^{1,a)}, Jing Lou

^{1}, Shaoping Quan

^{2,b)}and Andrew S. H. Ooi

^{3}

### Abstract

An initially streamwise rotating droplet released into a uniform cross flow is studied numerically. The computations are performed using a finite volume Navier–Stokes solver which employs a moving mesh interface tracking scheme to locate the interface. With a large initial Weber number (*We* _{ i } = 40) the streamwise rotating droplet flattens along the free stream direction more quickly as rotation rate () increases, and leads to a dramatic increase in the dynamic drag coefficient (*C* _{ D }/*A**, where *A** is the dimensionless frontal area). On the other hand, for *We* _{ i } = 4 and 0.4 at , the flattening of the droplet is less pronounced and the droplet even restores to spherical shape, hence, *C* _{ D }/*A** decreases sharply. The dynamic drag coefficient even becomes negative for *We* _{ i } = 4 and 0.4 at . At the largest deformation, the droplet can be classified into three major shapes: biconvex, convex-concave, and biconcave. One dominant feature of the wake downstream of the droplet is the formation and convection of vortex rings. The shape and deformation of the droplet is dependent not only on the size of the vortex ring, but also upon the free stream dynamic pressure and droplet pressure. The detachment of vortex ring in the wake leads to a substantial drag reduction, and this detachment occurs at *Re* ≈ 28.

Special thanks to ARAP projects at Agency for Science, Technology and Research (A*STAR), Singapore.

I. INTRODUCTION

II. PROBLEM SETUP AND NUMERICAL FORMULATION

III. RESULTS AND DISCUSSION

A. Droplet topology

B. Time-histories of frontal area, droplet angular velocity, and drag coefficient

C. Instantaneous flow field

IV. CONCLUSIONS

### Key Topics

- Fluid drops
- 145.0
- Vortex rings
- 16.0
- Rotating flows
- 14.0
- Viscosity
- 12.0
- Reynolds stress modeling
- 10.0

## Figures

The coordinate systems set up for an initially rotating droplet at angular velocity, , suddenly accelerated by a uniform free flow, *U* _{∞}. The axis of rotation is aligned in the free stream (*z*-) direction.

The coordinate systems set up for an initially rotating droplet at angular velocity, , suddenly accelerated by a uniform free flow, *U* _{∞}. The axis of rotation is aligned in the free stream (*z*-) direction.

(*y*, *z*)-plane cut view of the droplet shape evolution and isometric view of the droplet shape at the moment with the largest deformation. (a) *We* _{ i } = 40 and , (b) *We* _{ i } = 40 and , (c) *We* _{ i } = 40 and and (d) *We* _{ i } = 0.4 and . The solid gray lines indicate earlier (*y*, *z*)-plane cut view of the initially rotating droplet. The solid black lines are the (*y*, *z*)-plane cut view of droplet shape at the moment with the largest deformation.

(*y*, *z*)-plane cut view of the droplet shape evolution and isometric view of the droplet shape at the moment with the largest deformation. (a) *We* _{ i } = 40 and , (b) *We* _{ i } = 40 and , (c) *We* _{ i } = 40 and and (d) *We* _{ i } = 0.4 and . The solid gray lines indicate earlier (*y*, *z*)-plane cut view of the initially rotating droplet. The solid black lines are the (*y*, *z*)-plane cut view of droplet shape at the moment with the largest deformation.

Time histories of the frontal area, *A**, for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

Time histories of the frontal area, *A**, for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

Time histories of the droplet angular velocity, , for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

Time histories of the droplet angular velocity, , for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

(a) Instantaneous angular velocity, ω of the droplet, versus angular momentum, *L*, and (b) axial length of the droplet, *a*, versus angular velocity. Case I: *We* _{ i } = 40, Case II: *We* _{ i } = 4, and Case III: *We* _{ i } = 0.4.

(a) Instantaneous angular velocity, ω of the droplet, versus angular momentum, *L*, and (b) axial length of the droplet, *a*, versus angular velocity. Case I: *We* _{ i } = 40, Case II: *We* _{ i } = 4, and Case III: *We* _{ i } = 0.4.

Dynamic drag coefficient time-histories (*C* _{ D }/*A**) for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

Dynamic drag coefficient time-histories (*C* _{ D }/*A**) for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

Dynamic drag coefficients versus Reynolds numbers for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

Dynamic drag coefficients versus Reynolds numbers for : (a) *We* _{ i } = 40, (b) *We* _{ i } = 4, and (c) *We* _{ i } = 0.4.

Instantaneous projected streamlines onto the (*y*, *z*)-plane for initial *Re* _{ i } = 40, *We* _{ i } = 40, and λ = η = 50 with reference frame moving with the droplet centroid.

Instantaneous projected streamlines onto the (*y*, *z*)-plane for initial *Re* _{ i } = 40, *We* _{ i } = 40, and λ = η = 50 with reference frame moving with the droplet centroid.

Instantaneous projected streamlines onto the (*y*, *z*)-plane for initial *Re* _{ i } = 40, *We* _{ i } = 4, and λ = η = 50 with reference frame moving with the droplet centroid.

Instantaneous projected streamlines onto the (*y*, *z*)-plane for initial *Re* _{ i } = 40, *We* _{ i } = 4, and λ = η = 50 with reference frame moving with the droplet centroid.

Instantaneous projected streamlines onto the (*y*, *z*)-plane for initial *Re* _{ i } = 40, *We* _{ i } = 0.4, and λ = η = 50 with reference frame moving with the droplet centroid.

Instantaneous projected streamlines onto the (*y*, *z*)-plane for initial *Re* _{ i } = 40, *We* _{ i } = 0.4, and λ = η = 50 with reference frame moving with the droplet centroid.

Instantaneous three-dimensional streamlines for initial *Re* _{ i } = 40, *We* _{ i } = 40, and λ = η = 50 with the reference frame moving with the droplet centroid.

Instantaneous three-dimensional streamlines for initial *Re* _{ i } = 40, *We* _{ i } = 40, and λ = η = 50 with the reference frame moving with the droplet centroid.

Instantaneous three-dimensional streamlines for initial *Re* _{ i } = 40, *We* _{ i } = 0.4, and λ = η = 50 with reference frame moving with the droplet centroid.

Instantaneous three-dimensional streamlines for initial *Re* _{ i } = 40, *We* _{ i } = 0.4, and λ = η = 50 with reference frame moving with the droplet centroid.

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