Modal Decomposition

Modal decompositions are a great tool to automatically identify and separate different features & phenomena in a flow. They can be used for many useful things:

  • They can give us a better understanding by separating flows into their different modes.
  • One can reconstruct the flow using just the dominant modes with major flow features and thereby filter away measurement noise and outliers, typically described by the higher order modes.
  • Modal decompositions can be used for data reduction by storing just the dominant modes with the highest energy content and later reconstruct the flow from the stored data.

Modal decompositions can also be used for stability analysis when Oscillating Pattern Decomposition, identify cyclic patterns in the flow along with frequencies and growth or decay rates. DynamicStudio features four different Modal decompositions, each doing different things:

  • Proper Orthogonal Decomposition (POD)
  • Multiscale POD (mPOD)
  • Fourier Modal Decomposition (FMD)
  • Oscillating Pattern Decomposition (OPD)

Proper Orthogonal Decomposition (POD)

The classic Modal decomposition in DynamicStudio is the POD. As with all modal decompositions, it separates the flow into different modes in both space (Topos) and time (Chronos). Each of these modes represents a certain type of flow feature like a vortex, for example. The modes are sorted by energy content with mode 0, the mean, first, followed by less and less energetic modes.

This chart shows the energy of the different modes and the accumulated energy of a flow. In this example the first 4 modes contain over 92% of the flows energy. The rest could be considered noise.

Classic POD does not need time-resolved input but may mix up different phenomena occurring at different frequencies in the same mode. Classic POD is available for images, scalars, 2D and 3D maps.

Multiscale Proper Orthogonal Decomposition (mPOD)

With time-resolved input, dominant frequencies in a flow can be identified in a frequency domain plot of the covariance matrix. The mPOD allows the user to separate output modes into distinct frequency bands. With just a few mouse clicks, phenomena that classic POD might have mixed up can thus be clearly separated from one another. mPOD is applicable for scalars, 2D and 3D vector maps.

The mPOD collaboration with the von Karman Institute, Belgium (VKI) published in Measurement Science and Technology won the publication’s Outstanding Paper Award for 2020 in the field of Fluid Mechanics.  

Covariance matrix showing several peaks and the created mask for the mPOD.

Fourier Mode Decomposition (FMD)

Comparison of the Chronos (black curves) and Spectrum (red curves) from the same mode of a POD (top), and mPOD (center), and FMD (bottom).

The FMD separates the modes by their frequency content. Thus all modes are frequency independent from each other, which is a great tool if you want to study the influence of e.g. shedding frequencies or impinging jets, where many different vortical structures are present, but with different frequencies.

The main differences of the different modal decomposition methods offered in DynamicStudio can be seen in the comparison figure: the plots of the chronos and the spectrum of a mode.

The black curves are the Chronos and the red ones the corresponding Spectra. The x-axis is the frequency of the Spectrum, or the time of the Chronos.

All spectra show a low frequency peak, but the result from Classic POD include high frequency content, clearly visible in both the spectrum and the Chronos itself. In the mPOD results most of the high frequency content has been filtered away and ensure a smooth Chronos and a ‘clean’ spectrum. FMD results go a step further and exclude all but the dominant frequency also visible in the first two. The high frequency content is accounted for by other modes.

Oscillating Pattern Decomposition (OPD)

With Classic POD Analysis as preprocessing of a time-resolved dataset, OPD performs stability analysis of the flow. Resulting modes have distinct frequencies plus exponential growth or decay rates to identify dominant flow structures. OPD Topos are complex and can be animated to visualize features such as traveling vortices in the flow. This add-on is ideal for investigating vortex shedding in shear regions, acoustic- or thermo-acoustic driven vortex formation, and vortex enhancement. A key reason for investing in a TR-PIV system is the ability to perform frequency domain analysis. OPD software is a dedicated tool for such analysis and therefore an indispensable complement to any TR-PIV system.

OPD analysis of a pulsed jet. e-fold time (i.e. decay rate) of the OPD modes plotted against their associated frequencies (Top Right). Three dominant OPD modes are highlighted and corresponding convected vortices are shown. The higher frequency modes are clearly harmonics.
Complex OPD mode for an oscillating disc in water, where the modal frequency corresponds to the 4th harmonic of the oscillation frequency. The oblique vortex shedding from the disc tip is clearly identifiable. From Ergin F.G. ”Modal Investigation of Vortex Ring Shedding from an Oscillating Disc” 2021, Experimental Techniques.

Data Sheets and Literature