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From coarse wall measurements to turbulent velocity fields through deep learning

Alejandro Güemes, Stefano Discetti, Andrea Ianiro, Beril Sirmacek, Hossein Azizpour, Ricardo Vinuesa

arXiv:2103.07387v2physics.flu-dyn

TL;DR

This paper examines whether coarse wall measurements can support reconstruction of turbulent wall fields and wall-parallel velocity fields. It applies SRGAN to both tasks and reports accurate wall-field enhancement at moderate downsampling and better flow reconstructions than FCN-POD at fd = 4 and 8.

  • Problem

    The paper addresses reconstructing high-resolution wall measurements and wall-parallel flow fields from coarse wall measurements.

    Source: §I · p. 2

  • Method

    The study applies SRGAN to wall-field resolution enhancement and to estimating wall-parallel velocity fields from wall measurements, comparing flow reconstruction with FCN-POD.

    Source: §I · p. 2, B. Super-resolution generative… · p. 2, C. POD-based fully-convolutional… · p. 3

  • Results

    SRGAN accurately enhances wall fields at fd = 4 and 8 and outperforms FCN-POD for flow reconstruction at fd = 4 and 8, while their errors converge at fd = 16.

    Source: §IV · p. 5, §V · p. 9

  • Takeaways & Limitations

    The authors identify potential for the approach in closed-loop control applications relying on non-intrusive sensing.

    Source: §V · p. 9

    The study trains on high-resolution DNS data, whose computational cost and requirements increase with Reτ and make such data impossible to obtain at real-life Reynolds numbers.

    Source: §V · p. 9

Abstract

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This work evaluates the applicability of super-resolution generative adversarial networks (SRGANs) as a methodology for the reconstruction of turbulent-flow quantities from coarse wall measurements. The method is applied both for the resolution enhancement of wall fields and the estimation of wall-parallel velocity fields from coarse wall measurements of shear stress and pressure. The analysis has been carried out with a database of a turbulent open-channel flow with friction Reynolds number $Re_{\tau}=180$ generated through direct numerical simulation. Coarse wall measurements have been generated with three different downsampling factors $f_d=[4,8,16]$ from the high-resolution fields, and wall-parallel velocity fields have been reconstructed at four inner-scaled wall-normal distances $y^+=[15,30,50,100]$. We first show that SRGAN can be used to enhance the resolution of coarse wall measurements. If compared with direct reconstruction from the sole coarse wall measurements, SRGAN provides better instantaneous reconstructions, both in terms of mean-squared error and spectral-fractional error. Even though lower resolutions in the input wall data make it more challenging to achieve highly accurate predictions, the proposed SRGAN-based network yields very good reconstruction results. Furthermore, it is shown that even for the most challenging cases the SRGAN is capable of capturing the large-scale structures that populate the flow. The proposed novel methodology has great potential for closed-loop control applications relying on non-intrusive sensing.

I. INTRODUCTION

Deep neural networks have been applied to several fluid-mechanics problems, while GANs and CNNs have been used to enhance flow-field resolution. This work extends GAN-based reconstruction to high-resolution wall measurements and wall-parallel flow fields from coarse wall inputs.

I. INTRODUCTION

  • DNN applications in fluid mechanics include improving Reynolds-averaged simulations, extracting turbulence theory, predicting temporal dynamics, and embedding physical laws.The introduction also notes applications across medicine, climate, wildlife ecology, physics, and sustainability.Source: §I · p. 1
  • GANs use competing generator and discriminator networks, and have been applied to enhance resolution in flow fields, including cylinder wakes and smoke data.Earlier flow-field GAN studies used paired high- and low-resolution data; unsupervised applications to homogeneous turbulence and channel flows were also reported.Source: §I · p. 1
  • Prior CNN methods directly reconstructed flow fields from wall shear and pressure, or combined CNNs and FCNs with proper orthogonal decomposition.Ref. 28 reported similar prediction accuracy through downsampling factor 4.Source: §I · p. 1
  • Ref. 29 reported that shallow neural networks outperformed POD for reconstructing a cylinder wake, sea-surface temperature, and decaying homogeneous isotropic turbulence.This provides an additional data-driven reconstruction approach alongside CNN and FCN methods.Source: §I · p. 2
  • The study proposes GAN-based recovery of high-resolution wall measurements and extends the approach to reconstruct wall-parallel flow fields from coarse wall inputs.The flow-field method is compared with FCN-POD, selected for its demonstrated ability to use low-resolution input information.Source: §I · p. 2

II. METHODOLOGY

The paper defines the coordinate and fluctuation notation used throughout, including the wall quantities supplied as reconstruction inputs.

