FUNDAMENTAL LAWS AND EQUATIONS

Kinematics

What is a fluid? Specification of motion

A fluid is anything that flows, usually a liquid ora gas, the latter being distinguished by its great rel-ative compressibility.

Fluids are treated as continuous media, and theirmotion and state can be specified in terms of thevelocity u, pressure p, density ρ, etc evaluated atevery point in space x and time t. To define the den-sity at a point, for example, suppose the point to besurrounded by a very small element (small com-pared with length scales of interest in experiments)which nevertheless contains a very large number ofmolecules. The density is then the total mass of all

the molecules in the element divided by the volumeof the element.

Considering the velocity, pressure, etc as func-tions of time and position in space is consistent withmeasurement techniques using fixed instruments inmoving fluids. It is called the Eulerian specification.However, Newton’s laws of motion (see below) areexpressed in terms of individual particles, or fluidelements, which move about. Specifying a fluidmotion in terms of the position X(t) of an individualparticle (identified by its initial position, say) iscalled the Lagrangian specification. The two arelinked by the fact that the velocity of such an ele-ment is equal to the velocity of the fluid evaluated atthe position occupied by the element:

. (1)

The path followed by a fluid element is called aparticle path, while a curve which, at any instant, iseverywhere parallel to the local fluid velocity vector

dXdt

= u X(t), t[ ]

INTRODUCTION TO FLUID DYNAMICS 7

SCI. MAR., 61 (Supl. 1): 7-24 SCIENTIA MARINA 1997

LECTURES ON PLANKTON AND TURBULENCE. C. MARRASÉ, E. SAIZ and J.M. REDONDO (eds.)

Introduction to Fluid Dynamics*

T.J. PEDLEY

Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Silver St., Cambridge CB3 9EW, U.K.

SUMMARY: The basic equations of fluid mechanics are stated, with enough derivation to make them plausible but with-out rigour. The physical meanings of the terms in the equations are explained. Again, the behaviour of fluids in real situa-tions is made plausible, in the light of the fundamental equations, and explained in physical terms. Some applications rele-vant to life in the ocean are given.

Key words: Kinematics, fluid dynamics, mass conservation, Navier-Stokes equations, hydrostatics, Reynolds number, drag,lift, added mass, boundary layers, vorticity, water waves, internal waves, geostrophic flow, hydrodynamic instability.

*Received February 20, 1996. Accepted March 25, 1996.

is called a streamline. Particle paths are coincidentwith streamlines in steady flows, for which thevelocity u at any fixed point x does not vary withtime t.

Material derivative; acceleration.

Newton’s Laws refer to the acceleration of a par-ticle. A fluid element may have acceleration bothbecause the velocity at its location in space is chang-ing (local acceleration) and because it is moving toa location where the velocity is different (convectiveacceleration). The latter exists even in a steady flow.

How to evaluate the rate of change of a quantityat a moving fluid element, in the Eulerian specifica-tion? Consider a scalar such as density ρ (x ,t). Letthe particle be at position x at time t, and move to x+ δx at time t + δt, where (in the limit of small δt)

. (2)

Then the rate of change of ρ following the fluid,or material derivative, is

(by the chain rule for partial differentiation)

(3a)

(using (2))

(3b)

in vector notation, where the vector ∇ρ is the gradi-ent of the scalar field ρ :

.

A similar exercise can be performed for eachcomponent of velocity, and we can write the x-com-ponent of acceleration as

(4a)

etc. Combining all three components in vector short-hand we write

(4b)

but care is needed because the quantity ∇u is notdefined in standard vector notation. Note that ∂u/∂tis the local acceleration, (u.∇)u the convectiveacceleration. Note too that the convective accelera-tion is nonlinear in u, which is the source of thegreat complexity of the mathematics and physics offluid motion.

Conservation of mass

This is a fundamental principle, stating that forany closed volume fixed in space, the rate ofincrease of mass within the volume is equal to thenet rate at which fluid enters across the surface ofthe volume. When applied to the arbitrary small rec-tangular volume depicted in fig. 1, this principlegives:

Dividing by ∆x ∆y ∆ z and taking the limit as thevolume becomes very small we get

+∆x∆y ρw[ ]z− ρw[ ]z+∆z( ).

+∆z∆x ρv[ ]y− ρv[ ]y+∆y( ) +

∆x∆y∆z∂ρ∂t

= ∆y∆z ρu[ ]x− ρu[ ]x +∆x( ) +

DuDt

=∂u∂t

+ (u.∇)u,

Du

Dt=∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z,

∇ρ =∂ρ∂x

,∂ρ∂y

,∂ρ∂z

⎛⎝⎜

⎞⎠⎟

=∂ρ∂t

+ u.∇ρ

=∂ρ∂t

+ u∂ρ∂x

+ v∂ρ∂y

+ w∂ρ∂z

=∂ρ∂x

δx

δt+∂ρ∂y

δy

δt+∂ρ∂z

δz

δt+∂ρ∂t

DρDt

= limδt→0

ρ(x + δx, t +δt) − ρ(x, t)δt

δx = u(x, t)δt

8 T.J. PEDLEY

FIG. 1. – Mass flow into and out of a small rectangular region ofspace.

(5a)

or (in shorthand)

(5b)

where we have introduced the divergence of a vec-tor. Differentiating the products in (5a) and using(3), we obtain

(6)

This says that the rate of change of density of a fluidelement is positive if the divergence of the velocityfield is negative, i.e. if there is a tendency for theflow to converge on that element.

If a fluid is incompressible (as liquids often are,effectively) then even if its density is not uniformeverywhere (e.g. in a stratified ocean) the density ofeach fluid element cannot change, so

(7)

everywhere, and the velocity field must satisfy

(8a)or

. (8b)

This is an important constraint on the flow of anincompressible fluid.

The Navier-Stokes equations

Newton’s Laws of Motion

Newton’s first two laws state that if a particle (orfluid element) has an acceleration then it must beexperiencing a force (vector) equal to the product ofthe acceleration and the mass of the particle:

force = mass × acceleration.

For any collection of particles this becomes

net force = rate of change of momentum

where the momentum of a particle is the product ofits mass and its velocity. Newton’s third law states

that, if two elements A and B exert forces on eachother, the force exerted by A on B is the negative ofthe force exerted by B on A.

To apply these laws to a region of continuousfluid, the region must be thought of as split up intoa large number of small fluid elements (fig. 2), oneof which, at point x and time t, has volume ∆V , say.Then the mass of the element is ρ (x,t) ∆V , and itsacceleration is Du/Dt evaluated at (x,t). What is theforce?

