Source-linked AI summary
Searching for turbulence models by artificial neural network
Masataka Gamahara, Yuji Hattori
TL;DR
LES lacks a universally best SGS model, motivating a search for new modeling relations. This paper tests ANN as a form-free mapping from GS flow fields to SGS stress using DNS channel-flow data. The predicted SGS stress correlation exceeded 0.7, while the learned model resembled the gradient model and remained below a two-parameter dynamic mixed model.
Problem
Existing SGS models have no universally superior choice across flows, while LES accuracy is difficult to validate when experimental or DNS results are unavailable.
Method
ANN is trained to establish a GS-flow-field-to-SGS-stress relation without assuming the function form, using DNS data from turbulent channel flow.
Results
The correlation between real SGS stress and ANN output exceeded 0.7, and ANN established a model similar to the gradient model.
Takeaways & Limitations
ANN can serve as a tool for finding a new SGS model, with correlations comparable to or larger than similarity models but smaller than a two-parameter dynamic mixed model.
Abstract
from arXiv · showhide
Artificial neural network (ANN) is tested as a tool for finding a new subgrid model of the subgrid-scale (SGS) stress in large-eddy simulation. ANN is used to establish a functional relation between the grid-scale (GS) flow field and the SGS stress without any assumption of the form of function. Data required for training and test of ANN are provided by direct numerical simulation (DNS) of a turbulent channel flow. It is shown that ANN can establish a model similar to the gradient model. The correlation coefficients between the real SGS stress and the output of ANN are comparable to or larger than similarity models, but smaller than a two-parameter dynamic mixed model.
I. INTRODUCTION
The introduction motivates using ANN to discover an SGS-stress model without prescribing its functional form, addressing limitations and stagnation in existing modeling approaches.
- Existing SGS models have no universally superior choice across flows, although dynamic and mixed models often outperform non-dynamic models.
- LES accuracy remains difficult to assess when experimental or DNS validation data are unavailable, despite that being when LES is most needed.
- Developing a substantially better SGS model is difficult because modeling possibilities are numerous and essentially new ideas have been lacking.
- The paper tests whether ANN can find a subgrid model as a first step toward extracting an SGS relation beyond an assumed human-designed form.
- Unlike prior ANN applications that mainly optimized model constants or addressed other quantities, this work targets the SGS stress from convective terms.
- The authors aim to establish a GS-flow-field-to-SGS-stress functional relation without assuming its form, an approach they describe as not previously attempted.
II. NUMERICAL METHODS
The numerical-methods outline defines SGS stress as the residual associated with filtering and trains ANN to relate the GS flow field to that stress using DNS-derived data.
- LES filters small fluctuations from a flow variable, leaving the resolved or GS flow field governed by filtered Navier–Stokes equations.
- The SGS stress tensor represents the residual effects of unresolved fluctuations and depends on the GS flow field for modeling.
- ANN is used to establish a functional relation between the GS flow field and SGS stress tensor.
- DNS supplies GS-flow inputs and SGS-stress training targets, after which ANN is trained and evaluated on DNS data excluded from training.
B. Direct numerical simulation
The study generates training and test data from DNS of turbulent channel flow using high-order spatial discretization, periodic homogeneous directions, and wall-normal grid refinement.
- Training and test data are obtained from DNS of a turbulent channel flow.
- The DNS uses a sixth-order compact scheme in the wall-normal direction and Fourier collocation in streamwise and spanwise directions with periodic boundaries.
- Non-uniform wall-normal grids resolve the boundary layers, while Poisson equations are solved efficiently in Fourier space.
- The friction Reynolds numbers simulated are Reτ = 180, 400, 600 and 800.
- DNS validity is checked through the mean-flow wall and log-law regions and fine vortical structures observed in prior work.
C. Preparing data for training and test
DNS data are filtered to produce coarse-grid GS inputs and SGS-stress targets, then divided into training and test data for feedforward ANN models. The study uses pointwise GS variables to predict each independent SGS-stress component separately.
- Training uses randomly selected streamwise positions and corresponding yz planes, while the whole dataset tests the trained ANN.
- DNS data are filtered with a top-hat function to generate coarse-grid GS flow fields used as ANN inputs.
- The SGS stress is calculated from DNS data and used as the ANN training target.
- The ANN uses input, hidden, and output layers with sigmoid activation and back-propagation training that minimizes output-target differences.
- Each SGS-stress component is modeled independently, requiring six ANNs for the tensor components.
- Inputs are pointwise GS quantities at the same location as the target SGS stress, with four tested variable sets incorporating strain, vorticity, and wall-normal position.
III. RESULTS
The results examine input-variable choices and ANN correlations with DNS SGS stress. Including vorticity improves some correlations, while stress components with small amplitudes remain difficult to approximate.
- Including Ω produces a little improvement, with {S, Ω, y} yielding three correlation coefficients above 0.7.
- The set {S, Ω, y} performs worse than {∇u, y} despite having essentially the same degree of freedom.
