data sets, the present set of DNS and PIV measurements calls for a comprehensive analysis for manifold discovery of multi-source turbulent flow data.
Transitional turbulent boundarylayer data sets.
$(a)$
Direct numerical simulation: part of the
$x{-}y$
sectional domain and subdomains
$(\textrm {i}-\textrm {iii})$
are shown.
$(b)$
Particle image velocimetry: three different Reynolds numbers are considered.
For the DNS, the zero-pressure-gradient flat-plate turbulent boundary layer is simulated by numerically solving the incompressible Navier–Stokes equations. The streamwise velocity field
$u$
is visualised in figure 1
$(a)$
. The size of the computational domain is
$(L_x,L_y,L_z)/\theta _{ {avg}} = (469, 53, 79)$
, where
$\theta_{\textit{avg}}$
is the streamwise-averaged momentum thickness, and
$L_x$
.
$L_y$
, and
$L_y$
are the streamwise, wall-normal, and spanwise extent of the computational domain. The recycling scheme by Lund et al. (Reference Lund, Wu and Squires1998) is used to generate the turbulence fluctuations at the inlet in which the recycling plane
$x_{\textit{ref}}/\theta _{\textit{avg}}$
is set to 375, where
$x_{\textit{ref}}$
is the location of the recycling plane for the inflow boundary condition. The friction Reynolds number
$Re_\tau = u_\tau \delta /\nu$
spans from 481 to 1024 between the inlet and outlet, covering a range of turbulence characteristics such as nonlinear interactions and momentum exchange across the flow. All the variables are normalised by the kinematic viscosity
$\nu$
, the boundary-layer thickness
$\delta$
at 99 % of the free-stream velocity, and the friction velocity at the wall
$y=0$
,
$u_\tau =(\nu {\partial \overline {u}^{z,t}}/{\partial y}|_{y=0})^{1/2}$
, where
$\overline {(\cdot )}^{z,t}$
represents the average over the spanwise direction and time. Details of the simulation set-up are provided in Towne et al. (Reference Towne2023).
Time-resolved planar velocity fields measured by PIV in a wind tunnel are collected at three different Reynolds numbers of
$Re_\tau = 605$
, 987 and 1373, as shown in figure 1
$(b)$
. The friction velocity for the PIV cases is computed by the Clauser method (Clauser Reference Clauser1956). The free-stream turbulence intensity is approximately 0.5 % and the boundary layer is developed in a nominally zero-pressure-gradient environment. Details on camera set-ups and test sections are available in Towne et al. (Reference Towne2023).
In the present analysis, we consider the streamwise velocity
$u$
for both DNS and PIV data sets. This is intended to examine whether the dominant feature of the turbulent boundary layer mainly driven by the streamwise velocity can be extracted across the different data sources by machine learning. While the DNS is performed in a three-dimensional domain, subdomains sampled from an
$x{-}y$
sectional field are used to align the data set-up with the PIV data, as shown in figure 1
$(a)$
. The subdomain set-up will be explained later.
Observable-augmented autoencoder composed of convolutional neural network (CNN) and multi-layer perceptron (MLP) (Fukami & Taira Reference Fukami and Taira2023).
To seek a manifold that captures the physical features from these numerical and experimental data of the turbulent boundary layer, a nonlinear autoencoder-based data compression (Hinton & Salakhutdinov Reference Hinton and Salakhutdinov2006) is performed. An autoencoder is trained to replicate the data between the input and output while possessing the bottleneck in the model architecture, as illustrated in figure 2. The data dimension at the bottleneck referred to as a latent vector
$\boldsymbol \xi$
is generally much smaller than that of the input or output data. Hence, the latent vector
$\boldsymbol \xi$
can be regarded as a compressed representation of the given data
$\boldsymbol q$
if the autoencoder
${\mathcal F}_{\textit{AE}}$
successfully reconstructs the data. With an encoder
${\mathcal F}_e$
and a decoder
${\mathcal F}_d$
, the aforementioned process is expressed as
\begin{align} {\boldsymbol q}\approx {\mathcal F}_{\textit{AE}}(\boldsymbol q) = {\mathcal F}_d({\mathcal F}_e({\boldsymbol q})), \,\,\, {\boldsymbol \xi } = {\mathcal F}_e(\boldsymbol q),\,\,\, {\boldsymbol q} \approx \widehat {\boldsymbol q} = {\mathcal F}_d({\boldsymbol \xi }), \end{align}
where
$\widehat {(\cdot )}$
denotes a reconstructed variable.
