fluid mechanics and the theory of flight

Download free eBooks at bookboon.com1.1 The continuum hypothesis he irst task is to introduce a suitable, general description of a luid, and then to develop an appropriate mathematical

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R.S Johnson

Fluid Mechanics and the Theory of Flight

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Download free eBooks at bookboon.com

Fluid Mechanics and the Theory of Flight

ISBN 978-87-7681-975-0

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2.2 Helmholtz’s equation (the ‘vorticity’ equation) 42

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4.7 Uniform low past a spinning circle (circular cylinder) 119

4.8 Forces on objects (Blasius’ theorem, 1910) 121

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5 Aerofoil heory 140

Appendix 2: Check-list of basic equations 184

Appendix 3: Derivation of Euler’s equation (which describes an inviscid luid) 186

Appendix 4: Kelvin’s circulation theorem (1869) 189

Appendix 7: MAPLE program for plotting Joukowski aerofoils 193

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Preface

his text is based on lecture courses given by the author, over about 40 years, at Newcastle University, to inal-year applied mathematics students It has been written to provide a typical course that introduces the majority of the relevant ideas, concepts and techniques, rather than a wide-ranging and more general text hus the topics, with their detailed discussion linked to the many carefully worked examples, do not cover as broad a spectrum as might be found in other, more wide-ranging texts on luid mechanics; this is a quite deliberate choice here hus the development follows that of a conventional introductory module on luids, comprising a basic introduction to the main ideas of luid mechanics, culminating in a presentation of complex-variable techniques and classical aerofoil theory (here are many routes that could be followed, based on a general introduction to the fundamentals of the theory of luid mechanics For example, the course could then specialise in viscous low, or turbulence, or hydrodynamic stability, or gas dynamics and supersonic low, or water waves, to mention just a few; we opt for the use of the complex potential to model lows, with special application to simple aerofoil theory.) he material, and its style of presentation, have been selected ater many years of development and experience, resulting in something that works well in the lecture theatre hus, for example, some of the more technical aspects are set aside (but usually discussed in an Appendix)

It is assumed that the readers are familiar with the vector calculus, methods for solving ordinary and partial diferential equations, and complex-variable theory Nevertheless, with this general background, the material should be accessible to mathematicians, physicists and engineers he numerous worked examples are to be used in conjunction with the large number of set exercises – there are over 100 – for which the answers are provided In addition, there are some appendices that contain further relevant material, together with some detailed derivations; a list of brief biographies of the various contributors to the ideas presented here is also provided

Where appropriate, suitable igures and diagrams have been included, in order to aid the understanding – and to see the relevance – of much of the material However, the interested reader is advised to make use of the web, for example, to ind pictures and movies of the various phenomena that we mention

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1 Introduction and Basics

We start with a working deinition: a luid is a material that cannot, in general, withstand any force without change of shape (An exception is the special problem of a uniform – inward – pressure acting on a liquid, which is a luid that cannot be compressed, so there is no change of volume.) his property of a luid should be compared with what happens

to a solid: this can withstand a force, without any appreciable change of shape or volume – until it fractures!

We take this fundamental and deining property as the starting point for a simple classiication of materials, and luids

viscous inviscid viscous inviscid

(real) (m odel/ (real) (model/

under a pressure of 100 atmospheres.)

All conventional luids are viscous; simply observe the various phenomena associated with the stirred motion of a drink

in a cup; e.g ater stirring, the motion eventually comes to a halt; also, during the motion, the particles of luid directly

in contact with the inner surface of the cup are stationary

In this study, we will eventually work, mainly, with a model luid that is incompressible his applies even to air – relevant

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1.1 The continuum hypothesis

he irst task is to introduce a suitable, general description of a luid, and then to develop an appropriate (mathematical) representation of it his involves regarding the body of luid on the large (macroscopic) scale i.e consistent with the familiar observation that luid – air or water, for example – appears to ill completely the region of space that it occupies:

we ignore the existence of molecules and the ‘gaps’ between them (which would constitute a microscopic or molecular model) his crucial idealisation, which regards the luid as continuously distributed throughout a region of space, is called the continuum hypothesis

