A minimal flow-elements model for the generation of packets of hairpin vortices in shear flows

2014 ◽  
Vol 747 ◽  
pp. 30-43 ◽  
Author(s):  
Jacob Cohen ◽  
Michael Karp ◽  
Vyomesh Mehta
Keyword(s):  
Transition Process ◽  
Vortex Sheet ◽  
Base Flow ◽  
Minimal Flow ◽  
Streamwise Vorticity ◽  
Hairpin Vortices ◽  
Robust Model ◽  
Flow Velocity Profile ◽  
Spanwise Vortex ◽  
Induced Velocity

AbstractPackets of hairpin-shaped vortices and streamwise counter-rotating vortex pairs (CVPs) appear to be key structures during the late stages of the transition process as well as in low-Reynolds-number turbulence in wall-bounded flows. In this work we propose a robust model consisting of minimal flow elements that can produce packets of hairpins. Its three components are: simple shear, a CVP having finite streamwise vorticity magnitude and a two-dimensional (2D) wavy (in the streamwise direction) spanwise vortex sheet. This combination is inherently unstable: the CVP modifies the base flow due to the induced velocity forming an inflection point in the base-flow velocity profile. Consequently, the 2D wavy vortex sheet is amplified, causing undulation of the CVP. The undulation is further enhanced as the wave continues to be amplified and eventually the CVP breaks down into several segments. The induced velocity generates highly localized patches of spanwise vorticity above the regions connecting two consecutive streamwise elements of the CVP. These patches widen with time and join with the streamwise vortical elements situated beneath them forming a packet of hairpins. The results of the unbounded (having no walls) model are compared with pipe and channel flow experiments and with a direct numerical simulation of a transition process in Couette flow. The good agreement in all cases demonstrates the universality and robustness of the model.

A numerical investigation of the rolling-up of vortex sheets

1980 ◽  
Vol 373 (1752) ◽  
pp. 67-91 ◽  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Equations Of Motion ◽  
Additional Term ◽  
Vortex Sheet ◽  
Time Step ◽  
Vortex Sheets ◽  
Numerical Instabilities ◽  
Sinusoidal Disturbance ◽  
Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

scholarly journals Reattachment streaks in hypersonic compression ramp flow: an input–output analysis

2019 ◽  
Vol 880 ◽  
pp. 113-135 ◽  
Author(s):  
Anubhav Dwivedi ◽  
G. S. Sidharth ◽  
Joseph W. Nichols ◽  
Graham V. Candler ◽  
Mihailo R. Jovanović
Keyword(s):  
Boundary Layer ◽  
Streamwise Velocity ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Streamwise Vorticity ◽  
Input Output ◽  
Output Analysis ◽  
Input Output Analysis ◽  
Boundary Layer Interaction ◽  
Compression Ramp

We employ global input–output analysis to quantify amplification of exogenous disturbances in compressible boundary layer flows. Using the spatial structure of the dominant response to time-periodic inputs, we explain the origin of steady reattachment streaks in a hypersonic flow over a compression ramp. Our analysis of the laminar shock–boundary layer interaction reveals that the streaks arise from a preferential amplification of upstream counter-rotating vortical perturbations with a specific spanwise wavelength. These streaks are associated with heat-flux striations at the wall near flow reattachment and they can trigger transition to turbulence. The streak wavelength predicted by our analysis compares favourably with observations from two different hypersonic compression ramp experiments. Furthermore, our analysis of inviscid transport equations demonstrates that base-flow deceleration contributes to the amplification of streamwise velocity and that the baroclinic effects are responsible for the production of streamwise vorticity. Finally, the appearance of the temperature streaks near reattachment is triggered by the growth of streamwise velocity and streamwise vorticity perturbations as well as by the amplification of upstream temperature perturbations by the reattachment shock.

scholarly journals Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

2011 ◽  
Vol 681 ◽  
pp. 116-153 ◽  
Author(s):  
NICHOLAS J. VAUGHAN ◽  
TAMER A. ZAKI
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Low Frequency ◽  
Finite Amplitude ◽  
Base Flow ◽  
Streamwise Vorticity ◽  
Mean Flow ◽  
S Wave ◽  
Flow Distortion ◽  
Tollmien Schlichting

The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the Tollmien–Schlichting (T–S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).

