Introduction
Motivation
Consequently, an understanding of fluid structure systems is vital in both the design and validation of many engineering systems. Often, fluid-structure interaction serves as the starting point for the analysis of such systems.
Periodic Behavior in Unsteady Flows
Thus, it will be shown that under certain circumstances, the forced system can indeed be transformed to resemble its unforced counterpart. Thus, relating a forced system to an analogous unforced system can also facilitate the understanding and characterization of forced regimes beyond the reach of experiment. one body in the wake of another, where the next flow can be characterized by a similar frequency ratio.
Complexity Reduction and Reduced-Order Modeling
Although the use of quasi-steady approximations is not unique to fluid mechanics, the underlying theme is unifying. One goal of the present study is to develop a framework that can be used to identify forcing regimes where the assumption of quasi-steady dynamics is appropriate.
Background
- Streamwise-Oscillating Cylinders
- Flow Around Surface-Mounted Hemispheres
- Koopman Analysis and Dynamic Mode Decomposition
Let x𝑘 denote the complete state of the system at time 𝑡 = 𝑘Δ𝑡, evolving according to the relationship. By expressing an observable value of the system asg(x), the Koopman operator,K, is defined by the relation.
Outline of Thesis
Consequently, the eddy trajectories corresponding to the first half of the forcing cycle show significant fluctuations in the transverse direction. Snapshots of the system at different stages in the forcing cycle are presented in Figure 1.
Methods
Experimental Facilities
- NOAH Free-Surface Water Channel
- Captive Trajectory System
- Benchtop Pulsatile Wind Tunnel
- Forcing Trajectories
A 4:1 constriction, located downstream of the flow conditioners, is used to increase the flow rate before entering the test section. The walls of the test section were constructed using clear acrylic for optical access.
Diagnostics
- Particle Image Velocimetry
- Fluorescent Dye Flow Visualization
This is the logical choice, since all the flow structures in the wake are generated by the cylinder. A color injection port was located at the upstream stagnation point of the cylinder and (vertically) in the center of the test section.
Data Analysis and Decomposition
- Phase Averaging
- Welch’s Method
- Dynamic Mode Decomposition
- Γ 2 Vortex Identification
- Sparse Sensor Placement
A Nikon D600 DSLR camera with a Nikon AF Nikkor 50mm f/1.8D lens was used to record images of the illuminated dye. Eigenvalues of A, denoted by 𝜆𝑖 and known as DMD eigenvalues, represent the growth rate and frequency of the corresponding DMD mode. Under certain circumstances, DMD can be shown to be an approximation of the Koopman Mode Decomposition (Drmac et al., 2018; Rowley et al., 2009).
The Galilean invariance of the Γ2 criterion makes it an excellent candidate for use in non-inertial frames, including those.
Quasi-Steady Time Scaling
- Quasisteadiness Parameter
- Time Scaling
- Implementation
Application of the Γ2 criterion to the oscillating cylinder's flow is shown at various stages in Fig. The characteristic shedding frequency of the hemisphere corresponds to the generation and shedding of hairpin vortices. Thus, the structure upstream of the hemisphere in the forcing mode captures the streamwise oscillation of the horseshoe vortex.
This is highlighted by the time trace of velocity in the wake of the scaled system, shown in Fig.
Characterization of the Streamwise-Oscillating Cylinder’s Wake . 27
Dynamic Mode Decomposition and Power Spectrum
Furthermore, all the DMD eigenvalues for the stationary cylinder (Fig. 3.5) lie on or inside the unit circle, resulting in DMD modes that oscillate periodically or decay with time. With the exception of the 𝑆𝑡 =0 mode, which is an extraction of the mean flow around the cylinder, the representative modes are clustered around the stationary shedding frequency. It can be inferred that the modes containing these vertical structures are associated with vortex shedding and will henceforth be referred to as "shedding modes." Furthermore, the discharge state in fig.
