5 results
Advective balance in pipe-formed vortex rings
- Karim Shariff, Paul S. Krueger
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- Journal:
- Journal of Fluid Mechanics / Volume 836 / 10 February 2018
- Published online by Cambridge University Press:
- 12 December 2017, pp. 773-796
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Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$, the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in-$\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ($\unicode[STIX]{x1D713}=0$) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.
A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage
- Karim Shariff, Roberto Verzicco, Paolo Orlandi
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- Journal:
- Journal of Fluid Mechanics / Volume 279 / 25 November 1994
- Published online by Cambridge University Press:
- 26 April 2006, pp. 351-375
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Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss & Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall & Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 – α1(β)/Res, for a large range of Res. Here Res is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then α1(β) varies weakly with core thickness β. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode (n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n. The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted π/n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial modes having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.
Analysis of the radar reflectivity of aircraft vortex wakes
- KARIM SHARIFF, ALAN WRAY
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- Journal:
- Journal of Fluid Mechanics / Volume 463 / 25 July 2002
- Published online by Cambridge University Press:
- 31 July 2002, pp. 121-161
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Radar has been proposed as a way of tracking wake vortices to reduce aircraft spacing and tests have revealed radar echoes from aircraft wakes in clear air. The mechanism causing refractive index gradients in these tests is thought to be the same as that for homogeneous and isotropic atmospheric turbulence in the Kolmogorov inertial range, for which there is a scattering analysis due to Tatarski. In reality, however, the structure of aircraft wakes has a significant coherent part superimposed with turbulence, about whose structure very little is known. This work adopts a picture of a coherent (in fact two-dimensional) wake to perform a scattering analysis and calculate the reflected power. In particular, two simple mechanisms causing refractive index gradients are considered: (A) radial pressure (and therefore density) gradient in a columnar vortex arising from the rotational flow; (B) adiabatic transport of atmospheric fluid within a descending oval surrounding a vortex pair. In the scattering analysis, Tatarski's weak scattering approximation is kept but the usual assumptions of a far field and a uniform incident wave are dropped. Neither assumption is generally valid for a wake that is coherent across the radar beam. For analytical insight, an approximate analysis that invokes, in addition to weak scattering, the far-field and wide cylindrical beam assumptions, is also developed and compared with the more general analysis. Reflectivities calculated for the oval (mechanism B) are within 2–13 dB m2 of the measurements (≈−70 dB m2) of MIT Lincoln Laboratory at Kwajalein atoll. However, the present predictions have a cut-off away from normal incidence which is not present in the measurements. This implies that the two-dimensional picture is not entirely complete. Estimates suggest that the thin layer of vorticity which is baroclinically generated at the boundary of the oval is turbulent and this may account for reflectivity away from normal incidence. The reflectivity of a vortex (mechanism A) is comparable to that of the oval (mechanism B) but occurs at a frequency (about 50 MHz) that is lower than those considered in all the experiments to date. This result may be useful because: (i) existing atmospheric radars (known as ST radars) already operate at this frequency and so the present prediction could be verified; (ii) rain clutter is not a problem at this frequency; (iii) mechanism A is more robust because it is independent of atmospheric stratification.
Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5
- STEPHEN E. GUARINI, ROBERT D. MOSER, KARIM SHARIFF, ALAN WRAY
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- Journal:
- Journal of Fluid Mechanics / Volume 414 / 10 July 2000
- Published online by Cambridge University Press:
- 10 July 2000, pp. 1-33
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A direct numerical simulation of a supersonic turbulent boundary layer has been performed. We take advantage of a technique developed by Spalart for incompressible flow. In this technique, it is assumed that the boundary layer grows so slowly in the streamwise direction that the turbulence can be treated as approximately homogeneous in this direction. The slow growth is accounted for by a coordinate transformation and a multiple-scale analysis. The result is a modified system of equations, in which the flow is homogeneous in both the streamwise and spanwise directions, and which represents the state of the boundary layer at a given streamwise location. The equations are solved using a mixed Fourier and B-spline Galerkin method.
Results are presented for a case having an adiabatic wall, a Mach number of M = 2.5, and a Reynolds number, based on momentum integral thickness and wall viscosity, of Reθ′ = 849. The Reynolds number based on momentum integral thickness and free-stream viscosity is Reθ = 1577. The results indicate that the Van Driest transformed velocity satisfies the incompressible scalings and a small logarithmic region is obtained. Both turbulence intensities and the Reynolds shear stress compare well with the incompressible simulations of Spalart when scaled by mean density. Pressure fluctuations are higher than in incompressible flow. Morkovin's prediction that streamwise velocity and temperature fluctuations should be anti-correlated, which happens to be supported by compressible experiments, does not hold in the simulation. Instead, a relationship is found between the rates of turbulent heat and momentum transfer. The turbulent kinetic energy budget is computed and compared with the budgets from Spalart's incompressible simulations.
A universal time scale for vortex ring formation
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- MORTEZA GHARIB, EDMOND RAMBOD, KARIM SHARIFF
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- Journal:
- Journal of Fluid Mechanics / Volume 360 / 10 April 1998
- Published online by Cambridge University Press:
- 10 April 1998, pp. 121-140
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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.