II. METHODOLOGY

  • The study uses x, y, and z for the streamwise, wall-normal, and spanwise directions, respectively.The corresponding instantaneous velocity fluctuations are denoted u, v, and w.Source: §II · p. 2
  • The instantaneous velocity fluctuations are denoted u, v, and w for the x, y, and z directions, respectively.These coordinate and velocity conventions apply throughout the paper.Source: §II · p. 2
  • The wall-input quantities are streamwise and spanwise wall-shear-stress fluctuations, τwx and τwz, and wall-pressure fluctuations, pw.These quantities are used to reconstruct wall-parallel fluctuating velocity fields.Source: §II · p. 2

A. Dataset description

The study uses a turbulent open-channel DNS database at Reτ = 180, with training and test samples and downsampled wall fields as inputs.

A. Dataset description

  • The database is a DNS of turbulent open-channel flow at friction Reynolds number Reτ = 180, generated with the pseudo-spectral code SIMSON.The domain is 4πh × h × 2πh, represented by 65 Chebyshev modes in the wall-normal direction and 192 Fourier modes in each other direction.Source: A. Dataset description · p. 2
  • Training uses 50,400 samples separated by Δt+ = 5.08, and testing uses 3,125 samples separated by Δt+ = 1.69.The superscript + denotes inner scaling based on friction velocity for velocities and viscous length for distances.Source: A. Dataset description · p. 2
  • Wall inputs comprise streamwise and spanwise shear stress and pressure fluctuations, downsampled by factors fd = [4,8,16].Each downsampling factor reduces the number of field points to fd^-2 of the original; velocity reconstruction is assessed at y+ = [15,30,50,100].Source: A. Dataset description · p. 2

B. Super-resolution generative adversarial networks

The SRGAN uses a generator to produce high-resolution fields and a discriminator to distinguish generated from real fields; its generator is trained with a perceptual loss.

B. Super-resolution generative adversarial networks

  • The SRGAN generator maps a low-resolution field to a high-resolution output, while the discriminator distinguishes generated outputs from real high-resolution fields.The generator uses 16 residual blocks, with log2(fd) sub-pixel convolution layers to increase resolution; these layers are removed for full-resolution wall inputs.Source: B. Super-resolution generative… · p. 2
  • The generator’s perceptual loss uses pixel-based mean-squared error between the high-resolution reference and generated fields as its content loss.The adversarial loss is weighted by λ = 10^-3.Source: B. Super-resolution generative… · p. 2
  • Figure 1 schematizes the SRGAN generator and FCN-POD architectures, identifying layer types by color and giving convolution kernel sizes and filter counts.The legend includes 2D convolution, parametric-ReLU, batch normalization, sub-pixel convolution, ReLU, and max-pooling layers.Source: Figure 1 · p. 3

C. POD-based fully-convolutional networks

FCN-POD reconstructs flow fields by predicting local POD coefficients from wall quantities and projecting them onto subdomain bases.

C. POD-based fully-convolutional networks

  • FCN-POD predicts a tensor of subdomain POD coefficients from wall quantities, then projects each subdomain’s coefficients onto its basis to recover the flow field.For Reτ = 180, fields are divided into 12 × 12 subdomains of 16 × 16 grid points.Source: C. POD-based fully-convolutional… · p. 3
  • The subdomain count is chosen so that 90% of flow kinetic energy is represented by O(10^2) POD modes suitable for convolutional filters.POD is performed separately in each two-dimensional subdomain.Source: C. POD-based fully-convolutional… · p. 3
  • For coarse inputs, the FCN-POD baseline removes log2(fd) pooling layers; separate models are used for each wall-normal distance.The method does not ensure continuity between subdomains, though convolutional layers have been reported to yield reasonably smooth fields.Source: C. POD-based fully-convolutional… · p. 3

III. ASSESSMENT OF RESOLUTION ENHANCEMENT FOR WALL MEASUREMENTS

SRGAN wall-field enhancement recovers almost all visible features at fd = 4 and 8; fd = 16 loses small-scale detail but retains the largest structures. Error and spectral analyses likewise show reduced performance at stronger downsampling, particularly for spanwise shear stress and pressure.

III. ASSESSMENT OF RESOLUTION ENHANCEMENT FOR WALL MEASUREMENTS

  • At fd = 4 and 8, SRGAN recovers almost all visible flow features; at fd = 16, small-scale detail is lost while the largest structures remain well represented.Figure 2 compares DNS fields, coarse inputs, and SRGAN predictions for streamwise and spanwise wall-shear stress and pressure fluctuations.Source: §III · p. 3, Figure 2 · p. 4
  • Mean-squared-error results are excellent for fd = 4 and decay from fd = 8 to 16; at larger downsampling factors, streamwise wall-shear-stress errors are lower than errors for the other wall quantities.The authors attribute the latter pattern to streamwise alignment in streamwise wall-shear-stress fluctuations.Source: §III · p. 3, Table I · p. 4
  • For τwz and pw, only fd = 4 captures the energy distribution; at fd = 16, even the 50% energy-content level is not recovered.The broader scale distribution of τwx is associated with a lower parametric space for the SRGAN architecture, helping explain its lower error relative to the other wall quantities.Source: §III · p. 4
  • The τwx high-energy peak containing 90% or more of its energy is captured at fd = 4 and 8, but fd = 16 misses it and attenuates total energy by 50%.The figure reports pre-multiplied two-dimensional spectra with contours at 10%, 50%, and 90% of maximum DNS power-spectral density.Source: §III · pp. 3–4, Figure 3 · p. 5
  • The proposed normalized downsampling factor accounts for inner-scaled grid spacing; the authors report a fair comparison with Ref. 21 despite differing Reτ.Raw fd is a pixel ratio and does not account for viscous lengths per pixel.Source: §III · pp. 4–5

