Flow Around Immersed Objects - Transport Process - Lecture Slides, Slides of Computer Science

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Flow Around Immersed Objects

Incompressible Flow

Goals

• Describe forces that act on a particle in a fluid.
• Define and quantify the drag coefficient for

spherical and non-spherical objects in a flow

field.

• Define Stokes’ and Newton’s Laws for flow

around spheres.

Forces

Dynamic

Fk results from the relative motion of the object and the fluid (shear stress)

Static

Fs results from external pressure gradient (Fp) and gravity (Fg).

M 2  M 1   F  Fk Fg Fp

Dynamic Forces

A drag force (friction) arises in situations in which moving fluids are in contact with a solid surface (recall pipe flow.

Fw is the kinetic force exerted on the solid wall or particle, A is the wetted surface area, K is the kinetic energy per unit volume, and f is the friction factor.

K

F A

f  w

 V 2 2 

w 



density velocity head

wall shear

f

Projected Area

The projected area used in the Fk is the area “seen” by the fluid.

Spherical Particle

A  4

2 D R

  

Projected Area

Cylinder

For objects having shapes other than spherical, it is necessary to specify the size, geometry and orientation relative to the direction of flow.

Axis perpendicular to flow Rectangle A  LD

Axis parallel to flow Circle

D^2

A



Drag Coefficient

For slow flow around a sphere and Re<





0

24 24

Re Du

Cd  

Recall:

2

2 F Cd A u^0 k

 

Fk  3  Du 0

Stokes’ Law for Creeping Flow Around Sphere

Why Different Regions?

As the flow rate increases wake drag becomes an important factor. The streamline pattern becomes mixed at the rear of the particle and at very high Reynolds numbers completely separate in the wake. This causes a greater pressure at the front of the particle and thus an extra force term due to pressure difference.

Total Force

The gravity and buoyancy forces on an object immersed in liquids do not generally balance each other and the object will be in motion.

Ft  Fk  Fg  Fb

 Fb

 Fg

What is the direction of Fk?

It is always opposite to the direction of particle motion

Equilibrium

When a particle whose density is greater than that of the fluid begins to fall in response to the force imbalance, it begins to accelerate (F=ma). As the velocity increases the viscous drag force also increases until all forces are in balance. At this point the particle reaches terminal velocity.

0  Fk  Fg  Fb

m g V g

u

F C A p p f

t

   d p^ f  ^ 

2

Settling Velocity

The settling (terminal) velocity of small particles is often low enough that the Reynolds number is less than unity (Cd = 24/Re).

 



 

2

D g

u

p p f t

 Re ^1

Between 1000<Re<200,000 Cd = 0.44 Newton’s Law applies

 

f

p p f t

gD u

 1. 75 Fk ^ Dput^ f

2 2

Note: Intermediate flow requires iteration

Criterion for Settling Regime

Reynolds number is a poor criteria for determining the proper regime for settling. We can derive a value K that depends solely on the physical parameters

 

13

2  

  

   

 (^) f p f p

g K D

K < 2.6 Stokes’ Law

K > 68.9 Newton’s Law

Example

2

2 F Cd A u^0 k

 

6 3

3 (^0 1). 192 10

1. 005 10

2. 2 1. 2 1    

     kg m s

u D kg m m s m Re 



Fig. 7.3 gives Cd ≈ 0.

Projected Area = DL = 10 m^2

  N

s

m

m

kg

Fk 10 m 998. 2 1. 2 2 , 515

2

2 2 3

 ^2   

Example

Estimate the terminal velocity of limestone particles (Dp = 0.15 mm,  = 2800 kg/m^3 ) in water @ 20°C.