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Pharos UniversityMECH 253 FLUID

MECHANICS II

Lecture # 5 INVISCID FLOWS

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1 Inviscid Flow Inviscid flow implies that the viscous effect is

negligible. This occurs in the flow domain away from a solid boundary outside the boundary layer at Re.

The flows are governed by Euler Equations

where , v, and p can be functions of r and t .

gvvv

pt

)(

0)(

vt

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7.1 Inviscid Flow On the other hand, if flows are steady but

compressible, the governing equation becomes

where can be a function of r

For compressible flows, the state equation is needed; then, we will require the equation for temperature T also.

gvv p)(

0)( v

RTp

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7.1 Inviscid Flow

Compressible inviscid flows usually belong to the scope of aerodynamics of high speed flight of aircraft. Here we consider only incompressible inviscid flows.

For incompressible flow, the governing equations reduce to

where = constant.

gvvv

pt

)(

0 v

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7.1 Inviscid Flow For steady incompressible flow, the governing eqt reduce further to

where = constant. The equation of motion can be rewrited into

Take the scalar products with dr and integrate from a reference at along an arbitrary streamline =C , leads to

since

)(2

)v( 2

vv

zgp

gvv p)(

0 v

constant22

v 22

gz

vpgz

p

Calong0)d( rvvr

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7.1 Inviscid Flow If the constant (total energy per unit mass) is the same for

all streamlines, the path of the integral can be arbitrary, and in the flow domain except inside boundary

layers.

Finally, the governing equations for inviscid, irrotational steady flow are

Since is the vorticity , flows with are called irrotational flows.

0 v

0 v

0 v

vω 0 v

constant2

v2

gzp

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7.1 Inviscid Flow

Note that the velocity and pressure fields are decoupled. Hence, we can solve the velocity field from the continuity and vorticity equations. Then the pressure field is determined by Bernoulli equation.

A velocity potential exists for irrotational flow, such that,

and irrotationality is automatically satisfied.

v

0 v

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7.1 Inviscid Flow

The continuity equation becomes

which is also known as the Laplace equation.

Every potential satisfy this equation. Flows with the existence of potential functions satisfying the Laplace equation are called potential flow.

02

2

2

2

2

22

zyx

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7.1 Inviscid Flow The linearity of the governing equation for the flow fields

implies that different potential flows can be superposed.

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

However, the pressure cannot be superposed due to the nonlinearity in the Bernoulli equation, i.e.

21 vvv 2121 )(

21 ppp

Potential FlowsIntegral Equations Irrotational Flow Flow Potential Conservation of Mass

Laplace Equation

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If restricted to steady two dimensional potential flow, then the governing equations become

E.g. potential flow past a circular cylinder with D/L <<1 is a 2D potential flow near the middle of the cylinder, where both w component and /z0.

7.2 2D Potential Flows

L

D

U

x

y

z

0

y

v

x

uv

0

ky

u

x

vv

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The 2-D velocity potential function gives

and then the continuity equation becomes

The pressure distribution can be determined by the Bernoulli equation,

where p is the dynamic pressure

7.2 2D Potential Flows

xu

yv

02

2

2

22

yx

constant)( 2221 vup

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7.2 2D Potential Flows For 2D potential flows, a stream function (x,y) can also be

defined together with (x,y). In Cartisian coordinates,

where continuity equation is automatically satisfied, and irrotationality leads to the Laplace equation,

Both Laplace equations are satisfied for a 2D potential flow

02

2

2

22

yx

02

2

2

2

yx

02

2

2

2

yx

yu

xv

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7.2 Two-Dimensional Potential Flows For two-dimensional flows, become:

In a Cartesian coordinate system

In a Cylindrical coordinate system

v

y

,x

v,ujiv vu and

r,

rv,ur iiv vu rr and

Taking into account:

xv

yu

,

0

0

yxyxy

v

x

u

y

v

x

u

Continuity equation

Irrotational Flow Approximation

For 2D flows, we can also use the stream function Recall the definition of stream function for planar (x-y)

flows

Since vorticity is zero,

This proves that the Laplace equation holds for the stream function and the velocity potential

Cylindrical coordinate system

In cylindrical coordinates (r , q ,z ) with 0 /-axisymmetric case

Taking into account:

zrv

rru

1

,1

01111

)()(

0)()(

zrzrrzrr

rzrr

vr

vr

z

ur

r

rv

z

rur

rv

z

ru

Continuity equation

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7.2 Two-Dimensional Potential Flows Therefore, there exists a stream function such that in the Cartesian coordinate system and

in the cylindrical coordinate system.

