Variational construction of tubular and toroidal streamsurfaces for flow visualization

Abstract

Approximate streamsurfaces of a 3D velocity field have recently been constructed as isosurfaces of the closest first integral of the velocity field. Such approximate streamsurfaces enable effective and efficient visualization of vortical regions in 3D flows. Here we propose a variational construction of these approximate streamsurfaces to remove the limitation of Fourier series representation of the first integral in earlier work. Specifically, we use finite-element methods to solve a partial-differential equation that describes the best approximate first integral for a given velocity field. We use several examples to demonstrate the power of our approach for 3D flows in domains with arbitrary geometries and boundary conditions. These include generalized axisymmetric flows in the domains of a sphere (spherical vortex), a cylinder (cylindrical vortex), and a hollow cylinder (Taylor-Couette flow) as benchmark studies for various computational domains, non-integrable periodic flows (ABC and Euler flows), and Rayleigh-Bénard convection flows. We also illustrate the use of the variational construction in extracting momentum barriers in Rayleigh-Bénard convection.

Figures (30)

• Figure 1: Meshes used in the computation of the benchmark generalized axisymmetric flows in the domain of a sphere (left), a cylinder (middle) and a hollow cylinder (right).
• Figure 2: Second smallest eigenvalues as functions of the number of elements used in the discretization of the generalized axisymmetric flows: spherical vortex (left), cylindrical vortex (middle) and Taylor-Couette flow (right).
• Figure 3: Contour plots of $\hat{H}_2$ for the spherical vortex obtained from finite-element methods with 62,105 elements (left panels) and $\psi$ in \ref{['eq:psi-sph']} (right panels), at cross section $x=0$ (upper panels) and $z=0$ (lower panels). Here, the solutions from the finite-element computation are denoted by FEM, and that of analytical expressions are denoted by Reference.
• Figure 4: Contour plots of isosurfaces for $\hat{H}_2$ of the spherical vortex obtained from finite-element methods with 62105 elements. Here we have $\hat{H}_2=0.08$ (left panels) and $\hat{H}_2=0.12$ (right panels). The black lines are streamlines from forward simulations with initial points on the isosurfaces.
• Figure 5: Contour plots of $\hat{H}_2$ for the cylindrical vortex obtained from finite-element methods with 30,888 elements (left panels) and $\psi$ in \ref{['eq:psi-cylin']} (right panels), at cross section $x=0$ (upper panels) and $z=0$ (lower panels). Here, the solutions from finite-element computation are denoted by FEM, and that of analytical expressions are denoted by Reference.