The finite width of the spectral peak and the distribution of multiple shedding modes around it in Fig.
Oscillating Cylinder
- Observed States in the Wake
With the exception of the mean state (𝑆𝑡 =0), the colored circles on the spectrum represent excretion states. The colored state labels correspond to the circles of the same color on the spectrum. Except for the zero-frequency mode and the mode at the forcing frequency, each of the colored points on the spectra shown in Fig.
These additional secretion modes arise as a result of the gradual modulation of the secretion frequency during the forcing period.
Vortex Dynamics
- Starting Vortex Generation
- Vortex Identification
The latter half of the strain cycle coincides with strong shedding resulting in the identification of numerous well-defined vortices (Fig. 3.20c). The behavior observed during the two halves of the forced cycle motivates the analysis of the vortex trajectories in the oscillating case differently. As described in Section 2.4, the parts of the forced cycle where Ω 1 are indicative of quasi-steady behavior similar to the unforced case.
It was also shown that the two dominant peaks observed in the survey point spectra corresponded to vortex shedding in regions of the forcing cycle corresponding to Ω 1.
Complexity Reduction of the Streamwise Oscillating Cylinder
Quasisteadiness
The quasi-constancy parameter, Ω, is considered first, as it characterizes the parts of the forcing cycle that correspond to quasi-steady dynamics. 0 = 0.21 at all instantaneous Reynolds numbers encountered during the forcing cycle (Fey et al., 1998). Thus, consideration of the corresponding phase portrait (Figure 4.1a) sheds light on the behavior observed during different parts of the forcing cycle.
This depends on the range of Reynolds numbers traversed during each part of the forcing cycle.
Quasi-Steady Time Scaling
Besides the presence of two peaks in Ω, Fig. 4.1b also an increase in asymmetry with𝑅 𝑒𝑞. While 𝑓¤ controls the location of the peaks and is controlled by the cylinder acceleration, 𝜏¤ also includes the instantaneous shedding frequency, which depends on the sense of the cylinder motion relative to the free stream, leading to the observed asymmetry. This is an important distinction because the broader peak in Ω corresponds to the most active portion of the forcing trajectory where shedding is stronger. In this context, "stronger" refers to the duration of the quasi-steady portion, strength of shear vortices, number of completed shedding cycles, and relative size of each spectral peak.
Thus, Ω not only provides insight into quasi-steady regions of the forcing cycle, it is also related to the relative strength of such regimes.
Quasi-Steady Shedding Regimes
4.7, quasi-steady time scaling leads to a reduction in spectral activity for the frequency range active in the unscaled system. This is in stark contrast to the spectra for unscaled cases (Fig. 3.9) which had local minima at the steady-state secretion frequency. Furthermore, measuring only a single forcing cycle in the experimental time series could also lead to a narrower peak for these cases.
Relative to the unscaled system, scaling also results in the extraction of fewer turn modes, each containing less noise and contamination from other structures in the flow.
Reduced-Order Flow Reconstruction
The location of the query point is indicated by ×. reconstructed velocity at the survey point, shown in fig. More importantly, however, the model shows that the quasi-steady assumption regarding the stationary and oscillating systems enables the generation of a physically meaningful model. It is reiterated that quasi-steady time scaling links the dynamics of the scaled and stationary systems.
After time scaling, the dynamics of the forced system is similar to the dynamics of the stationary system, which is confirmed using DMD.
Limitations of Time Scaling
In addition to the wake dynamics described above, the cycle of pulsating forcing also leads to the formation of a horseshoe vortex upstream of the hemisphere. On the other hand, the relatively flat part of the time trace near the minimum free-stream velocity corresponds to the absence of wake shedding due to upward motion of the trailing arc vortex. Specifically, the time trace shows a contraction in the unsteady segment of the forcing cycle, which results in an effective reduction in the effect of the rear bow vortex.
Similarly, footprints of shedding are present in the reconstruction at phases corresponding to the movement of the bow vortex upstream.
Complexity Reduction of the Flow Around a Hemisphere in Pul-
Flow Structures in the Absence of Pulsatility
Although hair vortices are due to the shedding of arched vortex tubes located at the hemisphere separation line, the velocity gradient (and corresponding shear) away from the wall causes the hairs to tilt downward and stretch as they convect away from the hemisphere. (Tamai et al., 1987). As a result, the leading part of each leg is slightly raised and away from the wall. 1987) note that multiple hairpins can coalesce into a single structure, resulting in a hair shedding frequency that is different from the frequency at which individual hairpins are formed in a hemisphere.
Examples include the production of secondary hairpin vortices in the wake on either side of the hemisphere due to the interaction between the high-momentum outer flow and the low-velocity flow lifted from the wall by the horseshoe vortex (Acarlar and Smith, 1987).
Analysis of the Hemisphere Wake with Pulsatile Forcing
- Hairpin Shedding Regime
- Phase-Locked Regime
- Dynamic Mode Decomposition
The resulting horseshoe vortex can be seen as the small region of vortices located just upstream of the hemisphere in the last two columns of figure. Beyond flow structures in the immediate vicinity of the hemisphere, the presence of the wall merits a discussion of the boundary layer. As the free stream velocity decreases, the velocity induced by the trailing arc vortex eventually exceeds the instantaneous free stream, resulting in the forward motion of the trailing arc vortex.
Furthermore, the time trace is devoid of the high-frequency fluctuations seen in the above regime, further verifying the absence of periodic hairpin outbursts.
Complexity Reduction via Time Scaling
Color labels in (b) correspond to circles of the same color in the spectrum. Thus, as a precursor to the time scaling analysis, a moving average filter was applied to the forcing profile shown in Fig. 5.3 to remove high-frequency measurement noise, then a fourth-order Gaussian of the form. Thus, this part of the pulsation forcing cycle corresponds to the regime most likely to exhibit stripping and wake dynamics similar to the steady flow case.
Further analysis of the scaled system elucidates the effect of time scaling on frequency modulated emission.
Sensor Placement
The clustering of sensors in the track indicates both its dynamic significance in the flow field as well as its spatial complexity. Although most sensors reside in the wake, several sensors are located elsewhere in the flow. In contrast to the unscaled case, Fig. 5.16b that almost all sensors are clustered in the track.
Furthermore, clustering almost all sensors in time scaling means that the total number of sensors is likely to be reduced, alluding to reduced system complexity.
Flow Reconstruction
Frequency-modulated secretion is especially evident in the corresponding time trace of velocity, shown in Figure. This is likely due to the presence of additional vortex structures riding on top of the trailing arc vortex. However, future work can explore the inclusion of additional modes in the reconstruction to solve this problem.
A snapshot of the flow reconstructed with only the mean and forcing modes is shown in Fig. 1.
Chapter Summary and Outlook
The occurrence of fluid-structure interaction in the world around us warrants a deeper understanding of the underlying physics. Pulsating forcing frequencies much lower than the unforced hairpin rejection frequency enabled the development of frequency-modulated hairpin displacement during various portions of the forcing cycle. In contrast to the unscaled case, time scaling led to the development of a spectral peak at the stationary shedding frequency.
This could lead to the development of a full dynamical model of the system that includes time scaling, which is an interesting question for future work.
Conclusion
Summary
In particular, dynamic mode decomposition (DMD), the main modal analysis technique used in the thesis, was outlined and the quasi-steady time scaling framework developed. Consequently, the vortex shedding and oscillatory behavior in the wake were suppressed at that stage of the forced cycle. It was shown that peaks in Ω corresponded to segments of quasi-steady trailing vortex shedding similar to the stationary cylinder.
In contrast, when the forcing frequency was of the same order as the ejection frequency, DMD extracted a single phase-locked structure in the wake, corresponding to the trailing arc vortex characterized by Carr et al.
Future Work
