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MECH 221 FLUID MECHANICS(Fall 06/07)REVIEW

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MECH 221 – Review

What Have You Learnt? 1. Fluid Statics

2. Fluids in Motions

3. Kinematics of Fluid Motion

4. Integral and Differential Forms of Equations of Motion

5. Dimensional Analysis

6. Inviscid Flows

7. Boundary Layer Flows

8. Flows in Pipes

9. Open Channel Flows

On coming week lectures

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MECH 221 – Review

Fluid Statics It is to calculate the fluid pressure when the

fluid is no moving

Shear stress is due to relative motion of fluid, so no shear stress and only normal stress (Pressure) acting on the fluid

The fluid pressure is only due to body force, Gravitational Force

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MECH 221 – Review

Fluid Statics Fluid pressure will increase when the position

of the fluid become deeper, we have following equation:

gdz

dp z

y

x

g

0

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MECH 221 – Review

Fluid Statics Total force acting on the surface become:

hdAgAppdAF atm

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MECH 221 – Review

Fluid In Motion (Inviscid Flow) 2 sets equations for solving fluid motion problems

Conservation of Mass

Conservation of Momentum

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

gss vvv

0ddVt S)t(V

sv

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MECH 221 – Review

Fluid In Motion (Inviscid Flow) By invoking the continuity equation, the

momentum equation becomes Euler’s equation of motion

Bernoulli equation is a special form of the Euler’s equation along a streamline

constantz 2

g2

vp

Along streamline incompressible flow

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MECH 221 – Review

Fluid In Motion (Inviscid Flow) A conical plug is used to regulate the air flow

from the pipe. The air leaves the edge of the cone with a uniform thickness of 0.02m. If viscous effects are negligible and the flowrate is 0.05m3/s, determine the pressure within the pipe.

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MECH 221 – Review

Fluid In Motion (Inviscid Flow) Procedure:

Choose the reference point From the Bernoulli equation

P, V, Z all are unknowns For same horizontal level, Z1=Z2

Flowrate conservation Q=AV

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MECH 221 – Review

Fluid In Motion (Inviscid Flow) From the Bernoulli equation,

)(2

2

2

zz level, lhorizontia same at the

Since,

z 2

z 2

21

2221

222

211

21

2

222

1

211

vvpp

g

v

g

p

g

v

g

p

g

v

g

p

g

v

g

p

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MECH 221 – Review

Fluid In Motion (Inviscid Flow) From flowrate conservation,

smv

smv

mrtA

mD

A

mrmtmDsmQ

vAvAQ

894.190251.0/5.0

034.120415.0/5.0

Therefore,

0251.0)02.0)(2.0(22

0415.04

23.0

4

2.0,02.0 ,23.0 ,5.0Given

2

1

22

222

1

3

2211

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MECH 221 – Review

Fluid In Motion (Inviscid Flow)

21

221

22

3

21

21

2221

565.148

)034.12894.19(2

184.10

0p point, reference becomes pSet

184.1 C,25 atm, air@1 standardFor

894.19 ,034.12

)(2

mNp

p

mkg

smvs

mv

vvpp

Sub. into the Bernoulli equation,

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MECH 221 – Review

Fluid In Motion (Viscous Flow) In the mentioned fluid motion is inviscid

flows, only pressure forces act on the fluid since the viscous forces (stress) were neglected

With the viscous stress, the total stress on the fluid is the sum of pressure stress ( ) and viscous stress ( ) given by:τ

pσ

τ pσσ

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MECH 221 – Review

Fluid In Motion (Viscous Flow) The substitution of the viscous stress into the

momentum equations leads to:

These equations are also named as the Navier-Stokes equations

bτ

)()( p)(t

vvv

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MECH 221 – Review

Dimensional Analysis The objective of dimensional analysis is to obtain

the key non-dimensional parameters that govern the physical phenomena of flows

After the dimensional analysis or normalization of the complicated Navier-Stokes equations (steady flow), the non-dimensional parameters are identified

The equations are reduced to simple equation and solvable analytically under certain conditions

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MECH 221 – Review

Dimensional Analysis By using proper scales, the variables, velocity

(u), pressure (p) and length (L) are normalized to obtain the non-dimensional variables, which are order one

*

U

gL

ULp

U

P *

g2

2

2ivvv ***

L Ppp U / /vv

direction nalgravitatio inr unit vecto *gi

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MECH 221 – Review

Dimensional Analysis For simplicity consider the case where the

gravitational force has no consequence to the dynamic of the flow, the Navier-Stokes equations becomes

UL

Re

,Re

1 2*2

** vvv*

p

U

P

,Re

1 2* ** vvv*

p

When Re >> 1

scale pressure as 2UP

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MECH 221 – Review

Inviscid Flow Vs. Boundary Layer Flow

where is the viscous diffusion length in an advection time interval of .

Here, measures the time required for fluid travel a distance L.

2

22

v

L

U/L

LULRe

force viscous

force inertia

U/L

U/Laτ

aU/L τ

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MECH 221 – Review

Inviscid Flow Vs. Boundary Layer Flow

When , inertia force is much greater than viscous force, i.e., the viscous diffusion distance is much less than the length L.

Viscous force is unimportant in the flow region of , but can become very important in the region of

near the solid boundary.

This flow region near the solid boundary is called an boundary layer as first illustrated by Prandtl.

