The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

The formation of vortex rings generated through impulsively started jets is studied experimentally. Utilizing a piston/cylinder arrangement in a water tank, the velocity and vorticity fields of vortex rings are obtained using digital particle image velocimetry (DPIV) for a wide range of piston stroke to diameter (L/D) ratios. The results indicate that the flow field generated by large L/D consists of a leading vortex ring followed by a trailing jet. The vorticity field of the leading vortex ring formed is disconnected from that of the trailing jet. On the other hand, flow fields generated by small stroke ratios show only a single vortex ring. The transition between these two distinct states is observed to occur at a stroke ratio of approximately 4, which, in this paper, is referred to as the ‘formation number’. In all cases, the maximum circulation that a vortex ring can attain during its formation is reached at this non-dimensional time or formation number. The universality of this number was tested by generating vortex rings with different jet exit diameters and boundaries, as well as with various non-impulsive piston velocities. It is shown that the ‘formation number’ lies in the range of 3.6–4.5 for a broad range of flow conditions. An explanation is provided for the existence of the formation number based on the Kelvin–Benjamin variational principle for steady axis-touching vortex rings. It is shown that based on the measured impulse, circulation and energy of the observed vortex rings, the Kelvin–Benjamin principle correctly predicts the range of observed formation numbers.

/pdf/a-universal-time-scale-for-vortex-ring-formation-1a8f67ypj8.pdf

A universal time scale for vortex ring formation

A particle image velocimetry system capable of accurately recovering the out-of-plane velocity component has been realized based on a stereoscopic viewing arrangement. To allow a large viewing angle with long focal length objective lenses, the angular displacement or Scheimpflug imaging configuration is employed in which the image, object and lens planes intersect in a common line. The varying magnification factor associated with this imaging configuration is accounted for using an accurate and simple-to-use calibration procedure based on solving the projection equations for each of the two cameras. A pair of high-resolution cameras, both capable of recording image pairs in the microsecond range, are synchronized to a pulsed Nd-YAG laser. By placing the cameras on either side of the light sheet the favourable light scattering characteristics of micron-sized seeding particles in forward scatter provide images at significantly higher illumination than at normal or backscatter viewing angles. Ultimately designed for use in industrial wind tunnels, the camera system is capable of working with non-symmetric arrangements. It has been successfully tested in a laboratory environment by imaging the unsteady flow field of a vortex ring passing through a laser light sheet. Adaptive processing software capable of dynamically adjusting the sample location of the interrogation windows to the local displacement vector significantly improves data yield. The algorithm requires only the selection of the final window/overlap size. The hierarchical interrogation approach permits the processing of images whose displacement dynamic range exceeds the interrogation window size.

https://iopscience.iop.org/article/10.1088/0957-0233/8/12/010/pdf

Stereoscopic Digital Particle Image Velocimetry for Application in Wind Tunnel Flows

I review the concept of optimal vortex formation and examine its relevance to propulsion in biological and bio-inspired systems, ranging from the human heart to underwater vehicles. By using examples from the existing literature and new analyses, I show that optimal vortex formation can potentially serve as a unifying principle to understand the diversity of solutions used to achieve propulsion in nature. Additionally, optimal vortex formation can provide a framework in which to design engineered propulsions systems that are constrained by pressures unrelated to biology. Finally, I analyze the relationship between optimal vortex formation and previously observed constraints on Strouhal frequency during animal locomotion in air and water. It is proposed that the Strouhal frequency constraint is but one consequence of the process of optimal vortex formation and that others remain to be discovered.

/pdf/optimal-vortex-formation-as-a-unifying-principle-in-5dz5ofbhs4.pdf

Optimal Vortex Formation as a Unifying Principle in Biological Propulsion

Dynamics of flame/vortex interactions

Using Laser Doppler Anemometry (LDA) and Digital Particle Image Velocimetry (DPIV), the physical properties of laminar vortex rings are investigated in the Reynolds-number range 830 ≤ Re ≤ 1650. The measured initial circulations of the vortex rings are found to agree well with corrected versions of the vorticity-flux (slug-flow) model proposed by Didden and Pullin. The DPIV and LDA data show excellent agreement regarding local velocities and vortex-ring circulations. The DPIV data depict the distribution of the vorticity and circulation in the core regions, where the resulting vorticity distributions are found to be self-similar Gaussian profiles. The propagation velocity of the vortex rings is well approximated by an analytical model of Saffman for large core sizes. In the asymptotic limit t → ∞, the trajectories are in excellent agreement with the exact Stokes-dipole solution of Cantwell and Rott.

