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Fluid Mechanics II

v w

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Outline

Types of Motion or Deformation of Fluid Elements

Irrotational Flow Approximation

Stream Function

Velocity Potential

Continuit E uation

Elementary Flows

Complex Flows

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The History of Potential Flow and Boundary Layer

In 1904 Ludwig Prandtl introduced a new concept, called theboundary layer : if a fluid flows past the leading edge of a flat

, .

layer, the effects of viscosity are too large to be ignored. Outside the boundary layer, the laws of perfect-fluid flow should be satisfactory.

The calculations were still very difficult, and so only approximatemathematical solutions were possible.

But this idea clarified numerous unex lained henomena and provided a much better intellectual basis for discussing complicatedflows.

, -are intimately tied together.

We will consider perfect-fluid or inviscid flows in this chapter, and

3

boundary layer in chapters 3 and 4.

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Deformation of a Fluid ElementsGeneral deformation of fluid element is rather complex; however, we can

break the different types of deformation or motion into a superposition of.

Velocity Angular VelocityLinear Strain Rate Shear Strain Rate

4

In order for these deformation rates to be useful in the calculation of fluid

flows, they must be expressed in terms of velocity and derivatives of velocity.

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2. Linear DeformationLinear strain rate is defined as the rate of increase in length per unit

length. It is expressed in Cartesian coordinates as

z w yv xu zz yy xx ,,

o o ec s suc as w res, ro s, an eams s re c w e pu e . en,they usually shrink in direction(s) normal to that direction.

This is also true for fluid elements. Therefore, for an incompressible flow,,

amount in other direction(s) to compensate.

6The shape does not change, linear deformation

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2. Linear DeformationVelocity gradients can cause deformation, stretching resulting in a changein volume of the fluid element.

For all THREE directions:

The volumetric strain rate isthe sum of the linear strainV

z y xdt zz yy xx

rates in three mutuallyorthogonal directions.

called as volumetric strain rate or volumetric dilatation rate or bulkstrain rate . It is positive if the volume increases.

7

e near e orma on s zero or ncompress e u s.

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3. Angular Motion/RotationAngular velocity or rate of rotation at a point is defined as the average

rotation rate of two initially perpendicular lines that intersect at that point:

Angular motion results fromcross derivatives.

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3. An ular Motion/RotationThe rotation of the element about the z -axis is the average of the angularvelocities :

- -

Counterclockwise rotation is considered positive

, ,

and

The three components gives the rotation vector:

Using vector identities, the rotation vector is one-half the curl of thevelocity vector:

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Vorticity and RotationalityThe definition of the rotation vector operation is the following:

The vorticity is twice the angular rotation:

These calculations were carried out for any rigid-body rotation.

Only circular motions with zero vorticity are irrotational, and hence arepotential flows .

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If , then there is no rotation, and the flow is said to be

irrotational .

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Vorticit and RotationalitIf the vorticity at a point in a flow is nonzero, the fluidarticles that ha ens to occu that oint in s ace is

rotating, and the flow in that region is called rotational .Physically, fluid particles in a rotational region of flow

rotate en over en as t ey move a ong n t e ow.

For the 2-D flow in the xy-plane, the axis of rotation must, . ., .

k uv

z

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Vorticit and RotationalitThe vorticity vector in cylindrical coordinates ( r , , z ):

z

r z r

r

z er r

er z

e z r

For 2-D flow in the r -plane:

z r e

ur

rur

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The Irrotational Flow A roximationIrrotational approximation: 0 V

We must keep in mind that the assumption of irrotationalityis an approximation, which may be appropriate in somere ions of a flow field but not in other re ions.

In general, inviscid regions of flow far away from solid wallsand wakes of bodies are also irrotational.

However, there are situations in which an inviscid region offlow may not be irrotational (e.g., solid-body rotation).

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Inviscid Flow: Irrotational Flow

Examples where inviscid flow theory can be used:

Viscous Region - Rotationalnv sc eg on rrotat ona

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Example

Example : Determine whether the following 2-D flows are rotational orirrotational:a u = - y, v = x;

(b) v = 0, w = 3 yz ;(c) u = 2 x, w = 2 z .

