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3 THREE-DIMENSIONAL FLOW
In this section we consider three-dimensional irrotational flows, in which the fluid velocity ( u , v , w ) possibly depends on all three spatial co-ordinates ( x , y , z ). In such a case, the streamfunction that was so useful for two-dimensional flows no longer exists, since the definition of the streamfunction in equation (2.1) requires the flow to be independent of one of the co-ordinates. This means that we can no longer use the idea of a complex potential. However, elementary sources continue to be useful. Indeed the panel method can be extended to three-dimensional bodies by using a distribution of point sources or doublets on the body surface instead of line vortices.
3.1 SIMPLE FLOWS
3.1.1 Point sources and sinks
The most elementary three-dimensional flow field is that due to the injection of fluid at a point. Suppose that fluid is injected at the origin at a volume flow rate m. There is no preferred direction and so, from symmetry, the resulting flow must be radially outwards. It is clearly a good idea to use spherical polar co-ordinates to analyse this flow.
Consider a spherical control surface of radius r , centred on the origin. Since the fluid is incompressible, the rate at which fluid crosses this control surface must be equal to the rate of volume injection:
ur π r = m
i.e. (^2) 4
r
m u π r
m > 0 is a source, m < 0 is a sink.
The velocity potential comes from integrating
r 4 2
m u r (^) r
∂φ ∂ π
to obtain
m r
Just as a superposition of line sources lead to the solution for flow about certain cylindrical bodies , so the superposition of point sources produces the flow field around certain bodies of revolution.
r
m
control surface: radius r surface area 4 π r^2
Question 22
A Rankine half-body of revolution is produced by a point source of strength m and a uniform flow U.
θ
r D far downstream
U S
Stagnation point
Find
(i) the distance from the source to the stagnation point.
(ii) D , the body diameter far downstream ( Hint : use continuity for the region bounded by the body streamline.)
(iii) the distance upstream of the stagnation point at which the body changes the flow velocity by less than 5%.
Express your answer in terms of D and compare it with the corresponding result for a two- dimensional half-body in section 2.2.
3.1.2 Doublet
Having determined the velocity potential for a point source, we can differentiate it with respect to x to obtain a new solution of Laplace's equation. The reason why this works is discussed in section 2.2.3.
Differentiating − m / 4 π r with respect to x , we obtain the new solution
( ) (^) ( )
( )
1 2
3 2
2 2 2
2 2 2
m m x x y z x r x mx
x y z
φ π π
π
∂ − ∂^ −
This is the velocity potential of a point doublet. To avoid confusion with the notation for the source, we will relabel m as μ. When rewritten in terms of spherical polar co-ordinates, with x = r cos θ , φ becomes
r
θ
μ φ
Spherical polar co-ordinates
x
N.B. φ is an angle – not a potential
Check φ not ψ
This is the solution of Laplace's equation which satisfies the correct boundary condition on the sphere. However, our experience, with the corresponding solution for the flow around a cylinder in Section 2.5, warns us to treat it with some caution! Equation (3.4) predicts a tangential velocity 3 s (^) r a 2
U = − u θ = = U sin θ (3.5)
on the sphere's surface. This says that the fluid velocity just outside the boundary layer decreases in the flow direction as the fluid flows over the downstream side of the sphere. This would be associated with a strong adverse pressure and the flow would separate. It is not possible to patch a boundary layer onto this 'outer' solution over the rear of the sphere. Separation leads to a wide wake, which displaces the outer streamlines. Consequently, the fluid velocities predicted by equation (3.4) are only a reasonable approximation to the real flow upstream and over the front of the sphere.
[Note that the fluid velocities specified in Question 1, on the acceleration of fluid particles due to a cricket ball, follow directly from equation (3.4) with θ = π.]
Question 23
Estimate the form drag on a hemisphere in a uniform flow, by assuming that the flow on the upstream side of the hemisphere is given by inviscid sphere theory and that the pressure in the rear equals the shoulder pressure.
What are the defects of this analysis?
Summary of Section 3. Simple 3-D flow fields
2
2 3 3
Source at origin 0 0 (^4 )
Doublet at origin is angle from ,^ ,^0 4 2 4 doublet axis
r
r
m m u , u , u r (^) r
cos cos sin u u u r r r
θ φ
θ φ
φ
π π
μ θ μ θ μ θ θ π π π
u
Hemisphere
rear pressure assumed equal to shoulder pressure
a
U ∞ p ∞
3.2 VORTICITY IN THREE-DIMENSIONAL FLOWS
In Section 2.4.4, we introduced the concept of a line vortex, in which vorticity is concentrated on an infinity long straight line. We found that the development of the line vortex with time has a particularly simple form in an inviscid two-dimensional flow and just convects with the local fluid velocity.
Even vorticity in three-dimensional flows satisfies conservation principles, which enables it to be followed as it evolves in time. In particular, vorticity is not created, so that if the vorticity is initially only non zero in a small region of the flow, it remains localised. Tracking the vortex motion is a useful tool to aid the understanding of complex fluid flows.