Body force and stress

The force on an element consists in general oftwo parts, a body force such as gravity exerted onthe element independently of its neighbours, andsurface forces exerted on the element by all the otherelements (or boundaries) with which it is in contact.The gravitational body force on the element ∆V isgρ (x, t) ∆V , where g is the gravitational accelera-tion. The surface force acting on a small planar sur-face, part of the surface of the element of interest,can be shown to be proportional to the area of thesurface, ∆A say, and simply related to its orientation,as represented by the perpendicular (normal) unitvector n (fig. 3). The force per unit area, or stress, isthen given by

∂u

∂x+∂v

∂y+∂w

∂z= 0

divu = 0

DρDt

= 0

DρDt

= −ρdivu.

∂ρ∂t

= −div ρu( )

∂ρ∂t

= −∂∂x

ρu( ) − ∂∂y

ρv( ) − ∂∂z

ρw( )

INTRODUCTION TO FLUID DYNAMICS 9

FIG. 2. – An arbitrary region of fluid divided up into small rectan-gular elements (depicted only in two dimensions).

FIG. 3. – Surface force on an arbitrary small surface element embed-ded in the fluid, with area ∆A and normal n. F is the force exerted

by the fluid on side 1, on the fluid on side 2.

(9a)

or, in shorthand,

F = σ≈ n (9b)

where σ≈ is a matrix quantity, or tensor, dependingon x and t but not n or ∆A. σ≈ is called the stress ten-sor, and can be shown to be symmetric (i.e. σyx= σxy,etc) so it has just 6 independent components.

It is an experimental observation that the stress ina fluid at rest has a magnitude independent of n andis always parallel to n and negative, i.e. compres-sive. This means that σxy= σyz= σzx= 0, σxx= σyy=σzz= −p, say, where p is the positive pressure (hydro-static pressure); alternatively,

σ≈ = –p I≈ (10)

where I≈ is the identity matrix.

The relation between stress and deformation rate

In a moving fluid, the motion of a general fluidelement can be thought of as being broken up intothree parts: translation as a rigid body, rotation as arigid body, and deformation (see fig. 4).Quantitatively, the translation is represented by thevelocity field u, the rigid rotation is represented bythe curl of the velocity field, or vorticity,

ω = curlu , (11)

and the deformation is represented by the rate ofdeformation (or rate of strain) e≈ which, like stress, isa symmetric tensor quantity made up of the sym-metric part of the velocity gradient tensor. Formally,

(12)

or, in full component form,

(13)

Note that the sum of the diagonal elements of e≈ isequal to div u.

It is a further matter of experimental observationthat, whenever there is motion in which deformationis taking place, a stress is set up in the fluid whichtends to resist that deformation, analogous to fric-tion. The property of the fluid that causes this stressis its viscosity. Leaving aside pathological (‘non-Newtonian’) fluids the resisting stress is generallyproportional to the deformation rate. Combining thisstress with pressure, we obtain the constitutive equa-tion for a Newtonian fluid:

σ≈ = –p I≈ + 2µ e≈ – 2/3µ div uI≈ (14)

The last term is zero in an incompressible fluid, andwe shall ignore it henceforth. The quantity µ is thedynamic viscosity of the fluid.

To illustrate the concept of viscosity, consider theunidirectional shear flow depicted in fig. 4 wherethe plane y=0 is taken to be a rigid boundary. Thenormal vector n is in the y-direction, so equations(9) show that the stress on the boundary is

From (14) this becomes

but because the velocity is in the x-direction onlyand varies with y only, the only non-zero component

F = 2µexy ,− p + µeyy ,µezy( ),

F = σ xy ,σ yy ,σ zy( ).

e≈

=

∂u

∂x

12

∂u

∂y+∂v

∂x

⎛

⎝⎜

⎞

⎠⎟

12

∂u

∂z+∂w

∂x⎛⎝⎜

⎞⎠⎟

12

∂v

∂x+∂u

∂y

⎛

⎝⎜

⎞

⎠⎟

∂v

∂y

12

∂v

∂z+∂w

∂y

⎛

⎝⎜

⎞

⎠⎟

12

∂w

∂x+∂u

∂z⎛⎝⎜

⎞⎠⎟

12

∂w

∂y+∂v

∂z

⎛

⎝⎜

⎞

⎠⎟

∂w

∂z

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

e≈

=12

∇u +∇uT( )Fz = σ zxnx + σ zyny + σ zznz

Fy = σ yxnx + σ yyny + σ yznz

Fx = σ xxnx + σ xyny + σ xznz

10 T.J. PEDLEY

FIG. 4. – A unidirectional shear flow in which the velocity is in thex- direction and varies linearly with the perpendicular component y : u = αy. In time ∆t a small rectangular fluid element at level y0 istranslated a distance αy0∆t, rotated through an angle α/2, anddeformed so that the horizontal surfaces remain horizontal, and the

vertical surfaces are rotated through an angle α.

of e≈ is . Hence

In other words, the boundary experiences a perpen-dicular stress, downwards, of magnitude p, the pres-sure, and a tangential stress, in the x-direction, equalto µ times the velocity gradient ∂u/∂y. (It can beseen from (9) and (14) that tangential stresses arealways of viscous origin.)

The Navier-Stokes equations

The easiest way to apply Newton’s Laws to amoving fluid is to consider the rectangular blockelement in fig. 5. Newton’s Law says that the massof the element multiplied by its acceleration is equalto the total force acting on it, i.e. the sum of the bodyforce and the surface forces over all six faces. Theresulting equation is a vector equation; we will con-sider just the x-component in detail. The x-compo-nent of the stress forces on the faces perpendicularto the x-axis is the difference between the perpen-dicular stress σ

xx evaluated at the right-hand face(x+∆x) and that evaluated at the left-hand face (x)multiplied by the area of those faces, ∆y∆z, i.e.

If ∆x is small enough, this is

The x-component of the forces on the faces per-pendicular to the y-axis is

and similarly for the faces perpendicular to the z-axis. Hence the x-component of Newton’s Lawgives

or, dividing by the element volume,

(15a)

Similar equations arise for the y- and z-components,and they can be combined in vector form to give

ρDu

Dt= ρgx +

∂σ xx

∂x+∂σ xy

∂y+∂σ xz

∂z.

ρ∆x∆y∆z( ) Du

Dt= ρgx( )∆x∆y∆z +

+∂σxx∂x

+∂σxy∂y

+∂σxz∂z

⎛

⎝⎜

⎞

⎠⎟ ∆x∆y∆z

σ xy y+∆y−σ xy y

⎛⎝

⎞⎠ ∆z∆x =

∂σ xy

∂y∆x∆y∆z,

∂σ xx

∂x∆x∆y∆z.

σ xx x +∆x−σ xx x( )∆y∆z.

