Streamwise decay of localized states in channel flow

Channel flow, the pressure driven flow between parallel plates, has exact coherent structures that show various degrees of localization. For states which are localized in streamwise direction but extended in spanwise direction, we show that they are exponentially localized, with decay constants that are different on the upstream and downstream sides. We extend the analysis of Brand and Gibson [J. Fluid Mech. 750, R1 (2014)] for stationary states to the case of advected structures that is needed here, and derive expressions for the decay in terms of eigenvalues and eigenfunctions of certain second order differential equations. The results are in very good agreement with observations on exact coherent structures of different transversal wavelengths.

Figure 1

Visualization of the two localized ECS $P{O}_{Eyz}$ (a) and $P{O}_E$ (b). The plots show isosurfaces of positive (red) and negative streamwise velocity and isosurfaces of the $Q$-vortex criterion (yellow) at $Q=0.01$ and $Q=0.001$. The entire computational domain is $402d$ long, but only a section of length $52d$ is shown in the figures. For both states the shown computational domain has a width of $2\pi$.

Figure 2

Localization properties of the state $P{O}_E$ at $\text{Re}=1100$. (a) Maximum $M$ of the absolute values of the velocity components is shown versus streamwise coordinate $x$. Black lines show exponential decays with decay constants obtained from the model for the upstream and downstream tails. (b) Maximum norm of the terms of the linearized equations which contribute to the change of the streamwise component are shown.

Figure 3

Determination of the decay exponent $\mu$. Panel (a) shows $f\left(\mu \right)$ vs $\mu$ for $\gamma =1.0$ and $c=0.85$. The zeros are marked by blue symbols. Circles and squares are used if the corresponding profile is symmetric and asymmetric, respectively. At $\mu =413$ (not shown) another zero with an asymmetric profile exists. Panel (b) shows the potential $V\left(y\right)$ for the downstream decay, for $\gamma =1.0, c=0.85$, and $\mu =57.26$ (blue), where a symmetric eigenfunction with eigenvalue 0 exists (red). Panel (c) shows the potential for the parameters $\gamma =1.0, c=0.85$, and $\mu =-48.97$ (blue) on the upstream side. Here, 0 is an eigenvalue with a symmetric eigenfunction (red), while the asymmetric eigenfunction (green) corresponds to an eigenvalue of 1.51. The dashed lines in (b) and (c) indicate the positions of the eigenvalues for the eigenfunctions of the same color.

Figure 4

Streamwise velocity profiles on the downstream (a) and upstream (b) sides for the states $P{O}_E$ (red) and $P{O}_{Eyz}$ (blue) at $\text{Re}=1100$ and $\text{Re}=1800$, respectively. Circles represent DNS data and solid lines are used for the model profiles. In addition, the dashed green lines show a Gaussian $exp(-\kappa y^2/2)$ in (a), and an Airy function in (b), as approximate eigenfunctions for the corresponding eigenvalue problems for $P{O}_E$. Since the profiles are symmetric to the centerline, only one half of the channel width is shown.

Figure 5

Decay constants for exact coherent states as a function of Reynolds number. (a) Upstream side for $P{O}_E$ (red) and $P{O}_{Eyz}$ (blue). The model predictions, which are obtained from the roots of Eq. (6), are shown as continuous lines, and the decay rates fitted to the DNS data as triangles and squares, respectively. (b) Same as (a) for the downstream side. (c) Average wave speed for both states. The variation in wave speed with $\text{Re}$ is responsible for the variation in decay constants.

Figure 6

Dependence of the decay constants $\mu$ on the spanwise wave number $\gamma$ for different wave speeds $c$ for the symmetric profile on the upstream (solid) and downstream side (dashed).