3 results
Droplet–turbulence interactions and quasi-equilibrium dynamics in turbulent emulsions
- Siddhartha Mukherjee, Arman Safdari, Orest Shardt, Saša Kenjereš, Harry E. A. Van den Akker
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- Journal:
- Journal of Fluid Mechanics / Volume 878 / 10 November 2019
- Published online by Cambridge University Press:
- 06 September 2019, pp. 221-276
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We perform direct numerical simulations (DNS) of emulsions in homogeneous isotropic turbulence using a pseudopotential lattice-Boltzmann (PP-LB) method. Improving on previous literature by minimizing droplet dissolution and spurious currents, we show that the PP-LB technique is capable of long stable simulations in certain parameter regions. Varying the dispersed-phase volume fraction $\unicode[STIX]{x1D719}$, we demonstrate that droplet breakup extracts kinetic energy from the larger scales while injecting energy into the smaller scales, increasingly with higher $\unicode[STIX]{x1D719}$, with approximately the Hinze scale (Hinze, AIChE J., vol. 1 (3), 1955, pp. 289–295) separating the two effects. A generalization of the Hinze scale is proposed, which applies both to dense and dilute suspensions, including cases where there is a deviation from the $k^{-5/3}$ inertial range scaling and where coalescence becomes dominant. This is done using the Weber number spectrum $We(k)$, constructed from the multiphase kinetic energy spectrum $E(k)$, which indicates the critical droplet scale at which $We\approx 1$. This scale roughly separates coalescence and breakup dynamics as it closely corresponds to the transition of the droplet size ($d$) distribution into a $d^{-10/3}$ scaling (Garrett et al., J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163–2171; Deane & Stokes, Nature, vol. 418 (6900), 2002, p. 839). We show the need to maintain a separation of the turbulence forcing scale and domain size to prevent the formation of large connected regions of the dispersed phase. For the first time, we show that turbulent emulsions evolve into a quasi-equilibrium cycle of alternating coalescence and breakup dominated processes. Studying the system in its state-space comprising kinetic energy $E_{k}$, enstrophy $\unicode[STIX]{x1D714}^{2}$ and the droplet number density $N_{d}$, we find that their dynamics resemble limit cycles with a time delay. Extreme values in the evolution of $E_{k}$ are manifested in the evolution of $\unicode[STIX]{x1D714}^{2}$ and $N_{d}$ with a delay of ${\sim}0.3{\mathcal{T}}$ and ${\sim}0.9{\mathcal{T}}$ respectively (with ${\mathcal{T}}$ the large eddy timescale). Lastly, we also show that flow topology of turbulence in an emulsion is significantly more different from single-phase turbulence than previously thought. In particular, vortex compression and axial straining mechanisms increase in the droplet phase.
Entry and exit flows in curved pipes
- Jesse T. Ault, Bhargav Rallabandi, Orest Shardt, Kevin K. Chen, Howard A. Stone
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- Journal:
- Journal of Fluid Mechanics / Volume 815 / 25 March 2017
- Published online by Cambridge University Press:
- 23 February 2017, pp. 570-591
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Solutions are presented for both laminar developing flow in a curved pipe with a parabolic inlet velocity and laminar transitional flow downstream of a curved pipe into a straight outlet. Scalings and linearized analyses about appropriate base states are used to show that both cases obey the same governing equations and boundary conditions. In particular, the governing equations in the two cases are linearized about fully developed Poiseuille flow in cylindrical coordinates and about Dean’s velocity profile for curved pipe flow in toroidal coordinates respectively. Subsequently, we identify appropriate scalings of the axial coordinate and disturbance velocities that eliminate dependence on the Reynolds number $Re$ and dimensionless pipe curvature $\unicode[STIX]{x1D6FC}$ from the governing equations and boundary conditions in the limit of small $\unicode[STIX]{x1D6FC}$ and large $Re$. Direct numerical simulations confirm the scaling arguments and theoretical solutions for a range of $Re$ and $\unicode[STIX]{x1D6FC}$. Maximum values of the axial velocity, secondary velocity and pressure perturbations are determined along the curved pipe section. Results collapse when the scalings are applied, and the theoretical solutions are shown to be valid up to Dean numbers of $D=Re^{2}\unicode[STIX]{x1D6FC}=O(100)$. The developing flows are shown numerically and analytically to contain spatial oscillations. The numerically determined decay of the velocity perturbations is also used to determine entrance/development lengths for both flows, which are shown to scale linearly with the Reynolds number, but with a prefactor ${\sim}60\,\%$ larger than the textbook case of developing flow in a straight pipe.
Oscillatory Marangoni flows with inertia
- Orest Shardt, Hassan Masoud, Howard A. Stone
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- Journal:
- Journal of Fluid Mechanics / Volume 803 / 25 September 2016
- Published online by Cambridge University Press:
- 19 August 2016, pp. 94-118
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When the surface of a liquid has a non-uniform distribution of a surfactant that lowers surface tension, the resulting variation in surface tension drives a flow that spreads the surfactant towards a uniform distribution. We study the spreading dynamics of an insoluble and non-diffusing surfactant on an initially motionless liquid. We derive solutions for the evolution over time of sinusoidal variations in surfactant concentration with a small initial amplitude relative to the average concentration. In this limit, the coupled flow and surfactant transport equations are linear. In contrast to exponential decay when the inertia of the flow is negligible, the solution for unsteady Stokes flow exhibits oscillations when inertia is sufficient to spread the surfactant beyond a uniform distribution. This oscillatory behaviour exhibits two properties that distinguish it from that of a simple harmonic oscillator: the amplitude changes sign at most three times, and the decay at late times follows a power law with an exponent of $-3/2$. As the surface oscillates, the structure of the subsurface flow alternates between one and two rows of counter-rotating vortices, starting with one row and ending with two during the late-time monotonic decay. We also examine numerically the evolution of the surfactant distribution when the system is nonlinear due to a large initial amplitude.