Lecture 8

Control Volume & Fluxes.Eulerian and Lagrangian Formulations

Resultant force applied to a volume of fluid

dxdydzgweight

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. (It is the unique formulation to describe the movement of bodies)

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

• In stationary systems eulerian derivative is null meaning that local production balances transport.

Case of velocity

In this flow:• Is there acceleration (rate of change of the velocity of a material system)?• How does momentum flux change between entrance and exit?• If the flow is stationary what is the local velocity change rate (eulerian derivative)?• 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?• Can we say that lagrangian description is better then eulerian description or vice-versa?

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 between X1 and X2 depends on decay rate while the slope of the curves increase with flow velocity.

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

dSn.vdVoldt

ddVol

t VC SCsystem

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 can change because some fluid flew in (SYS1) and other flew out (SYS2) and also because properties of those systems have changed in time.

Rates of change

t

BB tSYS

ttSYS

t

BB tvc

ttvc

In a material system:

Inside the control volume:

tSYSt

vc BB 2

SYS 2 was coincident with CV at time t: SYS 2SYS 1

SYS 3CV

outflowflowinBB ttSYS

ttvc

2

At time t+∆t:

SYS 2SYS 1SYS 3

CV

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

dVtt

dVdV

t

BB

ttt

tvc

ttvc

Identically for the material System

dB

dV dVB

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 surfacesystemvc

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:

Total derivative

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.

Evolution Equation

dAn.vdVdt

ddV

t surfacesistemavc

The rate of change inside the system is the (Production-Destruction) + (diffusion exchange). Designating Production – Destruction by (Sources – Sinks) and knowing that:

dAn.Diffusion

dAnnvSSdVt surface

ivc

..0

Differential Equation

dAnnvSSdVt surface

ivc

..0

..0 vsst i

iii

iiii

i

iiiii

ssxxdt

d

ssxxx

vt

xxxvss

t

0

0

0