- All four input sets give correlation coefficients above 0.7 for τ11, the largest SGS-stress component.
- Correlation coefficients are evaluated between DNS SGS stress and ANN predictions, averaged in the streamwise and spanwise directions.
B. How successful is the learning?
For Reτ = 180, the ANN reproduces DNS SGS-stress distributions and their average and rms amplitudes at a plane where component fluctuations are nearly largest. The reported agreement indicates successful learning for this case.
- The comparison uses Reτ = 180 and the plane y = 0.1, where rms amplitudes of all components are nearly largest.
- ANN reproduces the DNS patterns in SGS-stress distributions fairly well after component values are normalized to [0, 1].
- Good agreement is observed between ANN and DNS for the average and rms amplitude of each SGS-stress component.
C. Basic features of learning
ANN learning improves with larger hidden layers but trades accuracy against computational cost. Performance is strongest for smaller filter sizes and generally weak near the wall.
- Training data: Across five tested training datasets, the difference between each correlation coefficient and the average is less than 0.043.
- Spatial dependence: Correlation coefficients are small in the near-wall region y+ < 10 but show little positional dependence outside it.The near-wall region includes the viscous sublayer and part of the buffer layer.
- Network size: Correlation coefficients exceed 0.7 when the hidden-layer neuron count n is at least 50.Increasing n improves approximation, but gains are slow for n ≥10.
- Network size: Larger hidden layers accelerate training-error decay and reduce the final error after 1000 iterations, while increasing calculation time.The choice of n therefore balances approximation accuracy and computational cost.
- Filter size: Successful learning occurs for ∆+ ≲20, corresponding to roughly 3–4 times the grid spacing.Correlation coefficients decrease quickly as filter size becomes large.
- Reynolds number: Correlation depends little on Reynolds number, although it is larger for Reτ = 400 than for the other three cases.The reported correlation coefficients are averaged over the whole domain.
D. Applicability to higher Reynolds numbers
The study tests whether ANN trained at Reτ = 180 can predict SGS stress at Reτ = 400, addressing the lack of DNS training data for high-Reynolds-number LES. The predicted fields reproduce DNS spatial patterns and achieve correlations above 0.7.
- Results: Despite differences in magnitude, ANN reproduces the spatial patterns present in the DNS data.
- Results: Correlation coefficients between DNS and ANN are larger than 0.7 for the higher-Reynolds-number prediction.
- Applicability: These results support using ANN trained at low Reynolds numbers for LES at high Reynolds numbers.
E. What ANN has learned?
ANN identifies a compact relation between the GS flow field and SGS stress that resembles the gradient model. Removing irrelevant inputs improves learning, while the ANN and gradient-model correlations are comparable and remain large.
- Input selection: ANN learning improves when irrelevant components of the input variables are eliminated.The study removes input components one at a time and checks whether correlation coefficients decline substantially.
- Model form: The preferred model form reflects that grid spacings in the y direction are much smaller than in the other two directions.
- Model comparison: The gradient model has high correlation with DNS, comparable to the correlation between ANN and DNS.
- Model comparison: ANN–gradient-model correlation coefficients are slightly smaller than those in the other two comparison rows but remain large.
- Inferred model: ANN establishes a model similar to the gradient model.The inferred structure is examined after identifying the input variables needed for accurate prediction.
IV. CONCLUDING REMARKS
ANN established a functional relation between the grid-scale flow field and SGS stress without assuming a functional form, using DNS-derived training and test data. Its predictions agreed spatially with filtered DNS data, produced correlations above 0.7, and resembled the gradient model, while remaining below a dynamic two-parameter mixed model.
- DNS data supplied training targets, and held-out DNS data were used to test the trained ANN.
- Correlation coefficients between ANN-predicted and DNS-derived SGS stress exceeded 0.7.
- ANN-predicted SGS stress had good spatial agreement with SGS stress obtained by filtering DNS data.
- ANN most likely established a model close to the gradient model because small-filter SGS stress is approximated by Leonard and cross-stress terms.
- Learning succeeded only for small filter sizes, approximately + ≲20, limiting the demonstrated operating range.
- ANN established a functional relation between the GS flow field and SGS stress without assuming its functional form.
- ANN correlations were comparable to or larger than similarity models but smaller than those of a dynamic two-parameter mixed model.
- The study trained tensor components separately and notes that symmetry should be incorporated when formulating a trained ANN.
L2-error
The cited figures examine L2 error, correlation coefficients, SGS-stress comparisons, and τ11 distributions across filter-size conditions. Correlations are averaged over the whole domain, but the supplied passages do not report numerical figure outcomes.
- Figure 10 presents the L2-norm error between ANN output and the training target as a function of an unspecified variable.
- The supplied figure-related passages do not provide numerical L2-error values or explicit plotted comparison outcomes.
- Figure 11 presents correlation coefficients between τ from DNS and another quantity, with the supplied caption fragment incomplete.
- The reported correlation coefficients are averaged over the whole domain and examined for dependence on filter size.