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3A1 Incompressible Flow

INVISCID FLOW

T.P. Hynes 2010

(Based on an earlier set of notes by A.P. Dowling and T.B. Nickels)

OVERVIEW

This course deals with incompressible, inviscid flows. At first sight these may seem severe restrictions. But that is not so. You may see from the 3A3 compressible flow lectures, that compressibility can be neglected in a steady flow whenever the Mach number is low (specifically whenever M^2 << 1 ). Of course, all real fluids are viscous. However, in high-Reynolds number flows, viscous effects are confined to within thin boundary layers near solid surfaces, to wakes and to mixing regions. When there is an adverse pressure gradient, separation may occur making the wake wide (see Figure 1b). Outside boundary layers, wakes and mixing regions, the flow can reasonably be considered to be inviscid.

The equations of motion have a particularly simple form in an incompressible, inviscid flow. They can be solved by straightforward analytical or computational techniques. These techniques are described in the course and are used to build up physical understanding of fluid flow through a range of practical examples.

To obtain the complete flow field, the outer inviscid solution must be 'patched' onto a boundary layer near the surface. This boundary layer flow is considered in the second part of 3A1. However, much of the practically relevant information can be deduced without analysing the boundary layer in detail. Since pressure does not vary significantly across a boundary layer, the analysis of the outer inviscid flow gives the pressure distribution over a surface. There are many practical reasons for interest in this surface pressure distribution. For example, we might wish to compare the lift produced by wings of different shapes. Alternatively, since an adverse surface pressure gradient can lead to separation in a high Reynolds number flow, a pressure distribution predicted from an inviscid solution will also give a warning of when separation is likely to occur. Of course, there are some things that cannot be determined from a solution of the inviscid equations of motion. For example, it cannot predict the skin-friction drag on a body, which depends crucially on viscosity and the matching to a boundary layer.

Boundary layer

Wake Equal pressures

Inviscid external flow

Figure 1a A streamlined body The boundary layers and wake are thin and inviscid theory will give excellent results for the flow outside these regions and for the pressure distribution over the body.

ASSUMED KNOWN FROM PART IB

Fluid Mechanics

Definition of Reynolds number.

In a high Reynolds number flow:

boundary layers are thin with pressure constant across the boundary layer, i.e. with negligible change in pressure perpendicular to the streamlines

the flow may be treated as inviscid outside boundary layers, wakes and mixing regions

an adverse pressure gradient may cause the flow to separate (adverse means pressure increasing in the direction of flow)

For a steady, inviscid incompressible flow, Bernoulli's equation gives (^12) 2

p V gz ρ

• • = constant along a streamline

Momentum integral equation (steady flow) for fluid within control volume V bounded by surface S

S

u ρ u .dS = F

where F is the total force acting on the fluid in V.

Vector Calculus

The total differential df of f ( x, y ) is

y (^) x

f f df dx dy x y

Similarly the total differential of a function g ( x, y,z,t ) is

y ,z ,t (^) x,z ,t x,y ,t x,y,z

g g g g dg dx dy dz dt x y z t

Definition of ∇ φ , ∇. u, ∇ × u, ∇^2 φ.

If curl u = 0 everywhere, u is called irrotational and has a scalar potential u = ∇ φ

Stokes Theorem: C S

^ u dl.^^ =
^ ω. dS where^ ω^ = ∇ ×^ u.

1. EQUATIONS OF MOTION IN AN INVISCID,

INCOMPRESSIBLE FLUID

1.1 Conservation of mass

Since mass is conserved, the rate of increase of mass within a volume is equal to the net rate at which mass crosses its bounding surface. Consider a small volume, with sides of

length δ x , δ y and δ z. We will denote the fluid density by ρ and the fluid velocity by

u = ( u,v,w ).

z

δ z

δ x

x

ρ u ρ u+ δ(ρ u)

δ y

Then rate of increase of mass within the volume = x y z t

Rate at which fluid crosses the shaded surface = ρ u δ y δ z.

Hence net rate at which fluid flows out across the bounding surfaces

( u ) y z ( v ) x z ( w ) x y x y z t

ρ δ ρ δ δ δ ρ δ δ δ ρ δ δ δ δ δ

Dividing by δ x δ y δ z

( u ) ( v ) ( w ) x y z t

In the limit δ x, δ y, δ z → 0 , δ ( ρ u ) / δ x tends to the derivative ∂ ( ρ u ) / ∂ x , etc. and

hence

( u ) ( v ) ( w ) x y z t

Throughout the rest of this course we will assume that the fluid is homogeneous and

incompressible. Then its density ρ is constant, and the equation of mass conservation

simplifies to

u v w x y z

i.e. ∇. u = 0 : the continuity equation (1.1)

Or in vector form Du u u u Dt t

Question 1.