The transformation between the two coordinate systems

r

,r

v,ur

x

,y

v,u

v

u

v

u ,

v

u

v

u rr

cossin

sin-cos

cossin-

sincos

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7.2 Two-Dimensional Potential Flows The potential function and the stream function are

conjugate pair of an analytical function in complex variable analysis. The conditions:

These are the Cauchy-Riemann conditions. The analytical property implies that the constant potential line and the constant streamline are orthogonal, i.e.,

and to imply that .

xyyx

and

u,v- v,u 0

Irrotational Flow Approximation

Irrotational approximation: vorticity is negligibly small

In general, inviscid regions are also irrotational, but there are situations where inviscid flow are rotational, e.g., solid body rotation

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7.3 Simple 2-D Potential Flows

Uniform Flow

Stagnation Flow

Source (Sink)

Free Vortex

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7.3.1 Uniform Flow

)( VU,v

yU

x

xV

y

VyUx VxUy

and

and

For a uniform flow given by , we have

Therefore,

Where the arbitrary integration constants are taken to be zero at the origin.

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7.3.1 Uniform Flow This is a simple uniform flow along a single direction.

22 vuU

Elementary Planar Irrotational FlowsUniform Stream

In Cartesian coordinates

Conversion to cylindrical coordinates can be achieved using the transformation

Proof with Mathematica

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7.3.2 Stagnation Flow For a stagnation flow, . Hence,

Therefore,

2 cos22

222 rB

)yx(B

xBy

y ,

yBx

x

By,Bx v

2sin 2

2rB

Bxy

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The flow an incoming far field flow which is perpendicular to the wall, and then turn its direction near the wall

The origin is the stagnation point of the flow. The velocity is zero there.

7.3.2 Stagnation Flow

x

y

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7.3.3 Source (Sink) Consider a line source at the origin along the z-

direction. The fluid flows radially outward from (or inward toward) the origin. If m denotes the flowrate per unit length, we have (source if m is positive and sink if negative).

Therefore,

mur r 2

ru

r

rr

mu

r r

02

and

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7.3.3 Source (Sink) The integration leads to

Where again the arbitrary integration constants are taken to be zero at .

rm

ln2

2

m

1,0,r

and

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A pure radial flow either away from source or into a sink A +ve m indicates a source, and –ve m indicates a sink The magnitude of the flow decrease as 1/r z direction = into the paper. (change graphics)

7.3.3 Source (Sink)

Elementary Planar Irrotational FlowsLine Source/Sink

Potential and streamfunction are derived by observing that volume flow rate across any circle is

This gives velocity components

Elementary Planar Irrotational FlowsLine Source/Sink

Using definition of (Ur, U)

These can be integrated to give and

Equations are for a source/sinkat the originProof with Mathematica

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7.3.4 Free Vortex Consider the flow circulating around the origin

with a constant circulation . We have: where fluid moves counter clockwise if is positive and clockwise if negative.

Therefore,

ur2

rru

r

ru

r r

20 and

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7.3.4 Free Vortex

2

rln2

1,0,r

and

The integration leads to

where again the arbitrary integration constants are taken to be zero at

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The potential represents a flow swirling around origin with a constant circulation .

The magnitude of the flow decrease as 1/r.

7.3.4 Free Vortex

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7.4. Superposition of 2-D Potential Flows Because the potential and stream functions

satisfy the linear Laplace equation, the superposition of two potential flow is also a potential flow.

From this, it is possible to construct potential flows of more complex geometry.

