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Register a new userStatistics of relative velocity for particles settling under gravity in a turbulent flow
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Authors
Akshay Bhatnagar
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We study the joint probability distributions of separation, $R$, and radial component of the relative velocity, $V_{rm R}$, of particles settling under gravity in a turbulent flow. We also obtain the moments of these distributions and analyze their anisotropy using spherical harmonics. We find that the qualitative nature of the joint distributions remains the same as no gravity case. Distributions of $V_{rm R}$ for fixed values of $R$ show a power-law dependence on $V_{rm R}$ for a range of $V_{rm R}$, exponent of the power-law depends on the gravity. Effects of gravity are also manifested in the following ways: (a) moments of the distributions are anisotropic; the degree of anisotropy depends on particles Stokes number, but does not depend on $R$ for small values of $R$. (b) mean velocity of collision between two particles is decreased for particles having equal Stokes numbers but increased for particles having different Stokes numbers. For the later, collision velocity is set by the difference in their settling velocities.
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We use direct numerical simulations to calculate the joint probability density function of the relative distance $R$ and relative radial velocity component $V_R$ for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, $D_2$. It was argued [1, 2] that the scale invariant part of the distribution has two asymptotic regimes: (1) $|V_R| ll R$ where the distribution depends solely on $R$; and (2) $|V_R| gg R$ where the distribution is a function of $|V_R|$ alone. The probability distributions in these two regimes are matched along a straight line $|V_R| = z^ast R$. Our simulations confirm that this is indeed correct. We further obtain $D_2$ and $z^ast$ as a function of the Stokes number, ${rm St}$. The former depends non-monotonically on ${rm St}$ with a minimum at about ${rm St} approx 0.7$ and the latter has only a weak dependence on ${rm St}$.
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