Experimental observation of hairpin auto-generation events in a turbulent boundary layer

2016 ◽  
Vol 795 ◽  
pp. 611-633 ◽  
Author(s):  
Y. Jodai ◽  
G. E. Elsinga
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Shear Layer ◽  
Transition Process ◽  
Wall Region ◽  
Generation Model ◽  
Streamwise Vortices ◽  
Hairpin Vortices ◽  
Time Resolved ◽  
Near Wall

Time-resolved tomographic particle image velocimetry experiments show that new hairpin vortices are generated within a fully developed and unperturbed turbulent boundary layer. The measurements are taken at a Reynolds number based on the momentum thickness of 2038, and cover the near-wall region below $y^{+}=140$, where $y^{+}$ is the wall-normal distance in wall units. Instantaneous visualizations of the flow reveal near-wall low-speed streaks with associated quasi-streamwise vortices, retrograde inverted arch vortices, hairpin vortices and hairpin packets. The hairpin heads are observed as close to the wall as $y^{+}=30$. Examples of hairpin packet evolution reveal the development of new hairpin vortices, which are created upstream and close to the wall in a manner consistent with the auto-generation model (Zhou et al., J. Fluid Mech., vol. 387, 1999, pp. 353–396). The development of the new hairpin appears to be initiated by an approaching sweep event, which perturbs the shear layer associated with the initial packet. The shear layer rolls up, thereby forming the new hairpin head. The head subsequently connects to existing streamwise vortices and develops into a hairpin. The time scale associated with the hairpin auto-generation is 20–30 wall units of time. This demonstrates that hairpins can be created over short distances within a developed turbulent boundary layer, implying that they are not simply remnants of the laminar-to-turbulent transition process far upstream.

The evolution of a localized vortex disturbance in external shear flows. Part 2. Comparison with experiments in rotating shear flows

1999 ◽  
Vol 379 ◽  
pp. 351-380 ◽  
Author(s):  
EDWIN MALKIEL ◽  
VLADIMIR LEVINSKI ◽  
JACOB COHEN
Keyword(s):  
Theoretical Model ◽  
Shear Flows ◽  
Laser Doppler Anemometry ◽  
Finite Amplitude ◽  
Base Flow ◽  
Mean Flow ◽  
Hairpin Vortices ◽  
Flow Parameters ◽  
Angular Velocities ◽  
Rotating Shear Flows

The evolution of artificially generated localized disturbances in the shape of hairpin vortices, in laminar axisymmetric rotating shear flows, is investigated experimentally. The results are compared with the predictions of a theoretical model (Levinski & Cohen 1995) with respect to the growth of such disturbances. Hairpin vortices were generated at the surface of the inner cylinder of an axisymmetric Couette apparatus, employing an injection–suction technique. The flow field was analysed from flow visualization using top and side views and by measurements of the mean and instantaneous velocity fields, carried out using laser Doppler anemometry and particle image velocimetry. An instability domain, within the range of base flow parameters where the flow is known to be linearly stable, was found. The marginal ratio between the angular velocities of the inner and outer cylinders beyond which the flow is stable to finite-amplitude localized disturbances agrees with the theoretical prediction based on the measured mean flow in the region of the disturbance. The dependence of the hairpin's inclination angle on the ratio between the two angular velocities is fairly well predicted by the theoretical model.

Vortex sheet roll-up revisited

2018 ◽  
Vol 855 ◽  
pp. 299-321 ◽  
Author(s):  
A. C. DeVoria ◽  
K. Mohseni
Keyword(s):  
Classical Problem ◽  
Critical Time ◽  
Vortex Sheet ◽  
The Self ◽  
Physical Consideration ◽  
Considerable Difficulty ◽  
Circulation Distribution ◽  
Continuous Object ◽  
Induced Velocity ◽  
Vorticity Transport

