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Concept# Strouhal number

Summary

In dimensional analysis, the Strouhal number (St, or sometimes Sr to avoid the conflict with the Stanton number) is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind. The Strouhal number is an integral part of the fundamentals of fluid mechanics.
The Strouhal number is often given as
where f is the frequency of vortex shedding, L is the characteristic length (for example, hydraulic diameter or the airfoil thickness) and U is the flow velocity. In certain cases, like heaving (plunging) flight, this characteristic length is the amplitude of oscillation. This selection of characteristic length can be used to present a distinction between Strouhal number and reduced frequency:
where k is the reduced frequency, and A is amplitude of the heaving oscillation.
For large Strouhal numbers (order of 1), viscosity dominates fluid flow, resulting in a collective oscillating movement of the fluid "plug". For low Strouhal numbers (order of 10−4 and below), the high-speed, quasi-steady-state portion of the movement dominates the oscillation. Oscillation at intermediate Strouhal numbers is characterized by the buildup and rapidly subsequent shedding of vortices.
For spheres in uniform flow in the Reynolds number range of 8×102 < Re < 2×105 there co-exist two values of the Strouhal number. The lower frequency is attributed to the large-scale instability of the wake, is independent of the Reynolds number Re and is approximately equal to 0.2. The higher-frequency Strouhal number is caused by small-scale instabilities from the separation of the shear layer.
Knowing Newton’s Second Law stating force is equivalent to mass times acceleration, or , and that acceleration is the derivative of velocity, or (characteristic speed/time) in the case of fluid mechanics, we see
Since characteristic speed can be represented as length per unit time, , we get
where,
m = mass,
U = characteristic speed,
L = characteristic length.

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The presence of aerodynamic vortices is widespread in nature. They can be found at small scales near the wing tip of flying insects or at bigger scale in the form of hurricanes, cyclones or even galaxies. They are identified as coherent regions of high vorticity where the flow is locally dominated by rotation over strain. A better comprehension of vortex dynamics has a great potential to increase aerodynamic performances of moving vehicles, such as drones or autonomous underwater vehicles. An accelerated flat plate, a pitching airfoil or a jet flow ejected from a nozzle give rise to the formation of a primary vortex, followed by the shedding of smaller secondary vortices. We experimentally study the growth, timing and trajectory of primary and secondary vortices generated from a rectangular flat plate that is rotated around its centre location in a quiescent fluid. We systematically vary the rotational speed of the plate to get a chord based Reynolds number \Rey that ranges from 800 to 12000. We identify the critical \Rey for the occurrence of secondary vortices to be at 2500. The timing of the formation of the primary vortex is \Rey independent but is affected by the plate's dimensions. The circulation of the primary vortex increases with the angular position $\alpha$ of the plate, until the plate reaches 30°. Increasing the thickness and decreasing the chord lead to a longer growth of the primary vortex. Therefore, the primary vortex reaches a higher dimensionless limit strength. We define a new dimensionless time $T^*$ based on the thickness of the plate to scale the age of the primary vortex. The primary vortex stops growing when $T^* \approx 10$, regardless of the dimensions of the plate. We consider this value to be the vortex formation number of the primary vortex generated from a rotating rectangular flat plate in a Reynolds number range that goes from 800 to 12000. When $\alpha$ > 30°, the circulation released in the flow is entrained into secondary vortices for $\Rey > 2500$. The circulation of all secondary vortices is approximately 4 to 5 times smaller than the circulation of the primary vortex. We present a modified version of the Kaden spiral that accurately predicts the shear layer evolution and the trajectory of primary and secondary vortices during the entire rotation of the plate.We model the timing dynamics of secondary vortices with a power law equation that depends on two distinct parameter: $\chi$ and $\alpha_{0}$.The parameter $\chi$ indicates the relative increase in the time interval between the release of successive secondary vortices.The parameter $\alpha_{0}$ indicates the angular position at which the primary vortex stops growing and pinches-off from the plate.We also observe that the total circulation released in the flow is proportional to $\alpha^{1/3}$, as predicted by the inviscid theory.The combination of the power law equation with the total circulation computed from inviscid theory predict the strength of primary and secondary vortices, based purely on the plate's geometry and kinematics.The strength prediction is confirmed by experimental measurements.In this thesis we provided a valuable insight into the growth, timing and trajectory of primary and secondary vortices generated by a rotating flat plate. Future work should be directed towards more complex object geometries and kinematics, to confirm the validity of the modified Kaden spiral and explore the influence on the formation number.

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Dynamic stall on airfoils negatively impacts their aerodynamic performance and can lead to structural damage. Accurate prediction and modelling of the dynamic stall loads are crucial for a more robust design of wings and blades that operate under unsteady conditions susceptible to dynamic stall and for widening the range of operation of these lifting surfaces. Many dynamic stall models rely on empirical parameters that need to be obtained from experimental or numerical data which limits their generalisability. Mere, we introduce physically derived time scales to replace the empirical parameters in the Goman-Khrabrov dynamic stall model. The physics-based time constants correspond to the dynamic stall delay and the decay of post-stall load fluctuations. The dynamic stall delay is largely independent of the type of motion, the Reynolds number and the airfoil geometry, and is described as a function of a normalised instantaneous pitch rate. The post-stall decay is independent of the motion kinematics and is related to the Strouhal number of the post-stall vortex shedding. The general validity of our physics-based time constants is demonstrated using three sets of experimental dynamic stall data covering various airfoil profiles, Reynolds numbers varying from 75 000 to 1 000 000, and sinusoidal and ramp-up pitching motions. The use of physics-based time constants generalises the Goman-Khrabrov dynamic stall model and extends its range of application.

Dario Floreano, Jun Shintake, Davide Zappetti, Timothée John Martin Peter, Yusuke Ikemoto

This paper presents a method to create fish-like robots with tensegrity systems and describes a prototype modeled on the body shape of the rainbow trout with a length of 400 mm and a mass of 102 g that is driven by a waterproof servomotor. The structure of the tensegrity robot consists of rigid body segments and elastic cables that represent bone/tissue and muscles of fish, respectively. This structural configuration employing the tensegrity class 2 is much simpler than other tensegrity-based underwater robots. It also allows the tuning of the mechanical stiffness, which is often said to be an important factor in fish swimming. In our robot, the body stiffness can be tuned by changing the cross-section of the cables and their pre-stretch ratio. We characterize the robot in terms of body stiffness, swimming speed, and thrust force while varying the body stiffness i.e., the cross-section of the elastic cables. The results show that the body stiffness of the robot can be designed to approximate that of the real fish and modulate its performance characteristics. The measured swimming speed of the robot is 0.23 m/s (0.58 BL/s), which is comparable to other fish robots of the same type. Strouhal number of robot 0.54 is close to that of the natural counterpart, suggesting that the presented method is an effective engineering approach to realize the swimming characteristics of real fish.

2020