Chapter III: Numerical Methods for Simulations and Data Analysis
3.2 Quasi-1D Model Numerical Implementation
We discretize the continuous operators in equations 2.90, 2.96, and 2.159 using the eigenfunctions described in either equation 2.81 or 2.84, for clamped-free or elas- tic translating boundary condition, respectively; and equation 2.154 for the span- wise flow rate expansion. All cases analyzed in this thesis are symmetric about the center-line, as shown in figure 2.1, such thatδ0 = 0, δÛ0 = 0, and for the cases that include the translating boundary condition,a¯0= 0andaÛ¯0= 0, are the equilibria the system is linearized about. The linear equations are developed using the MATLAB symbolic toolbox, the spatial derivative operators are calculated analytically with respect to each basis expansion,gi(x), or equilibrium channel shape,he(x), and the integral operators in coefficients A.1 - A.8 are calculated with MATLAB’s adap- tive step quadratureintegralfunction, with an absolute tolerance of 10−10 and relative tolerance of10−6. The Galerkin projection operator step in equations 2.97 and 2.99 is calculated using Simpson’s quadrature rule [65]. The number of basis functions kept in the expansion is 6 for all cases, was found, through trial and error, to converge the first two beam eigenvalues (lowest frequency modes) to within a tolerance of10−2in convective time units.
3.2.1 Leakage Flow in Constant Channel Verification
We verify the algorithm’s implementation by comparing our results to those of Na- gakura and Kaneko [33], who carried out a similar analysis for a cantilever in a constant, symmetric channel. They employed nonlinear operators 2.7 and 2.32 with he = h¯ = constant, andξx = 1for all Reynolds numbers, with the same definition for f as in equation 2.29. The work of Wu and Kaneko [45] also provides another
reference, where the authors compared numerical simulations to the original linear stability work [33], and use the same data set for their comparison. The Euler- Bernoulli beam equation is employed with viscous damping,
ρshb∂2
∂t2δ(x,t)+ξiωiρshb∂
∂tδ(x,t)+E I ∂4
∂x4δ(x,t)= pbot1 (x,t) −ptop1 (x,t), (3.1) where ξi is the modal damping coefficient and ωi are the angular frequencies of each mode of the undamped beam (solution to the homogeneous PDE 2.85). This is different from the model proposed in section 2.3.2, and is only used in this section to compare with existing results. Nagakura and Kaneko [33] parameters are shown in table 3.1. Although they report that the cantilever was made from bronze, they provide no reference for their chosen value of modal damping coefficientξi = 0.01, for alli. There also is no discussion around other forms of material damping that may dominate the behavior of the system (especially at low flow rates), as discussed by Banks and Inman [59].
Table 3.1: Nagakura and Kaneko [33] constant channel dimensional parameters.
.
Parameter Value Units
ρf 1.20 kg/m3
µf 1.80e-05 Pa*s ρs 8.78e+03 kg/m3
E 1.10e+11 Pa
ξi 1.0e-02 ND
hb 2.0e-04 m
b 1.0e-01 m
L 2.0e-01 m
h¯ 2.5e-03 m
qx0 0 - 5e-2 m2/s
ζin 1 ND
ζout 0 ND
The flow rate per unit span is varied from 0 to 5.0e-2 [m2/s] in the simulations.
We compare two data sets extracted and replotted from [33] and [45]: the spectrum evolution of the system as qx0 is increased for the parameters in table 3.1, and the neutral stability line, or flutter boundary, characterized by critical dimensional velocityU¯cr = qx0crh¯ as a function of dimensional throat size h. Figure 3.1 shows¯
-100 -50 0 50 100
Re(6i) [1/s]
0 100 200 300 400 500 600
Im(6i)[rad/s]
Nagakura et al. (1991) Model BC Forcing Mode
Mode 1 Mode 2 Mode 3 Mode 4 6iat 0.ow rate
Figure 3.1: Comparison of eigenvaluesλi as a function of flow rate per unit span qx0to simulation results in [33]. The plot shows the path asqx0is increased.
good agreement between the eigenvalues of the four beam modes for the range of flow rates. The current work is plotted fromqx0= 0, while [33] from a small finite value not disclosed in the article.
In our results, the flutter boundary is computed as the interpolated U¯cr where the first eigenvalue λbecomes marginally stable (Re[λ]= 0). The corresponding crit- ical angular frequency ωcr = Im[λ] |U¯
cr. A detailed description of the method is in section 4.2.4. This is in contrast with the energy formulation in [33] used to characterize the marginally stable point, whereU¯cr corresponds to its value when no energy is added or removed from to the beam. Figures 3.2 and 3.3 show good agreement between the critical quantities of the model and our implementation. The flutter boundary inU¯cris not an injective function ofh¯ because the boundary repre- sents a crossing from stable to unstable dynamics, but also from unstable to stable for a subset of parameters. That is seen specificallyh¯ ≈ 0.0025m values in figures 3.2 and 3.3.
The agreement shown in figures 3.1, 3.2, and 3.3 gives us confidence that the cou- pled equations have been implemented correctly, and that the numerical algorithm produces results similar to others in the literature. Our model predicts the exper- iments as well as previous models [33, 45] do, most notably capturing the mode
0 0.002 0.004 0.006 0.008 0.01 7h [m]
0 5 10 15 20
7Ucr[m/s]
Unstable
Stable
Current Quasi-1D Model Nagakura et al. (1991) - Model Nagakura et al. (1991) - Experiment
Figure 3.2: Comparison of critical velocity U¯cr as a function of throat size h¯ to experimental and simulation results in [33] and [45].
0 0.002 0.004 0.006 0.008 0.01 7h [m]
0 100 200 300 400 500 600
!cr[rad/s]
Unstable
Stable
Current Quasi-1D Model Nagakura et al. (1991) - Model Nagakura et al. (1991) - Experiment
Figure 3.3: Comparison of critical velocity U¯cr as a function of throat size h¯ to experimental and simulation results in [33] and [45].
switching seen in the transition from lower to higher ωcr in figure 3.3: as h¯ de- creases, progressively higher frequency modes bifurcate through a Hopf type into instability. A detailed study on these dynamics follows in section 4.2.
3.3 Fluid-Structure Immersed-Boundary Direct Numerical Simulation