cbe 150a – transport spring semester 2014 flow around immersed objects incompressible flow

CBE 150A – Transport Spring Semester 2014

Flow Around Immersed ObjectsIncompressible Flow


Goals

• Describe forces that act Describe forces that act on a particle in a fluid.on a particle in a fluid.

• Define and quantify the Define and quantify the drag coefficient for drag coefficient for spherical and non-spherical and non-spherical objects in a flow spherical objects in a flow field.field.

• Define Stokes’ and Define Stokes’ and Newton’s Laws for flow Newton’s Laws for flow around spheres.around spheres.


Flow Around ObjectsThere are many processes that involve flow through a porous medium such as a suspension of particles:


Flow Around ObjectsThere are many processes that involve flow around objects - some are more interesting than others:


ForcesDynamic

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).

pgk FFFFMM 12


Dynamic Forces

For flow around a submerged object a drag coefficient Cd is defined:

2

20uAC

F dk

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).


Projected Area

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

Spherical Particle

A4

22 DR


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 LDA

Axis parallel to flow Circle4

2DA


Drag Coefficient

The drag coefficient, like the friction factor in pipes depends on the Reynolds number

0Du

Re

D is particle diameter or a characteristic length and ρ and μ are fluid properties.


Drag Coefficient

For slow flow around a sphere and Re<10

0

2424

DuReCd

Recall:2

20uAC

F dk

03 DuFk Stokes’ Law for Creeping Flow Around Sphere


Drag CoefficientReCRe d 2410 44.01000 dCRe


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


Drag Coefficient Adjustment

Russian Shkval Torpedo


Determining Flow Fields


Videos


Static ForcesStatic 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

gVgmF pppg 1P

ghPP f 12

gV

ghAPPAF

fp

fb

12

bF

gF


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

bgkt FFFF

bF

gF

What is the direction of Fk?

It is always opposite to the direction of particle motion


EquilibriumWhen 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.

bgk FFF 0

gVgmu

ACF fppt

fpd 20

2


Terminal VelocityGeneral Expression: If the particle has a uniform density, the particle mass is Vpp and

gVu

AC gfpt

fpd 2

02

fd

pfpt C

Dgu

3

4

Use: Falling ball viscometer to measure viscosity


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

18

2 gDu fppt

1Re

Newton’s Region: Between 1000<Re<200,000 Cd = 0.44

f

fppt

gDu

75.1 ftpk uDF 22055.0

Note: Intermediate flow requires iteration


Criterion for Settling RegimeReynolds 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

31

2

fpf

p

gDK

K < 2.6 Stokes’ Law

Newton’s LawK > 68.9


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.

smkgx

mkg

water

water

3

3

10005.1

2.998

smu 2.10


Example

2

20uAC

F dk

63

30 10192.1

10005.1

12.12.998

smkg

msmmkgDuRe

Fig. 7.3 gives Cd ≈ 0.35

Projected Area = DL = 10 m2

Ns

m

m

kgmFk 515,22.12.99810

2

35.02

22

32


ExampleEstimate the terminal velocity of limestone particles (Dp = 0.15 mm, = 2800

kg/m3) in water @ 20°C.


Example

Guess Re = 4 Cd = 16.0 – from Figure 7.6

s

m

mkg

mmkg

sm

ut 015.02.9980.163

105.12.99828008.94

3

432

2.210005.1

015.02.998105.13

34

smkg

smmkgmRe


Example

Guess Re = 2 Cd = 22

Ut = 0.013 m/s Re = 1.9


10 Minute ProblemTiger 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 gramsDensity = 1.16 g /cm3

Seawater data: Density = 1.025 g /cm3

Viscosity = 0.01 g / cm sec

cbe 150a – transport spring semester 2014 flow around immersed objects incompressible flow

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