F. Kaplanski

A model for the formation of “optimal” vortex rings taking into account viscosity

The evolution of a viscous vortex ring from thin to thick-cored form is considered using an improved asymptotic solution, which is obtained after impressing a spatially uniform drift on the first-order solution of the Navier–Stokes equations. The obtained class of rings can be considered as the viscous analog solution to the Norbury vortices and classified in terms of the ratio of their initial outer radius to the core radius. The model agrees with the reported theoretical and experimental results referring to the post-formation and the formation stages. By using the matching procedure suggested earlier and the obtained properties of the viscous vortex ring, it is found that when the length-to-diameter aspect ratio L∕D reaches the limiting value 4.0 (“formation number”), the appropriate values of the normalized energy and circulation become around 0.3 and 2.0, respectively. An approach that enables to predict the “formation number” is proposed.

Vortex ring-like structures in gasoline fuel sprays under cold-start conditions

Abstract A phenomenological study of vortex ring-like structures in gasoline fuel sprays is presented for two types of production fuel injectors: a low-pressure, port fuel injector (PFI) and a high-pressure atomizer that injects fuel directly into an engine combustion chamber (G-DI). High-speed photography and phase Doppler anemometry (PDA) were used to study the fuel sprays. In general, each spray was seen to comprise three distinct periods: an initial, unsteady phase; a quasi-steady injection phase; and an exponential trailing phase. For both injectors, vortex ring-like structures could be clearly traced in the tail of the sprays. The location of the region of maximal vorticity of the droplet and gas mixture was used to calculate the temporal evolution of the radial and axial components of the translational velocity of the vortex ring-like structures. The radial components of this velocity remained close to zero in both cases. The experimental results were used to evaluate the robustness of previously developed models of laminar and turbulent vortex rings. The normalized time, , and normalized axial velocity, , were introduced, where t init is the time of initial observation of vortex ring-like structures. The time dependence of on was approximated as and for the PFI and G-DI sprays respectively. The G-DI spray compared favourably with the analytical vortex ring model, predicting , in the limit of long times, where α = 3/2 in the laminar case and α = 3/4 when the effects of turbulence are taken into account. The results for the PFI spray do not seem to be compatible with the predictions of the available theoretical models.

Reynolds-number effect on vortex ring evolution in a viscous fluid

It is known that the cross section of the vortex ring core takes an approximately elliptical shape with increasing Reynolds number. In order to model this feature, the functional form of a vortex ring solution of the Stokes equations is modified so as to be able to model higher Reynolds number rings. The model introduces two nondimensional parameters that govern the shape of the vortex core:λ ⩾ 1 and β ⩾ 1. Based on this modification, new expressions for the translation velocity, energy, circulation, and streamfunction are derived for a wide range of section ellipticity that are specific to such vortices. To validate the model, the data adapted from the numerical study of vortex ring at Reynolds number Re = 1400 performed by Danaila and Helie [Phys. Fluids 20, 073602 (2008)], is used. In this case, the appropriate values of λ and β are calculated by equating the normalized energy Ed and circulation Γd of the theoretical vortex to the corresponding values obtained from the numerical data. The model provide...

Spray dynamics as a multi-scale process

The analysis of the processes in sprays, taking into account the contribution of all spatial and temporal scales, is not feasible in most cases due to its complexity. The approach used in most applications is based on separate analysis of the processes at various scales, and the analysis of the link between these processes. This approach is demonstrated for the analysis of spray break-up and penetration in Diesel engine-like conditions, and vortex ring-like structures in gasoline engine-like conditions. The conventional WAVE, TAB, stochastic and modified WAVE (taking into account transient effects) models are reviewed. It is pointed out that the latter model leads to the prediction of spray penetration in Diesel engine-like conditions closest to the one observed experimentally. In gasoline engine-like conditions, spray penetration is often accompanied by the formation of vortex ring-like structures, the spatial scale of which is comparable with the scale of spray penetration. The general expression of the velocity of the vortex ring centroid can be simplified for short and long times, the latter simplification being particularly simple and useful for engineering applications. The thickness of the vortex ring is expressed as l = atb, where a is an arbitrary constant and 1/4 ≤ b ≤ 1/2. The cases when b = 1/2 and b = 1/4 refer to laminar and turbulent vortex rings respectively. The model is compatible with the observation of vortex ring-like structures in gasoline engine-like conditions.

Reynolds-number effect on vortex ring evolution

An analytical model describing a vortex ring for low Reynolds numbers (Re) proposed previously by Kaplanski and Rudi [Phys. Fluids,17, 087101 (2005)], is extended to a vortex rings for higher Reynolds numbers. The experimental results show that the vortex ring core takes the oblate ellipsoidal shape with increasing Re. In order to model this feature, we suggest an expression for the vorticity distribution, which corrects the linearized solution of the Navier‐Stokes equation, with two disposable nondimensional parameters λ and β governing the shape of the vortex core, and derive the new expressions for the streamfuction, circulation, energy and translation velocity on the basis of it. The appropriate values of λ and β are calculated by equating the nondimensional energy Ed and circulation Гd of the theoretical vortex to the corresponding values obtained from the experimental or numerical vortex ring. To validate the model, the data adapted from the numerical study of a vortex ring at Re = 1400 performed by...

Vortex Ring-like Structures in a Non-evaporating Gasoline-fuel Spray: Simplified Models versus Experimental Results

The results of recent developments of analytical vortex ring models and the applications of these models to interpretation of the experimentally observed dynamics of vortex ring–like structures in gasoline sprays, under non-evaporating conditions, are summarised. Analytical formulae in the limit of small Reynolds numbers (Re), are compared with numerical solutions. Particular attention is focused on the generalised vortex ring model in which the time evolution of the thickness of the vortex ring core L is approximated as ݐ ௕ , where a and b are constants (1 ≤ b ≤ 1/2). This model incorporates both the laminar model for b=1/2 and fully turbulent model for b=1/4. The values of velocities in the region of maximal vorticity, predicted by the generalised vortex ring model, are compared with the results of experimental studies of fuel droplets distributed in vortex ring-like structures in two gasoline injectors, under cold-start, engine-like conditions. Liquid iso-octane at a temperature of 22 °C was injected at a frequency of 1 Hz and a pressure of 100 bar (direct injection) and 3.5 bar (port injection) into air at atmospheric pressure and a temperature of 20 °C. Phase Doppler Anemometry was performed over a fine measurement grid that covered the whole spray. The decaying phase of fuel injection showed the most clearly defined vortex rings. The identification of their locations in each time step permitted the determination of the velocities of their displacement in the axial and radial directions. Although the radial component of velocity in both these regions is equal to zero, the location of both changes with time. This leads to an effective radial velocity component; the latter depends on b. Most of the values of the axial velocity of the vortex rings lie between the theoretically predicted values corresponding to the late stage of vortex ring development and b=1/4 (fully developed turbulence) and 1/2 (laminar case).