An incompressible, inviscid fluid flows steadily past a sphere of radius a. The fluid

velocity along the streamline A − B is given by u = u x i ( ) = U ( 1 + a^3 x^3 ) i , where U

is the upstream velocity far ahead of the sphere.

x

y

a A B

Determine the acceleration experienced by fluid particles as they flow from A to B and sketch the variation of this acceleration with x. What is the maximum deceleration of the fluid particles as they flow along this line and where does it occur?

Answer 0 610_. U_^2 / a at x = −1.205 a.

Fairly large accelerations often occur in fluid flows. If we apply Question 1 to a cricket ball of radius a = 40 mm with a velocity U = 100 mph ≈ 45 ms−^1. Then the maximum acceleration of a fluid particle along the streamline in front of the ball is 0.610 × (45)^2 / (40 × 10 −^3 ) ≈ 31 km s−^2. This is an acceleration of roughly 3000 times that of gravity. In many situations the accelerations experienced by fluid particles are very large and these are obviously accompanied by large forces. It is therefore not surprising that the force due to gravity is often negligible in comparison with the aerodynamic forces!

1.2.2 Differential form for the momentum equation

Newton's law of motion applied to the fluid particles occupying a volume V , says "mass × acceleration = applied force", i.e.

V

Du dV F Dt

^ ρ^ =

fluid volume V

F

The forces acting on an inviscid fluid have a particularly simple form - they are just those due to gravity and pressure. Applying Newton's law to the inviscid fluid currently

occupying a small box with sides of length δ x , δ y and δ z , we obtain from the balance of

forces in the z -direction.

z

δ z

δ x

δ y x

mg

p

p+ δ p

y

z^ (^ )

Dw x y z F p x y p p x y g x y z Dt

Divide by δ x δ y δ z

Dw p g Dt z

In the limit δ z → 0 , δ p / δ z tends to the partial derivative ∂ p / ∂ z and

Dw p g Dt z

Similarly, the x -component of momentum balance is

Du p Dt x

while the y -component is

Dv p Dt y

In vector form

Du p g Dt

Equations (1.1) and (1.3) supply one scalar and one vector equation and are formally sufficient (with appropriate boundary conditions) to determine the unknowns p (scalar) and u (vector). However, due to the nonlinear term u. ∇ u in the particle acceleration, the

momentum balance equation (1.3) is hard to solve in its current form. We will now go on to see how, in many cases, a much simpler equation can be derived from it.

Upstream irrotational flow

n

Outside the boundary layer inviscid, irrotational flow

u = ∇ φ , ∇^2 φ = 0

n. u = 0

Note: Not to scale − boundary layer thickness greatly exaggerated

Inside boundary layer u increases from 0 to inviscid flow value

1.3.3 Bernoulli's equation for inviscid, incompressible, irrotational flow

In an irrotational flow, the momentum equation (1.3) has a simpler form. When

∇ × u = 0 , 2

u. u V

, where V = | u |. (This result is proved in Appendix A).

Writing u = ∇ φ the momentum equation becomes

V p gz t

Rearranging

(^1 ) 2

p V gz t

i.e.

(^12) 2

p V gz t

= constant throughout the entire flow (1.7)

In Part IB, Bernoulli's equation was derived for a steady inviscid, incompressible flow.

Then 2

p V gz

• • was found to be a constant along a streamline. When the flow is

also irrotational, it is equal to the same constant throughout the entire flow field. It is also possible to extend Bernoulli's equation to unsteady flows by including the additional term

∂ φ ∂ t as shown in equation (1.7).

Hence, when the upstream flow is irrotational the rather cumbersome momentum equation (1.3) simplifies to say:

(a) that the flow remains irrotational throughout the inviscid regions of the flow (see Section 1.3.1) and

(b) that in these regions the pressure is related to the velocity and its potential through the form of Bernoulli's equation in (1.7).

Summary of Section 1

You should know:

1. when and where it will be a reasonable approximation to treat a flow as (a) inviscid and (b) incompressible,
2. that the equation of mass conservation in an incompressible fluid is ∇. u = 0 ,
3. that the momentum equation in an inviscid fluid is

Du p g Dt

4. that an inviscid, incompressible fluid flow that is initially irrotational remains irrotational,
5. that in an irrotational, incompressible fluid flow

(a) u = ∇ φ with ∇^2 φ = 0

(b) 2

p V gz t

constant throughout the entire flow.