The classical problem of roll-up of a two-dimensional free inviscid vortex sheet is reconsidered. The singular governing equation brings with it considerable difficulty in terms of actual calculation of the sheet dynamics. Here, the sheet is discretized into segments that maintain it as a continuous object with curvature. A model for the self-induced velocity of a finite segment is derived based on the physical consideration that the velocity remain bounded. This allows direct integration through the singularity of the Birkhoff–Rott equation. The self-induced velocity of the segments represents the explicit inclusion of stretching of the sheet and thus vorticity transport. The method is applied to two benchmark cases. The first is a finite vortex sheet with an elliptical circulation distribution. It is found that the self-induced velocity is most relevant in regions where the curvature and the sheet strength or its gradient are large. The second is the Kelvin–Helmholtz instability of an infinite vortex sheet. The critical time at which the sheet forms a singularity in curvature is accurately predicted. As observed by others, the vortex sheet strength forms a finite-valued cusp at this time. Here, it is shown that the cusp value rapidly increases after the critical time and is the impetus that initiates the roll-up process.

scholarly journals Osborne Reynolds pipe flow: Direct simulation from laminar through gradual transition to fully developed turbulence

2015 ◽  
Vol 112 (26) ◽  
pp. 7920-7924 ◽  
Author(s):  
Xiaohua Wu ◽  
Parviz Moin ◽  
Ronald J. Adrian ◽  
Jon R. Baltzer
Keyword(s):  
Large Scale ◽  
Base Flow ◽  
Small Scale ◽  
Bypass Transition ◽  
Hairpin Vortex ◽  
Direct Simulation ◽  
Hairpin Vortices ◽  
Fully Developed Turbulence ◽  
The Core ◽  
Developed Turbulence

The precise dynamics of breakdown in pipe transition is a century-old unresolved problem in fluid mechanics. We demonstrate that the abruptness and mysteriousness attributed to the Osborne Reynolds pipe transition can be partially resolved with a spatially developing direct simulation that carries weakly but finitely perturbed laminar inflow through gradual rather than abrupt transition arriving at the fully developed turbulent state. Our results with this approach show during transition the energy norms of such inlet perturbations grow exponentially rather than algebraically with axial distance. When inlet disturbance is located in the core region, helical vortex filaments evolve into large-scale reverse hairpin vortices. The interaction of these reverse hairpins among themselves or with the near-wall flow when they descend to the surface from the core produces small-scale hairpin packets, which leads to breakdown. When inlet disturbance is near the wall, certain quasi-spanwise structure is stretched into a Lambda vortex, and develops into a large-scale hairpin vortex. Small-scale hairpin packets emerge near the tip region of the large-scale hairpin vortex, and subsequently grow into a turbulent spot, which is itself a local concentration of small-scale hairpin vortices. This vortex dynamics is broadly analogous to that in the boundary layer bypass transition and in the secondary instability and breakdown stage of natural transition, suggesting the possibility of a partial unification. Under parabolic base flow the friction factor overshoots Moody’s correlation. Plug base flow requires stronger inlet disturbance for transition. Accuracy of the results is demonstrated by comparing with analytical solutions before breakdown, and with fully developed turbulence measurements after the completion of transition.

Three-dimensional vorticity, momentum and heat transport in a turbulent cylinder wake

2016 ◽  
Vol 809 ◽  
pp. 135-167 ◽  
Author(s):  
J. G. Chen ◽  
Y. Zhou ◽  
T. M. Zhou ◽  
R. A. Antonia
Keyword(s):  
Heat Flux ◽  
Heat Transport ◽  
Large Scale ◽  
Three Dimensional ◽  
Heat Diffusion ◽  
Small Scale ◽  
Stream Velocity ◽  
Streamwise Vorticity ◽  
Spanwise Vortex ◽  
D 20

