1. Flow Around Immersed Objects
Incompressible Flow

2. 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.

3. Flow Around Objects
There are many processes that involve flow through a porous medium such as a suspension of particles:

4. Flow Around Objects
There are many processes that involve flow around objects - some are more interesting than others:

5. 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).

6. Dynamic Forces
For flow around a submerged object a drag coefficient Cd is defined:
U0 is the approach velocity (far from the object), ρ is the density of the fluid, A is the projected area of the particle, and Cd is the drag coefficient analogous to the friction factor in pipe flow (keep this in mind).

7. Projected Area Spherical Particle
The projected area used in the Fk is the area “seen” by the fluid.
Spherical Particle

8. 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.
Cylinder
Axis perpendicular to flow
Rectangle
Axis parallel to flow
Circle

9. Drag Coefficient
The drag coefficient, like the friction factor in pipes depends on the Reynolds number
D is particle diameter or a characteristic length and ρ and μ are fluid properties.

10. Drag Coefficient Recall: Stokes’ Law for Creeping Flow Around Sphere
For slow flow around a sphere and Re<10
Recall:
Stokes’ Law for Creeping Flow Around Sphere

11. Drag Coefficient

12. 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 thus causing a greater pressure at the front of the particle and thus an extra force term due to pressure difference. At very high Reynolds numbers completely separate in the wake.
Streamline separation

13. Drag Coefficient Adjustment
Russian Shkval Torpedo

14. Determining Flow Fields

15. Videos

16. Static Forces
Static forces exist in the absence of fluid motion. They include the downward force of gravity and the upward force of buoyancy that results from the gravity induced pressure gradient in the z-direction

17. Total Force What is the direction of Fk?
The gravity and buoyancy forces on an object immersed in liquids do not generally balance each other and the object will be in motion.
What is the direction of Fk?
It is always opposite to the direction of particle motion

18. 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.

19. Terminal Velocity
General Expression: If the particle has a uniform density, the particle mass is Vprp and
Use: Falling ball viscometer to measure viscosity

20. Settling Velocity
Stokes’ Region: The settling (terminal) velocity of small particles is often low enough that the Reynolds number is less than unity (Cd = 24/Re).
Newton’s Region: Between 1000<Re<200,000 Cd = 0.44
Note: Intermediate flow requires iteration

21. 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
K < 2.6
Stokes’ Law
K > 68.9
Newton’s Law

22. Example
A cylindrical bridge pier 1 meter in diameter is submerged to a depth of 10m in a river at 20°C. Water is flowing past at a velocity of 1.2 m/s. Calculate the force in Newtons on the pier.

23. Example
Fig. 7.3 gives Cd ≈ 0.35
Projected Area = DL = 10 m2

24. Example
Estimate the terminal velocity of limestone particles (Dp = 0.15 mm, r = 2800 kg/m3) in 20°C.

25. Example
Guess Re = 4
Cd = 16.0 – from Figure 7.6

26. Example
Guess Re = 2
Cd = 22
Ut = m/s
Re = 1.9

27. 10 Minute Problem
Tiger Woods is practicing putting golf balls on a cruise ship, he makes a slight miscalculation and the ball rolls off the “green” and falls into the ocean. Assuming the ball quickly attains its terminal velocity and the descent is defined by the Newton’s law flow regime, how long does it take the ball to hit the ocean floor 300 m below ?
Golf ball data: Diameter = 43 mm
Weight = 45 grams
Density = 1.16 g /cm3
Seawater data: Density = g /cm3
Viscosity = 0.01 g / cm sec