Source and Sink Doublet Source in Uniform Stream 2-D Rankine Ovals Flows Around a Circular Cylinder

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7.4.1 Source and Sink Consider a source of m at (-a, 0) and a sink of m at (a, 0)

For a point P with polar coordinate of (r, ). If the polar coordinate from (-a,0) to P is and from (a, 0) to P is

Then the stream function and potential function obtained by superposition are given by:

11 ,r

22 ,r

1212 lnln22

rrm

, m

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7.4.1 Source and Sink

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7.4.1 Source and Sink Hence,

Since

We have

12

1212 tantan1

tantantan

2tan

m

22

sin22tan

ar

ar

m

ar

r

ar

r

cos

sintan

cos

sintan 12 and

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7.4.1 Source and Sink We have

By

Therefore,

22

1- sin2tan

2 ar

arm

cos2cossin

cos2cossin22222

1

222222

arararrr

arararrr

cos2

cos2ln

2 22

22

arar

ararm

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7.4.1 Source and Sink The velocity component are:

sin2

sin

sin2

sin

2

cos2

cos

cos2

cos

2

2222

2222

arar

r

arar

rmv

arar

ar

arar

armu

Elementary Planar Irrotational FlowsDoublet

A doublet is a combination of a line sink and source of equal magnitude

Source

Sink

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7.4.1 Source and Sink

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7.4.2 Doublet The doublet occurs when a source and a sink

of the same strength are collocated the same location, say at the origin.

This can be obtained by placing a source at (-a,0) and a sink of equal strength at (a,0) and then letting a 0, and m , with ma keeping constant, say 2am=M

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7.4.2 Doublet For source of m at (-a,0) and sink of m at (a,0)

Under these limiting conditions of a0, m , we have

cos2

cos2

cos2lnlim

sin2sin2tanlim

22

22

0a

221-

0a

r

a

arar

arar

r

a

ar

ar

and ar

arm

22

1- sin2tan

2

cos2

cos2ln

2 22

22

arar

ararm

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7.4.2 Doublet Therefore, as a0 and m with 2am=M

The corresponding velocity components are

r

M

2

cosand

r

M

2

sin

22

sin

r

Mu

and 22

cos

r

Mun

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7.4.2 Doublet

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7.4.3 Source in Uniform Stream

Assuming the uniform flow U is in x-direction and the source of m at(0,0), the velocity potential and stream function of the superposed potential flow become:

2sin

2

ln2

cosln2

mUr

mUy

rm

Urrm

Ux

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7.4.3 Source in Uniform Stream

2m

ψ 2m

ψ

0ψ

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7.4.3 Source in Uniform Stream The velocity components are:

A stagnation point occurs at

Therefore, the streamline passing through the

stagnation point when .

The maximum height of the curve is

sin2

cos Ur

ur

mU

rur

and

U

mrs

2

and

Urm

ss 2

2

ms

and as r U

mrh 0

2sin

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7.4.3 Source in Uniform Stream

For underground flows in an aquifer of constant thickness, the flow through porous media are potential flows.

An injection well at the origin than act as a point source and the underground flow can be regarded as a uniform flow.

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7.4.4 2-D Rankine Ovals The 2D Rankine ovals are the results of the

superposition of equal strength sink and source at x=a and –a with a uniform flow in x-direction.

Hence,

12

12

2sin

lnln2

cos

mUr

rrm

Ur

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7.4.4 2-D Rankine Ovals Equivalently,

22

1

22

22

sin2tan

2sin

cos2

cos2ln

2cos

ar

armUr

arar

ararmUr

sin2

sin

sin2

sin

2

cos2

cos

cos2

cos

2

2222

2222

arar

r

arar

rmv

arar

ar

arar

armu

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7.4.4 2-D Rankine Ovals The stagnation points occur at

where with corresponding .

0

12

1

2

1

2

s

ss

y

Ua

m

a

xa

U

max

i.e., ,

0v 0s

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7.4.4 2-D Rankine Ovals The maximum height of the Rankine oval is

located at when ,i.e.,

which can only be solved numerically.

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,r 0 s

a

r

m

Ua

a

r

a

r

ar

armUr

o0

2tan1

2

1

02

tan2

2

0

220

010

or

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7.4.4 2-D Rankine Ovals

rs

ro

ro

rs

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7.4.5 Flows Around a Circular Cylinder

Steady Cylinder

Rotating Cylinder

Lift Force

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7.4.5.1 Steady Cylinder Flow around a steady circular cylinder is the

limiting case of a Rankine oval when a0.