IV. PREDICTION OF TURBULENT FLOW FIELDS FROM COARSE WALL MEASUREMENTS

SRGAN improves wall-parallel velocity reconstructions over FCN-POD at downsampling factors 4 and 8, while the advantage disappears at factor 16. Its spectral and instantaneous predictions preserve energetic and large-scale flow structures across many cases, but recover fewer small scales farther from the wall or as wall data become coarser.

IV. PREDICTION OF TURBULENT FLOW FIELDS FROM COARSE WALL MEASUREMENTS

  • For fd = 4 and fd = 8, SRGAN has lower mean-squared error than FCN-POD; at fd = 16, their errors collapse as coarse wall data provide less information.The wall data contain streamwise fluctuations with characteristic lengths as small as ∼10ℓ* at fd = 1, increasing to ∼160ℓ* at fd = 16; fd = 4 and fd = 8 recover DNS small scales, but fd = 16 does not.Source: §IV · pp. 5–6
  • SRGAN captures most small-scale fluctuations at y+ = 50 where FCN-POD loses them at y+ = 30; at y+ = 100, SRGAN no longer recovers them.At y+ = 15, fd = 8 and fd = 16 capture most DNS flow features, as does fd = 8 at y+ = 30.Source: §IV · p. 5, Figure 4 · p. 6
  • Compared with Ref. 28, this study targets the entire energy spectra rather than the first 10 POD modes, making small-scale structures part of the prediction task.The authors note that coarse wall data omit small scales and the modulation effect of large scales.Source: §IV · pp. 6–7
  • At fd = 16 and y+ = 100, neither method recovers wall-normal or spanwise spectral scales above 10% of DNS energy; at y+ = 50, this also occurs for spanwise predictions without SRGAN.At fd = 16, both methods have the same mean-squared error.Source: §IV · p. 7
  • Because fluctuation attenuation can inflate instantaneous-field error, the authors use fractional spectral error to assess reconstruction by scale wavelength.Equation (4) is related to the linear coherence spectrum; Figure 7 displays Rab = 0.5 contours against DNS spectral densities.Source: §IV · p. 7, Figure 7 · p. 8
  • At y+ = 50 and fd = 8, SRGAN resembles DNS filtered to retain scales with Rab < 0.5, reducing mean-squared error from 0.603 to 0.317.The filtered comparison uses a spanwise cutoff wavelength of approximately λz+ ≈ 100.Source: §IV · pp. 7–8
  • SRGAN recovers a wider range of energetic streamwise and spanwise wavelengths for all three velocity fluctuations through fd = 8, compared with FCN-POD.Figure 6 compares predicted spectra with DNS contours across velocity components, wall distances, and downsampling factors.Source: §IV · p. 7, Figure 6 · p. 8

V. SUMMARY AND CONCLUSIONS

The study applies SRGAN to enhance coarse wall measurements and reconstruct wall-parallel velocity fields, reporting gains over FCN-POD and recovery of large-scale patterns. Its training and evaluation use turbulent open-channel DNS, which limits the demonstrated setting.

V. SUMMARY AND CONCLUSIONS

  • SRGAN significantly improves flow reconstruction over FCN-POD with full-resolution wall data, while increasing fd clearly degrades performance.The study also evaluates whether large-scale flow patterns remain recoverable.Source: §V · p. 9
  • For wall-field enhancement, SRGAN produces accurate reconstructions at fd=[4,8], while fd=16 misses small-scale contributions but preserves the large-scale footprint.The reconstructed large-scale patterns agree with filtered DNS references.Source: §V · p. 9
  • The proposed normalized downsampling factor ˜fd accounts for the fraction of viscous length per pixel, and the authors identify closed-loop control as a potential application.The study reports that SRGAN predictions agree with large-scale patterns in filtered DNS references.Source: §V · p. 9
  • Training used high-resolution DNS data, whose computational cost increases with Reτ and makes such data unobtainable at real-life Reynolds numbers.The authors identify transfer learning or experimental data as alternatives to explore, noting that experiments may be noisy and lower in spatial and temporal resolution.Source: §V · p. 9

DATA AVAILABILITY STATEMENT

The study data are available from the corresponding author upon reasonable request, and supporting code is openly available on GitHub.

DATA AVAILABILITY STATEMENT

  • Study data are available from the corresponding author upon reasonable request; supporting code is openly available on GitHub.The passage provides a Zenodo DOI for the code repository.Source: DATA AVAILABILITY STATEMENT · p. 9
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