If you’ve studied turbulence you may have heard some of these phrases:
“When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." -Werner Heisenberg
“Does Turbulence need God?” -William K. George
Questions regarding the nature of God and origin of man aside, turbulence has puzzled our greatest minds for generations. Quite an unfortunate fact of life considering that much of the technology and devices we have deeply incorporated into our lifestyles deal with turbulent flow.
Since the advent of machine learning, and more precisely the recent burst of interest in academia and industry in machine learning, there has been a reinvigorated thrust for innovation in the turbulence modeling community as well.
Figure 1: Video of combustion with turbulence present
Blog mission statement:
As the landscape of methodologies and tools continue to trend digital in industrial equipment design, manufacturing, and operation, the availability of useful data is further increasing. This, in addition to more accessible sophisticated machine learning algorithms, is a beautiful recipe for advancing the state-of-the-art in simulation. This blog serves to address one of the most challenging aspects of CFD simulation (turbulence modeling) and frame it in the context of active research areas where machine learning is trying to help. If all goes as intended, the blog will be approachable to those new to turbulence modeling and machine learning, while still being informative for experts. Simcenter STAR-CCM+ and Monolith AI represent two software which fuse high-fidelity complex physics simulation and no-code machine learning software tailored towards engineering simulation.
Some founding and influential literature will be outlined in a table below to start. Then, some topics from such will be cherry-picked and explored in more detail. The first topic will be that of predicting the anisotropic nature of turbulence, and the second on the scales of turbulence (with an example exemplifying how that may impact separation predictions). These examples will strive to relate a sample of the machine learning predictions to their roots (spatially) in the CFD domain.
Figure 2: Some visualizations of the flow and turbulence in common canonical flow configurations
The idea of step-reductions in simulation run time, or augmented accuracy in simulating turbulent flow, has the research and industry communities in a frenzy of excitement. As of very recently (summer 2021), professor Karthik Duraisamy (University of Michigan) hosted a symposium on model-consistent data-driven turbulence modeling . Proceedings, agenda, and notes provided by professor Duraisamy here (link). There were sessions for: model consistency via field inversion, via integrated inference and learning, and also with evolutionary/symbolic techniques, among other things. Furthermore, important topics like benchmarking and emerging techniques were also covered. I highly recommend the reader to review this wealth of knowledge from seminal leaders in the field.
Another exciting development in the community is that from professor Weymouth (Southampton University). Also in summer 2021, professor Weymouth published a model which uses deep learning based upon the spanwise-averaged Navier–Stokes equations . This approach strives to reduce the computational expense associated with the highly three-dimensional nature of turbulence through a dimensionality reduction (averaging). These spanwise-averaged equations contain closure terms which are modelled using a convolutional neural network. For the published use case, model predictions yield 90-92% correlation with the original 3-D system while taking only 0.5% of the CPU time. In the image below, from top to bottom, the 3-D CFD solution is compared with the machine learning-based prediction as well as the 2-D CFD solution, respectively.
Figure 3: Comparisons of 3D, 2D, and machine learning predictions from Weymouth et al. 
The table below presents a succinct list of studies which apply machine learning class methods to fluid mechanics problems, specifically, utilizing results generated from simulation-based investigations. Some of the seminal works in this field are only recent to the last 5 years, making this a very exciting time for both the research community and industry.
Yarlanki from the IBM research group in 2012 was perhaps one of the earlier contributors on the relevant subject matter to the present article, whereby his study  conducted applied neural networks to a HVAC system, with the specific goal of predicting a temperature distribution. While not taking a physics-informed approach in using the machine learning tool, there are many similarities in his approach to others which followed. Lauret  applied a neural network architecture to predict gas diffusion in an atmospheric flow not long after in 2014.
In 2015, Julia Ling with the Sandia National Laboratories  published a major work which applied both neural networks and random forests to a suite of canonical problem. This is a major milestone when considering the research presented herein, on several accounts, and would go on to directly influence the directions taken in future studies [14, 15, 17, 17, 19]. First, it was among the first to apply both neural networks as well as random forest regression methods to computational fluid dynamics simulation data. Ling’s study showed that random forest methods, an extension of decision tree methods, have comparable accuracy to neural networks, and in fact provide their own unique advantages. One such advantage is the ability to acquire data from multiple sources, whereby each data set is not necessarily required to provide exactly the same features. This could allow for a data scientist to utilize one particular LES study which reported a set of features/results, and also utilize data from a separate LES study which only provided say 80% of the same variables, with some additional unique variables reported.