F = µ∂u

∂y,− p,0

⎛⎝⎜

⎞⎠⎟

exy =12∂u

∂y

INTRODUCTION TO FLUID DYNAMICS 11

FIG. 5. – Normal and tangential surface forces per unit area (stress) on a small rectangular fluid element in motion.

(15b)

The equations can be further transformed, usingthe constitutive equation (14) (with div u = 0) and(13) to express e≈ in terms of u, to give for (15a)

(16a)

Similarly in the y- and z-directions:

(16b)

(16c)

In these equations, it should not be forgotten that Du/Dt etc are given by equations (4).

Finally, the above three equations can be com-pressed into a single vector equation as follows:

(16d)

where the symbol is shorthand for

Equations (16a-c), or (16d), are the Navier-Stokesequations for the motion of a Newtonian viscousfluid. Recall that the left side of (16d) represents themass-acceleration, or inertia terms in the equation,while the three terms on the right side are respec-tively the body force, the pressure gradient, and theviscous term.

The four equations (16a-c) and (8b) are four non-linear partial differential equations governing fourunknowns, the three velocity components u,v,w, andthe pressure p, each of which is in general a functionof four variables, x, y, z and t. Note that if the densi-ty ρ is variable, that is a fifth unknown, and the cor-responding fifth equation is (7). Not surprisingly,such equations cannot be solved in general, but theycan be used as a framework to understand thephysics of fluid motion in a variety of circum-stances.

A particular simplification that can sometimes bemade is to neglect viscosity altogether (to assumethat the fluid is inviscid). Conditions in which this is

permitted are discussed below. When it is allowed,however, we can put µ = 0 in equations (16) andthese are greatly simplified.

For quantitative purposes we should note the val-ues of density and viscosity for fresh water and airat 1 atmosphere pressure and at different tempera-tures:

Temp Water Air (dry)ρ(kgm-3) µ(kgm-1s-1) ρ(kgm-3) µ(kgm-1s-1)

0˚C 1.0000 x 103 1.787 x 10-3 1.293 1.71 x 10-5

10˚C 0.9997 x 103 1.304 x 10-3 1.247 1.76 x 10-5

20˚C 0.9982 x 103 1.002 x 10-3 1.205 1.81 x 10-5

Boundary conditions

Whether the fluid is viscous or not, it cannotcross the interface between itself and another medi-um (fluid or solid), so the normal component ofvelocity of the fluid at the interface must equal thenormal component of the velocity of the interfaceitself:

(17a)

where U is the interface velocity. In particular, on asolid boundary at rest,

n.u = 0 (17b)

In a viscous fluid it is another empirical fact thatthe velocity is continuous everywhere, and in partic-ular that the tangential component of the velocity ofthe fluid at the interface is equal to that of the inter-face - the no-slip condition. Hence

u = U (18)

at the interface (u = 0 on a solid boundary at rest).There are boundary conditions on stress as

well as on velocity. In general they can be sum-marised by the statement that the stress F (eq.9)must be continuous across every surface (not thestress tensor, note, just σ≈ .n), a condition that fol-lows from Newton’s third law. At a solid bound-ary this condition tells you what the force per unitarea is and the total stress force on the boundaryas a whole is obtained by integrating the stressover the boundary (thus the total force exerted bythe fluid on an immersed solid body can be calcu-lated).

un = Un or n.u = n.U

∂ 2

∂x2 +∂ 2

∂y2 +∂∂z2 .

∇2

ρDuDt

= ρg −∇p + µ∇2u

ρDw

Dt= ρgz −

∂p

∂z+ µ

∂ 2w

∂x2 +∂ 2w

∂y2 +∂ 2w

∂z2

⎛⎝⎜

⎞⎠⎟ .

ρDv

Dt= ρgy −

∂p

∂y+ µ

∂ 2v

∂x2 +∂ 2v

∂y2 +∂ 2v

∂z2

⎛

⎝⎜

⎞

⎠⎟

ρDu

Dt= ρgx −

∂p

∂x+ µ

∂ 2u

∂x2 +∂ 2u

∂y2 +∂ 2u

∂z2

⎛⎝⎜

⎞⎠⎟ .

ρDuDt

= ρg + divσ≈

12 T.J. PEDLEY

When the fluid of interest is water, and theboundary is its interface with the air, the dynamicsof the air can often be neglected and the atmospherecan be thought of as just exerting a pressure on theliquid. Then the boundary conditions on the liquid’smotion are that its pressure (modified by a small vis-cous normal stress) is equal to atmospheric pressureand that the viscous shear stress is zero.

CONSEQUENCES: PHYSICAL PHENOMENA

Hydrostatics

We consider a fluid at rest in the gravitationalfield, with a free upper surface at which the pressureis atmospheric. We choose a coordinate system x, y,z such that z is measured vertically upwards, so gx =gy = 0 and gz = -g, and we choose z = 0 as the levelof the free surface. The density ρ may vary withheight, z. Thus all components of u are zero, andpressure p = p

atm at z = 0. The Navier-Stokes equa-tions (16) reduce simply to

Hence(19)

or, for a fluid of constant density,

:

the pressure increases with depth below the free sur-face (z increasingly negative).

The above results are independent of whetherthere is a body at rest submerged in the fluid. If thereis, one can calculate the total force exerted by thefluid by integrating the pressure, multiplied by theappropriate component of the normal vector n, overthe body surface. The result is that, whatever theshape of the body, the net force is an upthrust andequal to g times the mass of fluid displaced by thebody. This is Archimedes’ principle. If the fluid den-sity is uniform, and the body has uniform density ρb,then the net force on the body, gravitational andupthrust, corresponds to a downwards force equal to

(20)

where V is the volume of the body. The quantity (ρb – ρ) is called the reduced density of the body.

Note that, for constant density problems in whichthe pressure does not arise explicitly in the boundaryconditions (e.g. at a free surface), the gravity termcan be removed from the equations by including it inan effective pressure, pe. Put

(21)

in equations (16) (with gx = gy = 0, gz= -g) and seethat g disappears from the equations, as long as pereplaces p.

Flow past bodies

The flow of a homogeneous incompressiblefluid of density ρ and viscosity µ past bodies hasalways been of interest to fluid dynamicists ingeneral and to oceanographers or ocean engineersin particular. We are concerned both with fixedbodies, past which the flow is driven at a givenspeed (or, equivalently, bodies impelled by anexternal force through a fluid otherwise at rest)and with self-propelled bodies such as marineorganisms.