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An ular Deformation/Shear Strain Now, we combine linear strain rate and shear strain rate into

shear strain tensor :

xw

z u

xv

yu

xu

21

21

wvwuw yw

z v

yv

yu

xv

zz zy zx

yz yy yxij

1121

21

z z y z x 22

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Conservation of Mass: Cartesian CoordinatesThe differential form of the equation for Conservation of Mass :

In vector notation, the equation is the following:

The Continuity EquationIf the flow is steady and compressible:

e ow s s ea y an ncompress e:

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Examples

Example : Assuming to be constant, do the following flows satisfy continuity?(a) u = -2 y, v = 3 x;

u = , v = xy;

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Examples

Example : Check the following incompressible flows for continuity anddetermine the vorticity of each:a v = r , vr = ;

(b) v = 0, vr = -5/ r .

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Stream Functions are defined for steady, incompressible, 2D flow.

2-D Continuity Equation :

Then we define the stream functions as follows:

Now, substitute the stream function into the continuity equation:

Any flow that satisfies streamfunction automatically satisfies thecontinuity condition.

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The slope at any point along a streamline:

Streamlines have constant , thus d = 0:

dz dydxwvu

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Stream Functions

Now, calculate the volumetric flow rate between streamlines:

The change in the value of the streamfunction is related to the volume rate of flow.

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Stream Functions in Cylindrical Coordinates

Incompressible, planar stream function in cylindrical coordinates:

ncompress e, ax symme r c s ream unc on n cy n r cacoordinates:

1 uru z r r

11r r u z r u z r

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StreamlinesA streamline is a line drawn

through the flow field in such amanner that the local velocity vectoris tangent to the streamline at every

oint alon the line at that instant.

The tangent of the streamline at a

given time gives the direction of theve oc y vec or. s ream ne oesnot indicate the magnitude of thevelocity.

The flow pattern shown by thestreamlines is an instantaneous

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.

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Streamlines

For real fluid flows, the fluid adjacent to the boundary of a solid bodydoes not move relative to body it sticks to the wall. So, in real fluids

But, the perfect fluid has no tendency to stick to walls because it hasno viscosity. So, the streamline adjacent to a solid body in perfect-fluidflow is one with finite velocity.

This leads to the idea that we may divide a perfect-fluid flow along a

streamline.

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Potential Flow

In the region outside the boundary layer, where the fluid may beassumed to have no viscosity, the mathematical solution takes on the

This form is analogous to the flow of heat in a temperature field or tothe flow of charge in an electrostatic field. All these flows obeyLaplaces equation under certain restrictions (for example: steady-statemass balance for a constant-density fluid).

,every velocity potential represents a potential flow . For example, = x2,

x2 + y2, e x, sin x do not satisfy Laplaces equation, so they cannot

constant-density fluid.

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Potential Flow: Velocity Potential

For irrotational flow, there exists a velocity potential:

Take one component of vorticity to show that the velocity potential is irrotational:

021 22

x y y x

We could do this to show all vorticity components are zero.

The flow must be irrotational if there is a velocity potential.

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Potential Flow: Velocity Potential

If the curl of a vector is zero, the vector can be expressed as the gradient of

Then, rewriting the u, v, and w components as a vector:

.

For irrotational, planar flow:

Now substitute the stream function:

Then, Laplaces Equation

Then for incompressible irrotational flow:

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Potential Flow: Velocit PotentialVector identity:

So, if , then

0 V

0 V V

If the curl of a vector is zero, the vector can be expressed as the gradientof a scalar function called potential function .

In fluid mechanics, vector V is the velocity vector, the curl of which is thevorticit vector, and thus we call as the velocit otential function .

For irrotational regions of flows:

V

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Potential Flow: Velocity Potential

Potential flows areirrotational vorticit iszero.If the vorticity is presente.g., oun ary ayer,

wake), then the flow

cannot be described byLaplaces equation.

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Laplacian Operator in Cartesian coordinates:

If a Potential Flow exists,

Laplacian Operator in cylindrical coordinates:

w t appropr ate oun ary

conditions, the entire velocityand pressure field can be

where the gradient in cylindrical coordinates, the gradient operator,

.

Then,

May choose cylindricalcoordinates based on thegeometry of the flow problem,

i.e., pipe flow.

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Lines of constant are streamlines :

Now, the change of from one point ( x, y) to a nearby point ( x + dx, y + dy):

Along lines of constant we have d = 0,

0

The equipotential lines are orthogonal to streamlines where they intersect.Lines of constant are called equipotential lines .

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The flow net consists of a family of streamlines and equipotential lines.

graphical flow situation.