Now, at every point (particle), we may deine a set of functions that describe the properties of the luid at that point:

In our initial considerations, we shall allow the density to vary, but we will soon revert to the appropriate choice for our incompressible (model) luid: ρ = constant Further, the three functions introduced above are certainly to be continuous

in both x and t for any reasonable representation of a physically realistic low

Note: his description, which deines the properties of the luid at any point, at any time – the most common one in

use – is called the Eulerian description he alternative is to follow a particular point (particle) as it moves in the luid, and then determine how the properties change on this particle; this is the Lagrangian description We shall write more

of these alternatives later

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We are now in a position to introduce two diferent ways of describing the general nature of the motion in a given velocity ield which represents a luid low.

1.2 Streamlines and particle paths

We assume that we are given the velocity ield u x ( , ) t (and how any particular motion is generated or maintained is, for the moment, altogether irrelevant); the existence of a motion is the sole basis for the following descriptions

1.2.1 A streamline is an imaginary line in the luid which everywhere has the velocity vector as its tangent, at any instant

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his set is oten expressed in the symmetric form d x d y d z

u = v = w .

Note that, in 2-space (x, y), we simply have

d d

= = − = − (at ixed t; x ≠ 0, t ≠ 0), and so

d y d x t

y = − x

∫ ∫ i.e t ln y = − ln x + constant.hus y xt = C (an arbitrary constant), and then at t = 1 we have simply xy = C (a family of rectangular hyperbolae; see igure)

Comment: Streamlines cannot cross except, possibly, where u = 0 (deining a stagnation point, where the low is stationary

or stagnant) because, at such points, the direction of the zero vector is not unique

1.2.2 A particle path is the path, x = X ( ) t , followed by a point (particle) as it moves in the luid according to the given velocity vector i.e

d

dt =

X

u;

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this is pure kinematics, determining X ( ) t given u X ( , ) t In component form, we have

d d

x u

t = ,

d d

y v

t = ,

d d

z w

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Note: A steady low is one for which the velocity ield is independent of time, and then the families of streamlines (SLs)

and particle paths (PPs) necessarily coincide (because

in the low his velocity ield is steady

Now the SLs are d d d

e ≡ u , where t is time and n is a

constant Find the streamlines for this low and the particle path which passes through (1,1) at t = 0 For what value of n will the two families of curves coincide ?

We have, for the PPs, d d

y t

n t

x x t

+

=

(1 )(1 )

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∫ ∫ (at ixed t) i.e ln x = y ent + const. or x = C exp ( y ent)

he two families coincide for steady low i.e n = 0

Comment: In the laboratory, it is sometimes convenient to observe streak lines; these are all the paths through a given

point, over an interval of time

1.3 The material (or convective) derivative

Let us consider some (scalar) property of the luid, labelled f ; in our representation of a luid, this will be the pressure,

or the density or a velocity component his will, in general, vary in position and time:

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Warning:

Do not think to write u ⋅∇ as ∇ ⋅u! Remember that ∇ is a diferential operator and so, in the former, it operates on whatever follows the ∇, and this is not u – it is some function e.g f.

Note: If we apply this operator to the velocity vector – which we might expect is the appropriate representation of the

acceleration of a luid particle – then we obtain

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1.4 The equation of mass conservation

A fundamental equation (not usually expressed explicitly in elementary particle mechanics) is a statement of mass conservation We can readily see the need for such an equation: the luid is, in general, in motion and can produce a mixing of regions of diferent densities Yet the total amount (mass) of material is presumably conserved; this total can change only if matter (material) is created or destroyed – and this will arise only if we allow e.g the conversion of mass into energy! We now derive the equation which ensures that mass is indeed conserved

Consider an imaginary (inite) volume V, bounded by a surface S, which is completely occupied by luid; we shall take V (and S) to be stationary in our chosen frame of reference (so that luid will cross S into and out of V) his igure shows the coniguration schematically:

where n is the outward unit normal on S, and ρ x ( , ) t and u x ( , ) t are given at every point in V and on S he total mass of all the luid in V, at any instant in time, is then

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Further, the net rate at which mass lows out of V across S is described in this igure:

∫ represents the double integral over S We now impose the condition that the only mechanism that produces

a change of mass in V is by virtue of material crossing S (into or out of V), thereby excluding the possibility of matter (mass) being created or destroyed at any points in V or on S; thus we require

∂ requires material to enter V across S.