Non-dimensionalisation: the Reynolds number

Consider a fixed rigid body, with a typicallength scale L, in a fluid which far away has con-stant, uniform velocity U∞ in the x-direction (fig.6). Whenever we want to consider a particularbody, we choose a sphere of radius a, diameter L= 2a. The governing equations are (8) and (16),and the boundary conditions on the velocity fieldare

u = v = w = 0 on the body surface, S (22)

at infinity. (23)u →U∞ ,v → 0, w → 0

pe = p + gρz

ρb − ρ( )Vg

p = patm − gρz

p = patm + g ρdzz

0∫

∂p

∂z= −ρg.

∂p

∂x=∂p

∂y= 0,

INTRODUCTION TO FLUID DYNAMICS 13

FIG. 6. – Flow of a uniform stream with velocity U∞ in the x-direc-tion past a body with boundary S which has a typical length scale L.

Usually the flow will be taken to be steady, ie

, but we shall also wish to think about devel-

opment of the flow from rest.For a body of given shape, the details of the flow

(i.e. the velocity and pressure at all points in thefluid, the force on the body, etc) will depend on U∞,L , µ and ρ as well as on the shape of the body.However, we can show that the flow in fact dependsonly on one dimensionless parameter, the Reynoldsnumber

(24)

and not on all four quantities separately, so onlyone range of experiments (or computations) wouldbe required to investigate the flow, not four. Theproof arises when we express the equations indimensionless form by making the following trans-formations:

Then the equations become: (8b):

; (25)

(16a), with replaced by (4a):

(26)

and there are similar equations starting with ∂v´/∂t´,∂w´/∂t´. The boundary condition (22) is unchanged,though the boundary S is now non-dimensional, soits shape is important but L no longer appears.Boundary condition (23) becomes

at infinity. (27)

Thus Re is the only parameter involving the physi-cal inputs to the problem that still arises.

The drag force on the body (parallel to U∞)proves to be of the form:

(28)

where A (proportional to L2)is the frontal area of thebody (πL2/4 for a sphere) and C

Dis called the drag

coefficient. It is a dimensionless number, computedby integrating the dimensionless stress over the sur-face of the body.

From now on time and space do not permit deriva-tion of the results from the equations. Results will bequoted, and discussed physically where appropriate.

It can be seen from (26) that, in order of mag-nitude terms, Re represents the ratio of the non-linear inertia terms on the left hand side of theequation to the viscous terms on the right. Theflow past a rigid body has a totally different char-acter according as Re is much less than or muchgreater than 1.

Low Reynolds number flow

When Re <<1, viscous forces dominate the flowand inertia is negligible. Reverting to dimensionalform, the Navier-Stokes equations (16d) reduce tothe Stokes equations

, (29)

where gravity has been incorporated into pe using

eq. (21). The conservation of mass equation div u =0, is of course unchanged. Several important con-clusions can be deduced from this linear set of equa-tions (and boundary conditions).

(i) Drag The force on the body is linearly relatedto the velocity and the viscosity: thus, for example,the drag is given by

(30)

for some dimensionless constant k (thus the dragcoefficient C

Dis inversely proportional to Re). In

particular, for a sphere of radius a, k = 3π, so

(31)

It is interesting to note that the pressure and theviscous shear stress on the body surface con-tribute comparable amounts to the drag. The netgravitational force on a sedimenting sphere ofdensity ρ

b, from (20), is (ρb-ρ)·4/3πa3g. This must

be balanced by the drag, 6πUsa, where Us is the

sedimentation speed. Equating the two gives

(32)Us =29

ρb − ρ( )ga2

µ

D = 6πµU∞a

D = kµU∞L

∇pe = µ∇2u

D = 12 ρU 2 ACD

u′→1, v′→ 0, w′→ 0

∂u′∂t ′

+ u′∂u′∂x′

+ v'∂u′∂y′

+ w'∂u′∂z′

=

= −∂p′∂x′

+1

Re∂ 2u′∂x′2

+∂ 2u′∂y′2

+∂ 2u′∂z′2

⎡

⎣⎢

⎤

⎦⎥

Du

Dt

∂u′∂x′

+∂v′∂y′

+∂w′∂z′

= 0

u′= u / U∞ , v′= v / U∞ , w′= w / U∞ , p′= p / ρU∞2 .

x′= x / L, y′= y / L, z′= z / L, t ′= U∞t / L,

Re =ρLU∞

µ,

∂∂t

≡ 0

14 T.J. PEDLEY

For example, a sphere of radius 10 µm, with density10% greater than water (ρ=103kg m-3, µ≈11kg m-1s-1)will sediment out at only 20 µm-1, whereas if the radiusis 100 µm, the sedimentation speed will be 2 mms-1.

(ii) Quasi-steadiness. Because the ∂/∂t term inthe equations vanishes at low Reynolds number, it isimmaterial whether the relative velocity of the body(or parts of it) and the fluid is steady or not. Theflow at any instant is the same as if the boundarymotions at that instant had been maintained steadilyfor a long time - i.e. the flow (and the drag force etc)is quasi-steady.

(iii) The far field. It can be shown that the farfield flow, that is the departure of the velocityfield from the uniform stream U∞, dies off veryslowly as the distance r from an origin inside thebody becomes large. In fact it dies off as 1/r,much more slowly, for example, than the inversesquare law of Newtonian gravitation or electro-statics. This has an important effect on particle -particle interactions in suspensions. Moreover,this far field flow is proportional to the net forcevector –D exerted by the body on the fluid, inde-pendent of the shape of the body. Thus, in vectorform, we can write

(33)

where

(34)

Measuring the far field is therefore one potentialway of estimating the force on the body.

The only exception to the above is the casewhere the net force on the body (or fluid) is zero,as for a neutrally buoyant, self-propelled micro-organism. In that case P is zero, the far field diesoff like 1/r2, and it does depend on the shape ofthe body and the details of how it is propellingitself.

(iv) Uniqueness and Reversibility. If u is asolution for the velocity field with a given veloci-ty distribution us on the boundary S, then it is theonly possible solution (that seems obvious, but isnot true for large Re). It also follows that –u is the(unique) velocity field if the boundary velocitiesare reversed, to –us. Thus if a boundary movesbackwards and forwards reversibly, all elements ofthe fluid will also move backwards and forwardsreversibly, and will not have moved, relative to thebody, after a whole number of cycles. Hence amicro-organism must have an irreversible beat inorder to swim.

(v) Flagellar propulsion. Many micro-organ-isms swim by beating or sending a wave downone or more flagella. Fig. 7 sketches a monofla-gellate (e.g. a spermatozoon). It sends a, usuallyhelical, wave along the flagellum from the head.This is a non-reversing motion because the waveconstantly propagates along. The reason thatsuch a wave can produce a net thrust, to over-come the drag on the head (and on the tail too) isthat about twice as much force is generated by a

P =-D

8πµ.

u − U∞ ≈1r

P +P.x( )x

r2⎡⎣⎢

⎤⎦⎥

INTRODUCTION TO FLUID DYNAMICS 15

FIG. 7. – (a) Sketch of a swimming spermatozoon, showing its position at two successive times and indicating that, whilethe organism swims to its left, the wave of bending on its flagellum propagates to the right. (b) Blow up of a small elementδs of the flagellum indicating the force components normal and tangential to it, proportional to the normal and tangential

components of relative velocity.

