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Flow characteristics in spin-up of a three-layer fluid

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Abstract

A numerical study is made of the spin-up from rest of a three-layer fluid in a closed, vertically-mounted cylinder. The densities in the upper layer ρ1 middle layer ρ2 and lower layer ρ3 are ρ3>ρ2>ρ1, and the kinematic viscosities are left arbitrary. The representative system Ekman number is small. Numerical solutions are obtained to the time-dependent axisymmetric Navier-Stokes equations, and the treatment of the interfaces is modeled by use of the Height of Liquid method. Complete three-component velocity fields, together with the evolution of the interface deformations, are depicted. At small times, when the kinematic viscosity in the upper layer is smaller than in the middle layer, the top interface rises (sinks) in the central axis (peripheral) region. When the kinematic viscosity in the lower layer is smaller than in the middle layer, the bottom interface rises (sinks) in the periphery (axis) region. Detailed shapes of interfaces are illustrated for several cases of exemplary viscosity ratios.

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Abbreviations

A:

Aspect ratio, R/H [—]

Fr:

Froude number, (ΩR)2/gH [—]

g:

Gravity [m/s2]

E:

Ekman number, v/Ωf R 2 [—]

H:

Height of cylinder [m]

h0 :

The original height at the initial state [m]

R:

Radius of cylinder [m]

Re:

Reynolds number, Ωf R 2/v [—]

p:

Dimensionless pressure [—]

u* :

Reference velocity, ΩR [m/s]

t:

Time [s]

(Vγ, Vϕ, Vz :

Dimensionless velocity components corresponding to (γ, —, z)

(γ, —, z):

Dimensionless (radial, azimuthal, vertical) coordinates in cylindrical system

(α, β, γ):

Indicators [ — ]

Ω:

Rotation rate [s-1]

υ:

Kinematic viscosity [m2/s]

υ1,υ2 :

Kinematic viscosity ratios, υ1 = υ1*/υ3* υ2 = υ2*/υ3* [—]

ρ:

Dimensionless density, ρ*/p3 [—]

ρ1,ρ2 :

Density ratios, ρ1 = ρ1/ρl ρ2 = ρ2/ρ3 [—]

τ:

Dimensionless time, tΩ [—]

ψ:

Dimensionless meridional stream function [—]

1:

Values in upper layer

2:

Values in middle layer

3:

Values in lower layer

f:

Final state

i:

Initial state

*:

Dimension values

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Authors and Affiliations

  1. Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, 305-701, Daejeon, Korea

    Evgeny Sviridov & Jae Min Hyun

Authors
  1. Evgeny Sviridov
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  2. Jae Min Hyun
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Correspondence to Jae Min Hyun.

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Sviridov, E., Hyun, J.M. Flow characteristics in spin-up of a three-layer fluid. J Mech Sci Technol 20, 271–277 (2006). https://doi.org/10.1007/BF02915829

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  • DOI: https://doi.org/10.1007/BF02915829

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