We now invoke the Divergence (Gauss’) heorem for the surface integral (where S bounds V), to produce

V

v t

∂ u is assumed continuous, and so the requirement that the integral of this expression always be zero [see the fundamental idea discussed in Exercise 11] gives

( ) 0

t

∂ + ∇ ⋅ =

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which is usually expressed [see the identities in Exercise 7] as

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A more interesting example, leading to an important, simple result used in elementary calculations for low along a pipe,

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here are two cases of interest: irst, for a viscous luid, both u and w are zero at the inner surface of the pipe (because there can be no low through the pipe, nor along the pipe), and so the evaluation on r = R z ( ) gives zero On the other hand, we might suppose that the luid can be modelled as inviscid (zero viscosity – no friction), in which case the luid

is allowed to low along the inside surface of the pipe (but, as before, not through it) In this case, we must have that the velocity vector is parallel to the pipe wall i.e ( ) ( )

∫ , the required result

In the special case (e.g a model) in which the velocity proile across the pipe is essentially independent of the radius (r), the integral produces the rule: speed×area = constant his type of low is usually referred to as uniform across a section,

as depicted for a real low which is nearly uniform across a section in the igure

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1.5 Pressure and hydrostatic equilibrium

We now introduce the initial ideas that will, eventually, lead to an equation of motion – the corresponding Newton’s Second Law – for a luid he irst stage is to discuss the forces that act on a luid; there are three (although we shall put one of these aside, for the moment):

• force due to pressure (force/area), exerted by the luid particles nearby

• internal friction (viscous forces) due to motion of other particles nearby

• external force (body force) that acts more-or-less equally on all luid particles e.g gravity

he irst two in this list are internal, local forces; in this discussion, we shall ignore any friction (and, in any event, there will be no motion, so friction cannot play any rôle) he pressure,p ( , ) x t , is deined at every point in the luid, and is independent of orientation (the luid is said to be isotropic) Under the action of pressure and a body force – gravity, perhaps – the luid is in equilibrium; we now construct the equation that describes this scenario

As before, let us consider an imaginary volume V, surface S, with outward normal n and totally occupied by luid Let

the body force acting on the luid be F x ( , ) t per unit mass; the pressure (due to the surrounding luid) acts on S

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(Note that the force, as expressed by the let-hand side, is force on.)

Again, we use the Divergence (Gauss’) heorem, to give (for the second term)

S V

p s = ∇ p v

∫ n ∫ (see Exercise 8),and so we obtain ( ) d

this is the equation of hydrostatic equilibrium (because water is a special case!)

Note that the density here, ρ, is not necessarily a constant: we have made no assumptions about ρ or the nature of the luid under discussion

Example 9

Hydrostatic equilibrium Given that the body force is due to (constant) gravity, so that F ≡ ( 0 , 0 , − g ), and that the pressure p = p0 on z = 0, ind p (z ) for an incompressible luid (i.e ρ = constant) in hydrostatic equilibrium

he governing equation is ∇ = F p ρ i.e p , p , p (0, 0, g )

p = p z hen p z ′ ( ) = − ρ g, and so p = p0− ρ gz

Comment: On the basis of the previous example, if z = 0 is the surface of the ocean, then the pressure increases linearly

with depth On the other hand, if z = 0 is the bottom of the atmosphere, then the pressure decreases linearly with height (but this is not a good model for the atmosphere – compressibility is important, with p = p ( ) ρ )

In this model, also note that the rate of increase/decrease is very diferent for water/air, because of the very diferent densities; for example, the pressure drops to about half an atmosphere at a height of about 5 5 ⋅ km in air, but it increases

by one atmosphere at a depth of about 10m in water

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1.6 Euler’s equation of motion (1755)