3.2.1. Kelvin's circulation theorem
The most important conservation principle associated with vorticity is Kelvin's circulation
theorem. This gives a condition to be satisfied by the circulation Γ = C
^ u dl. around any
closed curve C moving with the fluid:
Kelvin's circulation theorem, in an inviscid flow with uniform density, the circulation around a closed loop which moves with the fluid is constant.
Stokes theorem shows that Γ is directly related to the flux of vorticity ω = ∇ × u through any surface S bounded by the curve C.
C S
Γ = u .dl = ω .dS
An alternative statement of Kelvin's theorem is therefore that the flux of the vorticity, through any surface moving with the fluid, remains constant.
To prove Kelvin's circulation theorem, we need to evaluate the material derivative DΓ / D t.
C C C
D D Du^ D dl u dl dl u Dt Dt Dt Dt
The momentum equation (1.3) enables us to relate the particle acceleration in the first integral on the right-side of equation (3.6) to the fluid forces. Provided that the flow on the curve C is inviscid, we can rewrite
C C
Du dl p g dl Dt
ρ ρ
This integral is zero because the forces due to pressure and gravity are conservative and no net work is done against them in moving around a closed curve. In detail
C
dS
ω
This integrates to the difference between
u at the end points of the path of integration and is
zero for a closed curve.
Returning to equation (3.6), we have shown that both terms on the right-hand side are zero, thereby proving Kelvin's circulation theorem:
D
Dt
Note that this proof of Kelvin's circulation theorem only requires that the flow is inviscid on the curve C (see equation (3.7)). It remains true even when there are viscous forces within (but not on) C.
There are several immediate consequences of this theorem. We will introduce them by reviewing and re-deriving some results we obtained earlier by different methods.
(i) An inviscid flow that is initially irrotational remains irrotational.
In Section 1.3.1, we used conservation of angular momentum and the fact that there are no shearing stresses in an inviscid fluid to derive this result, but it follows immediately from Kelvin's circulation theorem.
Suppose that at some time the vorticity is non zero somewhere in the fluid. Then, by virtue of
Stokes theorem ( )
C S
Γ = u .dl = ω .dS it would be possible to select some closed curve around
which the circulation was non zero.
But this would contradict Kelvin's theorem, because the circulation around this fluid curve must initially have been zero when the vorticity was zero. Hence, the assumption that the vorticity is non zero at a later time is wrong.
Γ ω^ non-zero implies^ Γ^ non-zero, contradicting Kelvin’s circulation theorem
(ii) The starting vortex
We mentioned in Section 2.7 that a starting vortex is shed when a lifting aerofoil accelerates from rest (see in particular Figure 4). We now see why this has to be so, and determine the strength of the starting vortex.
a
b
c
d (^) e
Figure 10. A lifting aerofoil and its starting vortex
Consider the aerofoil at a time t after the start of its motion as illustrated in Figure 10. Viscous
forces are confined to the boundary layers on the aerofoil and to its wake, which ends at the
starting vortex. The regions influenced by viscosity are shaded in Figure 10. Now, the fluid
curve abcd is large enough to have been clear of all these regions since the start of the motion
and we can apply Kelvin's circulation theorem to it. As the fluid and the aerofoil were initially at
rest, the circulation around abcd must be zero at the subsequent time t. The circulation around
the curve aecd must therefore be equal and opposite to the circulation around abce. The
circulation around the aerofoil is equal and opposite to the strength of the starting vortex.
iii) 0
D
Dt
= in a two-dimensional inviscid flow
We derived this result from the momentum equation in Section 2.4.4 (see equation (2.17)). But it also follows directly from Kelvin's circulation theorem.
In a flow with no velocity in the z -direction, mass conservation ensures that the area of any fluid surface in the x-y plane is unchanged as the surface deforms in time.
Kelvin's circulation theorem, applied to a small curve C encircling an area dS, can be written in the form
D
dS Dt
Since ( ) 0
D
dS Dt
= this simplifies to
D
Dt
In a two-dimensional inviscid flow, vorticity convects unchanged with the local fluid velocity. This leads to a relative (not absolute) reduction in shear in an accelerating flow.
dS
C
In a general three-dimensional flow, the vorticity changes not only due to convection, but also due to the effects of stretching and tilting. These are most easily accounted for by investigating the motion of a vortex line.
3.2.2. Vortex lines and the Helmholtz laws of vortex motion
A vortex line is a curve which at any particular time t is parallel to the local vorticity (just as a streamline is parallel to the local fluid velocity). The vortex lines which pass through a simple closed curve C form the boundary of a vortex tube.
The Helmholtz laws state that, for an inviscid fluid flow with uniform density,
vortex lines move with the fluid
S Γ = (^) ω .dS is the same for all cross-sections S of a vortex tube, and is independent of
time.
To prove (1)
Consider a material surface which at time t= 0 coincides with the surface of a vortex tube. Initially, the vorticity is everywhere tangential to this surface. We will show that as time proceeds and the surface moves with the fluid particles, the vorticity remains tangential.
Consider a closed curve C 1 on the surface enclosing an element of area S 1. The circulation around C 1 is zero, because
C 1 (^) S 1 ^ u .dl^ =^ ω .dS
since ω and dS are perpendicular.