To promote the manifold identification capturing the development of turbulent boundary layers, we consider a friction Reynolds-number-based augmentation in performing the present nonlinear autoencoder-based compression, as illustrated in figure 2. In the formulation of a variable-based augmentation (Fukami & Taira Reference Fukami and Taira2023), an autoencoder is constructed to estimate an observable variable (
$Re_\tau$
in the present study) from the latent vector
$\boldsymbol \xi$
through a subnetwork depicted with a pink box in figure 2 while compressing the data such that
\begin{align} {\boldsymbol w}^* = \textrm {argmin}_{\boldsymbol w}[||{\boldsymbol q}-\hat {\boldsymbol q}||_2 + \beta ||Re_\tau - \widehat {Re_\tau }||_2], \end{align}
where
$\boldsymbol w$
denotes the weights inside the autoencoder and
$\beta$
balances the flow field and Reynolds number loss terms. Here, the weights inside the main autoencoder and the subnetwork are simultaneously optimised. As an observable variable needs to be accurately estimated to minimise the above cost function, the optimal weights
${\boldsymbol w}^*$
are found to capture the coherent relationship between the given data
$\boldsymbol q$
and an observable in the latent space, which has been used in a range of aerodynamic examples (Fukami & Taira Reference Fukami and Taira2023; Liu et al. Reference Liu, Beckers and Eldredge2024; Tran et al. Reference Tran, Fukami, Inada, Umehara, Ono, Ogawa and Taira2024; Mousavi & Eldredge Reference Mousavi and Eldredge2025; Eldredge & Mousavi Reference Eldredge and Mousavi2025). This study uses
$\beta = 0.05$
based on the L-curve analysis (Hansen & O’Leary Reference Hansen and O’Leary1993), which finds an appropriate regularisation parameter of the cost function. The case with
$\beta = 0.01$
is also considered to observe how the latent description is modified through the observable augmentation. The current formulation can take not only an observable but also parameters derived from observables for variable-based augmentation, as performed in this study that uses
$Re_\tau$
measured from the friction velocity
$u_\tau$
.
Choosing the friction Reynolds number
$Re_\tau$
is natural in our case as it characterises flow features in the wall-normal direction such as the balance between the near-wall region dominated by viscous effects and the outer region where turbulence is fully developed across the boundary layer. In other words, the primary features of the turbulent boundary layer in both the
$x$
and
$y$
directions can be incorporated in identifying a manifold that represents the flow characteristics across multiple data sources. Although not considered in this study, the momentum thickness Reynolds number
$Re_\theta$
may also provide a similar result in the manifold identification for the present case as it increases across the streamwise direction with the friction Reynolds number.
Representative reconstructed fields through nonlinear autoencoder compression across the latent dimension
$n_{\boldsymbol \xi }$
. The DNS (
$Re_\tau = 897$
) and PIV (
$Re_\tau = 605$
) fields are shown. The
$L_2$
reconstruction error
$\varepsilon$
is reported underneath each field.
For the present data-driven analyses, both DNS and PIV data are sampled such that the subdomain size in the streamwise and wall-normal directions becomes
$L_{x,\textit{ML}}/\delta \in [1, 3]$
and
$L_{y,\textit{ML}}/\delta \approx 1$
, respectively. The DNS data are uniformly interpolated in the wall-normal direction to align the set-up of the PIV data. These collected data are then resized to be
$N^2_{\textit{ML}} = 128^2$
, where
$N_{\textit{ML}}$
is the number of grid points for the resized data used in the data-driven analysis. As the current study aims to find a manifold assessing the data similarities across different sources, the present observable-augmented autoencoder is trained with both DNS and PIV snapshots together. We consider 19 000 snapshots for the present analysis, including 10 000 DNS samples composed of 10 subdomains per snapshot across 1000 time-resolved frames and 3000 PIV snapshots for each of three Reynolds numbers. We use 70 % of the snapshots for training and the remaining 30 % for validation with random splitting of the data set. The encoder and decoder parts of the autoencoder are constructed by convolutional neural networks (LeCun et al. Reference LeCun, Bottou, Bengio and Haffner1998) while the subnetwork is composed of multi-layer perceptrons (Rumelhart et al. Reference Rumelhart, Hinton and Williams1986), analogous to the original observable autoencoder network in Fukami & Taira (Reference Fukami and Taira2023). A sample code (https://github.com/kfukami/Observable-CNN-AE) can be referred to for further details on the present autoencoder model.
