Chapter IV: Two-Dimensional Modeling and Simulation Results
4.2 Cantilever in Constant Channel Flow
4.2.2 FSI DNS Discretization and Data Analysis
The FSI DNS Eulerian mesh is uniform in ˆx and ˆy with grid spacing ∆x∗ = ∆xL similar to the verification cases in section 3.3.1. Because the parametric study spans a wide range of ReL and hˆ values, with over 4000 simulations are carried out, the
∆x∗is automatically determined by the most restrictive of three conditions: the grid Reynolds number Re∆x ≤ 2; the minimum number of grid elements inhˆ is 20; the minimum number of elements on the beam surface is 160,
∆x∗ =min 2
ReL, hˆ 20, 1
160
. (4.2)
The Lagrangian grid spacing is always ∆s∗ = 2∆x∗, as suggested in [68]. The time step size ∆t∗ is determined by holding the CF L = ∆x∆t∗∗ = 0.2 for the ∆x∗ that satisfies the criteria 4.2. These conditions were determined by trial and error to capture least 200 time steps per beam oscillating cycle for all results. The grid Reynolds number chosen captures fluid advection and diffusion terms well. The resulting∆x∗,∆t∗combination produces results within acceptable wall-time for the number of simulations ran in this study. We explore the effect of grid refinement from criteria 4.2 in our results in section 4.2.3.
Data Analysis
To help clarify results in subsequent sections, we step through a representative ex- ample of the FSI DNS data to obtain critical flutter values. The relevant dimension- less parameters for the clamped-free boundary conditions are derived in section 2.3.6, and defined in table 2.3 (reproduced here for reference).
The beam displacement response is analyzed for each simulation in the series. We show in figure 4.2 a segment of the beam tip displacement over time for the two kˆ values in the series immediately adjacent to the stability boundary: kˆ = 4.17as stable and kˆ = 4.11as unstable. The figure also shows the least-squares fit to the exponential function coefficientζthat captures the envelope of the amplitude decay and growth of each segment, respectively. Their power spectra are shown in figure 4.3, with respective peak angular frequencies labeledωmax.
Table 4.1: Table of fluid structure non-dimensional parameters in FSI DNS grid convergence study.
Parameter Value ˆ
m 100
kˆ [3 - 4.5]
hˆ 0.05
ReL 200
hˆ2ReL 0.5
∆x∗ 0.0025
0 10 20 30 40
j"t$ -6
-4 -2 0 2 4 6
/p=L
#10!6 ^k= 4:171=!0:011
Ae1j"t$ /jp
(a) Stablekˆ =4.17.
0 10 20 30 40
j"t$ -1.5
-1 -0.5 0 0.5 1 1.5
/p=L
#10!5 ^k= 4:11,1= 0:011
/pj Ae1j"t$
(b) Unstablekˆ =4.11.
Figure 4.2: Beam tip amplitude time segment for FSI DNS results in table 4.1 for stable and unstable kˆ values.
100
! 10!10
10!5 100 105
PSD
^k= 4:17,!max= 3:77
(a) Stablekˆ =4.17.
100
! 10!10
10!5 100 105
PSD
^k= 4:11,!max= 3:77
(b) Unstablekˆ =4.11.
Figure 4.3: Beam tip amplitude power spectrum for FSI DNS results in table 4.1 for stable and unstablekˆ values.
Each DNS result is processed according to the procedure in section 3.4.1. We now plot the result of the DMD procedure as the physically meaningful spectra of kˆ = 4.11 and kˆ = 4.17, once again, along with their tip displacement fit parameter ζ, ωmaxpairs in figure 4.4. Theλtracked corresponds toλ=maxλiRe[λi]and can be seen as the ones closest toζ andωmaxpair at eachk.ˆ
-0.06 -0.04 -0.02 0 0.02
Re[6i] 0
20 40 60 80
Im[6i]
DMD ^k= 4:17 [1; !max] ^k= 4:17 DMD ^k= 4:11 [1; !max] ^k= 4:11
Figure 4.4: Comparison of dynamically significant DMD spectra (shown for Im[λ]> 0) and fit pair(ζ, ωmax)forkˆ =4.11andkˆ = 4.17.
3 3.5 4 4.5 -0.5
0 0.5
R e ( 6 )
3 3.5 4 4.5
k ^
02 4 6
Im ( 6 )
Figure 4.5: DMDλfor simulation series in table 4.1.
Figure 4.5 shows λ values obtained through this process for all 24 simulations in the series in table 4.1.
The flutter boundary critical values, kˆcr, is calculated as the linearly interpolated result between the stable and unstable (superscripts j and j + 1, respectively) kˆ values corresponding to the stable and unstable values of Re[λ]at Re[λ]=0,
kˆcr =−Reh λ(j)i
∆kˆ
∆λRe
+ kˆ(j). (4.3)
where∆kˆ = kˆ(j+1)−kˆ(j) and∆λRe =Reλ(j+1)
−Reλ(j)
. Similarly, the critical angular frequency, Im[λ]cr, is the interpolated value of Im[λ]across the boundary at Re[λ]= 0,
Im[λ]cr =−Reh
λ(j)i ∆λIm
∆λRe
+Imh λ(j)i
, (4.4)
where∆λIm = Im λ(j+1)
−Im λ(j)
. For the data set in table 4.1, kˆcr = 4.14and Im[λ]cr =3.78.
To illustrate the FSI DNS solution as the beam amplitude increases in time, we select a representative unstable parameter set from table 4.1, where kˆ = 3.07 <
4.14 = kˆcr, as seen in figure 4.5. The cantilever tip time evolution is shown in figure 4.6 with∆t∗ =5E-4, with exponential growth rate∼0.4.
0 5 10 15 20 25 30
j"t$ -0.05
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
/p=L
Figure 4.6: FSI DNS cantilever tip amplitude time evolution for case in table 4.1 at kˆ =3.07.
The x velocity profile within the channel at different snapshots in t∗ = j∆t∗ are shown in figure 4.7. The cantilever can be seen to still remain close to its equilib- rium att∗ =7.5. A slight shape change can be seen att∗ = 20, and quickly evolves of the next 5 convective time units to tip displacements close to the channel walls.
The snapshot at t∗ = 25 illustrates how the restriction of the upper channel over most of the length of the cantilever corresponds to higher axial velocities. This can also be seen toward the final 10% of the cantilever length on the lower channel as well. A resemblance to the second orthogonal Euler-Bernoulli beam mode becomes apparent aftert∗ =22.5. This can be seen in the normalized DMD mode shown in figure 4.8 in both the real and imaginary parts of the model shape.
Hence, the system shown becomes unstable at a single eigenvalue corresponding to a single mode shape. The critical parameters when the stable-to-unstable transi-
tion occurs is acquired via simulation over a range of parameters and interpolating between the set that captures negative and positive growth rates.
(a)t∗=7.5.
(b)t∗=20.
(c)t∗=22.5.
(d)t∗=25.
Figure 4.7: FSI DNSxvelocity contour snapshots at different convective time units t∗for case in table 4.1 and kˆ = 3.07.
0 0.2 0.4 0.6 0.8 1
x=L -1
-0.5 0 0.5
/=j/j1
FSI DNS: ^k= 3:07,U$= 5:71
Real Part Imaginary Part
Figure 4.8: Real and imaginary parts of unstable FSI DNS mode for case in table 4.1 atkˆ = 3.07.