We now take the representation of forces, as developed in §1.5, and let this be the resultant force acting on a luid that is

in motion (Note that, using this system of forces, there is no internal friction – viscosity – which will be included later;

in the absence of friction, we usually call this model luid an ideal luid.)

he application of Newton’s Second Law, which is required to balance the force against the rate of change of momentum, can be done in a very simple-minded way; this is the option we choose in this presentation A mathematically more complete derivation is given in Appendix 3

Consider a (small) parcel of luid, of volume ∆ V; the force acting on this parcel, based on the details given for the case

ρ∆ u

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hus we obtain the (approximate) equation

which is Euler’s equation of motion (1755) [L Euler (1707-1783), Swiss mathematician, regarded as the ‘father of luids’.]

When the material derivative is written out, this equation becomes

and correspondingly for the other two components

Comment: We observe that we have 4 (scalar) equations (the three components of Euler and the equation of mass

conservation) for the 5 unknowns: u v w p , , , , ρ his system is closed by prescribing the nature of the luid e.g

constant

ρ = (incompressibility) or p = p ( ) ρ (for certain gases)

In addition, we require appropriate boundary conditions (and also initial data for unsteady lows) Typically, we expect information about the velocity and/or pressure at the boundary of the luid

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which is identically satisied, with p = − ρ gz + const.

Another, more physically interesting problem (now in cylindrical coordinates), is provided by the next example

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Example 11

Spinning luid An incompressible luid is rotating at constant angular speed, ω, in a cylindrical vessel; it is otherwise in equilibrium under the action of (constant) gravity Show that the surface (which is at constant atmospheric pressure) takes the form of a paraboloid

In cylindrical coordinates ( , , ) r θ z , we have u = (0, ω r , 0) (see igure), and so Euler’s equation reduces to

An important inal observation, before we move on – and which is explored in Exercise 35 – is the following he governing equations are the same, whether an object is moving at constant speed through a luid, or the luid lows at this same constant speed past a ixed object his implies that the situation in the laboratory – low past an object in a wind tunnel, for example – can correspond precisely with the same object lying through the air his property of the

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Exercises 1

1 Algebra (relevant to gases) Given that t p = ρ RT and that p = k ργ (where R, k and γ are positive constants with

2

1<γ < ), ind: (a) T in terms of ρ; (b) T in terms of p [Here, p is pressure, T is temperature and ρ is density.]

2 More algebra (for gases) Repeat Ex.1 (a), (b), for the more accurate model

5 Diferential equations I Solve the diferential equation d y d x = v u, given u and v as follows, where a and t are constants :

(a) u=ax, v=2ay; (b) u=-4ay, v=ax; (c) u=xt, v=-yt; (d) u=xt, v=-y

Now use suitable sotware (e.g MAPLE) to plot

(e) for problem (a), the three curves which pass through (1,1), (1,2) and (1,3), respectively, for 0≤ ≤x 3, all

on one graph;

(f) for problem (d), the three curves which pass through (1,1), (2,1) and (3,1), respectively, for 0 5⋅ ≤ ≤x 5, all on one graph, for each of t = 0 1 2 , ,

6 Diferential equations II Solve the pair of diferential equations d d x t = u, d y d t = v, where t is now a variable, for

u and v as given in Ex.5, with the conditions

(a) & (c) x = x0, y = y0 at t = 0; (b) x = y = 1 at t = 0; (d) x = x0, y = y0 at t = 1

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(e) for problem (a), the three paths ( ( ), ( ))x t y t , with a=1,x0 =1,y0 =1 2 3, , , respectively, for 0≤ ≤t 1

all on the same graph;

(f) for problem (d), the three paths ( ( ), ( ))x t y t , with y0=1,x0 =1 2 3, , , respectively, for 0≤ ≤t 2, all on the same graph

7 Some diferential identities Given that φ(x) is a general scalar function, and that u (x ) and v (x ) are general valued functions, use any appropriate method to show that

8 Two integral identities A volume V is bounded by the surface S on which there is deined the outward normal unit

vector, n Given that φ (x ) is a general scalar function, use Gauss’ theorem (the ‘divergence theorem’) to show that

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,

where φ is an arbitrary function [Use the same idea as in Ex.8.]

10 Diferentiation under the integral sign Given

∫

=

) (

) , ( )

(

x b

x a

dy y x f x

d d

[It is helpful to introduce the primitive of f ( x , y ) at ixed x: that is g ( x , y ) = ∫ f ( x , y ) dy.]

(a) Verify that this formula recovers a familiar and elementary result in the case :

h

8 :

99 ; = << , , and then simplify further given that t wz = r2 − z2

exp S X

11 Show that, if

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for arbitrary (i.e all) values of a and b, then f ( x ) ≡ 0

[Hint: you may write f x( )= g x′( ), although other, more general methods of proof are possible.]

12 Streamlines and particle paths In the following problems, the velocity components of a low (represented in rectangular

Cartesian coordinates x ≡ ( x , y , z ), u ≡ ( u , v , w ) and t time) are given; ind the streamlines in each case, and the particle path which passes through x ≡ ( x0, y0, z0) at t = 0 (Here, k, c and ω are constants.)

(a) u = kx v , = − ky w , = 0; (b) u = 2 xt v , = − 2 yt w , = 0;

(c) u = x − t v , = − y w , = 0; (d) u = xt v , = − y w , = 0;

(e) u = 2 x t v / , = − y t w / , = 0; (f) u = xy2 t v = t y w =

0 / , / , ;

(g) u = ky v , = − + kx kct w , = 0; (h) u = kx v2 = ky w2 = − k x + y z

2

(i) u = 0 , v = − z + cos( ω t ), w = y + sin( ω t ) for ω ≠ ±1;

(j) see (i) with ω = 0

13 Steady lows I

(a) Determine which of the lows discussed in Ex.12 are steady.

(b) for problem Ex.12(a): the three streamlines which pass through (1,1), (1,2) and (1,3), respectively, for

0 5 ⋅ ≤ ≤ x 5, all on the same graph;

(c) see (b); the three particle paths, for k = 1, which pass through (1,1), (2,1) and (3,1), respectively, at t = 0, for 0 ≤ ≤ t 1 (all on the same graph);

(d) for problem Ex.12(c): the three streamlines, at t = 1, which pass through (2,1), 2,2) and (2,3), respectively, for 1 5 ⋅ ≤ ≤ x 10, all on the same graph;

≤ ≤

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Is this a steady low? Find the streamlines for this low

15 SLs and PPs I he velocity components of a low are e S2t xα −2,3t3αy−1X, where α > −1 3 is a constant Find the streamlines for this low and the particle path which passes through (1, 1) at t = 0 State (without performing a calculation) the value of α for which the families of streamlines and particle paths coincide

16 SLs and PPs II See Ex 15; repeat this for r x S 2eαt, y−1 2e αtX

17 SLs and PPs III See Ex 15; repeat this for r Sαt− ,x y t2 αX with α ≠ −1, where the particle path passes through (0, 1) at t = 0

18 Acceleration of a luid particle he velocity vector which describes the motion of a particle (point) in a luid is

) ,

( t x

u

u = , so that the particle follows a path deined by

d d

the material derivative

19 Material derivative I (a) A luid moves so that its velocity vector, written in rectangular Cartesian coordinates, is

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) exp(

) 2 ( ) 2 exp(

) , , , ( x y z t x2 t2 y2 z2 t2

What is the constant value of f on a particle? (his will involve arbitrary constants that arise in the integration process.)