1Re

)(O )L(O

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MECH 221 – Review

Inviscid Flow Vs. Boundary Layer Flow

Flow in the region outside the boundary layer where viscous force is negligible is inviscid. The inviscid flow is also called the potential flow.

U

Boundary layer flowPotential flow

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MECH 221 – Review

Inviscid Flow Inviscid flow implies that the viscous effect is

negligible. The governing equations are Continuity equation and Euler equation.

We introduce a potential function, which is automatically satisfy the continuity equation

v

02

2

2

2

2

22

zyx

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MECH 221 – Review

Inviscid Flow The continuity equation becomes Laplace

equation. The flow is described by Laplace equation is called potential flow

For 2D potential flows, a stream function (x,y) can also be defined together with (x,y)

xyyx

and

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MECH 221 – Review

Inviscid Flow

If 1 and 2 are two potential flows, the sum =(1+2) also constitutes a potential flow

We can combine certain basic solutions to obtain more complicated solution

+ =

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MECH 221 – Review

Inviscid Flow

Uniform Flow Stagnation Flow Source (Sink) Free Vortex

Source and Sink

DoubletSource in

Uniform Stream2-D Rankine

OvalsFlows Around a

Circular Cylinder

Basic Potential Flows

Combined Potential Flows

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MECH 221 – Review

Inviscid Flow

For stagnation flow,

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MECH 221 – Review

Boundary Layer Flow The thin layer adjacent to a solid boundary is

called the boundary layer and the flow inside the layer is called the boundary layer flow

Inside the thin layer the velocity of the fluid increases from zero at the wall (no slip) to the full value of corresponding potential flow.

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MECH 221 – Review

Boundary Layer Flow There exists a leading edge for all external

flows. The boundary layer flow developing from leading edge is laminar

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MECH 221 – Review

Boundary Layer Flow When we normalize the governing equations with

Re underneath the viscous term and resolve the variables of y and v inside the boundary flow, the non-dimensional normalized variables are selected:

V

vv

U

uu

yy

L

xx

L

,,,

V be the scale of v in the boundary layer

L is viscous diffusion layer near the wall (boundary layer)

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MECH 221 – Review

Boundary Layer Flow These results in the boundary layer equations that

in dimensional form are given by:

0

yx

vu

2

2

y

u

x

p

y

uv

x

uu

y

p

0

Continuity:

X-momentum:

Y-momentum:

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MECH 221 – Review

Boundary Layer Flow A boundary layer flow is similar and its velocity

profile as normalized by U depends only on the normalized distance from the wall:

i.e.,

yx

Uy

x

2/1

gU

u

(*)

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MECH 221 – Review

Boundary Layer Flow By introduce a stream function

The boundary layer equation in term of the similarity variables becomes:

)(' fU

yu

fxvU 2

1

f ff '' as and at 100

02 ''''' fff

(**)

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MECH 221 – Review

After we solve this ordinary equation, we obtain a solution of

We first find the value of by Equ. (*) based on coordinate of x and y, then find out the value of by checking the solution table in the reference. Finally the u at x and y is calculated by Equ. (**)

Therefore, we obtain following results:

Boundary Layer Flow

5

U

vx

)(' f

)(' f

x

w

U

Re

332.0 2

x

fCRe

664.0

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MECH 221 – Review

Boundary Layer Flow Laminar boundary layer flow can become

unstable and evolve to turbulent boundary layer flow at down stream. This process is called transition

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MECH 221 – Review

Boundary Layer Flow Under typical flow conditions, transition usually

occurs at a Reynolds number of 5 x 105

Velocity profile of turbulent boundary layer flows is unsteady

A good approximation to the mean velocity profile for turbulent boundary layer is the empirical 1/7 power-law profile given by

71

y

U

u

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MECH 221 – Review

Boundary Layer Flow

51

Re

37.0

xx

512

Re

0577.0

2/x

wf

UC

41

200225.0

U

vUw

For turbulent boundary layer, empirically we have

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MECH 221 – Review

Boundary Layer Flow The net force, F, acting on the body

The resultant force, F, can be decomposed into parallel and perpendicular components. The component parallel to the direction of motion is called the drag, D, and the component perpendicular to the direction of motion is called the lift, L.

...... sb

shear

sb

pressure

sb

ddd FFFF

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MECH 221 – Review

If i is the unit vector in the body motion direction, then magnitude of drag FD becomes:

For two-dimensional flows, we can denotes j as the unit vector normal to the flow direction, FL is the magnitude of lift and is determined by:

Boundary Layer Flow

)( ss

..

itni dAdApF w

sb

D F

jtnj )( ss

..

dAdApF w

sb

L F

Pressure Drag

Friction Drag

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MECH 221 – Review

The drag coefficient defined as

For uniform flow passing a flat plate and no pressure gradient is zero and no flow separation, :

Boundary Layer Flow

22 /AU

FC D

D

Re

072.05

1

L

DC

vUL

CL

L

D ReRe

328.1 whereLaminar Friction Drag

Turbulent Friction Drag

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MECH 221 – Review

Boundary Layer Flow The pressure drag is usually associated with

flow separation which provide the pressure difference between the front and rear faces of the body

For low velocity flows passing a sphere of diameter D, the drag coefficient then is expressed as:

D

DD AU

FC

Re

24

2/2

direction flow the in sphere the of area projected the is where 42 /πDA