Ideal Flow

1. Ideal Fluids

An ideal fluid is one which is incompressible, and has zero viscosity.

Though no real fluid satisfies these criteria, there are situations in which viscosity of gases and liquids, and compressibility effects in gases, have little effect, and the theory of the ideal fluid can give an accurate prediction of the real flow.

For example, for a real fluid flowing past and around a stationary object, ideal theory works well outside the boundary layer.

2. Steady Two-Dimensional Flow

In this unit we shall consider only steady, two-dimensional flow, in which:

• the fluid velocity at any point remains constant with time;

• the direction and magnitude of the fluid velocity will in general vary in the x- and y-directions, but not in the z-direction.

3. Streamlines, Pathlines and Streaklines

A streamline is a line in the flow such that the velocity of each particle on the line is tangential to the line.

A pathline is the path traced out by one particle of the fluid.

A streakline joins all the particles which have passed a particular fixed point in the flow.

For steady, ideal flow, all the above are the same, and we need use only the term streamline.

4. The Stream Function ψ (“psi”)

The stream function is defined so that it has a constant value along a streamline.

To one (arbitrarily chosen) streamline we assign the value ψ = 0.

If the velocity at a point on a streamline is w m/s, then

δψ ≈ w δnwhere δψ is the change in stream function between two adjacent streamlines, and δn is the normal distance between them.

ψ

ψ + δψ

w

δn

nδδψw ≈

/s)(m/functionstreaminincrease=sstreamline adjacent between(m) / distancenormal=

(m/s)/velocityfluid=

2δψnδ

w

Note the units of δψ : m2/s.

δψ is numerically equal to the discharge, δQ in m3/s, across an area measuring δn × 1m (perpendicular to δn

and w.)

nδwδψ ≈

In the limit, as δn → 0, wndψd

nδδψ =→

If point 1 is any point on the ψ = 0 streamline, and point 2 is any point on a streamline where the stream function is ψ ( ≠ 0), then:

∫2

1= dnwψ

(see next slide)

w

dn

∫2

1= dnwψ

0=ψ

1

2

For a few special cases of flow geometry, it is possible to find a path 1 → 2 between the 0-streamline and the ψ-streamline which satisfies the following conditions:

• the path is perpendicular to the flow velocity at every point

• the flow velocity is the same at every point in the path

In these special cases,

nwdnwψ ×==2

1∫where n is the distance measured along the path 1 → 2, and w is the constant velocity perpendicular to the path.

This is illustrated in the following examples.

4.1 Uniform flow with velocity u in the x-direction

ψ = u y

(Note the sign convention we are using:

Looking in the direction of n, if the flow is to the right then ψ increases.)

w = un = y

ψ = w n = u y

ψ = 0

4.2 Uniform flow with velocity vin the y-direction

w =

v

n = -x

Ψ =

0

ψ = -v xΨ

= v(-x

)

4.3 Uniform flow with velocity wmaking angle α with the x-axis

w

n

ψ = w n

ψ = 0

α

...... can be considered as the result of combining two flows: (see next slide)

uniform flow with velocity u = w cos α in the x-direction

PLUS

uniform flow with velocity v = w sin α in the y-direction

w

αu = w cos α v

= w

sin

α

v =

w sin

α

n = -x

ψ =

0

ψ =

v(-x

)

u = w cos αn

= y

ψ = u y

ψ = 0

wn

ψ = w n

ψ = 0

α

+

= ψ = w n

ψ = u y – v x

4.4 Line Source

Flow is radially outwards. The “line” is at right angles to the plane of flow, so is seen as a point in the diagram.

The strength of the source, m, is the discharge in m3/s per m length of the line source: ie the units of m are m2/s, the same as ψ.

r w = m/(2πr)

⇒ w = m/(2πr)

discharge per unit width = m

area per unit width = 2πr

velocity = (discharge)/(area)

r

n = r θ

θw = m/(2πr)

ψ = 0

ψ = w

n

For a line source of strength m:

ψ = w n⇒ ψ = [m/(2πr)]x(r θ)

⇒ ψ = m θ /(2π)

4.5 Uniform Flow + Line Source

At any point in the field, ψ = u y + m θ /(2π)

ψ = 0

ψ = u y

ψ = 0

ψ =

m θ/

(2π)

θ

Note:

in the equation ψ = u y + m θ /(2π), θ = tan -1 (y/x)

x

θ

Y

Lines of constant ψ = u y + m θ /(2π) look like this:

flow from source

… so these streamlines represent the combination of uniform parallel flow with flow from a line source.

unif

orm

par

alle

l flo

w

We can identify the stagnation point where the two flows cancel, and the stagnation or dividing streamlines which pass through this point.

stagnation point

dividing streamlines

Outside the dividing streamlines, this is a good model of flow meeting the front of a rounded body, shaped like the two dividing streamlines in the right hand half of the picture

*W J M Rankine 1820-1872: professor of Engineering, University of Glasgow, from 1855

This shape is called a “Rankine* Half-Body”.

