Large Strouhal Numbers Research Articles

Propulsion enhancement of flexible plunging foils: Comparing linear theory predictions with high-fidelity CFD results

The fluid–structure interaction of a flexible plunging hydrofoil immersed in a current is solved numerically to analyze its propulsion enhancement due to flexibility at Reynolds number 10000. After validating with available experimental data, the code is used to assess analytical predictions from a linear theory. We consider large stiffness ratios, with high thrust enhancement by flexibility, and small mass ratios appropriate for underwater propulsion. The maximum thrust enhancement is observed at the first natural frequency, accurately predicted by the linear theory algebraically. The magnitude of the maximum thrust is over-predicted by the theory as the flapping amplitude increases. For large Strouhal numbers the flow becomes aperiodic, which for large enough amplitudes happens at frequencies below the natural frequency. But even at these Strouhal numbers, the linear theory predicts quite well the frequency of maximum thrust enhancement and optimal propulsive efficiency. We conclude that the linear theory constitutes a reliable and useful guide for the design of underwater flexible flapping-foil thrusters, and we provide a practical chart to easily select the optimal flapping frequency as a function of the actuation point, the stiffness and the mass ratios of the hydrofoil.

Vortex-induced vibration of flexible pipe fitted with helical strakes in oscillatory flow

Helical strakes are widely used to suppress vortex-induced vibration (VIV) in offshore engineering. However, the VIV response and suppression efficiency of the straked pipe remain unclear in an oscillatory flow. In this paper, an experimental study of VIV for a flexible pipe fitted with helical strakes was conducted in an oscillatory flow with the Keulegan-Carpenter (KC) number varying from 21 to 165 and maximum reduced velocities ranging from 4 to 12. The effects of the helical strakes on the VIV response, Strouhal number, suppression efficiency and fatigue damage in an oscillatory flow were investigated. The results show that the suppression efficiency and fatigue damage reduction ratio are not as ideal in oscillatory flow as those in steady flow. Moreover, the VIV dominant frequency is obviously magnified and a large Strouhal number is observed, reaching 0.439 under a lower reduced velocity of 4. As the reduced velocity further increases to 6, 8, 12, two branches of Strouhal numbers are clearly seen. The dominant frequency decreases at the first branch with a small KC number and increases at the second branch with a large KC number.

Interaction of wake disturbance with compressible transitional boundary layers in a low-pressure turbine cascade under rotor-stator interaction

Transition-process-resolving numerical simulations of compressible transitional flows subjected to rotor-stator interaction in a low-pressure turbine cascade were performed. Cases of small and large Strouhal numbers (St) of blade passage, and with and without isotropic free-stream turbulence (IFST) are compared. Cascade geometry is T106, and the computations are conducted at its design point condition. A higher-order finite difference method based on sixth-order compact difference and tenth-order filtering is employed. Variation in St and the intensity of IFST changes the frequency of turbulence inflow, the convection and propagation of turbulence, and the interaction of disturbance with boundary layers in the blade passage. The changes result in different pressure fluctuation in the passage. Because pressure fluctuation can convey information away from the passage, it is expected to be used for diagnosing boundary layer states around the blade indirectly away from the passage.

Mass transfer enhancement and surface functionalization in digital microfluidics using AC electrowetting: the smaller, the better

This paper addresses modelling and computation of stirring and chemical transport at drop scale due to shape oscillations of small amplitude induced by coplanar AC electrowetting on dielectrics (EWOD). An axisymmetric flow inside a drop deformed by standing capillary waves is promoted by coplanar EWOD in the oscillating regime ( $$\omega \sim$$ 10–500 Hz). This can be seen as a new stirring/mixing mechanism at microscale provided that a deformable fluid interface is available. The latter requirement is fully satisfied in biomedical applications of digital microfluidics where droplets of volume ~1 μL are commonly handled. A steady streaming model at large Strouhal number is taken into account in order to investigate time-dependent chemical transport in a sessile drop under oscillating EWOD. Numerical calculations based on finite element method are performed to predict the transport of a biochemical solute with possible affinity for drop surface and supporting solid surface as well. A dedicated weak form is also derived to take into account diffusional transport along the liquid surface. Mass transfer enhancement is investigated by considering the variance of the chemical concentration in the transient regime. Special attention is given to investigate mass transfers and surface ageing by considering the respective roles of the Thiele number, the Damköhler number, the Peclet and Surface Peclet numbers.