On the evolution of laminar vortex rings

An experimental study is presented that examines the interaction of a vortex ring with a free surface. The main objective of this study is to identify the physical mechanisms that are responsible for the self-disconnection of vortex filaments in the near-surface region and the subsequent connection of disconnected vortex elements to the free surface. The understanding of those mechanisms is essential for the identification and estimation of the appropriate spatial and temporal scales of the disconnection and connection process. In this regard, the velocity and vorticity fields of an obliquely approaching laminar vortex ring with a Reynolds number of 1150 were mapped by using Digital Particle Image Velocimetry (DPIV). The evolution of the near-surface vorticity field indicates that the connection process starts in the side regions of the approaching vortex ring where surface-normal vorticity already exists in the bulk. A local strain rate analysis was conducted to support this conclusion. Disconnection in the near-surface tip region of the vortex ring occurs because of the removal of surfaceparallel vorticity by the viscous flux of vorticity through the surface. Temporal and spatial mapping of the vorticity field at the surface and in the perpendicular plane of symmetry shows that the viscous flux is balanced by a local deceleration of the flow at the surface. It is found that the observed timescales of the disconnection and connection process scale with the near-surface vorticity gradient rather than with the core diameter of the vortex ring.

/pdf/experimental-studies-of-vortex-disconnection-and-connection-ic7mmp7os5.pdf

Experimental studies of vortex disconnection and connection at a free surface

The spatiotemporal evolution of a turbulent vortex ring with an initial Reynolds number of 7500 is experimentally investigated using the technique of digital particle image velocimetry. The flow is initially characterized by the laminar/turbulent transition via azimuthal bending instabilities. After transition, the shedding of vorticity from peripheral regions of the ring is found to be responsible for the formation of a wake region. This shedding process results in the staircase‐like decay of the circulation and propagation speed of the vortex ring.

/pdf/on-the-decay-of-a-turbulent-vortex-ring-18flv94rqz.pdf

On the decay of a turbulent vortex ring

The interaction of turbulent vortex rings that approach a clean water surface under various angles is experimentally investigated. The temporal evolution of the vortex rings with an initial Reynolds number of Re_0 = 7500 is characterized by the laminar/turbulent transition and asymptotic relaminarization of the flow. Using the shadowgraph technique, two major flow cases were identified as a result of the vortex-ring/free-surface interaction: a trifurcation case that results from the interaction during the transition stage, and a bifurcation case that evolves during the fully-developed turbulent stage. In contrast to the laminar interaction, the turbulent bifurcation pattern is characterized by the reconnection and mutual interaction of many small-scale structures. Simultaneous digital particle image velocimetry (DPIV) and shadowgraph measurements reveal that the evolution of the small-scale structures at the free surface is strongly dominated by the bifurcation pattern, which in turn is a consequence of the persisting laminar sublayer in the core regions of the reconnected turbulent vortex loops.

/pdf/turbulent-vortex-ring-free-surface-interaction-411eamx2t8.pdf

Turbulent Vortex Ring/Free Surface Interaction

The oblique interaction of a turbulent vortex ring with a clean water surface is experimentally investigated during the transition stage using Digital Particle Image Velocimetry (DPIV) and the shadowgraph technique to map the surface velocity and deformation field simultaneously. The transitional vortex-ring/free-surface interaction leads to the formation of a trifurcation pattern at the free surface. Similar to the laminar flow case, the vortex ring initially bifurcates into two symmetric and separately connected vortex loops. The turbulent break-up of those vortex loops results in the formation of longitudinal wake vortices that symmetrically connect to the surface and eventually lead to a trifurcation pattern. In the absence of large-deformation surface waves, the simultaneous DPIV and shadowgraph measurements reveal good agreement between the surface vorticity and deformation field for small- and large-scale vortical structures. The simultaneous measurement technique is not restricted to the qualitative shadowgraph visualization, but can be easily extended to quantitative methods such as grating-imaging techniques, Color Schlieren, or Color Surface Mapping (CSM) techniques.

A. Weigand

Papers

On the evolution of laminar vortex rings

Experimental studies of vortex disconnection and connection at a free surface

On the decay of a turbulent vortex ring

Turbulent Vortex Ring/Free Surface Interaction

Simultaneous mapping of the velocity and deformation field at a free surface