2D Cylinder – Artur Palha

Problem definition

An important aspect of flow simulations is the accurate calculation of forces, specifically lift and drag. Therefore, we apply the hybrid solver to the flow around an impulsively started cylinder at $latex \mathrm{Re}=550$ and determine the forces acting on the cylinder. This test case problem has been extensively analysed in the literature, for example [Koumoutsakos et al. 1995] and [Rosenfeld et al. 1991], and these results will serve as a reference for the assessment of the hybrid solver, since one is a pure vortex particle solver and the other one is a pure Eulerian grid solver.

The configuration of the hybrid domain is presented below:

The domain decomposition for the impulsively started cylinder. (Not to scale) — The domain decomposition for the impulsively started cylinder. (*Not to scale*)

The Lagrangian domain, $latex \Omega_{L}$, covers the whole fluid domain and the Eulerian domain, $latex \Omega_{E}$, is confined to a small region around the cylinder. The parameters used in this simulation are presented in the next table:

Results

The contour plots of vorticity, comparing the hybrid results to the FE ones, are presented below:

Plot of the vorticity field at $latex t=[10,20,30,40]$, comparing the hybrid simulation (left) with the FE simulation (right).

We can see that the two solvers give very similar results. Regarding the drag and lift, we can say that the hybrid solver is capable of reproducing both the FE results and the results of [Koumoutsakos et al. 1995], see the next figure:

Evolution of the drag coefficient during the initial stages $latex t=0$ to $latex t=4$ with total drag coefficient $latex C_d$ (solid), pressure drag coefficient $latex C_{d_{pres}}$ (dashed) and friction drag coefficient $latex C_{d_{fric}}$ (dotted). The figure compares results of hybrid simulation (blue), FE simulation (red) and reference data (black) of [Koumoutsakos et al. 1995].

The differences exist mainly in the first instants of the simulation. It is important to highlight that increasing $latex h_{bdry}$ from $latex 0.1R$ to $latex 0.2R$ improves considerably the results, see the next figure:

Comparison of total drag coefficient $latex C_{d}$ for $latex d_{bdry} \in \{0.1R,0.2R\}$.

We have noted that this parameter is important and further research should be done in order to find its optimal value.

A longer time simulation, $late t\in[0,40]$, was performed and the lift and drag compared to the results of [Rosenfeld et al. 1991]:

Evolution of the lift and drag coefficients from $latex t=0$ to $latex t=40$ with artificial perturbation [Lecointe et al. 1984]. The figure compares hybrid (blue), FE only (red), and the reference data (black) from [Rosenfeld et al. 1991].

The hybrid solver follows very well both the reference results of [Rosenfeld et al. 1991] and the FE simulation up to the onset of the vortex shedding, $latex t\approx 5s$. After that, all results stop having a good match but all remain within the same bounds and show similar frequency, as expected.

Another aspect of the hybrid method, which is inherited from the vortex particle method, is its automatic adaptivity, where computational elements exist only in the support of vorticity, as opposed to standard grid solvers where the computational elements exist in the whole computational domain, see figures below:

Computational elements of hybrid flow solver.

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References

Koumoutsakos, P., & Leonard, A. (1995). High-resolution simulations of the flow around an impulsively started cylinder using vortex methods. Journal of Fluid Mechanics, 296(1), 1–38. doi:10.1017/S0022112095002059

Rosenfeld, M., Kwak, D., & Vinokur, M. (1991). A fractional step solution method for the unsteady incompressible navier-stokes equations in generalized coordinate systems. Journal of Computational Physics, 94(1), 102–137. doi:10.1016/0021-9991(91)90139-C

Lecointe, Y., & Piquet, J. (1984). On the use of several compact methods for the study of unsteady incompressible viscous flow round a circular cylinder. Computers & Fluids, 12(4), 255–280.