The transport of momentum and heat in the turbulent intermediate wake of a circular cylinder is inherently three-dimensional (3-D). This work aims to gain new insight into the 3-D vorticity structure, momentum and heat transport in this flow. All three components of the velocity and vorticity vectors, along with the fluctuating temperature, are measured simultaneously, at nominally the same point in the flow, with a probe consisting of four X-wires and four cold wires. Measurements are made in the ($x$,$y$) or mean shear plane at$x/d=10$, 20 and 40 at a Reynolds number of$2.5\times 10^{3}$based on the cylinder diameter$d$and the free-stream velocity. A phase-averaging technique is developed to separate the large-scale coherent structures from the remainder of the flow. It is found that the effects of vorticity on heat transport at$x/d=10$and$x/d=20{-}40$are distinctly different. At$x/d=10$, both spanwise and streamwise vorticity components account significantly for the heat flux. At$x/d=20$and 40, the spanwise vortex rollers play a major role in inducing the coherent components of the heat flux vector, while the ribs are responsible for the small-scale heat diffusion out of the spanwise vortex rollers. The present data indicate that, if the spanwise-velocity-related terms are ignored, the estimated values of the production can have errors of approximately 22 % and 13 % respectively for the turbulent energy and temperature variance at$x/d=40$, and the errors are expected to further increase downstream. A conceptual model summarizing the 3-D features of the heat and momentum transports at$x/d=10$is proposed. Compared with the previous two-dimensional model of Matsumura & Antonia (J. Fluid Mech., vol. 250, 1993, pp. 651–668) or MA, the new model provides a more detailed description of the role the rib-like structures undertake in transporting heat and momentum, and also underlines the importance of the upstream half of the spanwise vortex rollers, instead of only one quadrant of these rollers, as in the MA model, in diffusing heat out of the vortex.

The aerodynamics of hovering insect flight. V. A vortex theory

1984 ◽  
Vol 305 (1122) ◽  
pp. 115-144 ◽  
Keyword(s):  
Correction Factor ◽  
Insect Flight ◽  
Vortex Sheet ◽  
Hovering Flight ◽  
Vortex Theory ◽  
Actuator Disc ◽  
Temporal Pressure ◽  
The Mean ◽  
Wake Model ◽  
Induced Velocity

A full derivation is presented for the vortex theory of hovering flight outlined in preliminary reports. The theory relates the lift produced by flapping wings to the induced velocity and power of the wake. Suitable forms of the momentum theory are combined with the vortex approach to reduce the mathematical complexity as much as possible. Vorticity is continuously shed from the wings in sympathy with changes in wing circulation. The vortex sheet shed during a half-stroke convects downwards with the induced velocity field, and should be approximately planar at the end of a half-stroke. Vorticity within the sheet will roll up into complicated vortex rings, but the rate of this process is unknown. The exact state of the sheet is not crucial to the theory, however, since the impulse and energy of the vortex sheet do not change as it rolls up, and the theory is derived on the assumption that the extent of roll-up is negligible. The force impulse required to generate the sheet is derived from the vorticity of the sheet, and the mean wing lift is equal to that impulse divided by the period of generation. This method of calculating the mean lift is suitable for unsteady aerodynamic lift mechanisms as well as the quasi-steady mechanism. The relation between the mean lift and the impulse of the resulting vortex sheet is used to develop a conceptual artifice - a pulsed actuator disc - that approximates closely the net effect of the complicated lift forces produced in hovering. T he disc periodically applies a pressure impulse over some defined area, and is a generalized form of the Froude actuator disc from propeller theory. The pulsed disc provides a convenient link between circulatory lift and the powerful momentum and vortex analyses of the wake. The induced velocity and power of the wake are derived in stages, starting with the simple Rankine-Froude theory for the wake produced by a Froude disc applying a uniform, continuous pressure to the air. The wake model is then improved by considering a ‘modified’ Froude disc exerting a continuous, but non-uniform pressure. This step provides a spatial correction factor for the Rankine-Froude theory, by taking into account variations in pressure and circulation over the disc area. Finally, the wake produced by a pulsed Froude disc is analysed, and a temporal correction factor is derived for the periodic application of spatially uniform pressures. Both correction factors are generally small, and can be treated as independent perturbations of the Rankine-Froude model. Thus the corrections can be added linearly to obtain the total correction for the general case of a pulsed actuator disc with spatial and temporal pressure variations. The theory is compared with Rayner’s vortex theory for hovering flight. Under identical test conditions, numerical results from the two theories agree to within 3%. Rayner presented approximations from his results to be used when applying his theory to hovering animals. These approximations are not consistent with my theory or with classical propeller theory, and reasons for the discrepancy are suggested.

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