/ )),

2 /(

3 ), 2 /(

, / ( − x t − y t z t f = x t + y t − z t

≡

20 Material derivative II Find a velocity ield, u ≡ ( u , v , w ), for which the property

) / /

( )

2

c z b y kt t

a

x

f = + + , w where a, b, c and k are constants, is constant on luid particles

21 Eulerian vs Lagrangian description he Eulerian description of the motion of a luid is represented by u ( t x , ), that

is, the velocity at any point and at any time he Lagrangian description follows a given particle (point) in the luid; the Lagrangian velocity is u ( x0, t ), where x = x0 labels the particle at t = 0

A particle moves according to the rule

)) exp(

), exp(

), 2 exp(

( ) , ,

≡

written in rectangular Cartesian coordinates, where the particle is at x = x0 ≡ ( x0, y0, z0) at time t = 0

(a) Find the velocity of the particle in terms of x0 and t – the Lagrangian description – and then show that the velocity ield can be written as s u ≡ ( 4 xt , − 2 yt , − 2 zt ) , which is the Eulerian description

(b) Now obtain the acceleration of the particle from the Lagrangian description

(c) Show that the Lagrangian acceleration (that is, following a particle) is recovered from D

22 Velocity and acceleration A particle starts from x ≡ ( , ) a b at t = 0, and moves according to

o x ≡ ( , ) x y = S a ( 1 + t ) ,2 b ( 1 + t )2X F Find the velocity and acceleration directly, and then ind an expression for the velocity ield (by eliminating a and b) and hence show that the acceleration is recovered from D u D t

23 Incompressible low I.(a) Determine which velocity ields given in Ex.12 represent incompressible lows.

(b) Repeat (a) for Ex.19, Ex.20 and Ex.21.

(c) What relation must exist between α β γ , , so that the velocity ield given in Ex.14 represents an incompressible low ?

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x

u = f (r ) where r = x = x2+ y2+ z2

and f is a scalar function Find the most general form of f(r) so that u represents an incompressible low

(b) With the same notation as in (a), ind the conditions necessary on the constants a, b and c which ensure

that at u≡(ax −2 r2,bxy,cxz)/r5 represents an incompressible low

(c) Repeat (b) for the velocity ield ld u ≡ ( x + ar , y + br , z + cr ) /{ r ( x + r )}

25 Incompressible low III A low is represented by the velocity ield

u ≡ −

−

8 :

d

y d

where d = x2+ y2.Show that this describes an incompressible low

26 Incompressibility IV A velocity ield is given by y u≡ f y zt z ytS , 2 , 2 X where t is time; ind f x y z t ( , , , ) for which this low is incompressible and which satisies f = 0 on x = 0 for all y, z, t

27 Mass conservation Show that

u ≡ ( α xt yt zt , − , − ) and ρ = x2exp( − α t2) + ( y2+ 2 z2) exp( t2)satisfy the equation of mass conservation for one value of the constant α; what is this value?

28 Beltrami low A Beltrami low is one for which the vorticity and velocity vectors are everywhere parallel Write

ω= ku (where k is a non-zero constant) and seek a velocity ield that is consistent with this equation and of the form

u ≡ ( ( , , ), ( , , ), ( )) u x y z v x y z w x ,but it is not necessary to ind a general solution – just ind any (non-zero) solution

29 Pipe low A pipe with a rectangular cross-section, − a ( x ) ≤ y ≤ a ( x ), − b ( x ) ≤ z ≤ b ( x ), with its centre-line along the x-axis, has a non-swirling, incompressible low through it Show that

a

a b

b

−

and hence recover the standard result (see §1.4, Example 8) for a low which is uniform across every section

30 Branching pipe A pipe, of cross-sectional area A, branches into two, one of area nA and the other of area mA he

speed of an incompressible luid at area A is u and at area nA it is v; ind the speed in the branch of area mA (Assume

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that the low is uniform at all sections away from the junction, and that the luid completely ills both the feed pipe and the two branch pipes, without leaks or other branches i.e mass is conserved.)

