In my previous post I discussed the important role that turbulence plays in the evolution of the atmosphere and ocean. Here I'll discuss turbulence in the context of the flow over aircraft wings and wind turbines. Specifically I'll show how the complexity of the flow increases as the object moves faster. I'll also show how to represent this complexity in a simple say using the data mining technique of "principal component analysis", also known as "proper orthogonal decomposition" in engineering, "singular value decomposition" in mathematics, or "empirical orthogonal functions" in geophysics.

Arguably the most important qunatity in the field of fluid dynamics is the Reynolds number, which is essentially the ratio of the momentum force to the viscous force. It is high for objects moving fast in a fluid of low viscosity (eg: air), and low for objects moving slowly in a very viscous fluid (eg: honey). Flows with higher Reynolds numbers are more complex and have a greater range of scales, that is the largest vortex in the flow is significantly bigger than the smallest voretx in the flow. I'll illustrate below how the complexity of the flow over an aerofoil (representative of an idealised aircraft wing or wind turbine blade) increases with Reynolds number.

The configuration that I am looking at is an aerofoil at an angle of 18 degrees with the flow moving from left to right in the movies below. All of the movies illustrated below are generated by post-processing the data resulting from computational fluid dynamics simulations, using conceptually the same approach as that discussed in my previous post on the simulations of the atmosphere and ocean. The volume of fluid surrounding the aerfoil is broken down into a series of grid boxes and the Navier-Stokes equations are solved at each position to determine the velocity and pressure throughout the fluid volume.

In fact the flow around an aerofoil at very low Reynolds numbers (or very slow moving aerofoils), does not change with time. The fluid is moving, but its velocity and presssure at each position is not changing. It is not until the aerofoil reaches a certain critical Reynolds number (or speed) that the flow begins to change with time, as illustrated in the movie below. Here the flow is changing in time and two-dimensional. It is coloured by vorticity, which is a measure of the rotation of the fluid. Red is rotating in the couter clockwise direction and blue is rotating in the clockwise direction.

I will now use principal component analysis to breaks down the flow into a series of "modes", which can be added together with varying weights to reconstruct each instant in time. One can also think of the method as a form of information compression, and has also been used for facial feature detection. For this particular flow 94% of the energy can be represented by the first two modes (or "facial features") illustrated below. Further details of this flow and its stability properties can be found in my Journal of Computational Physics paper in reference [1] and in my PhD thesis in reference [2] listed below.

mode 1 |

mode 2 |

As the Reynolds number increases the flow transitions for an unsteady two-dimensional flow, to an unsteady three-dimensional flow illustrated in the movie below. This movie is illustrating three-dimensional surfaces defining the boundary of the complex vortex structures. They are coloured by rotation in the flow direction. The total data set is 17.5Gb.

Here the first two modes represent the two-dimensional aspects of the flow and capture only 65% of the energy.

mode 1 |

mode 2 |

The next two modes capture the three-dimensional aspects of flow. Further details of this flow and the associated modes can be found in my PhD thesis.

mode 3 |

mode 4 |

As the Reynolds number is increased further the flow becomes even more complex with many smaller vortices being generated, as illustrated in the movie below. This total data set is 35Gb.

Here the first two modes now represent only 45% of the total energy. What is interesting here, is that the flow is so complex that the large scale vortices are hidden amongst the forest of small scale vortices. The principal components, however, are able to extract the large features from the data. Further details on this flow and the modal decomposition can be found in my Journal of Fluid Mechanics paper in reference [3].

mode 1 |

mode 2 |

It is clear that as the Reynolds number increases and the flow becomes more complex, the first two modes represent less and less of the total energy. This also means that more modes are required to represent a given percentage of the total energy.

Principal Component Analysis has many applications and can be used to extract key features from any data set be it physical, biological or socio-economic.

**References:**

[1] Kitsios, V., RodrÃguez, D., Theofilis, V., Ooi, A. & Soria, J., 2009, BiGlobal stability analysis in curvilinear coordinates for massively separated lifting bodies,

*Journal of Computational Physics*, Vol. 228, pp 7181-7196. [link]
[2] Kitsios, V., 2010, Recovery of fluid mechanical modes in unsteady separated flows, PhD Thesis, The University of Melbourne. [PDF]

[3] Kitsios, V., Cordier, L., Bonnet, J.-P., Ooi, A. & Soria, J., 2011, On the coherent structures and stability properties of a leading edge separated aerofoil with turbulent recirculation,

*Journal of Fluid Mechanics*, Vol. 683, pp 395-416. [link]
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