Another significant benefit analyzing and predicting results with random forest regression methods is the ease of constructing the study to yield physically meaningful and interpretable results. Equivalently, this is termed physics informed machine learning, and allows for the machine learnt predictions of flow field variables to be interpreted with traditional engineering principles. For example, some of Ling’s inputs (e.g. variations of vorticity) quantify the swirling and anisotropic nature of the flow throughout the domain, with the algorithm’s predicted output being the uncertainty to expect from a RANS simulation. From basic CFD fundamentals, the limitations of RANS models are known, and do not counteract or conflict with the results from Ling’s machine learning results. This has clear value, as now the machine learning algorithms developed can be physically interpreted and regulated, as appropriate.
Several authors apply machine learning algorithms to canonical flow configurations, such as turbulent channel flows, to study the nature of isotropic homogenous turbulence [4, 9, 18]. These studies target the Reynolds stresses as the output parameter to be learned and define a variety of input features to associate the relationship thereof. The Reynolds stresses are a suitable choice, as they are one of the most transferable variables to quantify and describe the nature of the turbulence, amongst different flow configurations. These studies employ artificial neural networks, field (Bayesian) inversion, and autonomic closure methods.
Some studies [5, 7, 12] take a more applied approach, seeking to define the lift and drag on a variety of NACA shapes, to learn the relationship between airfoil shape and flow configuration to the output of these engineering quantifies. These studies apply field inversion techniques, convolutional neural networks, radial basis function neural networks, and traditional artificial neural networks.
The last category in this literature survey, summarized in the table, is that from the Stanford research group . These works consider film cooling application for machine learning methods, to improve computational fluid dynamics predictions. Specifically, the constitutive components and equations which are used to predict the turbulent flux of scalar values, embedded in the turbulence modeling framework (e.g. a basic gradient diffusion hypothesis, a constant value for turbulent Prandtl number throughout the flow field, turbulent diffusivity etc.). While having little to do with the actual hydrodynamics, this is an important facet of the model, as the distribution of temperature in a film cooling environment is often the chief result of interest for a designer. In fact, previous Stanford investigations have shown that the concluded that the turbulent diffusivity definition was the single largest error source. To address this important concern, Milani  has constructed machine learning algorithms to predict the turbulent diffusivity throughout the flow field conducted on LES and DNS data sets, on film cooling jets and canonical cases (e.g. wall mounted cube). Their results showed promise, as not only did the magnitude of the film effectiveness in the near-field injection from the machine learning predictions better approach the experimental ground truth data, but also the shape of the laterally averaged film effectiveness curves changed concavity to better match the experimental results.
Deep Dives on Specific Topics
Predicting Anisotropy in Turbulence
Whether it be a property of a material, a temperature field, or an arbitrary characteristic of a flow-field, the assumption of isotropy is seldom strictly correct. Often in engineering design work the assumption of isotropy is used simply because there is no better option available. Turbulence, unfortunately, is not an exception. Considering disturbances in a fluid flow which generate turbulence (think of shearing), they are also often not isotropic. Even circumstances like grid generated turbulence which uniformly promotes turbulence would quickly become anisotropic if subject to acceleration downstream (e.g. grid generated turbulence upstream of an asymmetric nozzle).
This anisotropic quality of the turbulence is quite cumbersome for eddy viscosity-based models to predict. Often, they take the approach to implicitly assume isotropic turbulence in their calculations for transport variables (turbulent kinetic energy); which from the preceding paragraph we now see as a departure from the truth.
Recent interest has skyrocketed in applying machine learning based approaches to learn the true nature of the anisotropy from higher fidelity simulations, such as Large Eddy Simulation (LES) which does not suffer from the same assumptions, and then somehow apply that knowledge back to lower fidelity eddy viscosity-based simulations. Ling and Wang were great leaders in this research effort, publishing numerous seminal works [21, 22, 23] which evaluated how well machine learning approaches could capture these distributions of anisotropy in the flow-field.
Aids for visualization of the anisotropy, both as contours (right) and in Lumley triangles (left), can be seen below from Hodges . For this domain of a film cooling jet, which is an applied variation of the classic jet in crossflow problem, scalar portrayal of an anisotropy ‘magnitude’ is shown as a contour within several wall normal planes. As the injected jet is deflected back towards the wall by the crossflow, the shearing between the streams indeed has higher levels of turbulence anisotropy, as well as near the wall where the gradients are highly anisotropic due to wall-blocking. As such, these regions would be difficult for eddy viscosity-based models to predict. Bear in mind that they are key regions influencing the resulting heat transfer at the wall and the aerodynamic performance; so quite important areas to get correct. The Lumley plots, with variations that are sometimes called Barycentric plots or turbulence triangles, strive to quantify whether the state of the turbulence is highly one-, two-, or three-dimensional in nature. If three-dimensional, that is equivalently isotropic.
Now let’s pivot: aside from the possible value in improving our modeling approaches with machine learning, there is also possible value in better understanding the physics we are simulating with tools borrowed from machine learning and data science. You can also think about feature engineering, which is a typical step when creating a machine learning model.
