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Deep unsupervised learning of turbulence for inflow generation at various Reynolds numbers
Junhyuk Kim, Changhoon Lee
TL;DR
Developing-flow simulations need realistic inflow, but existing approaches can be costly or depend on ad hoc assumptions. This paper trains a GAN on channel-flow DNS and combines it with an RNN for time-varying generation. The GAN generates instantaneous fields at various Reynolds numbers after training on three, while the RNN-GAN is presented as a synthetic inflow generator for turbulent channel flow.
Problem
Developing-flow simulations need realistic inflow, while recycling methods require costly auxiliary simulations and synthetic methods rely on ad hoc assumptions.
Source: §1 · p. 2Method
The authors train a GAN on channel-flow DNS fields and combine it with an RNN to generate time-varying flow.
Source: §1 · p. 2, §4 · p. 22Results
After training on three Reynolds numbers, the GAN generated instantaneous fields at various Reynolds numbers, though not statistically perfectly beyond the trained range.
Source: §4 · pp. 21–22Takeaways & Limitations
The authors conclude that the RNN-GAN can serve as a synthetic inflow generator for turbulent channel flow and may extend to boundary-layer or other simulations.
Source: §4 · p. 22The current model has about 17 million trainable parameters and successful training takes about three weeks on a single GPU.
Source: §3.2 · p. 21
Abstract
from arXiv · showhide
A realistic inflow boundary condition is essential for successful simulation of the developing turbulent boundary layer or channel flows. Recent advances in artificial intelligence (AI) have enabled the development of an inflow generator that performs better than the synthetic methods based on intuitions. In the present work, we applied generative adversarial networks (GANs), a representative of unsupervised learning, to generate an inlet boundary condition of turbulent channel flow. Upon learning the two-dimensional spatial structure of turbulence using data obtained from direct numerical simulation (DNS) of turbulent channel flow, the GAN could generate instantaneous flow fields that are statistically similar to those of DNS. Surprisingly, the GAN could produce fields at various Reynolds numbers without any additional simulation based on the trained data of only three Reynolds numbers. This indicates that the GAN could learn the universal nature of Reynolds number effect and might reflect other simulation conditions. Eventually, through a combination of the GAN and a recurrent neural network (RNN), we developed a novel model (RNN-GAN) that could generate time-varying fully developed flow for a long time. The spatiotemporal correlations of the generated flow are in good agreement with those of the DNS. This proves the usefulness of unsupervised learning in the generation of synthetic turbulence fields.
1. Introduction
The paper develops an unsupervised GAN-based inflow generator because existing methods trade computational cost against ad hoc design assumptions, while supervised autoencoder approaches have representation and initialization drawbacks. It extends the GAN with an RNN to generate time-varying flow across Reynolds numbers.
1. Introduction
- The GAN is developed to provide inflow turbulence statistically similar to fully developed channel-flow DNS fields.A realistic inflow boundary condition is needed for developing-flow simulations.Source: §1 · p. 2
- Recycling inflow methods can preserve spatial and temporal correlations but require costly auxiliary simulations, while synthetic methods rely on ad hoc assumptions.Synthetic eddy size, for example, must be selected carefully to avoid a long transition section.Source: §1 · p. 2
- Supervised autoencoder inflow generation can produce blurred or poorly represented turbulence and requires a fully developed field as its initial input.The cited prior work trained on channel-flow DNS at Reτ = 180.Source: §1 · p. 2
- Training a GAN on three Reynolds numbers and combining it with an RNN yields time-varying flow generation across Reynolds numbers without requiring an initial input field.The authors describe this generation as possible for a long time.Source: §1 · p. 2
2. Simulations for data collection
The authors ran fully developed heated-channel DNS to collect training fields for unsupervised learning. They used three Reynolds numbers and collected four flow variables on cross-sectional planes, with augmentation to expand the training data.
2. Simulations for data collection
- Fully developed turbulent channel DNS with a temperature field supplied data for an unsupervised generator; pressure gradient drives streamwise flow and wall temperature difference induces wall-normal heat flux.The governing equations include continuity, Navier–Stokes, and energy equations.Source: §2 · p. 3
- The dimensionless variables scale coordinates, velocities, and temperature by channel half-width, friction velocity, and half the wall temperature difference; Reτ and Pr are the input parameters.Reτ is defined using friction velocity, channel half-width, and kinetic viscosity; Pr is the kinetic-viscosity to thermal-diffusivity ratio.Source: §2 · p. 3
- The simulations used three Reynolds numbers, periodic streamwise and spanwise boundaries, and no-slip, constant-temperature wall conditions.A pseudo-spectral method handled horizontal directions and a Chebyshev-tau method the wall-normal direction.Source: §2 · p. 3
- The training fields contain streamwise, wall-normal, and spanwise velocities plus temperature on y–z planes at Reτ = 180, 360, and 540.For each Reynolds number, 20,000 fields were collected over more than 18,000 wall time units at intervals of 0.9.Source: §2 · p. 3, Figure 1 · p. 4
- Mirror augmentation quadrupled the data, and spanwise random phase shifts were also applied; preprocessing normalized variables and fixed generated fields at 192 × 192.The training y-grid count was matched to 192.Source: §2 · pp. 3–4
3. Unsupervised learning for inflow generation
The section examines GAN-based generation of turbulent inflow fields, extending instantaneous generation across Reynolds numbers with an RNN-GAN for long, time-varying flows. It describes comparisons with DNS and the models’ scope, including generation across Reynolds numbers and expanded spanwise domains, alongside limitations in spatial correlation and Reynolds-number extrapolation.