This becomes the superposition of a uniform parallel flow with a doublet in x-direction.

Under this limit and with M=2a. m=constant,

is the radius of the cylinder.

2

1

0 2

U

Mrr s

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7.4.5.1 Steady Cylinder The stream function and velocity potential

become:

The radial and circumferential velocities are:

sin12

sinsin

cos12

coscos

2

2

2

2

r

rUr

r

MUr

r

rUr

r

MUr

o

o

and

sin1cos12

20

2

20

r

rUU

r

rUU r and

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7.4.5.1 Steady Cylinder

ro

Examples of Irrotational Flows Formed by Superposition

Flow over a circular cylinder: Free stream + doublet

Assume body is = 0 (r = a) K = Va2

Examples of Irrotational Flows Formed by Superposition Velocity field can be found

by differentiating streamfunction

On the cylinder surface (r=a)

Normal velocity (Ur) is zero, Tangential velocity (U) is non-zero slip condition.

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Rotating Cylinder The potential flows for a rotating cylinder is the free vortex

flow. Therefore, the potential flow of a uniform parallel flow past a rotating cylinder at high Reynolds number is the superposition of a uniform parallel flow, a doublet and free vortex.

Hence, the stream function and the velocity potential are given by

rr

rUr

r

rUr

ln2

sin1

2cos1

2

20

2

20

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Rotating Cylinder The radial and circumferential velocities are given by

rr

rU

ru

2sin1

2

20

cos1

2

20

r

rU

rur

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Rotating Cylinder

The stagnation points occur at

From

0 ssr uu

0cos

0cos12

20

s

os

s

rr

r

rU

:B Case

:A Case

0sru

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Rotating Cylinder

2

12

22

41

4sin

14

02

sin2

Urryrx

Urry

Ur

rU u

rr

oosos

osos

o

oss

os

:A Case

when exits only Solution

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Rotating Cylinder

sr real positivefor implies which

with sign :B Case

Ur

UrUrr

r

rr

rU u

s

ss

ss

1

04

2

12

000

2

20

144

02

10

1sin0cos

68

Rotating Cylinder

The stagnation points occur at

Case 1:

Case 2:

Case 3:

14 0

Ur

14 0

Ur

14 0

Ur

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Rotating Cylinder

Case 1:

14 0

Ur

2

12

0000 41

4

Urr

x

Urr

y ss

and

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Rotating Cylinder Case 2:

The two stagnation points merge to one at cylinder surface where .

14 0

Ur

00 r,y,x ss

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Rotating Cylinder Case 3:

The stagnation point occurs outside the cylinder

when where . The condition of

leads to

Therefore, as , we have

14 0

Ur

ss ry 0u2

2

12

0000

144

UrUrr

r

r

y ss

14 0

Ur

1

2 00

Urr

ys

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Rotating Cylinder

Case 3: 14 0

Ur

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Lift Force

The force per unit length of cylinder due to pressure on the cylinder surface can be obtained by integrating the surface pressure around the cylinder.

The tangential velocity along the cylinder surface is obtained by letting r=ro,

0

0 2sin2

0r

Ur

urr

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Lift Force The surface pressure as obtained from Bernoulli

equation is

where is the pressure at far away from the cylinder.

0p

22

2sin2

00

2

2

Up

rU

p

p

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Lift Force

Hence,

The force due to pressure in x and y directions are then obtained by

22

02

2

0

20 4

sin2

sin412 UrUr

Up p

2

drp drp dpFFCCyx ]sincos[ 00000 jiji sF

UdrpF drpF yx 0

2

0 00

2

0 0 sin0cos and

ji drd o sincos swhere

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Lift Force The lift on rotating bodies is called the Magnus effect. the lift force is due to the circulation around the body.

An airfoil without rotation can develop a circulation around the airfoil when Kutta condition is satisfied at the rear tip of the airfoil.

Therefore, The tangential velocity along the cylinder surface is obtained by letting r=ro:

This forms the base of aerodynamic theory of airplane.