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This project aims to study some cases related to boundary layer transition, a well known phenomenon in Fluid Mechanics but still hard to predict on real cases.

The core of the repository is the implementation of methods that can help predict the boundary layer point of transition on 2D geometries, such as aerodynamic airfoils. It is relevant to point out that these methods work well under certain geometries or conditions, so bear in mind about its limitations and range of application.

Dynamic mode decomposition (DMD)

The dynamic mode decomposition (DMD) considers the flow to be made up of an overlapping of dynamic modes, each one associated to a specific mechanism or fluid structure that develops within the flow. Each of these can be analyzed and understood in an isolated and simple way if a flow mode decomposition is carried out. As far as boundary layer transition is concerned, the linear stability theory (LST) accomplishes transition prediction by evaluating the spatial growthrate of the TS waves, along with the eN criteria. The DMD method aims for the same objective, but it is based on a completely different physico-mathematical model in comparison with the LST, in terms of the spatial growthrate measurement of the disturbances associated to the TS waves. Once it is applied, it is possible to analyse not only flow transition but also all the flow mechanisms in isolation so that its behaviour and its development is properly understood.

Thus, in this repository one of the multiple versions of DMD is implemented for a very particular case of an airfoil. The case taken is a natural laminar flow airfoil NLF0416 with these freestream conditions:

• M = 0.1
• Re = 4000000
• AoA = 1 deg

Furthermore, some validation data about this particular case is available at

A transition prediction method for flow over airfoils based on high-order dynamic mode decomposition, Chinese Journal of Aeronautics, Mengmeng Wu, Zhonghua Han, Han Nie, Wenping Song, Soledad Le Clainché, Esteban Ferrer, Volume 32, Issue 11, 2019, Pages 2408-2421, ISSN 1000-9361.

These results will be used as a validation of the tool and methods developed in this repository.

The way the analysis is performed is as follows:

1. It is assumed that the case is 2D; if it's 3D there's a big chance of introducing a lot of noise to the DMD method, due to the three dimensional structures developed in such configuration. The case has been solved with the open-source CFD software SU2, meshed with Hypermesh and Paraview has also been used to ease the postprocessing.
2. Once the case is solved, the postprocessing begins with the generation of the grid whose points constitute the so called DMD snapshots that will be fed to the DMD solver afterwards. These points are located along the chord of the airfoil and arranged normal to the wall, at the x considered point. For example, the following figures show some snapshots arranged at the segment '0010' (covering from the leading edge to the 10% of the airfoil chord):

3. Having generated all the points of the grid, it is time to interpolate the solution between the nodes of the mesh and the points of the DMD grid, since the snapshots that will be fed to the DMD solver will carry not only spatial information of the snapshot (where it is located) but also the flow magnitudes at each point of each snapshot. This interpolation stage is performed using a Paraview macro, that takes all the points generated in Stage 2 and interpolates the solution to them.
4. Once the interpolation is done, it is necessary to structure the data coming out of Paraview and arrange it in a snapshot format. After this structuring is done, all the snapshots are fed to the DMD solver and the output is a list of the eigenvalues associated to the fluid structures (eigenvectors) developed in the flow.

In the following figures there is a comparison of the eigenvalues in the unit circle provided by the DMD method and the eigenvalues provided at the reference paper, for the segment of the airfoil '0020', that is, from the leading edge to the 20% of the chord of the airfoil.

One of the greatest things about the DMD approach is the freedom that the user has to set the snapshot, in terms of:

• Which variables to choose to be used for the analysis, the so called DMD variables.
• The arrangement of the snapshots. This refers to the chordwise disposal secuence of the planes. For example, for a given longitudinal snapshot spacing ds, "x1" means that there is one snapshot placed every ds measured distance; "x2" means that from snapshot to snapshot there is an arc distance of 2ds, and so on. The aim of defining different arrangements is to be able to capture real flow structures (and not spurious ones). The arrangments can be seen as a way of filtering the information that is fed to the DMD solver.

In the following example, the DMD variables are the two components of the flow velocity, {u, w}, and three snapshots arrangement have been applied: 1 snapshot every ds, 2 snapshots every ds and 3 snapshots every ds.

Even though there is a high dispersion in the results, there is some coincidence of eigenvalues using different snapshots arrangement. This could point to the capture of the same associated flow structure.

Boundary layer transition based on thickness growth criteria

Among all the developed criteria to assess boundary layer transition, there is one based on the computation of two magnitudes associated to the development of the boundary layer, the displacement thickness (delta 1) and the momentum thickness (delta 2), which are defined in the following way:

The criteria states that if the Reynolds number based on any of these two characteristic lengths exceeds a threshold, then the transition is likely to occur. As they are two different lengths, there are two Reynolds numbers needed to be monitored to check whether any of them exceeds their respective threshold limit:

Therefore, this module of the repository implements a method to compute the indicators defined above so as to obtain the evolution of the Reynolds numbers based on each of them. Thresholds for these magnitudes must be provided in order to be able to know the exact x-location where transition occurs, if it does. These values depend on the problem, but they can be taken from experiments or validation studies that fit well to the case.

So far, this module re-uses part of the scripting done for the previous (DMD) module in the sense that for the computation of the boundary layer thickness, it is necessary to measure the velocity evolution in the region close to the wall. Therefore, many of the methods implemented in the DMD module are used here to compute these magnitudes.

It is required first to define a grid/mesh of points close to the wall to capture the evolution of the longitudinal component of the boundary layer velocity, namely u. Once the grid is generated, an interpolation is done to retrieve the magnitude of u at that set of points in order to compute the integrals defined above.

There are other magnitudes that can also be monitored, such as temperature or pressure: