N22A-T016 TITLE: DIGITAL ENGINEERING - Data-Driven Hypersonic Turbulence Modeling Toolset

OUSD (R&E) MODERNIZATION PRIORITY: Artificial Intelligence (AI)/Machine Learning (ML);Hypersonics

TECHNOLOGY AREA(S): Air Platforms;Weapons

OBJECTIVE: Formulate, implement, and validate data-driven turbulence models for Reynolds Averaged Navier-Stokes (RANS) closure applicable to hypersonic flows with favorable pressure gradients, adverse pressure gradients, shock wave/turbulent boundary layer interaction (STBLI), and high heat flux.

DESCRIPTION: Hypersonic weapons are exposed to harsh operating environments requiring careful calculation of turbulent boundary layers to accurately estimate heat transfer and design thermal protection systems. Given the wide range of altitudes and velocities hypersonic vehicles operate in, the Navy requires a flexible modeling approach. However, direct numerical simulation data, let alone flight test or even wind tunnel experimental data, is expensive to develop and covers only very specific flight profiles. Faster, cheaper modeling approaches are needed to enable design for entire mission profiles. Modeling approaches, such as RANS equations that are well established for incompressible flow, provide inconsistent results, deviating by more than 50% from data when modeling relevant hypersonic flows, especially for STBLI [Refs 2, 3]. The principal problem lies in the models used to determine Reynolds Shear Stresses and turbulent heat flux required to close the RANS equations; existing methods are inadequate for hypersonic flow.

Over the last decade, improvements have been made in the development of data-driven techniques to close the RANS equations. Application of machine learning (ML) provides a powerful extension to empirical and semi-empirical methods common for developing and tuning closure models. ML allows application of much larger data sets with higher accuracy, removing some of the need for assumptions in traditional closures. These approaches typically use available Direct Numerical Simulations (DNS) or Large Eddy Simulations (LES) data sets to train ML models that can then be used on flows for which no high fidelity, scale-resolved results are available. Wang et al. [Ref 4] have improved on legacy RANS closures in square ducts with varying Reynolds number and flows with massive separation with varying Reynolds number and varying geometry. Wang et al. [Ref 5] extended the technique to hypersonic flat plate turbulent boundary layers and obtained substantial improvements over RANS on Mach 8 flow, even using only Mach 2 DNS results; even better results were obtained from an aggregate of Mach 6 and Mach 2 models. Wang’s [Ref 5] results point to the potential applicability of data-driven approaches to improve RANS modeling for more generalized hypersonic flow fields. Not only have these approaches been able to provide more accurate modeling, they also can be used to quantify uncertainty [Ref 1]. Uncertainty quantification is especially important for ML and other empirical approaches, which can experience losses in accuracy away from design conditions.

These data-driven applications are, however, not straightforward. Developing these models requires addressing such problems as defining input and output flow field variables for ML that have physical significance, are normalized, and have Galilean invariance [Ref 6]. Additionally, ML on DNS data cannot be used to simply replace terms in the RANS models, as ill-conditioning of the RANS equations and errors in mean flow quantities will result [Ref 1]. ML approaches are commonly used to predict discrepancies between RANS and DNS data [Refs 1, 4, 5] to train the model to predict the discrepancies between RANS calculations and DNS data throughout the flow field, but how this information is used to improve predictions of quantities of interest (such as heat transfer or separation region location) varies. These discrepancies can be used to adjust existing closure models [Ref 1], adjust model parameters [Ref 10], or to correct Reynolds Stress terms [Refs 4, 5]. Added to this is the general difficulty of ML in determining the scope of applicability of results, amplified in studying hypersonic flow by variations in Mach number, Reynolds number, flow geometry, and shock geometry that can substantially change the character of flow.

Data driven approaches offer great potential for improving the speed and accuracy of existing hypersonic turbulence models, but product development must take into account the facts that (1) ML corrections to RANS models apply only to a range of flight profiles and vehicle geometries, (2) we must know when a particular ML model loses accuracy due to a change in flow configuration, and (3) ML models can be developed using a wide range of training sets with different choices as to which ML approach (i.e., random forest, neural network, etc.) and different approaches to using the model data to obtain quantities of interest.

PHASE I: Formulate and assess methodologies to improve RANS turbulence models for hypersonic flows using data driven approaches. Specifically, we are seeking a proof of concept for an add-on compatible with existing CFD codes. Significant improvements in the prediction of heat transfer, skin-friction and pressure in attached and separated hypersonic flows are required. Validation against relevant hypersonic experimental data and DNS will be a key consideration towards successful phase transition. The analysis must show that the proposed methodology improves agreement with existing datasets over a wide range of relevant flow conditions. Develop a Phase II plan.

