Aula Teórica 7

Volume de Controlo e fluxos.Formulação Euleriana e Lagrangeana

Resultant force applied over a volume of fluid

dxdydzgPeso

dyy

dyyyx y

u

y

yxy y

u

General movement equation

ij

i

i

i gx

u

x

p

dt

du

2

2

ij

i

ij

ij

ii gx

u

x

p

x

uu

t

u

dt

du

2

2

This equation holds for a material system with a unit of mass. It is written in a LagrangianFormulation, i.e. one has to follow that portion of fluid in order to describe its velocity.That is not easy for us since we use to be in a fix place observing the flow, i.e. we are in an Eulerian reference .

Lagrangian vs Eulerian Descriptions

• Both describe time derivatives. • Lagrangian approach describes the rate of change of a

property in a material system, i.e. follows material as it moves.

• Eulerian describe the rate of change in one point of space.

• Lagrangian derivative registers changes independently of velocity. Eulerian registers changes also if fluid moves.

• In stationary systems local production balances transport.

Case of velocity

• Is there acceleration (rate of change of the velocity of a material system)?• How do momentum flux change between entrance and exit?• If the flow is stationary what is the local velocity change rate?• How does momentum inside the control volume change in time?• How does pressure vary along the flow?• What is the relation between momentum production and the divergence of momentum

fluxes?• What is the interest of using a lagrangian and an eulerian description?

Concentration

kcdt

dc

Fecal Bacteria dies in the environment according to a first order decay, i.e. the number of bacteria that dies per unit of time is proportional to existing bacteria. This process is describe by the equation:

C

t

C0

ktecc 0 This is a lagrangian formulation. This solution describes what is happening inside a water mass whether is moving or not.What happens in an Eulerian description?

Eulerian descriptionLet’s consider a river where the contaminated water would be moving as a patch (without diffusion)

t1

t2 t3

t4

Concentration decays as the patch moves. Time series in points x1 and x2 would be:

X1X2

C

t

X1

X2

Maximum concentration difference depends on decay rate while difference in time to show up and showing time reduce as flow velocity increases.

kcx

cu

t

c

Lagrangian vs Eulerian

• Examples of videos illustrating the difference between eulerian and lagrangian descriptions (not always very clear)

http://www.youtube.com/watch?v=zk_hPDAEdII&feature=related

http://www.youtube.com/watch?v=mdN8OOkx2ko&feature=related

Reynolds Theorem

• The rate of change of a property inside a material system is equal to the rate of change inside the control volume occupied by the fluid plus what is flowing in, minus what is flowing out.

dSnvdVoldt

ddVol

dt

d

VC SCsistema

.

Demonstration of Reynolds Theorem

SYS 2SYS 1

SYS 3

Let’s consider a conduct and 3 portions fluid (systems), SYS 1, SYS3 and SYS 3 that are moving.

Let’s consider a space control volume (not moving) that at time “t” is completed filled by the fluid SYS 2

CV

SYS 2SYS 1SYS 3

CV

Time = t

Time = t+∆t

Between time= t and time =(t+∆t) inside the control volume properties changed because some fluid flew in (SYS1) and other flew out (SYS2) and also if properties of those systems have changed.

Rates of change

t

BB tSYS

ttSYS

t

BB tvc

ttvc

tSYSt

vc BB 2

outflowflowinBB ttSYS

ttvc

2

In a material system:

Inside the control volume:

SYS 2 was coincident with CV at time t:

At time t+∆t:

Computing the budget per unit of time and using the specific property (per unit of volume)

dB

dV

dVB

outflowflowin

t

BB

t

BB tSYS

ttSYS

tvc

ttvc

22

How much is flowing in and out?

dAnvadvB .

The Mass discharge is the integral of the mass flowing per unit of area integrated over the area.

The volumetric discharge is the integral of the volume flowing per unit of area integrated over the area.

The Mass flowing per unit of area is the volume per unit of area times the mass per unit of volume.

dAnvadvB .

Using flux and specific property definitions integral equations can be written

dB

dV dVB

dVtt

dVdV

t

BB

ttt

tvc

ttvc

dVdt

d

t

BB tSYS

ttSYS 22

dAnvoutflowflowin .

If material is flowing in the internal product is negative and if is flowing out is positive. As a consequence:

And finally

dAn.vdVdt

ddV

t surfacesistemavc

dAnvdVt

dVdt

d

surfacevcsystem

.

Or:

If the Volume is infinitesimal

exitentrance AnvAnvVdt

dV

t

..

dAnvdVdt

ddV

t surfacesystemvc

.

k

k

xu

Vdt)(d

Vdt

)V(ddt)(d

Vdt

)V(d

But:

33

12

11

3213321

22312231

1321132

212121 333

xxx

xx

xxx

k

k

vxxvxx

vxxvxx

vxxvxx

x

uxxx

dt

dxxx

txxx

Dividing by the volume:

Becomes:

And thus:

Derivada total

jjk

k vxx

v

dt

d

t

k

k

j

j

x

v

x

v

tdt

d

jj xv

tdt

d

The Total derivative is the rate of change in a material system (Lagrangian description) ;The Partial derivative is the rate of change in a control volume (eulerian description) ;The advective derivative account for the transport by the velocity.