Velocity decreasese ve oc y s nverse y

proportional to the spacing between streamlines.

along this streamline.

Velocity increases.

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Stream Function and Velocit PotentialThe stream function isdefined by continuity; theLaplace equation forresults from irrotationality .

The velocity potential isdefined by irrotationality;

results from continuity . 021

22

x y y x

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Potential Flow: Plane Potential FlowsVelocity components for steady, incompressible, irrotational, 2D regions offlow in terms of velocity potential and stream function in various

Planar, Cartesian:

Planar, Cylindrical:

Planar, Cartesian:

x symme r c, y n r ca : z r z r

Planar, Cylindrical:

11

40

r r u

z r u z r Axisymmetric, Cylindrical:

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Incom ressible? Continuit e uation?Stream function exists? Irrotational? Velocity

potential exists?

Incompressible? Satisfycontinuit e uation?

Irrotational? Velocityotential exists?

Stream function

exists?

0 V 0or 0 2 2

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ExampleExample : A velocity potential in 2D flow is . Find the stream

function for this flow.

22 y x y

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ExampleExample : In a 2D incompressible flow, the fluid velocity components are

given by and . Show that the flow satisfied thecontinuit e uation and obtain the ex ression for the stream function.

y xu 4 x yv 4

If the flow is potential, obtain also the expression for the velocity potential.

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Example

Example : The 2D flow of a non-viscous, incompressible fluid in the vicinity ofthe 90 corner is described by the stream function . 2sin2 2r

.

sin/

Ar

46

cosr

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,

website:

http://simscience.org/fluid/green/potential.html

Next we will learn how the velocity fields of someelementar and com lex flows can be ex ressed interms of stream function and velocity potential.

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.Question: Do 1 = Ax and 2 = Ax + By satisfy Laplaces equation?

For 1 = Ax , u = 1/ x = A, v = 1/ y = 0So, 1 describes a uniform, steady flow of velocity A in the positive xdirection. This might be the description of a wind blowing over the ocean ata steady, uniform velocity of A.

For 2 = Ax + By, u = 2/ x = A, v = 2/ y = B

So, 2 describes a uniform, constant-velocity flow with velocity ( A2 + B2)1/2,.

These uniform flows are not of much practical interest alone, but they can

48

e com ne w t ot er ows to so ve more nterest ng pro ems .

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.For Uniform Flow in an arbitrary direction, :

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mvr2

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mvr r 2

. Now, obtain the stream function for the flow:

0

Integrating to obtain the solution:

The streamlines are radial lines and the equipotential

rln2

lines are concentric circles centered about the origin :

lines

lines

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.If m is positive, the flow is source ; if m is negative, the flow is

.

(the strength of the source or sink, the volume rate offlow emanating from the line (per unit length), where Q is flow rate,

L

Qm

and L is height).

These flows are of practical significance in the petroleum

stratum.

The e uations u = m/2 r , v 0 show that the radial flowvelocity becomes infinite at r = 0 ( mathematical singularity ); thus,this equation cannot describe any real flow at r = 0.

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ExamplesExample : A nonviscous, incompressible fluid flows between wedge-shaped

walls into a small opening. The velocity potential (in m 2/s), whicha roximatel describes this flow is = -2 ln r . Determine thevolume rate of flow (per unit length) into the opening.

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3. Vortex Flow

In vortex flow the streamlines are concentric circles, and the equipotentiallines are radial lines .

where K is a constant, namely the strength of the vortex .Solution:

The sign of K determines whether the flow rotatesclockwise (-) or counterclockwise (+).

In this case, ,

The tangential velocity varies inversely withthe distance from the origin. lines

lines

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e or g n encoun ers a s ngu ar y becoming infinite.

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3. Vortex Flow

Rotation refers to the orientation of a fluid element and not the pathfollowed by the element. The elements deform to maintain a constantorientation.

In general flow there is both deformation and rotation.

An ideal flow is one that has no viscosity and is incompressible.

If an ideal flow is initially irrotational, it will remain irrotational.

Two vortices: free vortex and forced vortex .

e sw r ng mo on o e wa er as ra ns rom a a u s s m ar othat of a free vortex , while the motion of a liquid contained in a tank isrotated about its axis with angular velocity corresponds to a forced vortex .