16 T.J. PEDLEY

FIG. 8. – Photographs of streamlines (a, b) or streaklines (c) for steady flow past a circular cylinder atdifferent values of the Reynolds number (M.Van Dyke, 1982): (a) Re <<1, (b) Re ≈ 26, (c) Re ≈ 105.

a

b

c

segment of the flagellum moving perpendicularto itself relative to the water as is generated bythe same segment moving parallel to itself. Thisfact forms the basis of resistive force theory forflagellar propulsion, which is a simple and rea-sonably accurate model for the analysis of flagel-lar locomotion.

Vorticity

The dynamics of fluid flow can often be mostdeeply understood in terms of the vorticity, definedby equation (11) and representing the local rotationof fluid elements. High velocity gradients corre-spond to high vorticity (see fig. 4). If we take thecurl of every term in the Navier-Stokes equation weobtain the following vorticity equation (in vectornotation):

(35)

where ν = µ/ρ is the kinematic viscosity of thefluid (assumed constant). This equation tells usthat the vorticity, evaluated at a fluid elementlocally parallel to ωω, changes, as that elementmoves, as a result of three effects, each repre-sented by one of the terms on the right hand sideof (35). The first term can be shown to be associ-ated with rotation and stretching (or compres-sion) of the fluid element, so that the direction ofωω remains parallel to the original fluid element,and increases in proportion as the length of thatelement changes. Such vortex-line stretching is adominant effect in the generation and mainte-nance of turbulence. It is totally absent in a two-dimensional (2D) flow in which there is novelocity component in one of the coordinatedirections (say z) and the variables are indepen-dent of z. The second term represents the effectof viscosity, and is diffusion-like in that vorticitytends to spread out from elements where it is highto those where it is low. The last term comesabout only in non-uniform (e.g. stratified) fluids,and can be important in some oceanographic sit-uations.

It can also be shown that, in a flow startedfrom rest, no vorticity develops anywhere untilviscous diffusion has had an effect there. As weshall see, the only source of vorticity, in such aflow and in the absence of the last term in (35),occurs at solid boundaries on account of the no-slip condition.

Higher Reynolds number.

It is convenient now to restrict attention to a 2Dflow of a homogeneous fluid past a 2D body such asa circular cylinder (fig. 8). In such a 2-D flow, withvelocity components u = (u,v,0), functions of x,y andt, the vorticity is entirely in the third, z, direction,and is given by

There is no vortex-line stretching, and the onlyeffect which can generate vorticity anywhere isviscosity. Let us suppose that the uniform streamat infinity is switched on from rest at the initialinstant. Initially there is no vorticity anywhere,and the initial irrotational velocity field is easy tocalculate. It satisfies all the governing equationsand all boundary conditions except the no-slipcondition at the cylinder surface. The predictedslip velocity therefore generates an infinite veloci-ty gradient ∂u/∂y and hence a thin sheet, of infinitevorticity at the cylinder surface. Because of vis-cosity, this immediately starts to diffuse out fromthe surface. At low values of Re, when viscosity isdominant and the convective term (u.∇)ωω in (35)is negligible, the diffusion is rapid, and vorticityspreads out a long way in all directions. An even-tual steady state is set up in which the flow isalmost totally symmetric front-to-back (fig. 8a);unlike the spherical case, the drag coefficient isnot quite inversely proportional to the Reynoldsnumber:

At somewhat higher values of Re, the (u.∇)ωωterm is not totally negligible, and once vorticity hasreached any particular fluid element it tends to becarried along by it as well as diffusing on to otherelements. Hence a front-to-back asymmetry devel-ops. For Re greater than about 5 the flow actuallyseparates from the wall of the cylinder, forming twoslowly recirculating flow regions (eddies) at therear. At still higher Re, it is observed that the eddiestend to break away alternately from the two sides ofthe cylinder, usually at a well-defined frequencyequal to about 0.42 U∞/a for Re ≥ 600, and steadyflow is no longer possible. At higher Re the wakebecomes turbulent (i.e. random and three-dimen-sional) and at Re≈ 2 ×105 the flow on the cylindersurface becomes turbulent.

CD =8π

Re log 7.4 / Re( ).

ω =∂v

∂x−∂u

∂y.

∂ω∂t

+ u.∇( )ω = ω .∇( )u+ν∇2ω +1ρ2 ∇ρ ∧∇p

INTRODUCTION TO FLUID DYNAMICS 17

Steady flows at relatively high Reynolds numberdo seem to be possible past streamlined bodies suchas a wing (or a fish dragged through the fluid), seefig. 9. Diffusion causes vorticity to occupy a(boundary) layer of thickness (νt)1/2 after time t.However, even a fluid element near the leading edgeat first will have been swept off downstream past thetrailing edge after a time t = L/U∞, where L is thelength of the wing chord. Hence the greatest thick-ness that the boundary layer on the body can have is

(36)

and it is easy to see that a steady state can developeverywhere on the body, with a boundary layer ofthickness up to δs, and a thin wake region, also con-taining vorticity, downstream. Note that the bound-ary layer of vorticity remains thin compared with thechord length if δs << L, i.e. Re >>1. In that case (andonly then) neglecting viscosity altogether, and for-getting about the boundary layer, is accurateenough, except in calculating the drag.

Drag on a symmetric body at large Reynoldsnumber. In order to estimate the force on a body itis necessary to work out the distribution of pressureδ s = νL / U∞( )1/ 2

,

18 T.J. PEDLEY

boundary layerwake

FIG. 9. – Sketch of boundary layer and wake for steady flow at high Reynolds number past a symmetric streamlined body.

p>p∞p=p∞

p<p∞

p>p∞

p<p∞

p<p∞

p>p∞p=p∞

p<p∞

p<p∞

A1

A2

S1 S2

(a)

(b)

FIG. 10. – Sketch of streamlines and pressures for flow past a circular cylinder. (a) Idealised flow of a fluid with no viscosity; (b) separatedflow at fairly high Reynolds number in a viscous fluid.

round the body. In a steady flow of constant densityfluid in which viscosity is unimportant (e.g. outsidethe boundary layer and wake of a body) equation(16d) can be integrated to give the result that thequantity

(37a)

along streamlines of the flow. Here z is measuredvertically upwards and |u| is the total fluid speed.This result is equivalent to the Newtonian principleof conservation of energy; equation (37a) is calledBernoulli’s equation. If we forget about the gravita-tional contribution, replacing p + pgz by the effec-tive pressure p

e(eq. 21), equation (37a) becomes

pe

= constant – (37b)

henceforth we just write p for pe. If the fluid speeds

up, the pressure falls, and vice versa, which is intu-itively obvious since a favourable pressure gradientis clearly required to give fluid elements positiveacceleration.

