Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow past a cylinder

Here proper orthogonal decomposition (POD) modal decomposition are performed for flow past a circular cylinder at supercritical Reynolds numbers by projecting this onto instability modes. The important task of modeling a cylinder wake by Stuart-Landau (SL) and the Stuart-Landau-Eckhaus (SLE) equation for instability modes is discussed, with the latter shown to be more consistent with multimodal pictures of POD and instability modes. The difficult task of finding the coefficients of the SLE equation is reported by taking a least squares approach for the reduced order model (ROM). The important aspect of the ROM is the choice of initial condition for the developed SLE equations, as these are stiff ordinary differential equations which are very sensitive to the choice of initial conditions. An accurate representation of enstrophy-based POD also reveals the presence of modes which occur in isolation (in comparison to modes that come in pairs) and the traditional approach of treating instability modes by SL or SLE equations does not work directly, which also reveals higher frequency variations. Quantifying effects of this mode by time-averaged Navier-Stokes equation (NSE) fail to show the variation of the phase of these isolated time-varying modes and this is captured here using direct numerical simulation (DNS) data by a multitime scale approach. A reconstructed 3-mode ROM solution and the disturbance vorticity from DNS match globally in the flow. The agreement between 3-mode SLE reconstruction and DNS also proves the consistency of the proposed method and helps explain the physical nature of the ensuing Hopf bifurcation following an instability.

Figure 1

First 16 amplitude functions of $\mathrm{Re}=100$ including the missing ones, as described in [1].

Figure 2

First 16 eigenfunctions of $\mathrm{Re}=100$ including the missing modes, as described in [1].

Figure 3

(a) Equilibrium amplitude of streamwise fluctuating velocity (rms) variation plotted against $\mathrm{Re}$ compared with computed data reported in [1]. (b) Time variation of disturbance vorticity at point along the centerline ($x=0.504$) in the near wake. (c) The fluctuating vorticity equilibrium amplitude plotted as function of $\mathrm{Re}$ for a point at $x=0.504$ in the near wake along the centerline.

Figure 4

Phasor plots of instability modes of $\mathrm{Re}=100$.

Figure 5

Phasor plots of instability modes of $\mathrm{Re}=150$.

Figure 6

Amplitude variation of the instability modes of $\mathrm{Re}=100$.

Figure 7

Amplitude variation of the instability modes of $\mathrm{Re}=150$.

Figure 8

Time integrated error contours for $0<t\le 396$ are plotted in the ($A_{20},A_{30}$) plane for different values of $A_{10}$ for $\mathrm{Re}=100$.

Figure 9

Extracted minimum integrated error plotted as a function of $A_{10}$ for $\mathrm{Re}=100$.

Figure 10

Optimal reconstruction for the three-mode case for $\mathrm{Re}=100$.

Figure 11

The slowly varying and rapidly varying components of the $T_1$ mode (POD mode-3 for $\mathrm{Re}=150$) (top two panels) and the phasor plot (bottom panel) of the corresponding instability mode.

Figure 12

Instability modes constructed for $\mathrm{Re}=150$ following the new representation of the $T_1$ mode.

Figure 13

Amplitude equation coefficients for $A_1$.

Figure 14

Amplitude equation coefficients for $A_{2m}$.

Figure 15

Amplitude equation coefficients for $A_{2d}$.

Figure 16

Amplitude equation coefficients for $A_3$.

Figure 17

Phase equation coefficients for ${\theta }_1$.

Figure 18

Phase equation coefficients for ${\theta }_{2d}$.

Figure 19

Phase equation coefficients for ${\theta }_3$.

Figure 20

Reconstructed instability mode amplitude variation with time for $\mathrm{Re}=100$.

Figure 21

Reconstructed $A_2$ mode amplitude variation with time for $\mathrm{Re}=100$.

Figure 22

Reconstructed phase variation with time of the instability modes for $\mathrm{Re}=100$.

Figure 23

Disturbance vorticity obtained from DNS reconstructed using the results from the SLE equations for $\mathrm{Re}=100$ at (0.5044, 0.0).

Figure 24

Disturbance vorticity obtained from DNS reconstructed using the results from the SLE equations for $\mathrm{Re}=100$ at (1.016, 1.016).