3.1. Generation of Instantaneous flow fields using GAN
- WGAN-GP improved turbulence fields over classical GAN, while minibatch-standard-deviation and normalization techniques helped training; progressive growing showed no noticeable benefit at 192 × 192 resolution.The model generated u, v, w, and T fields, with streamwise vorticity ωx added to the discriminator; a statistical constraint [27] on spanwise energy spectra further improved results, with best tested λ1 = 10.Source: §3.1 · pp. 4–5
- At Reτ = 180, GAN-generated mean and rms profiles, pointwise correlations, and normalized streamwise-vorticity rms agreed well with DNS.The generated fields also represented small-scale structures so well that they were difficult to distinguish from DNS fields.Source: §3.1 · pp. 6–7, Figure 3 · p. 7, Figure 4 · p. 8
3.2. RNN-GAN for time-varying flow generation
- The RNN-GAN advances a latent vector over time with IndyLSTM driven by normally distributed random noise, while a temporal discriminator evaluates sequences of generated flow fields.The compressed latent update requires relatively little memory and computation; Dtime uses 3D convolutions to extract temporal variation.Source: §3.2 · pp. 9, 13
3.1. Generation of Instantaneous flow fields using GAN
- Fields generated at Reτ = 270, 450, and 720 reproduced near-wall structures despite training only at Reτ = 180, 360, and 540, though profile accuracy weakened above Reτ = 540.The reported results suggest near-wall structures in wall coordinates are similar across Reynolds numbers, while mean and mean-square velocity profiles become inaccurate beyond the trained range.Source: §3.1 · p. 9, Figure 6 · p. 11, Figure 7 · p. 12
3.2. RNN-GAN for time-varying flow generation
- To promote stationary long-term output, the RNN was trained over 500 steps and a discriminator checked whether latent outputs and their moments followed a normal distribution.The paper notes that short-step training does not guarantee stationarity beyond the training interval and that early RNN outputs can be transient.Source: §3.2 · p. 13
- Training only the RNN caused temporal correlations to decay, so the model also trained the pretrained generator using spatial, temporal, and latent-distribution discriminators and statistical constraints.IndyLSTM replaces fully connected recurrent dependence with elementwise products to help prevent vanishing and exploding gradients, and performed better than traditional LSTM in the authors’ test.Source: §3.2 · pp. 13–15
- After 10,000 steps, RNN-GAN maintained statistically stationary flow with near-wall structures and energy spectra close to DNS, while the AE produced poorly resolved fields and deviant spectra.The spectra differed slightly at high wavenumbers; the resolved spatial correlations were considered adequate to prevent rapid dissipation of turbulence in a main simulation.Source: §3.2 · pp. 15, 18, Figure 9 · p. 16, Figure 10 · p. 17
- Generated velocity and temperature temporal correlations agreed favorably with DNS, including distinct near-wall and center behavior, and the DNS's spurious periodic rebound disappeared.The disappearance indicates that training-data simulations need not greatly expand the streamwise domain to avoid that periodicity.Source: Figure 11 · p. 17, §3.2 · p. 18
- RNN-GAN's spatial correlations were slightly worse than instantaneous GAN's, Reynolds-number extrapolation was weaker than interpolation, and training required about three weeks on one GPU for approximately 17M parameters.More training data, hyperparameter tuning, physical constraints, or a better extrapolation scheme were proposed as possible improvements; reducing feature maps could lower cost but might reduce inflow quality.Source: §3.2 · pp. 18, 21
- RNN-GAN captured Reynolds-number effects in wall-coordinate rms profiles and generated realistic near-wall structures at Reτ = 450 and 720 beyond the trained Reτ = 360, though statistics were not exact DNS matches.It also reproduced the decrease in temperature rms with increasing Reτ and showed statistically good agreement with DNS for streamwise vorticity.Source: §3.2 · p. 18, Figure 12 · p. 19, Figure 13 · p. 20
- Different RNN noise inputs yielded partially different flows after 10 steps and effectively uncorrelated flows after 100 steps, providing stochastic inlet variation.Using the same noise input through step 10,000 reproduced the same field.Source: §3.2 · p. 18, Figure 14 · p. 20
- RNN-GAN generated a twice-wider spanwise field by combining independent latent-space images, with no observed artifacts at their junction, though uncaptured large-scale motions remained unavailable.The passage notes that larger motions absent from the DNS training data cannot be represented.Source: §3.2 · p. 18, Figure 15 · p. 21
4. Conclusions
The RNN-GAN generates time-varying turbulent channel-flow fields at various Reynolds numbers, with temporal correlations and spanwise energy spectra in good agreement with DNS. Its instantaneous GAN component reproduces DNS statistics across Reynolds numbers, though fields beyond the training range are not statistically perfect.
