The paradigm of field inversion applied to a wall bounded jet using PIV measurements as reference data

Background

To this date, simulating the dynamics of a fluid remain extremely expensive for most practical design problems. The large range of length and time scales to be resolved makes it especially computationally heavy. Therefore different ways of filtering are being used to reduce the computational time of such simulations, which introduce additional assumptions limiting their accuracy. An example is the Reynolds averaged Navier Stokes approach where time averaged variables are solved and the influences of fluctuations are modelled. While using filtered approaches some modelling has to be done in order to close the system. And this is where some inaccuracies are introduced.

In engineering applications the standard is still the RANS approach for CFD modelling. Most commonly the so called two equation models are used. Very common RANS two equation models are k-epsilon and k-omega models, which were introduced in 1972 and 1941 respectively. Numerous adaptation have been proposed in order to improve these models such as in 1994 by Menter and in 2008 by Wilcox. As can be seen these models are fairly old but are still used today.

The use of CFD in the aerospace design process is severely limited by the inability to accurately and reliably predict turbulent flows with significant regions of separation. More complex RANS models have been proposed such as the Reynolds Stress Transport method, where the Reynolds stresses are modelled directly. However, currently RST models are not commonly used as they lack robustness and are occasionally less accurate.

More accurate modelling approaches include LES and combinations of LES and RANS. However these models involve significantly higher computational times and are often not practical to use in engineering applications.

Managing the vast amounts of data generated by current and future large-scale simulations will continue to be problematic and will become increasingly complex due to changing HPC hardware. Therefore it seems that CFD for engineering applications has reached its plateau. 

However in recent years more and more research has been done in the applications of machine learning. The capabilities of ML has increased rapidly and are now also used in closure modelling. Along side are the continuously improving experimental capabilities which allow for much higher resolution information.

The combination can be used for data driven techniques to improve upon current RANS models. The models used in CFD modelling always have been data driven in a sense. Where models are derived from theory and a set of model coefficients are used to tune the response of the model with available experimental data. 

Calibration methods for model coefficients have been proposed over the last decade. Methods include least squares optimisation and Bayesian procedures to infer values for the model coefficients. However errors due to assumptions made in the model still remain.

Also in the recent work of Deltares such problems arise. Research has been done in water flow behind an underflow gate. The flow phenomena that occurs closely resembles that of a wall bounded jet. The reason such research has been performed is to better predict the turbulent behaviour of the jet down stream in order to predict possible damage to sediment.

In their work experiments have been performed to acquire PIV data of velocity field. This has been compared to their CFD results. Multiple CFD simulations have been performed with different levels of fidelity. Also several gate openings, changing the effective Reynolds number have used to compare the experimental results of PIV and CFD. They concluded that although the velocity field solution of CFD is good enough for engineering practices, all CFD simulations show a mismatch in the area of the shear layer between the jet and the main flow. Also the correlation of turbulent kinetic energy from the CFD results is fairly poor.

Previous Work

In the work of Parish and Duraisamy the paradigm of field inversion is proposed. Here instead of calibrating the modelling coefficients a corrective field is used to effectively address the modelling deviancy. A functional relationship for the corrective field and the solution is sought to be found using ML techniques. The ML model is ultimately used to be able to predict the corrective field for a problem outside of the data set. This paradigm thus consists of two steps, first field inversion using the Bayesian framework is used to infer corrective fields for a sufficient large number of problems. After that ML is used to generate a predictive model based on flow features of the solution. 

In order to infer the values of highest probability for the corrective field, inverse methods are proposed. The method proposed uses Bayesian inversion, which includes an optimisation process. As the problem consists of a large number of variables normal optimisation processes are too computational expensive. Therefore, the adjoint method is proposed to compute gradients. 

In later work of van Korlaar this paradigm further extended to principal flow cases such as periodic hills. Also, the continuous adjoint is proposed in order to work seamlessly with the common CFD solver OpenFoam. It was found that the corrective term could accurately be predicted for unseen higher Reynolds number cases.

Research Objective

For this research it is proposed to extend upon the work done by Parish and Duraisamy and van Korlaar. The paradigm of field inversion is going to be applied on the wall bounded jet, occurring behind the underflow weir, as researched by Deltares.

Therefore the research objective is going to be:

To improve the RANS closure modelling for predicting wall bounded jet flows behind a weir extending on the paradigm of field inversion and machine learning using experimental PIV data as reference data.

From this objective several research questions can be formulated. The questions sought to be answered in this research are:

• Can the paradigm of field inversion be applied on more complex problems like wall bounded jet flows behind weirs?
  • Is a corrective term capable of improving a complex k-omega simulation with respect to a PIV baseline? And to what extend?
  • Can the continuous adjoint formulation be used on more complex and unstructured grids?
• Can experimentally obtained data from PIV be used for the field inversion paradigm?
  • Is it possible to apply field inversion with courser reference data?
  • Is it possible to apply field inversion with limited reference data?

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