Finite Element Techniques for Fluid Flow

VISCOUS

INCOMPRESSIBLE FLOW PROBLEMS 299

9.4 PRESSURE AND VELOCITIES APPROACH

Let us consider again the variational statement of equation

(9.7),

i.e.

SSS[

{-pôe

v

- e

v

ôp +

Tijôetj}

+ ρ{-^ + -^\ δυ

ί

= IJ{p

n

Sv

m

+ pMdS (9.21)

s

p

One can now substitute for τ

ι;

from equation (9.3) to obtain

J J J \\ -P

àe

v

- e

v

δρ + 2μβ

υ

ôe

u

- \μβ

ν

ôe

v

- pb

t

δν

ί

+

P

i

v

J

v

u

+ ff M

dK

- |J

{P-

*>»

+ Λ

δυ

°ϊ

dS = 0

for arbitrary δρ,δν^ Note that the pressure acts as a Lagrangian

multiplier in order to impose the continuity condition.

Consider now that

p=4>V\ ν,=φ

Γ

ν7 (ι =

1,2,3)

(9.23)

where p" is the element nodal pressure matrix and v? the nodal

velocity matrix. We ar ...