For the example pictured above, the same simulation data can be processed with the popular t-stochastic distributed neighbor embedding (t-SNE) algorithm. In making a t-SNE plot, the outputs the machine learning model is trying to predict are represented by the handful of discrete colors used. The segregation, or clustering, of each color depends on how well the features selected can be correlated to said outputs. For example- since we know smoking habits, dietary preference, and exercise patterns (these termed as the ‘inputs’) strongly correlate to cancer outcomes, a t-SNE plot showing binary classification of cancer condition (yes- and no- each receiving their own respective colors, termed as the ‘outputs’) would result in strongly segregated and separated clusters for each color. However, if we incorporated things we know are uncorrelated, such as favorite sports team, then the resulting plot and data clusters would be overlapping without clear zones of segregated colors.
Back to the film cooling flow example – in the above t-SNE plot we see each color represents a bin of anisotropy level (0-0.1, 0.1-0.2, …, 0.9-1.0). The inputs used were a large collection of fluid mechanics and turbulence parameters (e.g. turbulence intensity, wall-based Reynolds number, etc.). For moderate levels of turbulent anisotropy, levels between 0.4 – 0.5 (shown in purple), one can see a relatively clear separation from the other levels. More interestingly, these points are lassoed with a dotted line are then mapped back into the wall-normal plains of the flow-field. How fascinating – there is clear segregation! These points on the t-SNE map reside within the jetting flow stream, enclosed between the mixing layer with the freestream and the wall. Interestingly enough, it also excludes the center of the jet (potential core) in the first plane (x/D=0), which is less anisotropy due to being shielded from the shear layers.
Additionally, this means that this physical region of the flow-field is well posed for the machine learning model – that the machine learning model should be capable to provide an accurate ‘correlation’ between these set of input features and the resulting anisotropy. How exciting!
Another classic problem, which is vital to many industries, is flow over a cylinder (or bank of cylinders). A similar set of features are plotted in the mid-section of the flow domain. Some of the features may be familiar, such as turbulent viscosity ratio, while others less so and more theoretical in nature (tensor invariants of strain rate tensor ). While t-SNE may be a new concept to many readers, a contrast can be made to Principal Component Analysis. When comparing either plot generated from the same data, towards predicting the anisotropic nature of the turbulence, it is clear each has something unique to offer in regard to analysis and interpretability.
Setting Physical Scales for Turbulence
Perhaps one of the most coveted holy grails in turbulence modeling is the correct prediction of flow separation and reattachment. Regarding some of the most popular RANS models used in industry over the last decade, CFD practitioners have a very limited set of knobs to turn to responsibly calibrate their turbulence modeling for their application. The transport equation for turbulent dissipation rate equation, shown below, is one such destination that receives this attention for tweaking and tuning. One can speculate the origin of this is due to the fact that the turbulent kinetic energy equation, which has physical terms for production, destruction, and diffusion, while for turbulent dissipation there is no equivalently sensible form. The turbulent dissipation rate, and analogously the specific dissipation rate for the k-Ω based models, is how the timescale of the turbulence is determined. Whereas the turbulent kinetic energy sets the velocity scale.
Of the many options available for customization and improvements, one example we can touch on is the Cε2 coefficient, which typically is a constant value simply determined by minimizing predictive error over a suite of canonical benchmark cases. As shown in the pictures below for several different definitions for this coefficient, by changing this constant value to functions involving physical terms such as the anisotropy, one can significantly (and responsibly) vary the velocity field through this cylinder bank. Since these cases are under identical conditions otherwise, the separation patterns vary simply due to these different Cε2 definitions used.
Now, by incorporating machine learning, the possibilities continue to be enticing as we see (pictured below) that machine learning models (red) can predict anisotropy distributions in foreign simulations which match that of the reference data (LES, blue). For these two images, which are Barycentric plots, two separate wall-normal traverses are made through the domain to sample the state of the turbulence anisotropy. Again, where each corner representing one-, two-, or three-dimensional turbulence.
Now the puzzle is coming together – in one hand we can see tools in our toolbelt for modifying our separation predictions based on our physical understanding of the flow, and in the other hand we see ways to leverage high fidelity data to train machine learning models which are extremely predictive for a wide breadth of cases. This represents just one of the many possibilities for implementing machine learning to augment our existing ways of modeling turbulence.
Links to Other Articles
Monolith AI and Simcenter STAR-CCM+ bring machine learning to CFD simulations, https://blogs.sw.siemens.com/simcenter/ai-for-cfd-simcenter/
4 Myths about AI in CFD, https://blogs.sw.siemens.com/simcenter/4-myths-about-ai-in-cfd/
 Duraisamy, K. (2021, July 27). Symposium on Model-Consistent Data-driven Turbulence Modeling. http://turbgate.engin.umich.edu/symposium/index21.html.