31 Hydrostatic equilibrium I A luid in (vertical) hydrostatic equilibrium satisies

d d

p

z = − ρ g (g constant) ; see Lecture Notes

(a) Given that p = k ργ, where k and γ are positive constants, and that p = p0, ρ = ρ0 on z = 0, ind )

(z

ρ and p (z ) for 1 < γ < 2 Given, further, that t p=ρRT ( (R constant), ind T (z ) – the temperature – and deduce that d T d z = constant

(b) Repeat (a) for γ = 1

(c) An ocean, in z ≤ 0, is modelled by the density variation ρ = ρ0( 1 − α z ), where α (presumably small

!) and ρ0 are positive constants Find p (z ), given that p = p0 on z = 0

(d) Repeat (c) for ρ = ρ0( 1 + α − z )

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(e) he atmosphere is modelled as a perfect gas, so t p = ρ RT ( (R constant), with the temperature gradient prescribed according to

d d

T z

where α is a positive constant Given that T = T0 (with h α gH / RT0 < 1 )) and p = p0 on z = 0, ind T (z ) and )

(z

p where both these functions are continuous on z = H What is the behaviour of your solution for z → ∞? [Comment: Typically, the temperature in the Earth’s atmosphere drops linearly by about 700C in the irst 11 km (the troposphere), and then remains roughly constant (in the stratosphere) up to about 35 km.]

(f) See (a); ind ρ (z ) (only) given that g is replaced by y g0 /( 1 + α z )2 (g0 and α positive constants) What is the signiicance of this choice for g ?

32 Hydrostatic equilibrium II A luid is at rest, in hydrostatic equilibrium; the luid is described p = k ρ, where k is a constant, with p = p0 and ρ ρ = 0 on z = 0 Determine k and then ind p z ( ), given that the body force is that associated with constant gravity (F ≡ ( , , 0 0 g − ))

33 Archimedes’ Principle A surface S encloses luid of volume V which contains a solid body of volume Vb (surface Sb )

he luid exerts a resultant pressure force, R, on Vb , given by y p s

(which is Archimedes’ Principle, if F = g).

34 Euler’s equation An incompressible (ρ = constant) low in two dimensions [x ≡ ( x , z )], with F ≡ ( 0 , − g ), satisies Euler’s equation For this low, the velocity is u ≡ ( u0, w ( x )) , where u0 is a constant, with w = 0 on x = 0 and

p = p0 on z = 0 Find the solution for w and p, and show that it contains one free parameter

35 Galilean invariance Consider an incompressible low which comprises, in part, a uniform low u = u0=constant Write u = u0+ U and hence ind the appropriate forms taken by the mass conservation and Euler equations, written

in terms of u

0 and U Now introduce a frame of reference that is moving at the constant velocity u

0 , by setting

U = U x $ ( $, ) t , p = $( $, ) p x t where x $ = x − u0t ( ( ≡ x − u t y0 , − v t z0 , − w t0 )).Show that the equations written in terms of U $ , p $ and x $ are identical to the original equations of motion

[his important property is known as ‘Galilean invariance’; it means, for example, that the constant velocity of an object moving through a stationary luid is identical to the constant velocity of the luid past a stationary object.]

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2 Equations: Properties and Solutions

We now investigate the governing equations (Euler and mass conservation) in a little more detail We shall describe some general (and important) properties of lows that will be useful in our later work, and that are relevant in certain types of studies of luid motions We also show how two integrals of the equation of motion can be derived – valid under slightly diferent modelling assumptions – which are quite signiicant in the application of these ideas to practical problems

2.1 The vorticity vector and irrotational low

A concept that permeates much of luid theory is the notion of vorticity It is an important property of a luid low, both in terms of what is observed in real lows and the rôle it plays in allowing theoretical headway As we shall see, this provides

a measure of the local spin or rotation exhibited by luid elements It is deined by

Vorticity has a simple interpretation, which we will show by examining a low which is purely 2D; the idea is readily extended

to 3D (but is then more diicult to represent diagrammatically) Consider the low in the (∆ ∆ x , y) neighbourhood of a general point (x, y), described by some general velocity ield:

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Here, we have approximated the velocity components near to (x,y) by invoking the simplest approximation provided by Taylor expansions; we assume, of course, that the velocity ield allows this approach he average angular speed, relative

to the origin (labelled (x, y) here, for any point in the 2D plane) is approximately

12

1

2

y x

a low with vorticity

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