Wake identification of stratified flows using dynamic mode decomposition

The wake behind a bluff body in the presence of a density gradient is characterized by competing effects of buoyancy, momentum, and ultimately viscosity as the flow decays. Consequently, there are a number of distinct flow regimes into which the wakes can be placed, and there is some interest in whether any given wake can be traced back to its source through interrogation of selected fluid mechanical properties in the wake. Here we use dynamic mode decomposition (DMD) to find modes that can be used to characterize and automate such a process. A custom-designed algorithm is proposed that can sort and classify stratified wakes based on a selection of the most energetic DMD modes. The approach is very, or partially successful, depending on the quality and dimension of the input data. The success of these first steps may be used to develop similar methods for more challenging and fully turbulent wakes, and also can serve as a guide for data-driven methods that require no prior knowledge of the flow structure.

Figure 1

Regime diagrams from CH93 (a) and LI92 (b). The four main regimes differentiated in (a) are 2D, quasi-2D; SLW, strong lee wave; T, transition (SKH, without K-H instability; KH, with K-H instability); and 3D, three-dimensional. In (b), LI92 distinguished six main regimes: $A_u$, unsteady, attached 2D vortices; $V_t$, 2D vortex shedding; $L$, lee-wave instability; $N$, nonaxisymmetric attached vortex; $S$, symmetric vortex shedding; $V$, nonsymmetric vortex shedding; and $T$, turbulent wake.

Figure 2

Schematic showing experimental setup. The annotated scale is based on a sphere with $D=4\mathrm{cm}$ to provide a relative scale of the experiment.

Figure 3

Temporal data reconfiguration where the colored box $A$ is moving in reference to the sphere. Dashed lines indicate column vectors of velocity field within the box $A.$

Figure 4

Meshing in simulation and the location of refinement levels. The computational domain is $(x,y,z)=\left(\right[-10,50],[-8,8],[-8,8\left]\right)D$ and the results in $\left(\right[1.5,15],[-2,2],[-2,2\left]\right)D$ was used. (a) Mesh for $\text{Re}=[200,300,500]$ has four refinement levels: $2^{-[2\cdots 5]}D$ (b) Mesh for $\text{Re}=1000$ has six refinement levels: $2^{-[2\cdots 7]}D.$

Figure 5

A regime diagram for stratified sphere wakes. The final result in colored dots combines experiments from CH93 (lighter colors) and LI92 (darker colors) with combined laboratory and numerical experiment. The five identified regimes are labeled and color coded as: Vortex street (VS, purple), Symmetric nonoscillatory (SN, red), Asymmetric nonoscillatory (AN, yellow), Planar oscillatory (PO, green), and Spiral mode (SP, blue).

Figure 6

A snapshot of the flow field $u$ in vertical and horizontal centerplane (left) and a dominant DMD mode of simulation in 3D (middle) and experiment in 2D center plane (right) illustrating various regimes observed: (a) R10F0.5 Vortex Street (R2F0.5 for the experiment), (b) R5F1 Symmetric nonoscillatory, (c) R3F4 Asymmetric Nonoscillatory, (d) R3F8 planar oscillatory (e) R10F8 spiral mode. The isolevel of the middle 3D plot is 30% of the maximum velocity component.

Figure 7

The plots on the left show the real and imaginary parts of $\lambda$, and on the right $\tilde{E}\left(\text{St}\right)$ is shown for each mode of the case $\text{Re}=500$ and $\text{Fr}=4$. The dark red corresponds to the mean mode. The green dot is the most dominant mode selected to be evaluated in Fig. 8. The grey area is considered noise, set below $\tilde{E}$ threshold and above $\text{St}$ threshold. (a) Simulation, (b) Experiment.

Figure 8

A flow chart of the wake classifier with 3D DMD input. The selected DMD mode from Sec. 3c1 is sorted into five regimes in the nonoscillatory branch (left) and oscillatory branch (right) based on the symmetry of the cross-stream velocities in that mode.

Figure 9

Success rate of a classifier with 3D numerical DMD input. The pie chart shows the classification type in color. The number above each filled circle is the the percent accuracy of the majority outcome.

Figure 10

A flow chart of the wake classifier with two, 2D-center-plane flow fields as input. Two dominant DMD modes of each center plane are evaluated where the mode for the vertical center plane is selected based on ${\text{St}}_w$ closest to ${\text{St}}_v$. The nonoscillatory branch uses the mode from the horizontal center plane, while the oscillatory branch uses the vertical centerplane for the first step, and horizontal centerplane for the second step. Green dotted line represents the wake centerline.

Figure 11

Classifier success rate with 2D numerical DMD input.

Figure 12

Classifier success rate with 2D experimental DMD input. Cases with red $\mathrm{X}$ have wake length scales larger than the FOV for regimes to be distinguishable.

Figure 13

Ritz values $\lambda$ (left), $\tilde{E}$ versus $St$ (center), $v(x,y,z=0)$ of a dominant oscillatory mode (indicated by an arrow) for the $\text{Re}=300,\text{Fr}=2$ case computed from (a) standard DMD ($E_r=100\%$ of $\tilde{E}$) and (b) TDMD ($E_r=91.6\%$). Modes colored in red are energetic modes selected from Sec. 3c1.