Cospectrum and Cross-spectrum in Turbulent Flow

Categories: Turbulence 2023-10-17

Turbulence Spectrum Calculation Guide

How to calculate the cospectrum and cross-spectrum in turbulent flow?

The cospectrum and cross-spectrum are fundamental tools in turbulence analysis, especially when examining the relationship between two fluctuating quantities in turbulent flows. In a broad sense, they provide a frequency decomposition of the covariance between two signals. Here’s how to compute them:

1. Definitions:

   • Let $u(t)$ and $v(t)$ be two fluctuating quantities (e.g., two components of velocity).
   • Their Fourier transforms are $U(f)$ and $V(f)$, respectively.

2. Fourier Transform: Compute the Fourier transforms of the signals using the Fast Fourier Transform (FFT). Many software packages like MATLAB, Python’s numpy, or others offer FFT capabilities.

   $U(f) = FFT(u(t))$

   $V(f) = FFT(v(t))$

3. Cross-spectrum: The cross-spectrum $S_{uv}(f)$ is given by the product of one signal’s Fourier transform and the complex conjugate of the other’s:

   $S_{uv}(f) = U(f) \times V^*(f)$

   where $V^*(f)$ is the complex conjugate of $V(f)$.

4. Cospectrum: The cospectrum $C_{uv}(f)$ is the real part of the cross-spectrum:

   $C_{uv}(f) = Re[S_{uv}(f)]$

   It represents the in-phase components of the signals.

5. Quadrature Spectrum (optional): The imaginary part of the cross-spectrum is the quadrature spectrum $Q_{uv}(f)$:

   $Q_{uv}(f) = Im[S_{uv}(f)]$

   It represents the out-of-phase components of the signals.

6. Normalization: Typically, the cospectrum and quadrature spectrum are normalized by the total variance of the signals. This gives a more interpretable result in terms of the contribution of different frequency bands to the total covariance.

7. Interpretation:

   • Positive values of the cospectrum at a given frequency indicate that the two signals are in phase at that frequency, contributing positively to their covariance.
   • Negative values mean the signals are out of phase, contributing negatively to their covariance.
   • The quadrature spectrum can give insights into the phase lag between the signals.

8. Windowing and Averaging:

   • To improve the statistical reliability of the spectra, it’s common to split the data into segments, compute spectra for each segment, and then average the results. This method is called “ensemble averaging”.
   • Before Fourier transforming, each segment should be multiplied by a window function (like a Hamming or Hanning window) to reduce spectral leakage.

Remember that interpreting turbulence spectra, especially the cross-spectrum and cospectrum, requires a good understanding of the physics of the flow and the interactions between different flow structures. The importance of the frequency domain representation lies in its ability to differentiate between fluctuations occurring at different scales or frequencies, giving insights into the underlying mechanisms of turbulence.

The physical meaning of cospetrum and cross-spectrum in turbulent flow

The cospectrum and cross-spectrum are statistical tools that provide insights into the interactions between different turbulent structures and scales. They’re especially important when analyzing the relationship between two fluctuating quantities in turbulent flows, like velocity components. Let’s delve into their physical meanings:

1. Cross-spectrum:
   • The cross-spectrum $S_{uv}(f)$ represents the relationship between two fluctuating quantities (e.g., $u$ and $v$ velocity components) in the frequency domain.
   • If you are examining two identical signals, the cross-spectrum becomes the power spectrum of the signal, showing the distribution of energy across frequencies.
2. Cospectrum:
   • The cospectrum $C_{uv}(f)$ is the real part of the cross-spectrum. It represents the in-phase relationship between the two signals as a function of frequency.
   • Physically, positive values of the cospectrum at a specific frequency indicate that fluctuations of the two signals at that frequency tend to occur simultaneously (in-phase). In turbulence, this can indicate that specific turbulent structures or events cause simultaneous changes in both quantities.
   • Negative values at a frequency mean the signals fluctuate in an anti-phase manner at that frequency; when one signal has a positive anomaly, the other tends to have a negative anomaly.
3. Physical Interpretation:
   • For example, in boundary layer flows, there might be large, coherent structures like vortices. If the cospectrum of the streamwise (u) and vertical (w) velocities is positive at a specific low frequency, this suggests that these structures are causing simultaneous upward and forward motions in the flow.
   • On the other hand, if the cospectrum is negative, it might indicate that when there’s an upward motion in the flow (positive w velocity), there’s a simultaneous slowing down of the streamwise motion (negative u velocity anomaly).
   • By examining the scale or frequency at which these interactions are strongest, researchers can gain insights into the size or timescale of the dominant structures causing these interactions.
4. Quadrature Spectrum (for completeness):
   • The quadrature spectrum is the imaginary part of the cross-spectrum and provides insights into the phase difference between two signals. If the quadrature spectrum is significant at a certain frequency, it indicates a consistent phase difference between the signals at that frequency, meaning one signal consistently leads or lags the other.
5. Importance in Turbulence Research:
   • Turbulence is inherently multiscale and chaotic. The interactions between different scales of motion are fundamental to the energy cascade process and the overall dynamics of turbulence. By analyzing the cospectrum and cross-spectrum, researchers can identify and quantify these interactions.
   • This becomes especially critical in studies related to turbulence modeling, atmospheric boundary layer flows, turbulent combustion, and any application where understanding the interplay between different scales or components of motion is crucial.

In summary, the cospectrum and cross-spectrum provide valuable insights into the frequency-dependent interactions between turbulent quantities. They help researchers dissect the complex dance of turbulent structures and their impacts on flow properties.