Backward facing step flow in axisymmetric coordinate with immersed boundaries.

These test cases study laminar axisymmetric through confined annular geometries with backward facing step modeled by immersed boundaries. The axisymmetric steady ﬂow of an incompressible Newtonian ﬂuid is solved in the \((r,y)\) plane. The dependence of reattachment length on Reynolds number ranging from 50 to 400 is presented and compared with the experiments and numerical solutions ([1], [2], [3]).

Configurations

The geometry is constructed based on the reference case [1] but in vertical direction, as shown in Fig.1.

Figure 1: Step cylinder geometry.

The boundary conditions for the system are

\(u = 0, v = 0 \) at the right and immersed boundaries,
\(u = 0, \frac{\partial{v}}{\partial{r}} = 0\) at the left axis of symmetry,
\(u = 0, v = 2 u_{mean} \left[1-\left(\frac{r}{R}\right)\right]^2\) at the bottom and
\(u = 0, \frac{\partial{v}}{\partial{y}} = 0\) at the top.

Runtime parameters

Navier-Stokes equation is used with HYPRE or MUMPS solvers. The second order centered advection scheme is used. First order volume penalization immersed boundary method is used.

Test case list

Label
backward_facing_step_axisym_2D.nts

Results

The nondimensionalized reattachment length \(L_{H}\) is almost linear for Reynolds number range covered by the study with various expansion ratio \(\beta\). The following figure compares the reattachment length for Re = 50, 100, 150 and 200 and an expansion ratio of 2 with references [1–3].

Figure 3: Comparions of the reattachment length for Re=100 and expansion ratio equal to 2.

The next table shows a spatial convergence order of one for several domain-averaged quantities.

Mesh	mean_pressure	Order	mean_velocity_u	Order	mean_velocity_v	Order	mean_velocity_magnitude	Order
64	-8.83402351e-02	n/a	5.02550612e-03	n/a	5.38946479e-01	n/a	5.49199039e-01	n/a
128	-8.99617909e-02	n/a	5.01111985e-03	n/a	5.40406307e-01	n/a	5.50865946e-01	n/a
256	-9.08452280e-02	+0.8762	5.00330500e-03	+0.8804	5.41229619e-01	+0.8263	5.51819015e-01	+0.8065
512	-9.13076517e-02	+0.9339	4.99928058e-03	+0.9574	5.41667023e-01	+0.9125	5.52328508e-01	+0.9035

References

[1] Hammad, K., Ötügen, M. & Arik, A PIV study of the laminar axisymmetric sudden expansion flow, E. Experiments in Fluids (1999) 26: 266.

[2] Macagno EO; Hung TK, Computational and experimental study of a captive eddy, J Fluid Mech (1967) 28: 43—64.

[3] Dağtekin İ, Ünsal M. Numerical analysis of axisymmetric and planar sudden expansion flows for laminar regime, Int J Numer Methods Fluids (2011) 65: 1133–44.