CE 150 1

CE 150Fluid Mechanics

G.A. Kallio

Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology

California State University, Chico

CE 150 2

Elementary Fluid Dynamics

Reading: Munson, et al., Chapter 3

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Inviscid Flow

• In this chapter we consider “ideal” fluid motion known as inviscid flow; this type of flow occurs when either

1) 0 (only valid for He near 0 K), or

2) viscous shearing stresses are negligible

• The inviscid flow assumption is often valid in regions removed from solid surfaces; it can be applied to many problems involving flow through pipes and flow over aerodynamic shapes

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Newton’s 2nd Law for a Fluid Particle

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Newton’s 2nd Law for a Fluid Particle• The equation of motion for a fluid

particle in a steady inviscid flow:

• We consider force components in two directions: along a streamline (s) and normal to a streamline (n):

gp

extgp

FFdt

Vdm

FFFFam

cos

sin

2

WFV

m

WFds

dVmV

pn

ps

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Newton’s 2nd Law Along a Streamline

• Noting that

we have:

, and

,sin ,

Vds

dpF

ds

dzVgWVm

ps

ds

dp

ds

dzg

ds

dVV

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Newton’s 2nd Law Along a Streamline• Integrating along the streamline:

• If the fluid density remains constant

• This is the Bernoulli equation

constant221 gzV

dp

s

streamlinea alongconstant

or

streamlinea alongconstant

221

221

gzVp

gzVp

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Newton’s 2nd Law Across a Streamline

• A similar analysis applied normal to the streamline for a fluid of constant density yields

• This equation is not as useful as the Bernoulli equation because the radius of curvature of the streamline is seldom known

constant 2

gzdnV

pn

)(

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Physical Interpretation of the Bernoulli Equation

• Acceleration of a fluid particle is due to an imbalance of pressure forces and fluid weight

• Conservation equation involving three energy processes:– kinetic energy

– potential energy

– pressure work

streamlinea alongconstant 221 gzVp

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Alternate Form of the Bernoulli Equation

• Pressure head (p/g) - height of fluid column needed to produce a pressure p

• Velocity head (V2/2g) - vertical distance required for fluid to fall from rest and reach velocity V

• Elevation head (z) - actual elevation of the fluid w.r.t. a datum

streamlinea alongconstant 2

2

zg

V

g

p

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Bernoulli Equation Restrictions

• The following restrictions apply to the use of the (simple) Bernoulli equation:1) fluid flow must be inviscid

2) fluid flow must be steady (i.e., flow properties are not f(t) at a given location)

3) fluid density must be constant

4) equation must be applied along a streamline (unless flow is irrotational)

5) no energy sources or sinks may exist along streamline (e.g., pumps, turbines, compressors, fans, etc.)

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Using the Bernoulli Equation• The Bernoulli equation can be

applied between any two points, (1) and (2), along a streamline:

• Free jets - pressure at the surface is atmospheric, or gage pressure is zero; pressure inside jet is also zero if streamlines are straight

• Confined flows - pressures cannot be prescribed unless velocities and elevations are known

22

221

212

121

1 gzVpgzVp

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Mass and Volumetric Flow Rates• Mass flow rate: fluid mass

conveyed per unit time [kg/s]

where Vn = velocity normal to area [m/s]

= fluid density [kg/m3]

A = cross-sectional area [m2]

– if is uniform over the area A and the average velocity V is used, then

• Volumetric flow rate [m3/s]:

A ndAVm

AVm

AVQ

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Conservation of Mass

• “Mass can neither be created nor destroyed”

• For a control volume undergoing steady fluid flow, the rate of mass entering must equal the rate of mass exiting:

• If = constant, then

222111

21

VAVA

mm

212211 or QQVAVA

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The Bernoulli Equation in Terms of Pressure

• Each term of the Bernoulli equation can be written to represent a pressure:

• pgh : this is known as the hydrostatic pressure; while not a real pressure, it represents the possible pressure in the fluid due to changes in elevation

)(constant 221

TpgzVp

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The Bernoulli Equation in Terms of Pressure• p : this is known as the static

pressure and represents the actual thermodynamic pressure of the fluid

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The Bernoulli Equation in Terms of Pressure

• The static pressure at (1) in Figure 3.4 can be measured from the liquid level in the open tube as pgh

• : this is known as the dynamic pressure; it is the pressure measured by the fluid level (pgH) in the stagnation tube shown in Figure 3.4 minus the static pressure; thus, it is the pressure due to the fluid velocity

221 V

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The Bernoulli Equation in Terms of Pressure

• The stagnation pressure is the sum of the static and dynamic pressures:

– the stagnation pressure exists at a stagnation point, where a fluid streamline abruptly terminates at the surface of a stationary body; here, the velocity of the fluid must be zero

• Total pressure (pT) is the sum of the static, dynamic, and hydrostatic pressures

212

112 Vpp

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Velocity and Flow Measurement• Pitot-static tube - utilizes the static

and stagnation pressures to measure the velocity of a fluid flow (usually gases):

/)(2 43 ppV

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Velocity and Flow Measurement• Orifice, Nozzle, and Venturi meters -

restriction devices that allow measurement of flow rate in pipes:

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Velocity and Flow Measurement

• Bernoulli equation analysis yields the following equation for orifice, nozzle, and venturi meters:

– Theoretical flowrate:

– Actual flowrate:

])/(1[

)(22

12

212 AA

ppAQideal

)1( CCQQ idealactual

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Velocity and Flow Measurement

• Sluice gates and weirs - restriction devices that allow flow rate measurement of open-channel flows:

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Velocity and Flow Measurement