Quasi-1D Flutter Boundary Comparison to Inviscid Model . 105

Chapter IV: Two-Dimensional Modeling and Simulation Results

4.2 Cantilever in Constant Channel Flow

4.2.5 Quasi-1D Flutter Boundary Comparison to Inviscid Model . 105

10^!1 10⁰ 10¹

^h²ReL

0 5 10 15 20

U$ cr

Unstable

Stable

FSI DNS Quasi-1D Model

(a)U_cr^∗ as a function ofhˆ²Re_L.

10^!1 10⁰ 10¹

^h²ReL

0 1 2 3 4 5

f$ scr

Unstable

Stable

FSI DNS Quasi-1D Model

(b) f_scr^∗ as a function ofhˆ²Re_L. Figure 4.25: Comparison of FSI DNS and quasi-1D model flutter boundary critical values athˆ = 0.125andM^∗ =0.01, case 2 in table 4.3.

0 0.2 0.4 0.6 0.8 1

x=L -0.5

0 0.5 1

/=j/j1

Quasi-1D Model: ^k= 0:41483,U^$= 15:5263

0 0.2 0.4 0.6 0.8 1

x=L -1

-0.5 0 0.5 1

/=j/j1

FSI DNS DMD: ^k= 0:39685,U^$= 15:8739 Real Part

Imaginary Part

Figure 4.26: Comparison of real and imaginary parts of unstable mode near flutter boundary from quasi-1D model (left) and FSI DNS (right) athˆ = 0.125,M^∗ = 0.01, andhˆ²ReL = 3. (case 2 in table 4.3).

the stability boundary shifts upwards. Yet ashˆ²Re_Lis further increased to 12.50 and thereafter the system is destabilized for M^∗ > 0.03, with the boundary eventually disappearing for 0.03 < M^∗ < 0.2 at hˆ²Re_L = 50. This means that no matter the stiffness of the system, the first mode is unstable if the beam is heavy enough.

The original stabilization trend for increasing hˆ²ReL remains true, however, for M^∗ < 0.03. Furthermore, as hˆ²Re_L becomes large, the quasi-1D flutter boundary appears to near the inviscid results acquired from [39], with the mode switching inflection nearly matching over allhˆ²Re_L boundaries shown.

Shoele and Mittal [39] conjectured based on earlier DNS studies [76] that Re_L ≈ 200was enough to consider the system inviscid at least over the hˆ values in their study. Though this may be true for hˆ > 0.125, the inviscid behavior boundary for hˆ =0.05appears to beReL ≈2×10⁴from figure 4.27. Predictions of the quasi-1D model indicate that the choice of Re_L for inviscid treatment is a strong function of h.ˆ

10

^!2

10

^!1

10

⁰

M

^$

0

5 10 15

U

$ cr

Unstable

Stable

Q1D ^h²ReL= 1:25 Q1D ^h²ReL= 2:50 Q1D ^h²ReL= 12:50 Q1D ^h²ReL= 25 Q1D ^h²ReL= 50

Inviscid - Shoele and Mittal (2015)

Figure 4.27: Comparison of flutter boundary for lowest frequency mode between different quasi-1D model (Q1D) hˆ²Re_L values and inviscid model by Shoele and Mittal [39] athˆ =0.05.

In light of these results and those in section 4.2.4, we utilize the quasi-1D model to

produce the complete flutter boundary for M^∗ = 0.01in figure 4.28 and M^∗ = 0.1 in figure 4.29. Both figures only show the flutter boundary for the lowest frequency mode branch. Three trends become apparent from these plots: first, as M^∗ in- creases, the lowest frequency mode is destabilized significantly; ashˆ²Re_Lincreases, the mode is stabilized. This stabilization is accelerated at higher h, for values forˆ hˆ²Re_L < 10. Figure 4.27 shows that opposite is true ashˆ²Re_L is increased further.

Lastly, ashˆ increases, the lower mode is stabilized.

Though f_scr^∗ remains within a narrow range for the lowest frequency mode, an inter- esting pattern arises as hˆ²Re_L and hˆ are varied. Bands of lower frequency appear alternating with higher frequency states in bothM^∗values. This indicates that these parameters have an effect on the frequency response, but it is much less pronounced than their effects on the stability boundary as judged byU_cr^∗.