In the case of flow past a symmetric body, (fig.10a), all streamlines start from a region of uniformpressure (p∞ say) and uniform velocity (U∞), so theconstant in (37b) is the same for all streamlines,p∞+1/2U2

∞. If viscosity were really negligible, thenthe flow round a circular cylinder would be sym-

metric (fig. 10a). At the front stagnation point S1,the point of zero velocity where the streamlinedividing flow above from flow below impinges, thepressure is high (p = p∞+1/2ρU2

∞), and this highpressure is balanced by an equally high pressure atthe rear stagnation point S2. The pressure at the sides(A1, A2 ) is low (p = p∞–3/2U2

∞). The net effect is thatthe hydrodynamic force on the cylinder is zero.

In a viscous fluid, as stated above, there is a thinboundary layer on the front half, in which thevelocity falls from a large value to zero, so the pres-sure distribution is similar to that described above;however the flow separates on the rear half andthings are very different. The reason for the separa-tion is that the adverse pressure gradient (the pres-sure rise), from A1 to S2 say, causes the low veloci-ty in the boundary layer to tend to reverse its direc-tion, and it is observed that separation occurs assoon as flow reversal takes place. In the separatedflow region (fig. 10b) the fluid velocity is low andthe pressure remains close to its value at the sides.Thus there is a front-to-back pressure differenceproportional to ρU2

∞, and the drag coefficient CD

(eq. 28) is approximately constant, independent ofRe as long as Re is large (see fig. 11). The directcontribution of tangential viscous stresses to thedrag is negligibly small, although it is the presenceof viscosity which causes the flow separation in thefirst place.

12ρ u 2;

p + pgz +12ρ u 2 = constant

INTRODUCTION TO FLUID DYNAMICS 19

0.050.10.31.03.07.942.080.0300.0

10-1

0.1

1

10

100

100 101 102 103 104 105 106

CD

Re=U∞Dν

D (mm)

FIG. 11. – Log-log plot of drag coefficient versus Reynolds number for steady flow past a circular cylinder. [The sharp reduction in CD

at Re ≈ 2 × 105 is associated with the transition to turbulence in the boundary layer]. Redrawn from Schlichting (1968).

Lift. For a symmetric streamlined body (fig. 9)flow separation occurs only very near the trailingedge, and direct viscous drag is more important.However, if such a streamlined body (or wing) istilted so that the oncoming flow makes an angle ofincidence with its centre plane, viscosity again hasan important effect. In general, a non-viscous flowpast a wing at incidence would turn sharply roundthe trailing edge, where the velocity would beextremely high and the pressure extremely low (fig.12a). As the flow starts up from rest, viscosity caus-es separation at the corner, a concentrated vortex isshed and left behind, and thereafter the flow isforced to come tangentially off the trailing edge: theKutta-condition (fig. 12b). In order to achieve thistangential flow, the velocity on top of the wing mustincrease and the velocity below must decrease. Itfollows from Bernoulli’s equation that the pressureabove the wing must fall, and that below rise, so atransverse force is generated. This is call lift andkeeps aircraft and birds in the air against gravity.The magnitude of the lift is also represented by a liftcoefficient C

L :

(38)

where S is the horizontal area of the wing. Like CD,

CL is approximately independent of Re for large Re .

Added mass. We have seen that the force on abody in an inviscid fluid is zero when the flow issteady. When the flow is unsteady, however, theforce is non-zero, because accelerating the body rel-ative to the fluid requires that the fluid also has to beaccelerated. Thus the body exerts a force on thefluid and so, by Newton’s third law, the fluid exertsan equal and opposite force on the body. In all cases,this force is equal in magnitude to the accelerationof the body relative to the fluid multiplied by themass of fluid displaced by the body (ρV in the nota-tion of eq. 20) multiplied by a constant, say β:

F=βρV dU/dt. (39)

For a sphere, β = 0.5; for a circular cylinder, β = 1.The quantity βρV is call the added mass of the bodyin question (recall that ρ is the fluid density). Thecorresponding force, given by (39), is called theacceleration reaction, or the reactive force.

Fish swimming. We have seen that flagellatessuch as spermatozoa swim by sending bendingwaves down their tails, and thrust is generatedthrough the viscous, resistive force. Inertia is negli-

gible because the Reynolds number is small. Formost fish, the Reynolds number is large, but never-theless many fish also swim by sending a bendingwave down their bodies and tails. In this case, how-ever, thrust is generated primarily by the reactiveforce associated with the sideways acceleration of theelements of fluid as they pass down the animal (rela-tive to a frame of reference fixed in the fish’s nose).Lighthill has developed a simple, reactive-forcemodel for fish swimming.

Flow in the open ocean

Water waves

The most obvious dynamical feature of theocean, to even a casual observer, is the presence ofsurface waves, of a variety of lengths and heights.Waves are mainly generated as a result of stressesexerted by the wind, although they can also arisethrough the impact or relative motion of foreignbodies such as rain drops or ships. Once generated,however, waves can propagate over large distancesand persist for long times, unaffected by the atmos-phere or solid bodies.

L =12

U∞2 SCL ,

20 T.J. PEDLEY

(a)

(b)

FIG. 12. – Flow past a streamlined body at incidence. (a) Idealisedflow of a fluid with no viscosity - large velocity and pressure gradi-ent round the trailing edge. (b) In a viscous fluid the flow mustcome smoothly off the trailing edge, which explains the generation

of lift (see text).

In a periodic wave motion, all fluid elementsaffected by it experience oscillations. Like alloscillations, such as that of a simple pendulum,these oscillations come about as an interactionbetween a restoring force, tending to restore a par-ticle to a nearby equilibrium position, and inertia,which causes the particle to overshoot each time itreaches its equilibrium position (in real systemsthere is also some viscous damping, which causesthe amplitude of the oscillations to die out after along time, if there is no further stimulation; weignore damping here). In the case of a simple pen-dulum (a mass suspended by a light string) theequilibrium state is one in which the string is ver-tical and the mass at rest, the restoring force isgravity and the inertia is the momentum of themass itself. In the case of water waves, the equilib-rium state has the free surface horizontal, therestoring force is again gravity (except for smallwavelengths, when surface tension is also impor-tant) and the inertia is the momentum of the fluid.Viscosity is negligible because there are no solidboundaries generating vorticity.

