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MECH 221 FLUID MECHANICS(Fall 06/07)

Chapter 3: FLUID IN MOTIONS

Instructor: Professor C. T. HSU

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MECH 221 – Chapter 3

3.1. Newton’s Second Law

The net force F acting on a matter of mass m leads to an acceleration a following the linear relation: F = ma

For a solid body of fixed shape, m is a constant and a is described along the trajectory of motion.

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MECH 221 – Chapter 3

3.1. Newton’s Second Law If r(t) represents the particle trajectory, the

velocity v(t) and acceleration are then given by:

v(t) = dr/dt ; a(t) = dv/dt = d2r/dt2

Therefore, F = mdv/dt

If m = m(t), then F = (mdv/dt)+(vdm/dt)=d(mv)/dt = dM/dt. where M = mv is the momentum.

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MECH 221 – Chapter 3

3.1. Newton’s Second Law

For fluids enclosed in a control volume V(t) which may deform with time along the trajectory of motion, it is only correct to use the fluid momentum M to describe the Newton’s second law

For M= , we have F = dM/dt =

)(tV

dVdt

dv

V(t)dVv

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MECH 221 – Chapter 3

3.2. Description of Fluid Flow

Fluid dynamics is the mechanics to study the evolution of fluid particles in a space domain (flow field). There are two ways to describe the flow field

Lagrangian description

Eulerian description

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MECH 221 – Chapter 3

3.2.1 Lagrangian Description

Given initially the locations of all the fluid particles, a Lagrangian description is to follow historically each particle motion by finding the particle locations and properties at every time instant.

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MECH 221 – Chapter 3

3.2.1 Lagrangian Description Therefore, if a specific fluid particle is initially

(t=t0) located at (x0, y0, z0), the Lagragian description is to determine (x(t), y(t), z(t)) and the fluid properties, such as

f(t) = f [x(t), y(t), z(t); t],v(t) = v [x(t), y(t), z(t); t] , etc.

given f0 = f (x0, y0, z0; t0),

v0 = v (x0, y0, z0; t0) etc.

Note that (x(t), y(t), z(t)) is a function of time.

(for any function f)

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MECH 221 – Chapter 3

3.2.2. Eulerian Description An Eulerian description of fluid flow is simply to

state the evolution of fluid properties at a fixed point (x, y, z) with time. Here (x, y, z) is independent of time. Hence,f(t) = f[x, y, z, t] , v = v [x, y ,z, t] etc.

Eulerian description is to observe the fluid properties of different fluid particles passing through the same fixed location at different time instant, while Lagrangian description is to observe the fluid properties at different locations following the same particle

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MECH 221 – Chapter 3

3.2.3. Relation between Lagrangian and Eulerian description

It is important to note that there is only one flow property at the same location with respect to the same time, i.e., f(x(t),y(t),z(t),t) = f(x,y,z,t). Therefore, the Lagrange differential with respective to dt,

which is equal to the total Eulerian differential:

tDt

fDf dd

dtt

fdz

z

fdy

y

fdx

x

fdf

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MECH 221 – Chapter 3

3.2.3. Relation between Lagrangian and Eulerian description

Hence, the total derivative is given by:

ft

f

t

f

z

fw

y

fv

x

fu

t

f

dt

dz

z

f

dt

dy

y

f

dt

dx

x

f

Dt

Df

v

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MECH 221 – Chapter 3

3.3. Equations of Motion for Inviscid Flow

Conservation of Mass

Conservation of Momentum

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MECH 221 – Chapter 3

3.3.1. Conservation of Mass Mass in fluid flows must conserve. The total mass in V(t) is given by:

Therefore, the conservation of mass requires thatdm/dt = 0.

where the Leibniz rule was invoked.

)(

/tV

dVdt

ddtdm

S

dt

dVdV

t)t(V

)(tV

dVm

S)t(V

ddVt

sv

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MECH 221 – Chapter 3

3.3.1. Conservation of Mass Hence:

This is the Integral Form of mass conservation equation.