As time proceeds, C 1 moves with the fluid particles and so, by Kelvin's circulation theorem, it still has zero circulation. Since the same argument can be applied to any curve on the surface we
must conclude that ω. dS is still zero. Hence a material surface
that initially coincides with the surface of a vortex tube remains tangential to the vorticity at all subsequent times. That is it remains a vortex tube and we have shown that vortex tubes move with the fluid. Since vortex lines are completely surrounded by vortex tubes, they too must move with the fluid. Thus completing the proof of (1).
C
Figure 12: Curve C on the surface of a vortex tube
Curve C 1 on the surface of the vortex tube
C
C 1
To prove (2)
Note that since ω = ∇ × u , ∇. ω = 0 ( ∇. ∇ × anything = 0 see Maths Data book). The flux
of ω is therefore conserved, i.e.
S Γ = (^) ω .dS is the same for all cross-sections S of a vortex tube.
Kelvin's circulation theorem says that Γ is independent of time.
There are two immediate consequences of this. Firstly it means that a vorticity tube cannot end inside a fluid. It must either be closed or end at a boundary. Vortex rings are examples of closed vortex tubes. Secondly, when the cross-sectional area of a vortex tube decreases, the constraint
that S ω .dS is constant means that^ ω^ increases.^ This is commonly referred to as vortex
stretching.
3.2.3. Vortex stretching
final
initial
swirl
flow
E
E
Figure 13 Swirling flow in a converging pipe
The streamlines converge as the fluid flows through the pipe contraction shown in Figure 13. This leads to a reduction in the cross-sectional area dS of the plug of fluid E, and a corresponding
increase in the axial vorticity, so that ω. dS is constant. This can be interpreted as a
consequence of conservation of angular momentum, since the angular velocity must increase when the radius of gyration decreases.
Since the volume of this incompressible fluid element is constant, the plug's length l is related to
its cross-sectional area by l dS = constant. Combining this with ω dS = constant, we find that
vorticity l
is independent of time (3.9)
where l is the length of a small fluid line element in the direction of the vorticity.
When this fluid element is stretched, the vorticity becomes more intense. High angular velocities can be achieved in bath-plug vortices and tornadoes by this mechanism. Alternatively, vortex shortening can be used to reduce vorticity and, for example, wind-tunnels have large area contractions upstream of their working sections to reduce spanwise vorticity.
vortex lines. They distort and are stretched as they convect past the cylinder. This stretching intensifies the vorticity and a strong "horse-shoe" shaped vortex is produced. The vorticity is particularly intense near the stagnation line on the cylinder. In the structural analysis of buildings, it is important to know the wind force induced by these vortices. When the cylinder represents a pillar or bridge pier in a river, the intense velocities associated with the horse-shoe vortex can erode the river bed and undermine the foundations.
3.2.5 Secondary flow around bends
When a flow changes direction, the vortex lines also change their orientation and this can lead to the phenomenon of secondary flow. Consider fluid flowing around the bend in a pipe shown in Figure 16. At entry the fluid near the centre of the pipe will be travelling the fastest, so that the fluid velocity will be greater at A than at B. Hence, there is a velocity gradient from B to A,
resulting in the azimuthal vorticity ω as shown. As the vorticity line CD is convected around the
bend to C' D' say, D' moves ahead of C'. This occurs for two reasons: the route around the bend on the inner arc DD' is shorter and secondly in the bend D travels faster than C because curved streamline effects make the pressure lower at D' lower than at C' and hence, from Bernoulli's equation, the speed is higher. The result is that the vortex line element C' D' now has a component along the pipe causing the fluid to swirl helically. The fluid has positive circulation about the mean direction of flow as sketched in Figure 16. The opposite effect occurs in the top half of the pipe, with the result that the flow at outlet contains two opposite swirling motions as shown in the figure.
Vorticity in the direction of the mean stream is called secondary vorticity , and the associated swirling flow which is to be superimposed on the mean (or primary) flow is called secondary flow. Secondary flow effects may persist as far 75 pipe diameters downstream of a bend. The spiralling of the fluid increases the velocities near the pipe walls and so enhances wall friction and heat transfer effects.
Figure 15 Distortion of vortex lines by a cylinder in a stream
Stagnation line
Successive positions of vortex line
Downward flow in lee
boundary layer vortex line
outlet
entry
u
A
D
B
C
D′
C′
u
Summary of Section 3.
Kelvin's circulation theorem 0
D
Dt
Helmholtz laws of vortex motion
(1) vortex lines move with the fluid.
(2) S Γ = (^) ω .dS is the same for all cross-sections of a vortex tube and is independent
of time.
Fluid motion causes
(1) Stretching of vortex lines which intensifies the local vorticity. The vorticity
w varies in proportion to the length l of a small section of the vortex line.
(2) Tilting of vortex lines which can lead to secondary flows with streamwise
vorticity.
Vortex dynamics can explain:
(1) the alteration in flow across changes in cross-sectional area, (2) horse-shoe vortices formed upstream of obstacles, (3) secondary flow in bends.
