Experiment AR-4

Incompressible Viscous Flow in a Pipe

1.0 Fluid mechanics fundamentals

1.1 Mach number

The Mach number for a gas flow is defined as the ratio of the flow speed to the local

speed of sound, c. For a perfect gas the speed of sound is given by the expression:

c = (γRT)

1/2

, (1)

where γ is the specific heat ratio C

p

/C

v

, R is the gas constant, and T is the absolute

temperature. For air, R=287 N·m/kg·K and γ=1.4. When M < 0.3, local changes in

density are less than 2% of the average density and the flow may be regarded as

incompressible.

1.2 Continuity equation

For steady one-dimensional flow in a duct, the mass flux ρV (kg·m

-2

·s

-1

) integrated

over the cross sectional area is:

dm/dt (kg/s) = ʃ ρV dA (2)

The fluid density is ρ and the fluid velocity is V. Because mass is conserved, the mass

flow rate, dm/dt, is constant along the flow. For an incompressible fluid, ρ is also

constant and the mass flow rate is the product of the density ρ and the mean velocity

V

mean

.

dm/dt = ρ ʃ V dA = ρ V

mean

A (3)

For a perfect gas

ρ=p/RT, (4)

where p is the local pressure.

1.3 Momentum equation

Figure 1 (below) is a sketch of a control volume for fluid moving through a constant area

duct with viscous shear forces exerted between the fluid and the duct walls.

p

2

p

1

ρ

2

V

2

2

ρ

1

V

1

2

2 1

Figure 1: Control volume for application of Newton’s second law (below)

For steady one-dimensional flow within a duct of constant area, the net change in

momentum for fluid flowing through a control volume situated between points 1 and 2

along the flow direction is given by the relationship

ʃ

A1

ρ

1

V

1

2

dA

1

- ʃ

A2

ρ

2

V

2

2

dA

2

= ʃ

A2

p

2

dA

2

– ʃ

A1

p

1

dA

1

+ ʃ

Ap

τ dA

P

Here the peripheral wall area is A

P

and A

1

, A

2

are the respective cross sectional areas

normal to the flow at 1 and 2.

For an incompressible fluid: ʃ

A1

(ρV

1

2

+p

1

)dA

1

= ʃ

A2

(ρV

2

2

+p

2

)dA

1

+ ʃ

Ap

τ dA

P

(5)

1.4 Kinetic energy

The kinetic energy “current” (J/s) of the fluid at a given position along the flow is given

by the kinetic energy flux, ½ ρV

3

, integrated over the cross-sectional area:

K.E. current = ʃ

A

½ ρV

3

dA

wall shear stress

τ

For an incompressible fluid,

K.E. current = ρʃ

A

½ V

3

dA = ½ ρ(V

3

)

mean

A (watts) (6)

1.5 Velocity profiles for fully developed flow in a cylindrical tube (circular pipe)

For a fully developed laminar flow in a pipe of radius R, the velocity profile is given by

the expression:

V(r) = V

max

[1-(r/R)

2

] (7)

For fully developed turbulent flow in a pipe of radius R, the velocity profile is often

represented with the “one-seventh power law”:

V(r) = V

max

(1-r/R)

1/7

(8)

1.6 Stagnation pressure

The Pitot tube brings the flow to rest isentropically to its stagnation (total) pressure

p

t

= p + ½ρV

2

(9)

The difference between total and static pressures is the “dynamic pressure” ½ρV

2

.

2.0 Experimental Apparatus

Measurements are made for a developing flow of an initially uniform flow entering a

long, cylindrical brass tube, shown schematically in the Fig. 2 below. The flow becomes

fully developed within about 40-50 tube diameters for turbulent flow and about 100 tube

diameters for laminar flow. The flow, forced through the tube by a fan at its entrance,

exhausts into the ambient room air at the exit of the tube. The fan speed is adjustable

by changing the voltage applied to the fan using a variable AC transformer (Variac).