PHASE II: Expand the capabilities and flow configurations of the add-on developed in Phase I. Emphasis should be placed on expanding the models to a wider range of flow geometries, Mach numbers, Reynolds numbers, wall temperature ratios and flight enthalpies. For instance, add ML models based on different training datasets and a variety of data-driven approaches to provide improved accuracy for different flow regimes. Generation of new DNS training datasets can be performed as needed to eliminate gaps in existing datasets. Inclusion of boundary layer transition effects (i.e., length and shape of the transition region and heat transfer overshoot) are needed to increase the applicably of RANS to flow with laminar, transitional and fully turbulent regions. Any new features should be assessed for accuracy.

PHASE III DUAL USE APPLICATIONS: Automate user choice in specific model and flow parameters. Apply uncertainty estimation methods such as those surveyed in Ref 1 to determine which of the expanded training sets, ML models, and closure methods (i.e., Reynolds Stress estimation, coefficients, closure models) will provide the best result for the particular flow profile under consideration, taking into account factors such as geometry, Mach number, Reynolds number, and target quantities of interest (i.e., separation region location and size, heat transfer, etc.). Provide an automated, flexible means of assessing turbulent boundary layers, especially in STBLI without requiring dedicated knowledge and experienced judgment needed to determine the ideal data and model for different flow problems. As with Phase II, specific details of breadth of flows that automation is applicable to and depth of accuracy and detail available, is left to assessment of market need and available developmental resources.

REFERENCES:

1. Duraisamy, Kathik et al. "Turbulence Modeling in the Age of Data." Annual Review of Fluid Mechanics, vol. 51, 2019, pp. 1-23. https://arxiv.org/pdf/1804.00183.pdf.
2. Holden, Michael et al. "Comparisons of Experimental and Computational Results from "Blind" Turbulent Shock Wave Interaction Study Over Cone Flare and Hollow Cylinder Flare Configurations." AIAA Aviation Conference, Atlanta, GA, 2014. https://cubrc.org/_iassets/docs/6_Wadhams_AIAA_Atlanta_SWTBI.pdf.
3. Georgiadis, Nicholas J. et al. "Status of Turbulence Modeling for Hypersonic Propulsion Flowpaths., NASA/TM-2012-217277. https://ntrs.nasa.gov/api/citations/20120008521/downloads/20120008521.pdf?attachment=true.
4. Wang, Jian-Xun et al. "A Physics Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies based on DNS Data." Physical Review of Fluids, March 2017. https://arxiv.org/pdf/1606.07987.pdf.
5. Wange, Jian-Xun et. al. "Prediction of Reynolds Stress in High Mach Number Turbulent Boundary Layers using Physics Informed Machine Learning." Theoretical and Computational Fluid Dynamics, Vol 33, 2019, pp. 1-29. https://arxiv.org/abs/1808.07752.
6. Wu, J.L. et al. "Physics- Informed Machine Learning Approach for Augmenting Turbulence Models—A Comprehensive Approach." Physical Review Fluids, Vol 3, 2018. https://arxiv.org/pdf/1801.02762.pdf.
7. Zhang, Chao et al. "Direct Numerical Simulation Database for Supersonic and Hypersonic Turbulent Boundary Layers." AIAA Journal, Vol 56, No. 11, 2018. https://arc.aiaa.org/doi/pdfplus/10.2514/1.J057296.
8. Duraismy, Karthik et al. "Augmentation of Turbulence Models Using Field Inversion and Machine Learning." AIAA SciTech Forum, 55th Aerospace Sciences Meeting, Grapevine Texas, Jan 2017. https://deepblue.lib.umich.edu/bitstream/handle/2027.42/143032/6.2017-0993.pdf?isAllowed=y&sequence=1.
9. Gnoffo, Peter et al. "Uncertainty Assessments of 2D and Axisymmetric Hypersonic Shock Wave- Turbulent Boundary Layer Interaction Simulations at Compression Corners." 42nd AIAA Thermophysics Conference, 27-30 June 2011, Honolulu, Hawaii. https://arc.aiaa.org/doi/pdf/10.2514/6.2011-3142.
10. Durbin, Paul. "Some Recent Developments in Turbulence Closure Modeling." Annual Review of Fluid Mechanics, Vol. 50, 2018, pp. 77-103. https://www.annualreviews.org/doi/abs/10.1146/annurev-fluid-122316-045020.

KEYWORDS: Turbulence modeling; data-driven; machine learning; ML; hypersonics; boundary layers; Reynolds-averaged Navier–Stokes equations; RANS; Direct Numerical Simulations; DNS; Large Eddy Simulations; LES