4. Conclusions
- After training on three Reynolds numbers, the GAN generated fields at various Reynolds numbers, but did not produce statistically perfect fields beyond the trained range.Extrapolation was less successful than interpolation within the trained range.Source: §4 · pp. 21–22
- The instantaneous GAN used WGAN-GP and an energy-spectrum statistical constraint; its generated fields had characteristics similar to DNS and statistical profiles in very good agreement.The authors suggest that underlying statistical characteristics of turbulence may help explain why unsupervised learning works.Source: §4 · p. 21
- The trained RNN-GAN generated fully developed flow fields for a long time at various Reynolds numbers, with time correlations and spanwise energy spectra in good agreement with DNS.The generated flow also varied stochastically, and its domain size could be varied.Source: §4 · p. 22
- The authors identify the model as a potential synthetic inflow generator for developing turbulent channel-flow simulations, and suggest the framework may extend to boundary layers or other simulations.The proposed extensions are possibilities, not results tested in this study.Source: §4 · p. 22
Appendix A. Effect of the statistical constraint
The wave-space energy-spectrum constraint improved generated-field quality and GAN learning speed; its strength affected transfer learning and required some tuning.
Appendix A. Effect of the statistical constraint
- With the statistical constraint, GANs trained faster and produced better-quality fields than without it; λ1 = 10 also better prevented sudden RNN-GAN transfer-learning deterioration than λ1 = 1.The strength effect was not large, and the optimal value depended on the statistic; the comparison was made after 30 000 iterations.Source: Appendix A · p. 22
Appendix B. Effect of data augmentation
Mirror and spectral augmentation improved how well the GAN represented channel-flow symmetry and homogeneity, and helped stabilize early training.
Appendix B. Effect of data augmentation
- Mirror augmentation was intended to reflect channel-flow symmetry, while spectral augmentation was intended to reflect statistical homogeneity.The study compared GANs trained with and without these schemes.Source: Appendix B · p. 22
- The augmented GAN represented the statistically symmetric and homogeneous characteristics better than the GAN without augmentation.The comparison used pointwise-averaged streamwise velocity after 100 000 training iterations.Source: Figure B.17 · p. 23, Appendix B · p. 24
Appendix C. Inflow generation based on the supervised learning
The supervised autoencoder can be recursively rolled out, but its generated fields lose small-scale detail and do not accurately reproduce DNS spatiotemporal correlations. Its use across Reynolds numbers is also limited by poorer rms profiles and the need for a developed DNS initial field.
Appendix C. Inflow generation based on the supervised learning
- The AE encodes the previous field into a low-dimensional representation, uses an MLP there, and decodes the result as the next field; training uses DNS input and an MSE loss.The encoder and decoder use convolution layers with downsampling and upsampling; the AE's compression is intended to help recursive prediction yield statistically stationary fields.Source: Appendix C · p. 24
- The comparison AE used a 768-node MLP hidden layer and was trained only on Reτ = 180 data; training ran for 300 000 iterations without observed overfitting.The authors selected 768 nodes as the best case among several tests.Source: Appendix C · pp. 24–25
- After one AE time step (t+ = 0.9), the generated field was poorly resolved, with small-scale variations not well represented; after 100 steps (t+ = 90.0), similarity to DNS was difficult to find.Source: Appendix C · p. 25
- Although recursive prediction was possible for a long time, the AE did not accurately represent DNS spatiotemporal correlations, and its results were sensitive to MLP hyperparameters.Using three previous steps as input overfit the first step and performed worse; other statistics also showed errors relative to the proposed model.Source: Appendix C · p. 25
- For generation at a different Reynolds number, the AE required a fully developed DNS initial field at that Reynolds number, and its multi-Reynolds-number training produced poorer rms profiles than training on one Reynolds number.The authors state that their unsupervised RNN-GAN solved these issues.Source: Appendix C · p. 25