In an oscillation of small amplitude, every parti-cle exhibits simple harmonic motion: its vertical dis-placement, say Y, from equilibrium, varies with timeaccording to the differential equation

(40)

The general solution for Y is a sinusoidal oscilla-tion of the form

where A and φ are constants (determined by initialconditions), the amplitude and phase respectively,and ω is the angular frequency of the oscillation(the frequency in Hertz is ω/2π). In the case of asimple pendulum, ω = (g/l)1/2 where l is the lengthof the string. In the case of simple water waves ofwave length λ = 2π/k (k is the wave number), in anocean whose depth is much greater than λ, wehave

ω = (gk)1/2 (41)

as long as surface tension is negligible.Suppose a parallel-crested (one-dimensional)

train of such waves is propagating in the x-direction.Then the displacement of the free surface will begiven by

(42)

again for constant A and φ. The speed of propagationof the wave crests, or phase velocity, is

(43)

Thus long waves (small k) travel more rapidly thanshort waves (large k). This explains why, when thewaves are generated by a localised disturbance, suchas a storm at sea, or a stone dropped in a pond, thelonger waves (swell) arrive at the shore first. In thiscase, the wave front travels at a different speed,called the group velocity, c

g:

(44)

so that wave crests, travelling faster, appear to ariseat the back of the packet of waves, and to disappearat the front.

When a water wave propagates, with its free sur-face given by (42), fluid elements at and below thesurface move in circular paths, and the amplitude oftheir motion falls off exponentially with depth belowthe surface: the amplitude is proportional to Aekz

when the undisturbed surface is at z = 0. Thus theamplitude is negligibly small at a depth of only halfa wavelength (kz = -π). This explains why the theo-ry of waves in very deep water works well in rela-tively shallow water, too, with depth h greater thanhalf a wavelength. When the waves are very long, orthe water very shallow, equation (41) is replaced by

(45)

Small amplitude wave theory is very useful,because the equations are linear and a generalmotion can be made up from the addition of manysinusoidal components such as (42) (a Fourier seriesor transform). At larger amplitudes, nonlineareffects become important and the theory becomesless general, although many interesting and impor-tant phenomena arise, such as wave breaking.

Internal waves

Although the water in the ocean is effectivelyincompressible, it does not have uniform densitybecause it is stratified on account of the variationwith depth of the pressure and, to a lesser extent, thetemperature and the salinity. The temperature/densi-ty distribution is marked usually by one or more

ω = gk tanh kh( )1/ 2

cg =dωdk

=12

g

k⎛⎝

⎞⎠

12,

c =ωk

=g

k⎛⎝

⎞⎠

1 2

.

η = Acos(ωt − kx − φ ),

Y = Acos ωt − φ( )

d 2Y

dt 2 + ω 2Y = 0.

INTRODUCTION TO FLUID DYNAMICS 21

thermoclines, in which the density gradient is steep-er than elsewhere. Whether the density gradient isuniform or locally sharp, less dense fluid sits, inequilibrium, above denser fluid. A disturbance tothis state causes some heavy fluid elements to riseabove their original level, and some light ones to fallbelow. As in the case of surface waves, gravity thenprovides a restoring force and internal gravity wavescan propagate. As for surface waves, a relation canbe calculated between the frequency and the wavenumber of such waves. For example, if there is asharp interface between two deep regions of fluidwith densities ρ1 (above) and ρ2, then equation (41)is replaced by

(46)

This can be seen to give much lower frequenciesthan (41) if (ρ2 – ρ1) is not large: if ρ2 – ρ1 = 0.1 ρ2 ,then the frequency given by (46) is 4.4 times small-er than that given by (41) (with ρ2 = ρ). The propa-gation speed is correspondingly smaller, too.

When the density gradient is uniform, with

(47)

where N is a constant with the dimensions of a fre-quency (the Brunt -Väisälä frequency), the situationis a bit more complicated, because internal waves donot have to propagate horizontally. Indeed, a wavewhose crests propagate at an angle θ to the horizon-tal, so that the displacement of a fluid element isgiven by

has a frequency ω given by

. (48)

However, the group velocity (velocity of a wavefront, or of energy propagation) is perpendicular tothe phase velocity, and in this case is given by thevector

(49)

Rotating fluids: geostrophic flows

Gravity waves are (mostly) small-scale phenom-ena for which the rotation of the earth is unimpor-tant. That is not the case with ocean currents and the

large-scale circulation of the oceans. To analysesuch motions, it is necessary to recognise that thenatural frame of reference is fixed in the rotatingearth, and the governing equations of motion have tobe changed accordingly. If viscosity is neglected, theequation of motion of a fluid in a frame of referencerotating with constant angular velocity Ω becomes(in place of (16d)):

(50)

Here g has been modified to include the small “cen-trifugal force’’ term, and we could also incorporateit into the pressure using (21). The additional term

is called the Coriolis force.Time does not permit a thorough investigation of

the dynamics of rotating fluids. We consider only aflow in which the Coriolis force is much larger thanthe other inertia terms and therefore must by itselfbalance the gradient in (effective) pressure: ageostrophic flow. For such a flow, (50) reduces to

(51)

Suppose the flow is horizontal: u= (u, v, 0 ), with zvertically upwards again. Then the horizontal com-ponents of the pressure gradient are given by

(52)

where Ωυ is the vertical component of the earth’sangular velocity (total angular velocity multipliedby the sine of the latitude). The pressure gradient isperpendicular to the velocity, or vice versa, indicat-ing that if there is a horizontal pressure gradient, thecorresponding geostrophic flow will be perpendicu-lar to it. This explains why the wind goes anticlock-wise round atmospheric depressions in the northernhemisphere (clockwise in the southern hemisphere).Similar flows occur in the oceans, although the bar-riers formed by the continents are impermeable,unlike in the atmosphere.

The condition for a steady flow to be geostroph-ic is that the inertia term (u.∇)u should be smallcompared with the Coriolis term. Thus if U is a typ-ical velocity magnitude, and L a length scale for theflow, the geostrophic approximation will be a goodone if

i.e. the Rossby number should be small:

U 2

L<< ΩvU,

∂p

∂x= −Ωυv,

∂p

∂y= +Ωυu

ρΩ∧u = −∇p.

ρΩ∧u

ρDuDt

+ ρΩ∧u = ρg −∇p.

cg =N

ksinθ sinθ ,0,− cosθ( ).

ω = N cosθ

y = Acos ωt − k x cosθ + z sinθ( )( )[ ],

g

ρdρdz

= −N 2 ,

ω 2 = gk ρ2 − ρ1( ) / ρ2 + ρ1( )[ ].