0ddVt S)t(V

sv

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MECH 221 – Chapter 3

3.3.1. Conservation of Mass Integral form of mass conservation equation

By Divergence theorem:

Hence:

)()]([

tVdV

tv

= 0

dVdtVS

)()( vsv

0)(

StVddV

tsv

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MECH 221 – Chapter 3

3.3.1. Conservation of Mass As V(t)→0, the integrand is independent of V(t)

and therefore,

This is the Differential Form of mass conservation and also called as continuity equation.

0 )(

vt

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MECH 221 – Chapter 3

3.3.2. Conservation of Momentum The Newton’s second law,

is Lagrangian in a description of momentum conservation. For motion of fluid particles that have no rotation, the flow is termed irrotational. An irrotational flow does not subject to shear force, i.e., pressure force only. Because the shear force is only caused by fluid viscosity, the irrotational flow is also called as “inviscid” flow

dt

dMF

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MECH 221 – Chapter 3

3.3.2. Conservation of Momentum For fluid subjecting to earth gravitational acceleration,

the net force on fluids in the control volume V enclosed by a control surface S is:

where s is out-normal to S from V and the divergence theorem is applied for the second equality.

This force applied on the fluid body will leads to the acceleration which is described as the rate of change in momentum.

V(t)S

dVρpd gsF

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MECH 221 – Chapter 3

3.3.2. Conservation of Momentum

where the Leibniz rule was invoked.

)t(V

dVdt

ddt/d vM

S

dt

dV)(dV)(

t)t(Vvv

S)t(V

ddV)(t

svvv

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MECH 221 – Chapter 3

3.3.2. Conservation of Momentum Hence:

This is the Integral Form of momentum conservation equation.

dVpdddV)(t S )t(VS)t(V

gss vvv

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MECH 221 – Chapter 3

3.3.2. Conservation of Momentum

Integral form of momentum conservation equation

By Divergence theorem:

Hence:

dV)(d)t(VS vvvv s

dVpdV()(t)t(V )t(V

][)][

g vvv

dVpdddV)(t S )t(VS)t(V

gss vvv

dVppd)t(Vs s

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MECH 221 – Chapter 3

3.3.2. Conservation of Momentum

As V→0, the integrands are independent of V. Therefore,

This is the Differential Form of momentum conservation equation for inviscid flows.

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MECH 221 – Chapter 3

3.3.2. Conservation of Momentum By invoking the continuity equation,

The momentum equation can take the following alternative form:

which is commonly referred to as Euler’s equation of motion.

0 )

v(

t

g

p)()(t

vvv

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MECH 221 – Chapter 3

3.4. Bernoulli Equation for Steady Flows Bernoulli equation is a special form of the Euler’s

equation along a streamline. For a first look, we restrict our discussion to steady flow so that the Euler’s equation becomes:

Assuming that g is in the negative z direction, i.e., g =- and using the following vector identity,

the Euler’s equation for steady flows becomes

g p)( vv

zg

)()(2

1)( vvvvvv

)(gv(2

1pvv

2 z )

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MECH 221 – Chapter 3

3.4. Bernoulli Equation for Steady Flows We now take the scalar product to the above equation

by the position increment vector dr along a streamline and observe that

Thus, the result leads to

The above equation now can be integrated to give

0gdv(ddp

2 z /2)

constantz 2

g2

vdp

(along streamline)

dfdf r 0d r]vv )([; (for any function f)

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MECH 221 – Chapter 3

3.4. Bernoulli Equation for Steady Flows For incompressible fluids where ρ = constant, we

have

For irrotational flows, everywhere in the flow domain and

constantz 2

g2

vp

(along streamline)

0 v

0gv(2

1p

2 z )