 Font, B., Weymouth, G. D., Nguyen, V. T., & Tutty, O. R. (2021). Deep learning of the spanwise-averaged Navier–Stokes equations. Journal of Computational Physics, 434, 110199.
 Cruz, Matheus Altomare. “MACHINE LEARNING TECHNIQUES FOR ACCURACY IMPROVEMENT OF RANS SIMULATIONS.“ Universidade Federal do Rio de Janeiro, 2018.
 Parish, Eric J., Karthik Duraisamy. “A paradigm for data-driven predictive modeling using ﬁeld inversion and machine learning.“ Journal of Computational Physics, 305 (2016): 758-774.
 Singh, Anand Pratap, Shivaji Medida, Karthik Duraisamy. “Machine-learning-augmented predictive modeling of turbulent separated ﬂows over airfoils.“ AIAA Journal, (2017): 2215-2227.
 Guo, Xiaoxiao, Wei Li, Francesco Iorio. “Convolutional neural networks for steady ﬂow approximation.“ Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, 2016.
 Zhang, Yao, Woong Je Sung, Dimitri N. Mavris. “Application of convolutional neural network to predict airfoil lift coefﬁcient.“ 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2018
 Yarlanki, S., Bipin Rajendran, and H. Hamann. “Estimation of turbulence closure coefﬁcients for data centers using machine learning algorithms.“ 13th InterSociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, IEEE, 2012.
 King, Ryan N., Peter E. Hamlington, Werner JA Dahm. “Autonomic closure for turbulence simulations.“ Physical Review, E 93.3 (2016): 031301.
 King, Ryan N., Peter E. Hamlington, Werner JA Dahm. “Autonomic closure for turbulence simulations. “ Physical Review E, 93.3 (2016): 031301.
 Maulik, Romit, Omer San. “A neural network approach for the blind deconvolution of turbulent ﬂows.“ Journal of Fluid Mechanics, 831 (2017): 151-181.
 Zhang, Weiwei, et al. “Machine learning methods for turbulence modeling in subsonic ﬂows over airfoils.“ arXiv preprint, arXiv:1806.05904 (2018).
 Ling, Julia, J. Templeton. “Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty.“ Physics of Fluids, 27.8 (2015): 085103.
 Ling, Julia, et al. “Uncertainty analysis and data-driven model advances for a jet-in-crossﬂow.“ Journal of Turbomachinery, 139.2 (2017).
 Weatheritt, Jack, et al. “A comparative study of contrasting machine learning frameworks applied to RANS modeling of jets in crossﬂow.“ ASME Turbo Expo 2017: Turbomachinery Technical Conference and Exposition, American Society of Mechanical Engineers Digital Collection, 2017.
 Milani, Pedro M., et al. “A machine learning approach for determining the turbulent diffusivity in ﬁlm cooling ﬂows.“ Journal of Turbomachinery, 140.2 (2018).
 Edeling, Wouter Nico, Gianluca Iaccarino, Paola Cinnella. “Data-free and data-driven RANS predictions with quantiﬁed uncertainty.“ Flow, Turbulence and Combustion, 100.3 (2018): 593-616.
 Beck, Andrea D., David G. Flad, Claus-Dieter Munz. “Deep neural networks for data-driven turbulence models.“ arXiv preprint, arXiv:1806.04482 (2018).
 Wang, Jian-Xun, Jin-Long Wu, Heng Xiao. “Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data.“ Physical Review Fluids, 2.3 (2017): 034603.
 Wu, Jin-Long, et al. “A priori assessment of prediction conﬁdence for data-driven turbulence modeling.“ Flow, Turbulence and Combustion, 99.1 (2017): 25-46.
 Ling, Julia, Andrew Kurzawski, and Jeremy Templeton. "Reynolds averaged turbulence modelling using deep neural networks with embedded invariance." Journal of Fluid Mechanics 807 (2016): 155-166.
 Ling, Julia, et al. "Uncertainty analysis and data-driven model advances for a jet-in-crossflow." Journal of Turbomachinery 139.2 (2017): 021008.
 Wang, Jian-Xun, et al. "A comprehensive physics-informed machine learning framework for predictive turbulence modeling." arXiv preprint arXiv:1701.07102 (2017).
 Hodges, Justin, and Jayanta S. Kapat. "Topology and Physical Interpretation of Turbulence Model Behavior on an Array of Film Cooling Jets." Turbo Expo: Power for Land, Sea, and Air. Vol. 58646. American Society of Mechanical Engineers, 2019.
 S. Pope, Journal of Fluid Mechanics 72, 331 (1975).