3. Results
We apply the current autoencoder with the friction Reynolds-number-based augmentation to the DNS and PIV data sets of the turbulent boundary layer. The reconstructed fields through the observable-augmented autoencoder with
$\beta = 0.05$
across different numbers of the latent dimension
$n_{\boldsymbol \xi }$
are presented in figure 3. As representative cases, a DNS snapshot at
$Re_\tau = 897$
and a PIV snapshot at
$Re_\tau = 605$
are shown. The
$L_2$
reconstruction error
$\varepsilon = ||{\boldsymbol q}-{\mathcal F}_{\textit{AE}}({\boldsymbol q})||_2/||{\boldsymbol q}||_2$
is reported under each of the reconstructed velocity fields. The error decreases as the latent dimension increases, corresponding to the low compression of velocity data. While large-scale structures can be represented only with
$n_{\boldsymbol \xi } \leqslant 256$
, fine-scale features are captured well across the field with
$n_{\boldsymbol \xi } \geqslant 512$
.
As a reference, we note that the linear POD requires more than 3000 modes to achieve the same reconstruction level as the present nonlinear autoencoder with
$n_{\boldsymbol \xi } = 512$
. In other words, the use of nonlinear activation functions inside the model improves compression in seeking low-order representations of the turbulent boundary-layer data sets. Based on the
$L_2$
error norm and the reconstruction of fine-scale structures, we hereafter choose the latent dimension of 512 for the discussions.
Next, we are interested in what can be captured through the present nonlinear compression of multi-source turbulent boundary-layer data. Let us perform POD on the compressed representation
$\boldsymbol \xi$
to examine the primary features extracted from the turbulent flows with the autoencoder. The coordinate composed of three dominant POD coefficients
$a_1-a_2-a_3$
is shown in figure 4. In addition to the cases of the observable autoencoder with
$\beta = 0.01$
and 0.05, a regular autoencoder without the Reynolds-number augmentation, i.e.
$\beta = 0$
, is also presented for comparison.
Cross-source manifold identified through the observable-augmented nonlinear autoencoder (AE). The coefficient space composed of the three dominant POD modes is shown. The
$L_2$
error
$\varepsilon$
averaged over the samples is reported underneath each coefficient space.
All three spaces learn the friction Reynolds-number dependence for the DNS data. The clear trend with respect to
$Re_\tau$
is observed in the
$a_1$
axis with
$\beta \leqslant 0.01$
and the diagonal direction on the
$a_1-a_2$
plane with
$\beta = 0.05$
. The main difference between the cases with and without the Reynolds-number augmentation is their low-order expression about the data source. In the space of a regular autoencoder, the PIV cases generally overlay the centre region of the DNS cases. In contrast, there is a clear distinction between the DNS and PIV cases by introducing the Reynolds-number-based augmentation. The
$a_3$
axis with
$\beta = 0.01$
and 0.05 captures the difference in the data source.
Noteworthy here is that the reconstruction performance over the three cases is almost the same, presenting the
$L_2$
error of 14 % as reported in figure 4. In other words, while all nonlinear autoencoders achieve similar compression of the turbulent flows, what they learn in the latent space from the data becomes different from each other. With the friction Reynolds-number-based augmentation, the current submanifold compactly represents the Reynolds number and the difference in the data source as their characteristics of given data sets. In addition, the latent expression with
$\beta = 0.05$
provides a continuous transition between the data sources of DNS and PIV, forming a V-shape submanifold of the DNS data. Since this submanifold extracts structural characteristics from high-resolution flow fields in addition to the Reynolds-number dependence, there are some variations of
$Re_\tau$
at
$-10\lt a_1\lt 0$
in the latent space with
$\beta = 0.05$
.
Machine-learning-based super-resolution reconstruction for DNS data of turbulent boundary layers. The
$L_2$
reconstruction error
$\varepsilon$
is reported underneath each field.
To further examine how the present manifold can be used for assessing data-driven techniques with multi-source turbulent flow data, we consider machine-learning-based super-resolution analysis of turbulent flows (Fukami et al. Reference Fukami, Fukagata and Taira2019). Super-resolution aims to reconstruct high-resolution data from the corresponding low-resolution signal, which has been examined for turbulent flow reconstruction (Fukami et al. Reference Fukami, Fukagata and Taira2023). The process of super-resolution reconstruction in the context of fluid flows is expressed as
${\boldsymbol q}_{\textit{HR}} = {\mathcal F}_{\textit{SR}}({\boldsymbol q}_{\textit{LR}})$
, where the high-resolution flow field
${\boldsymbol q}_{\textit{HR}}$
is reconstructed from the low-resolution flow data
${\boldsymbol q}_{\textit{LR}}$
with a machine-learning model
${\mathcal F}_{ \textit{SR}}$
.