22 T.J. PEDLEY

(53)

If the Rossby number is large, the earth’s rotationcan be neglected. Note that the Rossby number isalways large at the equator, where Ω

v = 0.

Hydrodynamic instability

A smooth, laminar flow becomes turbulent as aresult of hydrodynamic instability. Small, randomperturbations are inevitably present in any real sys-tem; if they die away again, the flow is stable, but ifthey grow large, the original flow becomes unrecog-nisable and is unstable. Usually, steady flows whichare slow or weak enough are stable, but they becomeunstable above some critical speed or strength.

The way to investigate instabilities mathemati-cally is to assume that the disturbances to the steadystate are very small and to linearise the equationsaccordingly. Thus, if the steady state velocity, pres-sure and density are given by u0(x), p0 (x) and ρ0 (x)(all functions of position, in general) it is postulatedthat, with the perturbation, we have

where u′, p′ and ρ′ are small. Then these are sub-stituted into the governing equations, and termsinvolving squares or products of small quantitiesare neglected, so the equations are linearised. Forexample, equation (7) which, with (3b), is

becomes

(54)

and the nonlinear term u′∇ρ′ is neglected. Afterlinearisation, it is usually possible to think of thedisturbance as made up of many modes in whichthe variables depend sinusoidally on one or morespace coordinates and exponentially on time, e.g.

(55)

(cf 42), where and we are using complexnumber notation. Such terms are substituted into the

equations, and it turns out that a solution of the sup-posed form exists only if σ takes a particular value.If that value has negative (or zero) real part, the dis-turbance dies away (or oscillates at constant ampli-tude); if it has positive real part it grows exponen-tially, indicating instability. If any disturbance of theform (55) (i.e. for any values of k and l) grows, thenthe flow is unstable, because in general all distur-bances are present, infinitesimally, at first.

Consider, for example, the case of two fluids ofdifferent densities, one on top of the other. We haveseen that the frequency of a disturbance ofwavenumber k is given by equation (46) if ρ2(thedensity of the lower fluid) is greater than ρ1.However, if ρ1 > ρ2, ω

2 as given by (46) is negative.But if we replace ω by iσ, σ2 is positive, σ is real,and the oscillation cosωt can be written as 1/2(eσt +e-σt).Thus exponential growth is predicted. Hencethe interface between a dense fluid and a less densefluid below it is unstable.

A similar analysis can be performed for a contin-uous density distribution, denser on top, caused by atemperature gradient, say, in a fluid heated frombelow. In this case the diffusion of heat (and hencedensity) must be allowed for, as well as conserva-tion of fluid mass and momentum. For example, ahorizontal layer of fluid, contained between tworigid horizontal planes, distance h apart and main-tained at temperatures T0 (top) and T0 + ∆T (bot-tom) is unstable if the temperature difference ∆T islarge enough. More precisely, instability occurs ifa dimensionless parameter called the Rayleighnumber Ra exceeds the critical value of 1708,where

(56)

Here α, ν and κ are fluid properties, the coefficientof expansion, the kinematic viscosity (µ/ρ) and thethermal diffusivity respectively. When instabilityoccurs, for values of Ra not much greater than 1708,the resulting motion is a regular array of usuallyhexagonal cells (fig. 13), with fluid flow up in thecentre of the cells and down at the edges. Such amotion is an example of thermal convection, calledRayleigh-Benard convection. When Ra is muchhigher than 1708, the cells themselves becomeunstable, the convection becomes very complicatedand eventually turbulent.

Rayleigh-Benard convection is an example inwhich instability of the original steady state leadsto another, regular, steady motion which itself goes

Ra =gα∆Th3

νκ

i = −1

ρ' = f z( )exp i kx + ly( ) + σt

∂ρ′∂t

+ u0.∇ρ′+u′.∇ρ0 = 0

∂ρ∂t

+ u.∇ρ = 0,

u = u0 x( ) + u' x, t( ),p = p0 x( ) + p' x, t( ),ρ = ρ0 x( ) + ρ' x, t( )

U 2

Ωv L<< 1.

INTRODUCTION TO FLUID DYNAMICS 23

unstable as Ra is increased, and turbulence resultsonly after a whole sequence of such instabilities, orbifurcations. Other systems do not seem to haveintermediate stable steady states, but there is arapid transition from laminar to turbulent flowwhen critical conditions are passed. Perhaps themost familiar and important of such flows are uni-directional (or approximately so) shear flows, suchas that depicted in fig. 4. Examples are flow in astraight pipe and flow in the boundary layer on arigid body or in the shear layer at the edge of therecirculation behind it. Flow in a circular pipe ofdiameter D normally becomes turbulent when theReynolds number

where u– is the cross-sectionally averaged velocity,exceeds a critical value of just over 2000. Flow in aboundary layer on a thin flat plate (an approximationto a streamlined body) becomes unstable when theReynolds number based on the free stream velocityand the boundary layer thickness δs (eq. 36) exceedsabout 244. Flow in a shear layer is more unstablestill, associated with the fact that the velocity profilecontains an inflection point.

When numbers are put in to formulae such asthose quoted above, it becomes clear that oceanicflows are necessarily turbulent. Hence the existenceof this course.

REFERENCES

Koschmieder, E.L. – 1974 Adv. Chem. Phys.26:177-212Schlichting, H. – 1968. Boundary layer theory. (6th ed.). McGraw-

Hill, New York.Van Dyke, M.- 1982. An Album of Fluid Motion, The Parabolic

Press, Stanford.

Further reading - on theoretical fluid dynamics

Acheson, D.J. – 1990. Elementary Fluid Dynamics, OxfordUniversity Press.

Batchelor, G.K. – 1967. An Introduction to Fluid Dynamics,Cambridge University Press.

Greenspan, H. – 1968. The Theory of Rotating Fluids , CambridgeUniversity Press.

Lighthill, J. – 1978. Waves in Fluids, Cambridge University Press.Phillips, O.M. – 1966. The Dynamics of the Upper Ocean,

Cambridge University Press.Turner, J.S. – 1973. Buoyancy Effects in Fluids, Cambridge

University Press.

- on biological fluid dynamics

Caro, C.G., Pedley, T.J., Schroter, R.C. and Seed, W.A. – 1978. TheMechanics of the Circulation, Oxford University Press.

Childress, S. – 1981. Mechanics of Swimming and Flying,Cambridge University Press.

Lighthill, J. – 1975. Mathematical Biofluid dynamics, SIAM.

(The above dates refer to the first editions)Re =Du

ν,

24 T.J. PEDLEY

FIG. 13. – Photograph of convection pattern for Rayleigh-Benard con-vection in a layer of fluid heated from below. (Koschmieder, 1974).