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F V d F d V

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Free Vortex and Forced VortexIrrotational Flow: Free Vortex Rotational Flow: Forced Vortex

Velocity Velocity

inward.increasesoutward.

i.e., waterdraining from a

i.e., a rotating tankfilled with fluid

Traveling from A to B, consider two sticks

Irrotational Flow: Rotational Flow: Rigid Body RotationInitially, sticks aligned, one in the flow direction,and the other perpendicular to the flow.

As the move from A to B the er endicular-

Initially, sticks aligned, one in theflow direction, and the other

perpendicular to the flow.

aligned stick rotates clockwise, while the flow-aligned stick rotates counter clockwise.

The average angular velocities cancel each other,

As they move from A to B the sticksmove in a rigid body motion, and thusthe flow is rotational.

55

thus, the flow is irrotational.

0

V V

2

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ree or ex an orce or ex

566.4 Vortex

A simple analogy can be made

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between flow A and a merry-go-

,and a Ferris wheel.

As children revolve around a,

the same angular velocity as thatof the ride itself. This is analogousto a rotational flow.

In contrast, children on a Ferriswheel always remain oriented inan upright position as they traceou e r c rcu ar pa . s sanalogous to an irrotational flow.

A simple analogy: ( a ) rotationalcircular flow is analogous to a

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roun a ou , w e rro a onacircular flow is analogous to a

Ferris wheel.

Tornadoes and Hurricanes

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Tornadoes and HurricanesA combined vortex flow is one in which there is a forced vortex at the core, and a free

vortex outside the core.

The minimum pressure at the vortex center can give rise to a secondary flow whichs pro uce y t e pressure gra ent n t e pr mary vortex ow.

In the region near the ground, the wind velocity is decreased due to the friction provided by the ground.

However, the pressure difference in the radial direction causes a radially inward flowadjacent to the ground, and upward flow at the vortex center.

center and outer edge:

p1 p0 = V max 2

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Circulation

Circulation ( ) gives a measure of the average of rate of rotation of fluid particles that are situated in an area that is bounded by a closed curved.

This concept is often useful when evaluating forces (such lift force)

developed on bodies immersed in moving fluids.It is defined as the line inte ral of the tan ential com onent of the

velocity ( V) around a closed curve fixed in the flow.

= 0 for irrotational flow.

If there are singularities enclosed within the curve, 0, for example:free vortex .

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Circulation: Free Vortex

For the free vortex :

= 0

The circulation is non-zero and constant for the free vortex :The velocity potential and the stream function for the free vortex can

be rewritten in terms of the circulation:

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Example

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Example

Example : A liquid drains from a large tank through a small opening. A vortexforms whose velocity distribution away from the tank opening can be

Determine an expression relating the surface shape to the strength of

)2/(

.

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4. Doublet Flow

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4. Doublet Flow

Combination of an equal Source and Sink pair.

Rearrange and take tangent,

Note, the following:

Substituting the above expressions,

an

en,

If a is small, then tangent of angle is approximated by the angle:

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4 D bl Fl

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4. Doublet Flow

Now, we obtain the doublet flow by letting the source and sink approach one

another ( a 0), and letting the strength increase ( m ).

K is the stren th of the doublet and is e ualis then constant.

to ma/ .

The corresponding velocity potential then is the following:

Streamlines of a Doublet:lines

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4. Doublet Flow

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4. Doublet Flow

If K is negative, the doublet is backward with the sink located at x = 0 -

(infinitesimally to the left of the origin) and the source located at x = 0 +

n n tes ma y to t e r g t o t e or g n . In t s case, t e stream nes areidentical in shape, but the flow is in the opposite direction.

Also note that the above equations were derived for a doublet in which thesource and the sink were placed on the x-axis. If, however, the source andsink are placed on the y-axis, the resulting doublet is oriented, andexpressions for the stream function and the velocity potential become

r K

cos

r s n

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Summary of Basic Flows

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y

66

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Because potential flows are governed by linear partial differential equations,

the solutions can be combined in su er osition .

If 1 and 2 are each solutions of the Laplace equation, then A 1, ( A+ 1),( 1+ ), and ( A 1+ 2) are also solutions,.

Thus, some of the basic and can be combined to yield a streamline thatrepresents a particular body shape.

The su er osition re resentin a bod can lead to describin the flow aroundthe body in detail.

The superposition is only valid for irrotatioanal flow fields for which the.