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MECH 221 – Chapter 3

3.4. Bernoulli Equation for Steady Flows Since, for dr in any direction, we have:

For anywhere of irrotational fluids

For anywhere of incompressible fluids

dfdf r

constantz 2

g2

vp

constantz 2

g2

vdp

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MECH 221 – Chapter 3

3.5. Static, Dynamic, Stagnation and Total Pressure

Consider the Bernoulli equation,

The static pressure ps is defined as the pressure associated with the gravitational force when the fluid is not in motion. If the atmospheric pressure is used as the reference for a gage pressure at z=0.

constantz 2

g2

vp

(for incompressible fluid)

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MECH 221 – Chapter 3

3.5. Static, Dynamic, Stagnation and Total Pressure

Then we have as also from chapter 2.

The dynamic pressure pd is then the pressure deviates from the static pressure, i.e., p = pd+ps.

The substitution of p = pd+ps. into the Bernoulli equation gives

zgps

constant 2

2

vpd

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MECH 221 – Chapter 3

3.5. Static, Dynamic, Stagnation and Total Pressure

The maximum dynamic pressure occurs at the stagnation point where v=0 and this maximum pressure is called as the stagnation pressure p0. Hence,

The total pressure pT is then the sum of the stagnation pressure and the static pressure, i.e., pT= p0 - ρgz. For z = -h, the static pressure is ρgh and the total pressure is p0 + ρgh.

od p2

vp

2

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MECH 221 – Chapter 3

3.6. Energy Line and Hydraulic Grade Line

In fact the Bernoulli equation also states that the energy density (per unit volume) possessed by the fluid particle is constant not only along a streamline but also at everywhere in fluid domain for irrotational flow. The energy consists of pressure energy (p), kinetic energy (ρv2/2) and gravitational potential energy (ρgz).

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MECH 221 – Chapter 3

3.6. Energy Line and Hydraulic Grade Line

It becomes simpler if this total energy is interpreted into a total head H (height from a datum) by dividing the Bernoulli equation with ρg such that

where p/ρg is the pressure head, v2/2g is the velocity head and z is the elevation head.

Hgg

p 2

z 2v

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MECH 221 – Chapter 3

3.6. Energy Line and Hydraulic Grade Line

A piezometric head is then defined as that consists of only the pressure and elevation heads, i.e.,

The variations of H and Hp along the path of fluid flow can be plotted into lines and are termed as “energy line” and “hydraulic grade line”, respectively.

It is noted that H is always higher than Hp and that a negative pressure (below atmospheric pressure) occurs when Hp is below the fluid streamline

zg

pH p

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MECH 221 – Chapter 3

3.6. Energy Line and Hydraulic Grade Line

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MECH 221 – Chapter 3

3.7. Applications of Bernoulli Equation

Pitot-Static Tube

Free Jets

Flow Rate Meter

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MECH 221 – Chapter 3

3.7.1. Pitot-Static Tube Pitot-static tube is a device that measures the

difference between h1 and h2 so that the velocity of the fluid flow at the measurement location can be determined from

)hh(g2 12 v

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MECH 221 – Chapter 3

3.7.1. Pitot-Static Tube

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MECH 221 – Chapter 3

3.7.2. Free Jets Free jets are the flow from an orifice of an

apparatus that converts the total elevation head h into velocity head, i.e.,

gh2 v

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MECH 221 – Chapter 3

3.7.3. Flow Rate Meter The commonly used flow rate meter is the

Venturi meter that determines the flow rate Q through pipes by measuring the difference of piezometric heads at locations of different cross-sectional areas along the pipe.

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MECH 221 – Chapter 3

3.7.3. Flow Rate Meter If hp1 and hp2 represent the piezometric heads at

section 1 and 2 with cross-sectional areas A1 and A2 respectively, we have:

2211vv AAQ

gh

gh

pp 2

v

2

v2

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1

22

and

2

12

21

2 )/(1

)(2 v

AA

hhgpp

2

12

21

2 )/(1

)(2

AA

hhgAQ pp

Then, Therefore,