High fidelity numerical simulation of airfoil thickness and kinematics effects on flapping airfoil propulsion

High-fidelity numerical simulations with the spectral difference (SD) method are carried out to investigate the unsteady flow over a series of oscillating NACA 4-digit airfoils. Airfoil thickness and kinematics effects on the flapping airfoil propulsion are highlighted. It is confirmed that the aerodynamic performance of airfoils with different thickness can be very different under the same kinematics. Distinct evolutionary patterns of vortical structures are analyzed to unveil the underlying flow physics behind the diverse flow phenomena associated with different airfoil thickness and kinematics and reveal the synthetic effects of airfoil thickness and kinematics on the propulsive performance. Thickness effects at various reduced frequencies and Strouhal numbers for the same chord length based Reynolds number (=1200) are then discussed in detail. It is found that at relatively small Strouhal number (=0.3), for all types of airfoils with the combined pitching and plunging motion (pitch angle 20°, the pitch axis located at one third of chord length from the leading edge, pitch leading plunge by 75°), low reduced frequency (=1) is conducive for both the thrust production and propulsive efficiency. Moreover, relatively thin airfoils (e.g. NACA0006) can generate larger thrust and maintain higher propulsive efficiency than thick airfoils (e.g. NACA0030). However, with the same kinematics but at relatively large Strouhal number (=0.45), it is found that airfoils with different thickness exhibit diverse trend on thrust production and propulsive efficiency, especially at large reduced frequency (=3.5). Results on effects of airfoil thickness based Reynolds numbers indicate that relative thin airfoils show superior propulsion performance in the tested Reynolds number range. The evolution of leading edge vortices and the interaction between the leading and trailing edge vortices play key roles in flapping airfoil propulsive performance.

Modeling the damping properties of perforated screens traversed by a bias flow and backed by a cavity at low Strouhal number

Acoustic absorption properties of perforated screens traversed by a bias flow and backed by a resonant cavity are examined at a new absorption regime. The reflection coefficient of these dampers can be canceled at specific frequencies for low Strouhal numbers St ⪡ 1 and Helmholtz numbers of order unity He ∼ 1 . This regime differs from the classical one where resonance is sought at low Helmholtz numbers He ⪡ 1 and large Strouhal numbers St ⪢ 1 . A theoretical analysis is carried out and analytical expressions are derived to determine conditions for optimal absorption in this regime. It is shown that the optimal bias flow velocity and back cavity length can be fixed separately. The former is determined by the plate porosity and the latter controls the peak absorption frequency. This operating regime at low Strouhal numbers enables to increase sound absorption at low frequencies over a larger frequency bandwidth around the peak absorption frequency than the one obtained with a damping system working at high Strouhal numbers. This is confirmed by experiments carried out on a set of dampers in the frequency range 100–1000 Hz. Predictions of the reflection coefficient, including effects of bias flow velocity and cavity depth, show good agreement with experimental data for small pressure perturbation levels. These elements can contribute to the development of robust dampers to control low frequency thermoacoustic instabilities in gas turbines and jet engines.

Mixing Enhancement by Microrotor in Step Channel

In this paper, the mixing enhancement of a micromixer consisting of a step channel and a shuttlecock rotor suspended in the step is numerically analyzed. Asymptotic mixing performance is investigated as a function of Strouhal and Peclet numbers by particle tracking simulation and the Eulerian approach. The simulation results show that the rotor creates downward and inward flows in behind the rotor paddles, whereas the upward and outward flows are produced in front of the rotor paddles. At a small Strouhal number, convective mixing is very poor. However, the mixing direction is rotated by 90 deg, which can reduce the mixing time by the square of the aspect ratio of the cross section of the channel. In contrast, at a relatively large Strouhal number, good convective mixing occurs. Quantitative analysis of mixing performance of the mixer demonstrates that the mixing structures are similar for the same Strouhal number and mixing is improved with increasing Strouhal number. The mixing efficiency of the mixer decreases linearly with increasing log of the Peclet number at a relatively large Strouhal number.

