Critical Density Triplets for the Arrestment of a Sphere Falling in a Sharply Stratified Fluid

Abstract

We study the motion of a rigid sphere falling in a two-layer stratified fluid under the action of gravity in the potential flow regime. Experiments at a moderate Reynolds number of approximately 20–450 indicate that a sphere with the precise critical density, higher than the bottom layer density, can display behaviors such as bounce or arrestment after crossing the interface. We experimentally demonstrate that such a critical sphere density increases linearly as the bottom fluid density increases with a fixed top fluid density. Additionally, the critical density approaches the bottom-layer fluid density as the thickness of density transition layer increases. We propose an estimation of the critical density based on the potential energy. Assuming the zero-layer thickness, the estimation constitutes an upper bound of the critical density with less than 0.043 relative difference within the experimental density regime 0.997 g/cm³ ∼1.11 g/cm³ under the zero-layer thickness assumption. By matching the experimental layer thickness, we obtain a critical density estimation with less than 0.01 relative difference within the same parameter regime.

Appendix

1.1 Numerical Method

In this section, we document the details of the numerical calculation of the drift and reflux contributions to the potential energy and associated issues.

As the sphere penetrates the interface and deforms it, there is considerable stretching of the mesh in the region around the sphere, due to the potential nature of the flow. The uniform mesh on the interface cannot resolve the dynamics efficiently. Hence, for simplicity, we adopt a nonuniform mesh, which takes the parameterization:

$$\displaystyle \begin{aligned} x(s)= \begin{cases} 0&s=0,\\ e^{\frac{s}{r_{1}}} & 0<s \leq r_{1}, y(s)=0, \\ k_{1}(s-r_{1})+1 & r_{1}<s, \end{cases} \end{aligned} $$

(20)

where r ₁ and k ₁ are constants selected to resolve the interface evolution profile, which varies for different initial position of the sphere and the duration of the evolution. The mesh points cluster exponentially at the neighborhood of zero and distributed uniformly when they are far away from zero. The exponential profile of the initial mesh in the immediate vicinity of the particle provides a high density of meshes where the stretching is maximum.

A fourth-order explicit Runge-Kutta method with typical step size Δt = 10⁻³ was used to compute the time evolution of the interface region as the sphere moved through the layers by solving the initial value problem with the velocity field provided in Eq. (1).

We approximate the interface by the cubic spline with the boundary condition “not-a-knot” to ensure fourth order accuracy in the interface tracking stage. The point of zero Lagrangian displacement (r ^∗(z _b), 0) is calculated by solving the root of the spline function.

The integrals in Eq. (9) are evaluated by the trapezoidal rule. To achieve higher accuracy, one can adopt the spline-based quadrature rules described in [26, 35]. The potential energy as a function of the sphere position is approximated by a fifth-order spline function. Since differentiation could introduce unexpected oscillations when the data is not smooth enough, instead of solving $\partial ^{2}_{z_{b}}P (z_{b}^{*})=0$ for the critical point $z_{b}^{*}$, we calculate $z_{b}^{*}$ by finding the minimum value of $\partial _{z_{b}}P$.

We verified that all numerical results were not sensitive to an increase of either spatial or temporal resolution, therefore establishing the convergence of the numerical scheme.