This enables measurements of flow parameters (velocity profiles for fully developed flow

at the tube exit and the static pressure distribution along the tube) for two or three rates

of discharge Q (m

3

/s).

uniform velocity profile fully developed velocity profile

inlet exhaust

Figure 2: Flow development near the entrance to a constant area cylindrical duct

Static pressures are measured at various wall taps positioned along the length of the

tube. A Pitot (total head) tube, mounted on a micrometer carriage, enables

measurements of the total (stagnation) pressure as a function of radial position across a

plane perpendicular to the flow located near the tube exit. The pressure measurements

are referenced to the static pressure at the tap located nearest the entrance of the tube

with use of several differential manometers of varying sensitivities. Rubber tubing is

used to connect one port of a manometer to one of the many wall taps or to the Pitot

tube, while the other manometer port is connected to the reference static pressure tap.

3.0 Measurements

3.1 With the fan operating at top speed, measure the static pressure distribution along

the length of the brass tube. Use these data to determine the pressure gradient, dp/dz,

along the tube. Be sure to record the power input, P=VI (volts x amps), to the fan.

3.2 Again with the fan at maximum speed, use the Pitot tube to measure the radial

distribution of stagnation pressure near the tube exit. Because the flow is axisymmetric,

the micrometer carriage is designed for measurements that do not encompass the full

diameter of the tube. Use these measurements to determine the radial variation in flow

velocity across the tube. Try to obtain measurements as close as possible to the inside

wall surface and record your best estimate of the separation between the wall and the

centerline of the Pitot tube at this location.

3.3 Repeat the measurements of 3.1 and 3.2 for one or two smaller discharge rates. Do

not reduce the applied voltage to the fan below values necessary to maintain a steady

flow. A very crude shutter exists at the fan entrance, apparently used in the past to vary

the flow discharge rate. This approach doesn’t yield predictable quantitative variations in

discharge rates and has long since been abandoned. For each discharge rate, again

be sure to record the power input to the fan.

4. Analysis

4.1 Calculate the Reynolds number, based on the mean velocity at the tube exit and the

tube diameter, for each discharge rate and determine whether the flow is laminar or

turbulent.

4.2 For the maximum fan speed, compare the experimental velocity profile, obtained

from the stagnation pressure measurements with the Pitot tube, to the prediction of Eq.

(8).

4.3 Use the measured power input to the fan at its maximum speed and your measured

velocity profile in 4.2 to determine the efficiency with which the electrical energy input is

converted to kinetic energy of the gas flow.

4.4 Assume that the flow has a uniform velocity distribution at the location of the static

pressure tap nearest the flow tube inlet. For the maximum discharge rate, use your

measured velocity profile near the tube exhaust and the continuity equation (2) to

calculate this uniform velocity.

4.5 Refer to the control volume of Figure 1 and work with your data for the maximum

discharge rate: Take position 1 to be that of the upstream reference tap with a static

pressure p

1

and the uniform velocity calculated in 4.4. Take position 2, with a static

pressure p

2

, to be where the downstream velocity profile is measured, Use the

momentum equation (5) to estimate how much the momentum of the fluid flow is

reduced by the wall shear force as it passes through the flow tube.

4.6 The “head loss” caused by wall shear forces for a fully developed flow through a

constant area circular (horizontal) pipe is defined as

h

ℓ

= Δp/ρ

where Δp is the pressure drop along a pipe length L. The “friction factor”, ƒ, is given by

the expression:

ƒ = h

ℓ

2(D/L)/(V

mean

)

2

(10)

Use your measurement of the pressure gradient dp/dz for the downstream one-half of

the tube to estimate Δp/L for a fully developed flow along a similar pipe of length L. Use

this to calculate the friction factor for such a pipe. The “Moody diagram”, available in

most elementary texts on fluid mechanics (see the reference below, p. 362), gives the

friction factors as a function of the Reynolds number, ρV

mean

D/μ, for pipes of varying

roughness. Compare your estimate of ƒ with that given on the Moody diagram for a

“smooth” pipe.

Reference:

Robert W. Fox and Alan T. McDonald, Introduction to Fluid Mechanics, John Wiley &

Sons, NY 1985.

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