Minimum enstrophy principle for two-dimensional inviscid flows around obstacles

Large-scale coherent structures emerging in two-dimensional flows can be predicted from statistical physics inspired methods consisting in minimizing the global enstrophy while conserving the total energy and circulation in the Euler equations. In many situations, solid obstacles inside the domain may also constrain the flow and have to be accounted for via a minimum enstrophy principle. In this work, we detail this extended variational formulation and its numerical resolution. It is shown from applications to complex geometries containing multiple circular obstacles that the number of solutions is enhanced, allowing many possibilities of bifurcations for the large-scale structures. These phase change phenomena can explain the downstream recombinations of the flow in rod-bundle experiments and simulations.

Figure 1

Sketch of the tranvserse flow fields observed in cross sections of an experimental $5\times 5$ rod-bundle flow after the passage of the coolant flow in the interstices between a mixing grid and the cylinder array. The flow is injected with a mostly axial velocity field as depicted by the bottom-left arrow. The cross sections of the flow shown here are respectively found at axial positions $z=3.4d_h$ (left) and $34d_h$ (right) away from the grid. While cropped here for visibility, the cylinders extend from far upstream of the mixing grid until its far wake downstream.

Figure 2

Sketch of the two domain geometries explored in this study: a circular annulus domain (left) with $R_0=0.45$ and $R_1=0.1$, and a square with two obstacles (right) of side length $L=1$ and obstacle radii $R_1=R_2=0.1$.

Figure 3

Representation of MEP solutions in the case of a ring with outer radius $R_0=0.45$ and inner radius of $R_1=0.1$ at different $a_1$. The total circulation is set at $\mathrm{\Gamma }=1$. (Left) $(\beta ,{\mathrm{\Gamma }}_1/\mathrm{\Gamma })$ plane. (Right) $(\beta ,{\mathrm{\Lambda }}^2)$ plane.

Figure 4

Interpolated ${(\beta ,{\mathrm{\Lambda }}^2)}_{{\mathrm{\Gamma }}_1/\mathrm{\Gamma }}$ phase diagram in the case of an annular domain with a central obstacle, for ${\mathrm{\Gamma }}_1/\mathrm{\Gamma }=-0.1$. The curved solid line corresponds to the continuous solution and the horizontal dashed line to the mixed solution associated with the first zero-mean eigenvector.

Figure 5

Bifurcation diagram showing the domination zones of the “mixed” (dark gray) and “continuous” (light gray) solutions in parameter space, in the case of the annular domain. It was obtained by scanning $E$ and ${\mathrm{\Gamma }}_1/\mathrm{\Gamma }$ values while keeping $\mathrm{\Gamma }$ constant. The global minimum of the enstrophy is obtained through $\beta a_1=0$ and is shown as a thick black curve.

Figure 6

Interpolated ${(\beta ,{\mathrm{\Lambda }}^2)}_{{\mathrm{\Gamma }}_1}$ phase diagram in the case of a square domain with two opposed obstacles and identical circulations. Here ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=0.1$. The curved and horizontal lines respectively correspond to the continuous solution and the mixed solution associated with the first zero-mean eigenvector.

Figure 7

Bifurcation diagram showing the domination zones of the “mixed” (dark gray) and “continuous” (light gray) solutions in parameter space, in the case of a square domain with two obstacles of equal circulation. It was obtained by scanning $E$ and ${\mathrm{\Gamma }}_1$ values while keeping $\mathrm{\Gamma }$ constant. The global minimum of the enstrophy is obtained through $\beta a_1=0$ and is shown as a thick black curve.

Figure 8

Mesh convergence of the phase diagram of the continuous and mixed solution in the case of a square domain geometry (left) and annular domain geometry (right). The dashed, dotted, and solid lines, respectively, correspond to lowest, intermediary, and highest mesh resolutions.

Figure 9

Phase diagram obtained in a square geometry using our numerical resolution method.