The velocity at any point in the composite field is the vector sum of thevelocities of the individual flow fields.

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1 Rankine Half Bod

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1. Rankine Half-BodThere will be a stagnation point, somewhere along the negative x-axis wherethe velocities due to the source and uniform flow are cancelled ( = ).

For the source: For the uniform flow: cosU vr For = , U v r

Then, for a stagnation point, at some x = -b (r = b ), = :

mU vr and

Now, the stagnation streamline can be defined by evaluating y at r = b , and= .

2

sin mUr

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1. Rankine Half-Body

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ySince m/2 = bU , it follows that the equation of the streamline passingthrough the stagnation point, and gives the outline of the Rankine half-

bod :

Then

For inviscid flow, a streamline can be replaced by a solid boundary. So,the source and uniform can be used to describe the flow around astreamlined body placed in a uniform stream half-body.The other streamlines can be obtained by setting = constant.

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Singularity (inside the body)

r

b

sin

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. an ne a - o yThe width of the half-body:

Total width = 2 b

The magnitude of the velocity ( V ) at any point in the flow:

Noting,

Knowing the velocity we can now determine the pressure field using theBernoulli Equation:

71 po and U are at a point far away from the body and are known.

1. Rankine Half-Body

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y

We wish to find the flow pattern around some arbitrary body.This is normally done by combining uniform flows, sources,s n s, e c.

When a combination is found that produces a streamline with,

streamline is a representation of the flow around the body.

ignored.

The sin ularit in the flow field source onl occurs inside

the body.

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1. Rankine Half-Body

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The velocity tangent to the surface of the body is not zero,i.e., the fluid slips by the boundary (as neglecting viscosity).So, all potential flows differ from the flow of real fluids

(considering viscosity) and do not accurately represent the. , ,

the velocity distribution will generally correspond to that

occur.

approximate that predicted from the potential flow theory sincethe boundar la er is thin and there is little variation of

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pressure through the boundary layer.

6.5 Half-body

Example

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Example : The shape of a hill arising from a plain can be approximated with the topsection of a half-body. The height of the hill approaches 60 m.

a When a 60 km/hr wind blows toward the hill what is the ma nitude of the air velocity at a point on the hill directly above the origin (point 2)?

(b) What is the elevation of point (2) above the plain and what is thedifference in ressure between oint (1) on the lain far from the hill and

point (2)? Assume an air density of 1.23 kg/m 3.

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2. Rankine Oval

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Rankine Ovals are the combination a source, a sink and a uniform flow, producing a closed body.

Stream function and velocity function describing the flow:

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2. Rankine Oval

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The streamline = 0 forms the surface of a body of length 2 l and width 2 h placed in a uniform flow.

Ua /m is lar e slender bodUa /m is small blunt shape body

The body half-length2/1

1

Uam

al

The bod half-width

76Iterative

ah

mUa

ah

ah

2tan121

2

2. Rankine Oval

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Ua /m is large slender bodyUa /m is small blunt shape body

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2. Rankine Ovalowns ream rom e po n o max mum

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owns ream rom e po n o max mum body width, the surface pressure increaseswith distance along the surface.

This condition (called adverse pressure

gradient ) typically leads to separation (not

from the surface, resulting in a large low pressure wake on the downstream side ofthe body.

The potential solution for the Rankine

outside the thin, viscous boundary layer andthe pressure distribution on the front part of

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t e o y.

6.9 Potential and viscous flow

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3.1. Flow Around a Stationar Circular C linder

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On the surface of the cylinder ( r = a ): vrs = 0

The maximum velocity occurs at the top and bottom of the cylinder,

magnitude of 2 U ( = /2). Why are the

The figure shows the patternof streamlines for this flow.

We disregard the doublet flow

s ream nes soclose here?No slip or slip?

on the inside of the circle r = aand imagine that a solidcylinder replaces this portionof the flow. A remarkablefeature is the symmetry of theflow upstream and

80

owns ream o e cy n er.

Why are the streamlines so far here?

. .

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. .

Pressure distribution on a circular cylinder found with the Bernoulli equation

Then substituting for the surface velocity:

on the front portion of the cylinder. Theactual surface pressures and ideal valuesagree for a distance up to = 60.

Flow separation on the back-half in thereal flow due to viscous effects causesdifferences between the theor and

81

experiment. So, the ideal flow is nolonger valid.