4.6 Line Sink

Flow is radially inwards. A line sink is the opposite of a line source!

For a line sink of strength m: ψ = -m θ /(2π)

4.7 Source and Sink (of equal strength)The diagram shows a source and a sink of equal strength m, placed on the x-axis, a distance 2b apart.

At point P, ψP = m θ1 /(2π) - m θ2 /(2π) where θ1 =tan-1(y/(x+b)) and θ2 =tan-1(y/(x-b))

b b

θ1θ2

x

x+bx-b

P

y

Lines of constant ψ = m θ1 /(2π) - m θ2 /(2π)

look like this:

… so these streamlines represent flow from a line source to a line sink.

sourcesink

4.8 Source and Sink (of equal strength) combined with Uniform Flow

At any point, ψP = u y + m θ1 /(2π) - m θ2 /(2π) where θ1 =tan-1(y/(x+b)) and θ2 =tan-1(y/(x-b))

b b

θ1θ2

x

x+bx-b

P

y

Outside the dividing streamlines, lines of constant ψ = u y + m θ1 /(2π) - m θ2 /(2π) look like this:

The “dividing streamlines” represent a shape called a “Rankine Oval”, and the streamlines outside represent flow of an ideal fluid round a solid of this shape.

Rankine Oval

4.7 Doublet A source (A) and a sink (B) of equal strength m are moved progressively closer together, at the same time increasing the strength, so that k = mb = constant.

b b

θ1 θ2

x

x+bx-b

P

y

θ

r

BA

As b → 0, both θ1 and θ2 → θ ; and both PA and PB → r.By the sine rule, sin(θ2 – θ1) = sin θ2 × 2b/(PA), so as b → 0, sin(θ2 – θ1) → sin θ × 2b/(PA) = (y/r) ×2b/(r).Since a small angle (in radians) is equal to its sine, this can be written: (θ2 – θ1) = 2by/r 2.Now the stream function for source and sink is given by:

b b

θ1 θ2

x

x+bx-b

P

y

θ

r

θ2 – θ1

BA

ψ = m θ1 /(2π) - m θ2 /(2π)

or ψ = (m /(2π))×(θ1 - θ2)

Hence ψ = (m /(2π))×(-2by/r 2)But b ×m = k, so:ψ = -(k /(2π)) × 2y/r 2,

or: ψ = -(k /π) × y/r 2,

or: ψ = -(k /π) × y/(x 2 + y 2)

A system consisting of a source and sink placed very close together is called a “Doublet”. The equations for stream function for a doublet are summarised below:

ψ = -(k /π) × y/r 2 , or:ψ = -ky /(πr 2) , or, since y = r sin θ and x 2 + y 2 = r 2,ψ = -k sin θ /(π r)

ψ = -k y /(π (x 2 + y 2))

(Remember: the source and the sink are 2b apart. Their strengths are m and –m respectively, and k = b ×m .)

4.8 Uniform Flow + Doublet

The stream function for this combination is given by:

ψ = u y - k y /(π (x 2 + y 2))

and, outside the dividing streamlines, lines of constant ψ look like this:

This time the “dividing streamlines” form a circle”, and the streamlines outside represent flow of an ideal fluid round a cylinder.

4.9 Free Vortex A free vortex is a region of fluid in which particles move in concentric circles, the velocity varying inversely as the radius: w = C/r, where C is a constant with units m2/s.

r1

r2w2 = C/r2

w1 = C/r1

As r → ∞, w → 0.

As r → 0, w → ∞

Vortex flow occurs in nature, eg in tornadoes, whirlpools etc; and also in man-made flows, eg vortices created in the wake behind objects moving in a fluid, flow in the casing of a centrifugal pump etc.

In a free vortex, (normal distance dn )= (increase in radius dr). The velocity varies with r, so we need to integrate to get ψ.

r0 rw =

C/r

w 0 = C

/r 0

ψ = C ln(r/r0)

ψ = 0

dψ = w dr ⇒ ψ

⇒ ψ

∫∫00

r

r

r

rdr

rCdrw ==

0rrC ln=

Note : r0 is the arbitrarily chosen radius at which ψ = 0. If r0 is chosen to be 1 unit,

then ψ rC ln=

Like all the other flows we have considered so far, free vortex flow is described (perhaps unexpectedly) as irrotational. The centre of a small element of fluid moves in a circle round the centre of the vortex, but the element does not rotate about its own centre.

In a given time, points on the inner radius move farther than those on the outer. The four-sided element shown alters in shape, but does not rotate. (Note the orientation of the diagonals in each position.)

4.10 Uniform Flow + Doublet + Free Vortex

The stream function for this combination is given by:

ψ = u y - k y /(π r 2) + C ln rAnd, outside the dividing streamlines, lines of constant ψ look like the figure on the right.

The “dividing streamlines” form a circle, and the streamlines outside represent flow of an ideal fluid round a rotating cylinder.