Chapter III: Numerical Methods for Simulations and Data Analysis
3.3 Fluid-Structure Immersed-Boundary Direct Numerical Simulation
3.3.1 Internal Flow Fluid-Structure-Interaction Verification
The algorithm has been verified extensively in [71] for external flows, where regimes of the standard and inverted flags 1 were explored and compared to results from other strongly-coupled fluid-structure solvers. In this section, we verify the algo- rithm for internal flows using the suggested benchmark of an elastic member in an internal, incompressible, laminar flow [38, 74–76].
The IB projection method as developed by Colonius and Taira [66, 67] uses a multi- domain approach to treat its far-field zero vorticity boundary condition. This means that the geometry solved as internal flow lives as part of a larger, quiescent fluid cell.
This verification exercise is in part to ensure that the inlet and equivalent outflow boundary conditions behave as other algorithms do when solving a similar problem.
The geometry consists of a cylindrical bluff body within a slightly asymmetric channel and a flexible cantilever (standard clamped flag) immediately downstream in its wake, as illustrated in figure 3.4. The cantilever is infinitesimally thin, but has a structural thickness according to its bending stiffness and mass ratio. Its equiv- alent numerical thickness will be discussed in further detail in section 4.2, but is inconsequential for cases in this section. The parabolic profile has an average ve- locityU¯ = Uc, which is the characteristic velocity for the Reynolds number ReD based on the cylinder diameter D. We consider the same two cases as in Shoele
1Standard refers to a flag with clamped or pinned leading edge, where as inverted the pinned or clamped boundary condition is at the trailing edge.
Figure 3.4: Illustration of FSI DNS validation geometry by [74].
and Mittal [76]: ReL = 350(ReD = 100) and ρρs
f = 10as case 1, and ReL = 700 (ReD = 200) and ρρs
f =1as case 2, both with the samek.ˆ
Table 3.2: Non-dimensional and mesh parameters for DNS FSI verification cases in terms of beam length L.
Parameter Case 1 Case 2
ReL 350 700
mˆ 0.5710 0.05710
kˆ 0.0218 0.0218
∆x∗ =∆x/L 0.00571 0.0028
∆t∗ =∆tUc/L 5.714E-4 2.86E-4 Re∆x = ReL∆x∗ 2 2 CF L =∆t∗/∆x∗ 0.3 0.3
Table 3.2 denotes the relevant meshing and non-dimensional parameters for each case in terms of the beam length L, per equations 3.2. The two-dimensional fluid grid has uniform spacing ∆x in ˆx and ˆy, and defined as ∆x∗ when normalized by L. The Lagrangian grids used as immersed boundaries on all bodies have a spacing
∆s∗ = 2∆x∗, including the cylinder, walls, inlet, and beam. There are a total of 88 Lagrangian points on the beam for cases 1 and 176 for case 2. The target grid Reynolds number Re∆x for all cases is 2, and the targetCF L = 0.3, with≈ 2600 time steps per beam tip oscillation cycle for case 1 and≈3500time steps per cycle
for case 2. This ensures that the fluid-structure algorithm is stable and resolved.
The total simulation time horizon is over 70Lconvective time units for cases 1 and and 20 for case 2. The beam tip has reached a steady amplitude and frequency for at least 8 cycles before results were measured in both cases.
FSI DNS Verification Results
Figures 3.5 and 3.6 show an instantaneous snapshot of the flow velocities when the cantilever is at its peak amplitude for cases 1 and 2, respectively. The wake of the cylinder is clearly visible in both cases, with ˆx velocities fastest at the cylinder top and bottom, and at the channel restriction point by cantilever tip. Stagnation points can be seen at the cylinder surface closest to the inlet, and the tip displacement is visibly higher for case 1 than case 2.
Figure 3.5: Case 1 (table 3.2) velocity contour plot when beam tip (i.e. trailing edge) is at its maximum amplitude.
Figure 3.6: Case 2 (table 3.2) velocity component contour plot when beam tip (i.e.
trailing edge) is at its maximum amplitude.
Figure 3.7 shows the length normalized ytip and xtip displacements of the beam (i.e. trailing edge) for the last two tip oscillation cycles in our simulations. T is the Lconvective time immediately before the last two oscillations. Our results are compared to those in the appendix of [76]. The length scale in all plots is L (as opposed to D) and the origin is at the leading edge of the beam, as illustrated in figure 3.4. The data acquired from [76] was scaled accordingly to reflect the correct length and time scale. The time history of tip displacements appear to be in fair agreement, both in ˆxand ˆy.
Table 3.3 with our appended values is reproduced from [76], notably with a rescal- ing of Las the reference length scale. The definition of measured values are
Am
L = max ytip
−min ytip
2L , St = f L Uc
, CD = 2Fx
ρfUc2L, (3.5) where f is the dimensional frequency of oscillation. Fair agreement between all val- ues is seen both case 1 and case 2. The general spread among measured quantities for case 1 is smaller than that of case 2, but our results appear to fall within the vari- ance of previous algorithms. These results give us confidence that our discretization
Table 3.3: Quantitative comparison of measured values for FSI DNS verification, with cases defined in table 3.2.
# Sources A/L S t CD
Case 1 Current FSI DNS 0.260 0.663 1.285 Shoele and Mittal [76] 0.254 0.665 1.197 Turek and Hron [74] 0.237 0.665 1.180 Bhardwaj and Mittal [75] 0.263 0.665 1.017 Tian et al. [38] 0.223 0.665 1.174 Case 2 Current FSI DNS 0.142 1.000 0.774 Shoele and Mittal [76] 0.125 0.945 0.709 Turek and Hron [74] 0.103 0.910 0.657 Bhardwaj and Mittal [75] 0.117 0.980 0.629 Tian et al. [38] 0.091 1.015 0.617
parameters and the boundary conditions replicate results that are consistent with the literature.
0 1 2 3 4 tUc=L!T
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
ytip=L
Shoele and Mittal (2014) Bhardwaj and Mittal (2012) Turek and Hron (2006) Current FSI DNS
(a) Comparison of case 1ytipdisplacement for last 2 cycles.
0 1 2 3 4
tUc=L!T 0.88
0.9 0.92 0.94 0.96 0.98 1
xtip=L
Shoele and Mittal (2014) Bhardwaj and Mittal (2012) Turek and Hron (2006) Current FSI DNS
(b) Comparison of case 1xtipdisplacement for last 2 cycles.
Figure 3.7: Comparison of beam tip displacement for case 1 with data set in ap- pendix of [76]. The data includes Shoele and Mittal (2014) [76], Bhardwaj and Mittal (2012) [75], and Turek and Hron (2006) [74].