Pressure Coefficient

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The pressure gradient influences flow patterns and pressure distributionsacting on bodies create forces.A common dimensionless group for describing the pressure distribution is

pressure coe c ent p :

202

1

0

V p p

C p

where p is the local pressure, p0 and V 0 are the free-stream pressure andvelocity.

82The points B and D are points of stagnation (C p = +1.0), and the minimum

pressure (C p = 3.0) occurs at the point C.

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From Eulers equation for pressure gradient and acceleration along a pathline,

sa t

t distance along a pathline ( p/ s < 0) favorable pressure gradient .

The fluid particle decelerates ( a t < 0) if the pressure increases with

.

83

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Flow separation occurs when the fluid pathlines adjacent to body deviatesfrom the contour of the body and produce a wake.

It tends to increase drag, reduce lift and produce unsteady forces leading

to structural failure (e.g., Tavoma Narrows Bridge in 1904).The rediction and control of se aration is continuin challen e forengineers involved with the design of fluid systems.

846.8 Circular cylinder with separation

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3.1. Flow Around a

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Cylinder

characteristics on a circular

cylinder: Wake

separation location.

velocity profiles at variouslocations on the cylinder,

(c) surface pressuredistributions for inviscid Turbulent or laminar data

86

ow an oun ary ayerflow.

matches better withirrotational flow

approximation? Why?

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3.1. Flow Around a Stationary Circular Cylinder

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Mathematically, this makes sense since the pressure distribution issymmetrical around the cylinder (because of the symmetric pressuredistribution the force on the front half cancels that one the rear half to roducezero drag).

However, in practice/experiment, we see substantial drag on a circularc linder when is laced in a movin fluid.

This discrepancy is known as dAlemberts Paradox , 1717-1783.

Potential theory incorrectly predicts that the drag on a cylinder is zero.

88

. .

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Flow around a rotating cylinder is approximated by the combination of a

The addition of the vortex changes the flow pattern everywhere, except at thecylinder surface and at infinity.

.

The streamlines that represents the cylinder is still a circle, but the values of thesurface velocity are changed.

.

r r a

Ur ln2

sin1 22

2

cos12

2

r a

Ur and

aU

r v

ar s

2sin2 On the surface of the cylinder ( r = a ):

89

The additional vortex only affects v s, but not vr .

3.2. Flow Around a Rotating Circular Cylinder

iTh i i h ( 0)

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If = 0 , then stag = 0 or ,Ua stag 4sin The stagnation points occur at = Stag where( v = 0):

. .,the front and rear of the cylinder.

- ,stagnation points occur at someother location on the surface as

gures an c .

If /(4 Ua ) > 1, then thes agna on po n s oca e away

from the cylinder. There is a portion of fluid that is trapped

90

next to the surface and continuallyrotates around the cylinder.

. .

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For the cylinder with circulation, the surface pressure is obtained from.

2

2

0 2sin221

21

aU pU p s

2

or

2220 4sin41

2 U aaU U p p s

Eq A

91

. .

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0 x F Substituting Eq A into F x, for the drag, and integrated, yields

Substituting Eq A into F y, for the lift, and integrated, yields

, .

U F y For cylinder with circulation, lift is developed equal to the product of fluid density,

Magnus ffect Lift on rotating bodies

ups ream ve oc y an c rcu a on.

The negative sign means that if U is positive in the positive x direction, and

Potential flow past a cylinder with circulation gives zero drag, but non-zero lift.

c rcu at on s pos t ve a ree vortex w t counterc oc w se rotat on , t e

direction is downward.http://www.grc.nasa.gov/WWW/K-12/airplane/cyl.html

92The equation relating lift force on airfoils to , U, and is called Kutta-Joukowski law .

. .

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Uniform flow towards +ve x-direction ( U =+ve), counterclockwise ( = +ve)

Low v and hi h P on to -half of c linder

Downward force (Fy = -ve)

n orm ow owar s ve x- rec on =+ve), clockwise ( = -ve)

Low v and high P on bottom-half of cylinder

Upward lift force (F = +ve)

93

3.2. Flow Around a Rotating Circular Cylinder

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From Eq.A the Ps distribution is studied. 2 scenerios;1. : No rotation of the cylinder ;flow is symmetrical topto bottom front to back on c linder

0 Ua

94

2. : Clockwise rotation of cylinder;flow is symmetrical

front to back, but not top to bottom.-0.25 Ua