The development of asymmetry for oscillatory flow within a tube containing sharp edge periodic baffles

This paper investigates the evolution of asymmetric patterns for oscillatory flow in a baffled tube. A numerical simulation for three-dimensional flows in an axisymmetric geometry was developed and compared with experimental results obtained using particle image velocimetry (PIV). Sharp edged baffles were used for both numerical simulations and experiments. From the numerical simulation, a stability map of the flow symmetry was obtained as a function of Reynold-Strouhal numbers. The simulations show that for all Strouhal numbers, the flow was axisymmetric at Reynolds numbers less than 100 and asymmetric at Reynolds numbers larger than 225. The flow was less stable to asymmetric disturbances at small or large Strouhal numbers when compared to St=1.0. In particular, the flow in the region St<0.5 and Re>100 was asymmetric. Two mechanisms for vortex instability transition into three dimensions has been identified. At small Strouhal numbers, the primary mechanism is a shear (Kelvin-Helmholtz) instability. At larger Strouhal numbers, the axisymmetry of the flow is broken because of the collision of travelling eddies that have been shed from opposite baffles. The numerical results are in general in agreement quantitatively with the experimental observations and both experiment and simulation assist in understanding the development of unsteadiness in periodic reversing flows.

Flexible Flapping Airfoil Propulsion at Low Reynolds Numbers

Water tunnel experiments on a flexible airfoil plunging with constant amplitude have been carried out for Reynolds numbers of 0 to 27000. Peaks in thrust coefficient at intermediate values of airfoil stiffness were observed at both zero and non-zero Reynolds numbers, indicating that a degree of flexibility is beneficial at low Reynolds numbers. Time-averaged velocity fields and momentum flux data revealed a broader, higher-velocity jet in cases of optimum airfoil stiffness. Stronger vortices, separated by a larger lateral distance, characterised the corresponding instantaneous velocity fields. The flexibility causes the airfoil to pitch passively; the phase angle of the pitch was found to lead the plunge. Pitch amplitude and trailing-edge amplitude were found to be single-valued functions of pitch phase angle. The shape characteristics of the airfoil could therefore be described by the pitch phase angle only. Thrust coefficient was found to be a function of only two parameters: Strouhal number and pitch phase angle. For each Strouhal number, a peak in thrust coefficient was observed at a particular value of the pitch phase angle. The optimum pitch phase angle was found to tend to a limit of 105±5 degrees at very large Strouhal numbers. A significant thrust benefit was observed over very stiff airfoils when the optimum flexibility is utilized.

Sensitive enhanced diffusivities for flows with fluctuating mean winds: a two-parameter study

Enhanced diffusion coefficients arising from the theory of periodic homogenized averaging for a passive scalar diffusing in the presence of a large-scale, fluctuating mean wind superimposed upon a small-scale, steady flow with non-trivial topology are studied. The purpose of the study is to assess how the extreme sensitivity of enhanced diffusion coefficients to small variations in large-scale flow parameters previously exhibited for steady flows in two spatial dimensions is modified by either the presence of temporal fluctuation, or the consideration of fully three-dimensional steady flow. We observe the various mixing parameters (Péclet, Strouhal and periodic Péclet numbers) and related non-dimensionalizations. We document non-monotonic Péclet number dependence in the enhanced diffusivities, and address how this behaviour is camouflaged with certain non-dimensional groups. For asymptotically large Strouhal number at fixed, bounded Péclet number, we establish that rapid wind fluctuations do not modify the steady theory, whereas for asymptotically small Strouhal number the enhanced diffusion coefficients are shown to be represented as an average over the steady geometry. The more difficult case of large Péclet number is considered numerically through the use of a conjugate gradient algorithm. We consider Péclet-number-dependent Strouhal numbers, S = QPe−(1+γ), and present numerical evidence documenting critical values of γ which distinguish the enhanced diffusivities as arising simply from steady theory (γ < −1) for which fluctuation provides no averaging, fully unsteady theory (γ ∈ (−1, 0)) with closure coefficients plagued by non-monotonic Péclet number dependence, and averaged steady theory (γ > 0). The transitional case with γ = 0 is examined in detail. Steady averaging is observed to agree well with the full simulations in this case for Q [les ] 1, but fails for larger Q. For non-sheared flow, with Q [les ] 1, weak temporal fluctuation in a large-scale wind is shown to reduce the sensitivity arising from the steady flow geometry; however, the degree of this reduction is itself strongly dependent upon the details of the imposed fluctuation. For more intense temporal fluctuation, strongly aligned orthogonal to the steady wind, time variation averages the sensitive scaling existing in the steady geometry, and the present study observes a Pe1 scaling behaviour in the enhanced diffusion coefficients at moderately large Péclet number. Finally, we conclude with the numerical documentation of sensitive scaling behaviour (similar to the two-dimensional steady case) in fully three dimensional ABC flow.