To recognise turbulent flow structures that have been learned as the difference between the DNS and PIV data sets on the manifold, we use the machine-learning-based super-resolution reconstruction in the following manner. The present super-resolution model is first trained with the DNS data only and then evaluated with the PIV data. We then examine how the super-resolved flow fields from PIV data are described on the identified manifold through the nonlinear autoencoder.
Application of machine-learning-based super-resolution model
${\mathcal F}_{\textit{SR}}$
trained with the DNS data to the PIV data sets. The reconstructed fields, the
$L_1$
difference field between the reconstructed and original PIV
$|\varepsilon _{L_1}| = |{\boldsymbol q}_{\textit{PIV}}-{\mathcal F}_{\textit{SR}}({{\boldsymbol q}_{\textit{LR,PIV}}})|$
, and the root mean square of streamwise velocity fluctuation
$u_{\textit{rms}}$
across the wall-normal direction are shown.
For successful super-resolution reconstruction of turbulent flows, the machine-learning model
${\mathcal F}_{\textit{SR}}$
should be carefully constructed to accommodate a range of spatial length scales while accounting for scale invariance of turbulent flow structures (Fukami et al. Reference Fukami, Goto and Taira2024a
). We use the interconnected hybrid downsampled skip-connection/multi-scale (DSC/MS) model (Fukami et al. Reference Fukami, Fukagata and Taira2023) based on CNN (LeCun et al. Reference LeCun, Bottou, Bengio and Haffner1998). This model is aimed at capturing rotational and translational invariance while incorporating multi-size filter operations that greatly assist in learning the correlation across a range of length scales in turbulent flows. We refer to Fukami et al. (Reference Fukami, Fukagata and Taira2023) and a sample code (http://www.seas.ucla.edu/fluidflow/codes.html) for further details on the present super-resolution model. In this study, the model is trained to reconstruct the high-resolution velocity field of size
$128^2$
from the corresponding low-resolution data of size
$8^2$
generated by average pooling (Fukami et al. Reference Fukami, Fukagata and Taira2019).
Let us assess the present super-resolution model with the DNS data, as shown in figure 5. The super-resolution model
${\mathcal F}_{\textit{SR}}$
is trained with 25 000 samples composed of 10 subdomains per snapshot across 2500 time-resolved DNS frames. The DSC/MS model accurately produces the fine-scale structures in the flow fields across the Reynolds number. The super-resolution model
${\mathcal F}_{\textit{SR}}$
with the DNS data is then tested with the PIV data, as presented in figure 6. Here, the PIV fields are downsampled to be the size
$8^2$
, and then these low-resolution PIV input data
${\boldsymbol q}_{ \textit{LR}, \textit{PIV}}$
are given to the model
${\mathcal F}_{\textit{SR}}$
. While the reconstructed fields are generally similar to the reference PIV field
${\boldsymbol q}_{\textit{PIV}}$
across the Reynolds number, their velocity profiles exhibit slight differences near the wall. This is further evident from the root mean square of streamwise velocity fluctuation
$u_{\textit{rms}}$
across the wall-normal direction shown in figure 6. Here, the
$u_{\textit{rms}}$
profile is plotted across
$y^+ = y u_\tau/\nu$
obtained from the PIV data set-up. The profile of the PIV data presents overestimation near the wall due to the reflection of the laser sheet on the surface in measuring the data (Towne et al. Reference Towne2023). This overestimation is corrected by the super-resolution model trained with the DNS data possessing a finer spatial resolution near the wall. The PIV cases of
$Re_\tau = 987$
and 1373 do not resolve the near-wall region of
$y^+ \approx 15$
, where the inner peak of
$u_{\textit{rms}}$
resides, even at the first grid point from the wall. In other words, the super-resolution model performs some nonlinear stretching of data in the wall-normal direction such that the inner peak at
$y^+\approx 15$
can be captured, despite not being resolved with PIV.
At last, we provide these super-resolved PIV data to the encoder
${\mathcal F}_e$
of the observable autoencoder with
$\beta = 0.05$
to examine the behaviour on the identified low-rank coordinate such that
${\boldsymbol \xi }_{\textit{SRPIV}} = {\mathcal F}_e({\mathcal F}_{\textit{SR}}({\boldsymbol q}_{{\textit{LR}, \textit{PIV}}}))$
. The coefficient space composed of the three dominant POD modes for the latent vector
${\boldsymbol \xi }_{\textit{SRPIV}}$
is shown in figure 7. Due to the correction near the wall, the super-resolved PIV data are projected between the DNS and PIV data points in the coefficient space. This suggests that the nonlinear autoencoder
${\mathcal F}_{\textit{AE}}$
learns the difference in the resolved length scales near the wall across the
$a_3$
direction to distinguish the data sources. Although the decoder does not receive
$Re_\tau$
as an input, the latent space of the observable-augmented autoencoder includes the information of
$Re_\tau$
as presented in figure 4, thereby reflecting the reconstructed inner peak from the data in the latent space as the projection between DNS and PIV. In other words, the identified manifold gains the feature of given turbulence data in a compact manner, enabling standardised evaluation of machine-learning analysis when considering multi-source turbulent flow data together.
