Solutions to aliasing in time-resolved flow data

Abstract

Avoiding aliasing in time-resolved flow data obtained through high-fidelity simulations while keeping the computational and storage costs at acceptable levels is often a challenge. Well-established solutions such as increasing the sampling rate or low-pass filtering to reduce aliasing can be prohibitively expensive for large datasets. This paper provides a set of alternative strategies for identifying and mitigating aliasing that are applicable even to large datasets. We show how time-derivative data, which can be obtained directly from the governing equations, can be used to detect aliasing and to turn the ill-posed problem of removing aliasing from data into a well-posed problem, yielding a prediction of the true spectrum. Similarly, we show how spatial filtering can be used to remove aliasing for convective systems. We also propose strategies to prevent aliasing when generating a database, including a method tailored for computing nonlinear forcing terms that arise within the resolvent framework. These methods are demonstrated using a nonlinear Ginzburg–Landau model and large-eddy simulation data for a subsonic turbulent jet.

Graphical abstract

Wavenumber-frequency spectra of the forcing component for the streamwise momentum computed on the lipline near the jet nozzle.

A: Spatial aliasing in the LES database

The present dataset calculated on an unstructured grid is mapped onto a cylindrical grid to compute the FT in the azimuthal direction in a robust manner. The dimensions of the structured grid should normally be determined to provide a resolution similar to that of the LES grid in order to avoid spatial aliasing. The original grid was refined near the nozzle in the axial direction, and around the shear layer in the radial and azimuthal directions [5]. The azimuthal refinement at the shear layer brings excessive interpolation around the jet axis and causes an increase in storage cost without any benefit. A solution to this problem is to determine the minimum number of points in azimuth to map the flow data onto the structured grid without aliasing. In previous studies in which the present dataset had been used to investigate the state variables, 128 points in azimuth had been used. A convergence analysis for aliasing in the azimuthal direction for both the state variables and the forcing terms around the shear layer is conducted using a single snapshot and is depicted in Fig. 18. The analysis reveals that aliasing in the state variables is negligible for the first azimuthal mode, while even the $m=0$ mode is aliased for the forcing terms. The same test is repeated on a cylindrical grid with 512 points in azimuth, which roughly corresponds to the number of points around the shear layer near the nozzle in the unstructured LES grid (see Fig. 19). It is seen that for the forcing terms near the nozzle, convergence is obtained for $f_{u_x}$ with 256 points while it is not the case for $f_p$. This shows that the LES grid resolution should be kept around the shear layer near the nozzle to avoid spatial aliasing in the azimuthal direction, but at the cost of quadrupling the size of the database.

Fig. 18

Azimuthal FT of $u_x$ (blue) and $f_{u_x}$ (orange) terms calculated at various axial positions on the lip line. Solid and dashed lines correspond to 128 and 64 grid points, respectively, in azimuthal direction. $u_x$ is scaled by a random factor to increase readability (color figure online)

Fig. 19

Azimuthal FT of $f_{u_x}$ (blue) and $f_{p}$ (orange) terms calculated at various axial positions on the lip line. Solid and dashed lines correspond to 512 and 256 grid points, respectively, in azimuthal direction (color figure online)

An alternative solution to the mapping problem is to low-pass filter the flow field in the azimuthal direction, and to downsample afterward obeying the Nyquist criterion. Azimuthal filtering can be applied either through a weighted moving-average filter, or by taking the azimuthal FT of the data, setting the mode numbers to be filtered to zero, and taking the inverse azimuthal FT. Since the data is already periodic, taking the FT does not cause any spectral leakage. Note that once filtered in the azimuthal direction, a direct conversion of the velocity field from Cartesian to cylindrical, or vice versa, is not valid any more since the conversion is not linear in the azimuthal direction. However, one can switch between the two velocity fields after taking the azimuthal FT.

Conversion of velocity from Cartesian to cylindrical coordinate system is performed as

$$\begin{aligned} u_r&= u_y \cos (\theta ) + u_z \sin (\theta ), \end{aligned}$$

(40)

$$\begin{aligned} u_{\theta }&= -u_y\sin (\theta ) + u_z\cos (\theta ), \end{aligned}$$

(41)

where $\theta$ is measured from the y-axis. Taking the Fourier transform (FT) of (40) and (41) in $\theta$ yields the following convolution expressions:

$$\begin{aligned} \hat{u}_r^{(i)}&= \hat{u}_y^{(i)}*\mathcal {F}\left( \cos (\theta )\right) + \hat{u}_z^{(i)}*\mathcal {F}\left( \sin (\theta )\right) , \end{aligned}$$

(42)

$$\begin{aligned} \hat{u}_{\theta }^{(i)}&= -\hat{u}_y^{(i)}*\mathcal {F}\left( \sin (\theta )\right) + \hat{u}_z^{(i)}*\mathcal {F}\left( \cos (\theta )\right) , \end{aligned}$$

(43)

where the superscript (i) denotes the azimuthal mode number. Using Matlab’s convention for FT, the FTs of $\cos (\theta )$ and $\sin (\theta )$ are given as

$$\begin{aligned} \mathcal {F}\left( \cos (\theta )\right)&= \frac{1}{2}\left( \delta (i-1)+\delta (i+1)\right) , \end{aligned}$$

(44)

$$\begin{aligned} \mathcal {F}\left( \sin (\theta )\right)&= \frac{i}{2}\left( \delta (i-1)-\delta (i+1)\right) . \end{aligned}$$

(45)

Then the convolution expressions given in (42) and (43) can be re-written as

$$\begin{aligned} \hat{u}_r^{(i)}&= \frac{1}{2}\left( \hat{u}_y^{(i-1)} + \hat{u}_y^{(i+1)}\right) + \frac{i}{2}\left( \hat{u}_z^{(i-1)} - \hat{u}_z^{(i+1)}\right) , \end{aligned}$$

(46)

$$\begin{aligned} \hat{u}_{\theta }^{(i)}&= -\frac{i}{2}\left( \hat{u}_y^{(i-1)} - \hat{u}_y^{(i+1)}\right) + \frac{1}{2}\left( \hat{u}_z^{(i-1)} + \hat{u}_z^{(i+1)}\right) , \end{aligned}$$

(47)

Similarly, conversion from cylindrical to Cartesian coordinates can be achieved using

$$\begin{aligned} \hat{u}_y^{(i)}&= \frac{1}{2}\left( \hat{u}_r^{(i-1)} + \hat{u}_r^{(i+1)}\right) - \frac{i}{2}\left( \hat{u}_{\theta }^{(i-1)} - \hat{u}_{\theta }^{(i+1)}\right) , \end{aligned}$$

(48)

$$\begin{aligned} \hat{u}_z^{(i)}&= \frac{i}{2}\left( \hat{u}_r^{(i-1)} - \hat{u}_r^{(i+1)}\right) + \frac{1}{2}\left( \hat{u}_{\theta }^{(i-1)} + \hat{u}_{\theta }^{(i+1)}\right) . \end{aligned}$$

(49)

Once the azimuthal modes of the velocity have been calculated in the transformed coordinates, one can perform an inverse FT in $\theta$ to reconstruct the filtered velocity field.