navier-stokes equations { 2d case - university of exeter · navier-stokes equations {2d case nse...

Navier-StokesEquations {2d case

NSE (A)

Equationanalysis

Laminar owbetween plates(A)

Flow downinclined plane(A)

Tips (A)

Navier-Stokes Equations { 2d caseSOE3211/2 Fluid Mechanics lecture 3

NSE (A)

Equationanalysis

Tips (A)

NSE (A)

� conservation of mass, momentum.

� often written as set of pde's

� di�erential form { uid ow at a point

� 2d case, incompressible ow :

Continuity equation :

+@uy@y

� conservation of mass

� seen before { potential ow

NSE (A)

Equationanalysis

Tips (A)

Momentum equations :

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

�+ fx

+ ux@uy@x

+ uy@uy@y

= �1

�@2uy@x2

+@2uy@y2

�+ fy

� (x and y cmpts)

� 3 variables, ux , uy , p

� linked equations

� need to simplfy by considering details of problem

NSE (A)

Equationanalysis

Tips (A)

The NSE are

� Non-linear { terms involving ux@ux@x

� Partial di�erential equations { ux , pfunctions of x , y , t

� 2nd order { highest order derivatives @2ux@x2

� Coupled { momentum equation involves p, ux , uy

Two ways to solve these equations

1 Apply to simple cases { simple geometry, simple conditions{ and reduce equations until we can solve them

2 Use computational methods { CFD(SOE3212/3)

NSE (A)

Equationanalysis

Tips (A)

The NSE are

� Non-linear { terms involving ux@ux@x

� Partial di�erential equations { ux , pfunctions of x , y , t

� 2nd order { highest order derivatives @2ux@x2

� Coupled { momentum equation involves p, ux , uy

Two ways to solve these equations

1 Apply to simple cases { simple geometry, simple conditions{ and reduce equations until we can solve them

2 Use computational methods { CFD(SOE3212/3)

NSE (A)

Equationanalysis

Tips (A)

Equation analysis (A)

Consider the various terms :

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

�+ fx

� change of ux at a point

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

�+ fx

ux@ux@x

+ uy@ux@y

� transport/advection term

� how does ow (ux ; uy )move ux?

� non-linear

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

�+ fx

@x� pressure gradient {usually drives ow

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

�+ fx

�@2ux@x2

+@2ux@y2

�� viscous term { e�ect ofviscosity � on ow

� has a di�usive e�ect

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

�+ fx

fx � external body forces { eg.gravity

NSE (A)

Equationanalysis

Tips (A)

Laminar ow between plates (A)

Fully developed laminar owbetween in�nite plates aty = �a

What do we expect from the ow?

� u = 0 at walls

� Flow symmetric around y = 0

� Flow parallel to walls

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

+ ux@uy@x

+ uy@uy@y

= �1

�@2uy@x2

+@2uy@y2

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

+ ux@uy@x

+ uy@uy@y

= �1

�@2uy@x2

+@2uy@y2

Flow parallel to walls { we expect

uy = 0;dp

dy= 0 and ux = ux(y)

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@y2

+ ux@uy@x

+ uy@uy@y

= �1

�@2uy@x2

+@2uy@y2

Flow parallel to walls { we expect

uy = 0;dp

dy= 0 and ux = ux(y)

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@y2

+ ux@uy@x

+ uy@uy@y

= �1

�@2uy@x2

+@2uy@y2

Flow fully developed { no change in pro�le in streamwise di-rection

@t= 0;

NSE (A)

Equationanalysis

Tips (A)

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@y2

+ ux@uy@x

+ uy@uy@y

= �1

�@2uy@x2

+@2uy@y2

Flow fully developed { no change in pro�le in streamwise di-rection

@t= 0;

NSE (A)

Equationanalysis

Tips (A)

So momentum equation becomes

0 = �1

dx+ �

d2uxdy2

Integrate once :

dx= ��

But at y = 0, duxdy

= 0 (symmetry), so C1 = 0.

Integrate again1

dx= ��ux + C2

But at y = �a, ux = 0, so

NSE (A)

Equationanalysis

Tips (A)

0 = �1

dx+ �

d2uxdy2

Integrate once :

dx= ��

But at y = 0, duxdy

Integrate again1

dx= ��ux + C2

NSE (A)

Equationanalysis

Tips (A)

0 = �1

dx+ �

d2uxdy2

Integrate once :

dx= ��

But at y = 0, duxdy

Integrate again1

dx= ��ux + C2

NSE (A)

Equationanalysis

Tips (A)

0 = �1

dx+ �

d2uxdy2

Integrate once :

dx= ��

But at y = 0, duxdy

Integrate again1

dx= ��ux + C2

NSE (A)

Equationanalysis

Tips (A)

0 = �1

dx+ �

d2uxdy2

Integrate once :

dx= ��

But at y = 0, duxdy

Integrate again1

dx= ��ux + C2

NSE (A)

Equationanalysis

Tips (A)

Final solution

ux(y) =1

2��

�y2 � a2

� dpdx

{ equation of a parabola

Also, remember that

� = �@ux@y

So from this we see that in this case

� = ydp

NSE (A)

Equationanalysis

Tips (A)

Flow down inclined plane (A)

{ Flow of liquid down inclined plane

Take x-component momentum equation

+ ux@ux@x

+ uy@ux@y

= �1

�@2ux@x2

+@2ux@y2

�+ fx

NSE (A)

Equationanalysis

Tips (A)

Note :

1 Steady ow

2 ux(y) only

3 No pressure gradient

4 fx = g sin�

Equation becomesd2uxdy2

= �g

�sin�

which we can integrate easilly.

NSE (A)

Equationanalysis

Tips (A)

Note :

1 Steady ow

2 ux(y) only

3 No pressure gradient

4 fx = g sin�

Equation becomesd2uxdy2

= �g

�sin�

which we can integrate easilly.

NSE (A)

Equationanalysis

Tips (A)

Boundary conditions :

� lower surface { ux(0) = 0

� upper surface { duxdy

Solution

�sin�

�hy �

NSE (A)

Equationanalysis

Tips (A)

Boundary conditions :

� lower surface { ux(0) = 0

� upper surface { duxdy

Solution

�sin�

�hy �

NSE (A)

Equationanalysis

Tips (A)

Tips (A)Most NSE problems will be time-independent. They willprobably only involve one direction of ow, and one coordinatedirection. They will probably be either pressure driven (so noviscous term) or shear driven (ie. viscous related, so nopressure term).

Thus, most NSE problems will lead to a 2nd order ODE for avelocity component (ux or uy ) as a function of one coordinate(x or y).

Thus we would expect to integrate twice, and to impose twoboundary conditions.

� A wall boundary condition produces a �xed value : eg.ux = 0.

� A free surface produces a zero gradient condition, eg.duxdy

navier-stokes equations { 2d case - university of exeter · navier-stokes equations {2d case nse...

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