Liquid convection effects on the pushing-engulfment transition of insoluble particles by a solidifying interface: Part I
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Liquid Convection Effects on the Pushing-Engulfment Transition of Insoluble Particles by a Solidifying Interface: Part II. Numerical Calculation of Drag and Lift Forces on a Particle in Parabolic Shear Flow S. MUKHERJEE, M.A.R. SHARIF, and D.M. STEFANESCU In this work, lift and drag forces acting on a particle in the close vicinity of a wall are calculated by numerically solving the incompressible two-dimensional Navier–Stokes equations. The flow field computations are done using the well-known Marker and Cell (MAC) method on a staggered grid. A parabolic shear flow at the inlet is assumed. The particle is assumed to be nonrotating and Magnus forces are not considered. The numerical results are compared to those obtained from analytical/ empirical expressions for drag and lift forces from different theoretical models. Reasonably good agreement has been found between the two approaches.
I. INTRODUCTION
THE calculation of the drag and lift forces acting on a particle that interacts with the solidification front is of paramount importance to models that describe the particle-to engulfment transition (PET). The analytical equations that have been proposed so far have severe limitations. In addition, the transport of particles and droplets plays an important role in combustion and other industrial processes. Therefore, it is important to investigate the effects of fluid shear on the fluid forces acting on a particle by both experimentation and numerical modeling. In this work, a two-dimensional numerical simulation has been done to calculate the drag and the lift forces acting on a particle in the close vicinity of a solid wall in a parabolic shear flow. The numerical scheme involves the solution of the two-dimensional incompressible Navier– Stokes equation using the well-known Marker and Cell (MAC) method[1] on a staggered grid. The numerical results are then compared with available analytical and empirical results. II. ANALYTICAL DRAG AND LIFT FORCES Fluid flow around a particle results in a pressure gradient across the particle diameter. This pressure gradient induces a drag force on the particle (form drag). In addition, shear stresses developing on the particle surface result in friction drag. The drag force can be expressed in the most general form as 1 rL Vc2 AP 2
[1]
where CD is the drag coefficient, L is the density of the liquid, Vc the characteristic velocity, and AP the frontal area S. MUKHERJEE, Graduate Research Assistant, Metallurgical and Materials Engineering, M.A.R. SHARIF, Associate Professor, Aerospace Engineering and Mechanics, and D.M. STEFANESCU, Cudworth Professor, Metallurgical and Materials Engineering, and Director, Solidification Laboratory, are with The University of Alabama, Tuscaloosa, AL 35487. Contact e-mail: [email protected] Manuscript submitted May 5, 2002. METALLURGICAL AND MATERIALS TRANSACTIONS A
FD 6phVP RP
[2]
where VP is the velocity of the particle relative to the liquid, RP is the particle radius, and is the dynamic viscosity of the fl
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