General Transport Equations

Dimensional analysis helps us to understand much about a

system, especially what is important and what is not;

however, if we are to be able to calculate things when

forming a design, we need to formulate equations.

Fluid mechanics are based on conservation or transport

equations. The things being transported by a moving fluid

are mass, momentum, and energy.

Exact equations can be derived by examining a small

volume of fluid. This will be done here for a Cartesian

system, but the same equations can be derived for an

arbitrary coordinate system.

Resources

• White, chapter 4, sections 4.1 to 4.6

• Alexandrou, chapter 5

• Virtual wind tunnel: http://raphael.mit.edu/Java/

• More links are available on the website

Use of general equations

• Not all systems fit into neat categories.

• Some systems are complicated.

• Many different factors that can change completely the

methods used to model a system and subsequent results.

• These factors include: unsteadiness (changes with time);

non-uniformity (changes in space); compressibility

(changes in density); stresses („viscosity‟); No. dimensions.

Example: turbofan engine

Non-uniform and everything changing: how can we cope

with it?

Source: http://en.wikipedia.org/wiki/Image:Turbofan_operation.png

Computational Fluid Dynamics

(CFD)

Meshing

Further information about CFD available via web site.

Gradients

x

f(x)

changes with x

x

f

Over small distances, the curve is

nearly straight, the gradient is

approximately constant and

How does CFD calculate flows?

• Divide the flow into very small elements.

• Over a small element, gradients are approximately equal to the

linear gradient over the small element.

• For each element, solve the simplified equations for the fluid

flow through them.

• Equations express conservation of mass, momentum, and

(sometimes) energy.

• Solution for one element feeds into the solution for the

neighbours.

• At the edge of the fluid flow, boundary conditions are

required.

• If the flow is unsteady (changes with time), initial conditions

are required.

Gradients in more than one

dimension

Gradient= tan

Gradient zero

Gradient between zero and tan

i.e. Gradient depends on direction and

therefore is a vector

Gradient operatorGradient is then represented by a vector operator , pronounced „grad‟. It is called an operator because it cannot appear on its own, it

always needs to operate on a variable e.g.

In a Cartesian coordinate system (different forms exist for other

coordinate systems- see handout.

Or, for a vector u=u(u, v, w),

There is also the Laplacian operator

One-dimensional mass transport

Summary

• Control volume small enough for all changes

over it to be approximately linear.

• Control volume small enough for the product of

changes to be negligible (δ20).

• Compare mass flow rate in with mass flow out.

• Compare result with rate of change of mass in the

control volume.

(Remember “rate” means gradient with time)

Three-dimensional mass transport

Fluxes

Mass is just one of the quantities that can be transported

by a fluid.

Volume flow rate of fluid through control volume in x-

direction= speed area = u y z

If is a quantity per unit volume transported by the fluid,

then its rate of transport in the x-direction= u y z.

The flux of (flow rate per unit area) in the x-direction=

u.

Therefore generally, the flux of a quantity per unit

volume transported by a fluid is

Different fluxes

e ueueEnergy

u uuuuMomentum

uuMass

Flux in

three-

dimensions

Flux in x-

direction

Quantity

per unit

volume

Conservation of Momentum

Newton‟s Second Law of Motion

Rate of change of momentum=sum of forces

Momentum flux per unit volume= uu

Rate of change of momentum per unit volume in the

control volume

Derive this from first principles as for the

conservation of mass.

Total derivative

The total, material, or substantial derivative is

given by

Necessary for field variables i.e. those that change with

time and position = (x,t).

Rate of

change of

temperature

= / t

Rate of change

of temperature

=

(dx/dt)( / x)x

Total derivatives

g

Act on all the fluid

within the control

volume

Body forces

Force per unit volume

Static forces

Act on all the surface

of the control volume.

Act whether the fluid

is moving or not.

Far face hidden.

p

p

p

p+( p/ x) x

p+( p/ y) y

xz

y

Direct stresses and shear stresses

Direct stresses act in a direction perpendicular to the

surface of a control volume. In a fluid they can be

static or dynamic.

Shear stresses act in a direction parallel to the surface

of a control volume. It acts as if to shear a control

volume. Only a dynamic force in a fluid i.e. the fluid

has to be moving.

xx

xy

xz

zx

zy

zz

yx

yy

yz

Act on the surface

of the control

volume only when

the fluid is moving

Note can be both direct

stresses and shear

stresses.

y

z

x

yx+ yx

yy+ yy

yz+ yz

xx+ xx

xy+ xy

xz+ xz

Far face hidden

Dynamic forces

Dynamic surface forces in the x-

direction

xx

yx

zx

x

y

First subscript refers to the face on which the stress is acting.

The second subscript refers to the direction of the stress.

Stress Tensor

The stress tensor is the combination of the static and dynamic

surfaces forces. In many engineering situations, this is the difficult

thing to specify e.g. turbulent flows, multiphase flows, non-

Newtonian fluids.

zzyzxz

zyyyxy

zxyxxx

p

p

p

σ

Equation for the conservation of

momentum

Advective

component

Static

surface

forces

Dynamic

surface

forces

Body

forces

Navier-Stokes relations

Assumes fluid is Newtonian i.e. stress strain and isotropic.

Practical momentum equations

• The Navier-Stokes equations are exact, but

unsolvable; therefore, they need to be simplified

for practical engineering calculations.

• Different systems require different

simplifications.

• As a rule of thumb, you want to have the simplest

equations possible to describe a particular

system.

Simplifications (1)• Viscosity ( )= constant. Not true if significant changes in

temperature.

• Density ( ) =constant i.e.the flow is incompressible. Not true if there are large temperature variations or the flow is fast moving.

Mass

Momentum

These are often the equations solved by CFD packages

and misnamed as the Navier-Stokes equations.

Simplifications (2)

• =0 i.e.fluid is inviscid. Means that forces owing

to viscosity are negligible, not viscosity is negligible.

gu

pDt

DEuler‟s equation

Simplifications (3)

• u/ t=0 i.e. flow is steady.

• Number of dimensions.

One-dimensional, steady, incompressible,

inviscid flow results in Bernoulli equation.

Show this to yourself.

Example

A confectionary manufacturer is designing a machine for the icing of biscuits, as shown in figure Q1. Starting from the general momentum equation describe the factors that have to be taken into account to obtain a working momentum equation, and why these are relevant to this particular situation. All relevant factors should be taken into account, but you should pay particular attention to the stress tensor.

Icing

Biscuit

Figure Q1

Complications

• Multiphase flows

• Non-Newtonian fluids

• Turbulence

In particular, the difficulty is stating and solving the

stress tensor.