(a)U_cr^∗ contours vs. hˆ²Re_L andh.ˆ (b) f_scr^∗ contours vs. hˆ²Re_L andh.ˆ Figure 4.28: Quasi-1D predicted critical flutter values as a function ofhˆ²Re_L andhˆ atM^∗ =0.01.

(a)U_cr^∗ contours as a function ofhˆ²Re_L andh. (b)ˆ f_scr^∗ contours as a function ofhˆ²Re_L and h.ˆ

Figure 4.29: Quasi-1D predicted critical flutter values as a function ofhˆ²Re_L andhˆ atM^∗ =0.1.

4.2.6 Elastic-Translating Boundary Condition in a Constant Channel

We now consider the elastic-translating boundary condition defined in section 2.3.3 for the quasi-1D model, and in section 3.3.2 for the FSI DNS. The problem ge- ometry is shown in figure 4.30, with its leading edge boundary condition specified as a simple harmonic oscillator, via non-dimensional parametersmˆ_bc, cˆ_bc, and kˆ_bc defined in table 2.4.

Uin= 1

L= 1

¯

a _¯a

¯h

h¯ 3¯h

Elastic-translating BC

Figure 4.30: Illustration of linear diffuser flow geometry for cantilevered beam (top), and elastically-mounted rigid beam (bottom).

We will, once again, explore agreement between the model and FSI DNS simu- lations that map the stability boundary in kˆ space. Our results are plotted in the convention of U^∗, as before, defined in equation 4.6. Given results from section 4.2.4, we restrictM^∗ = 0.02, hˆ = 0.125, hˆ²ReL = 0.5as the beam parameters that critical properties are well captured by the clamped-free quasi-1D model, but at the

“boundary” of criterion 3, wherehˆ 1criterion.

Since the clamped-free boundary condition is the limiting case where the boundary stiffness, kˆ_bc → ∞, we choose kˆ_bc = 0.01to probe the dynamics in the opposite limit of low flexure stiffness. Once again, kˆ is the bifurcation parameter, and the kˆcr (orU_cr^∗) is mapped as a function of mˆbc as defined in table 2.4. The structural damping parameter cˆbc = 0 for all cases. Table 4.4 shows the case run in this section, withmˆ_bc andkˆ parameter ranges shown. Parameter and spatial grid details for can be found in table C.3 in appendix C.

Table 4.4: Table of cases for constant channel flow simulations with elastic- translating boundary conditions. Parametersmˆ_bc andkˆ are varied.

Case # hˆ²Re_L Re_L mˆ hˆ kˆ_bc mˆ_bc kˆ

1 0.5 32 50 0.125 0.1 [ 7 - 1800] [ 0.42 - 41.67 ] Figure 4.31 shows the critical values for cases in table 4.4. The stability boundary in U^∗agrees well between the FSI DNS and quasi-1D model. In particular, the mode branching that occurs asmˆbc decreases appears to be well captured. Similarly, the critical frequency trend is replicated, though with a slight under prediction by the model. A representative mode is shown in figure 4.32 formˆ_bc = 100, also showing good qualitative agreement between model and DNS DMD results.

By spanning 3 orders of magnitude in mˆ_bc with a low boundary stiffness value (kˆ_bc = 0.01), results from figure 4.31 indicate that the model also provides a good approximation of the dynamics when the elastic-translating boundary condition is introduced into the system.

(a)U_cr^∗ as a function ofmˆ_bc.

10⁰ 10² 10⁴

^

mbc

0 1 2 3 4 5

f$ scr

Unstable

Stable FSI DNS

Quasi-1D Model

(b) f_scr^∗ as a function ofmˆ_bc. Figure 4.31: Comparison of FSI DNS and quasi-1D model flutter boundary critical values athˆ = 0.125,M^∗ =0.02, andhˆ²ReL =0.5for all cases in table 4.4.

Figure 4.32: Comparison of quasi-1D model (left) and FSI DNS (right) normalized unstable mode shapes athˆ = 0.125,M^∗ =0.02,mˆ_bc =100.