Diffusion of scalar and vector fields in compressible turbulent media

Abstract Processes of scalar and vector fields (temperature, particle number density, magnetic field etc.) propagation in compressible strongly turbulent media are considered. The integral equation of the fields transportation and its diffusion approximation are obtained. The diffusion coefficient and corresponding coefficient of magnetic field generation (α-effect) are expressed in terms of the velocity correlators. The correlators of high rank are taken into account. A new mathematical approach is suggested which allows the theory to extend up to the large Strouhal number. It is shown that the turbulent diffusion coefficient can be negative, which implies the clustering of the admixture particles in patches (the local increasing of the temperature and magnetic fields). Explicit expressions for correlators of the scalar fields in turbulent media are obtained.

Streaming flows generated by high-frequency small-amplitude oscillations of arbitrarily shaped cylinders

Small-amplitude harmonic oscillations of arbitrarily shaped cylinders are considered both experimentally and theoretically. For the theoretical model, the flow regime is separated into inner and outer regions. In the inner region, the flow is governed by the classical Stokes boundary layer equation. In the outer region, the full Navier–Stokes equation for the steady streaming flow is solved numerically by using a finite difference method coupled with conformal mapping techniques. Numerical results of streaming, a nonlinear response to harmonic motion, show complicated flow schemes. Experimental results confirm the existence of such flows. Streaming flow around a sharp corner of a square cylinder is investigated through numerical calculation and experimental flow visualization. The absence of any vortex shedding on the time scale of the streaming flow is noted. These results suggest that in the limit of a small amplitude of oscillation, or equivalently large Strouhal numbers, sharp-edged bodies experience attached flow in the mean sense.

Instability of a gas jet in liquid

The linear stability of a jet submerged in a fluid of different density is analyzed, as a first step in studying the formation of bubbles caused by the injection of air into water. Numerical results for both the temporal and spatial growth rates for the axisymmetric mode have been obtained for different Weber numbers. For a jet submerged in the same fluid, correspondence between the temporal and spatial growth rates has been found for large Strouhal numbers. However, such a correspondence for an air jet in water, for which there is a drastic difference in fluid density, could not be found.

Equation of nonsteady transonic flows of an ideal gas

At the present time, there are several different equations for describing a transonic, nonsteady, irrotational flow of an ideal perfect gas ([1], Table 1), depending on the ratios between the small characteristic parameters of the flow. In order to extend the range of application of these equations, a composite equation is formed from them (for example, the equations for small and large Strouhal numbers in the theory of oscillation of a wing are combined). In this paper, a more general equation is obtained for the plane flow of this class by means of natural orthogonal coordinates ψΨ (family of equipotential lines and streamlines) without the use of ɛ estimates, for which the equation by comparison with the composite equation contains a new nonlinear term. Accurate solutions of the equation are found, describing nonsteady transonic flows in plane nozzles; one of them describes the process of the establishment of a design cycle in Laval nozzles with immovable walls.

Parametric method in the theory of a periodic boundary layer with large strouhal numbers

A universal equation for the Linn problem on a periodic boundary layer is obtained and solved numerically with certain values of the parameters.

Hydrodynamic interaction of cascades of plates during their relative motion

The problem of the hydrodynamic interaction taking place between the profiled plates of a (turbine) cascade is considered. This problem has hitherto been little studied — existing papers [1, 2] consider the model of effectively steady-state flow, the flow of the liquid at any instant of time being regarded as free from eddies, while time plays the part of a parameter; however, under practical conditions the field of velocities around moving cascades changes rapidly with time, so casting doubt on the validity of the steady-flow mode. The model proposed in this paper allows for high-frequency pulsations in the flow. The approximate solution here derived determines the interference between the profiles on the basis of a limiting solution with infinitely large Strouhal numbers. A typical calculation is presented.