We emphasise that the present finding of the manifold is achieved by the observable augmentation with
$Re_\tau$
that characterises the balance of turbulent flow features between the near-wall and outer regions while providing information on the development of turbulent boundary layers. This selection of observable variables capturing the flow features in both the
$x$
and
$y$
directions is based on our foundational understanding of turbulent boundary-layer flows. Rather than naively applying data-science techniques without consideration, appropriately incorporating prior knowledge into the model design is essential to learn multi-source turbulent flow data with nonlinear machine learning.
Assessment of super-resolved PIV fields on the identified coordinate. The coefficient space composed of the three dominant POD modes is shown.
4. Concluding remarks
This study considered nonlinear machine learning to seek a low-rank manifold that captures the characteristics of multi-source turbulent flow data. With an example of transitional turbulent boundary-layer data from both numerical and experimental sources, we found that a low-order subspace is identified through nonlinear autoencoder-based compression with an observable augmentation. In the identified space, the Reynolds-number dependence and the difference in the resolved length scale in a flow snapshot are compactly represented. The observable-augmented autoencoder provides a continuous transition between numerical and experimental data in the latent subspace while extracting structural features of turbulent flows in a low-order manner, which differs from what can be achieved through a standard classifier. Furthermore, the present coordinate enables the comparison of the data even with different spatial resolutions from multiple sources, which is challenging with a traditional norm. We showed that the super-resolved experimental flow fields that have no comparable solutions can be assessed in the identified subspace through projection. The current findings further support the importance of considering prior knowledge for data-driven studies in learning multi-source turbulent flow data.
The present manifold learning across multi-source turbulent flow data may be considered for analysis at a range of Reynolds numbers by carefully preparing the data sets with respect to their spatiotemporal resolution and Reynolds numbers. To make the identified subspace more robust, multi-fidelity data at different Reynolds numbers obtained from numerical simulations such as large-eddy simulations and detached eddy simulations could be incorporated. As experiments can typically achieve higher Reynolds numbers compared with numerical simulations, one can consider manifold learning with the data sets composed of DNS data at low Reynolds numbers and PIV data at high Reynolds numbers. The resulting manifold with such data pairs would learn the characteristics of spatially high-resolved DNS at low Reynolds numbers and that of spatially low-resolved PIV at high Reynolds numbers. Leveraging such a latent space, super-resolution reconstruction with respect to not only spatial resolution but also Reynolds number could be performed through the decoder part of the observable-augmented autoencoder.
While this study primarily used flow field data obtained by DNS and PIV, sensor measurements could be further integrated into the coordinate identification process as either input/output data or variables used for the subnetwork-based augmentation. The denoising capability of autoencoder-based techniques also encourages multi-source data analysis of turbulent flows in seeking low-rank representations (Smith et al. Reference Smith, Fukami, Sedky, Jones and Taira2024). Although the current analysis used POD to identify the primary characteristics in the latent subspace, sensitivity analysis (Mo et al. Reference Mo, Traverso and Magri2024) and geometric analysis with magnification factors (Kelshaw & Magri Reference Kelshaw and Magri2024) may also be helpful to examine the latent space characteristics. Depending on the physics and flow of interest, a careful choice of turbulent flow data given into nonlinear machine-learning models is necessary as well as preparing appropriate learning formulations to embed prior knowledge of turbulence. With this in mind, we can study multi-source turbulent flow data with nonlinear machine-learning techniques. The proposed approach to seeking a manifold of multi-source turbulent flow data offers guidance on how to examine turbulence characteristics across data sets with varying spatiotemporal data resolutions, measurement techniques and length scales, enhancing the reliability of cross-method comparisons and comprehensive data-fusion analyses of turbulence.
Funding
K.T. thanks the support from the US Air Force Office of Scientific Research (FA9550-21-1-0178) and the US Department of Defense Vannevar Bush Faculty Fellowship (N00014-22-1-2798). Part of the machine-learning analysis was performed on Delta GPU at the National Center for Supercomputing Applications (NCSA) through the ACCESS program (Allocation PHY230125).
Declaration of interests
The authors report no conflict of interest.
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