Study of Incompressible Flow Over a Backward Facing Step Using a Triangular Penalty Element
The purpose of this study is to assess the numerical efficiency of the penalty formulation and the Newton-Raphson solution strategy for solving a particular problem of laminar flow over a backward facing step. A 7-node triangle with 14 df is employed to o
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U grad U. + 1/p grad p 1
Div • U - p/A • 0
v t:. u.1
= f.
1
on V
l/A + 0
with relevant boundary conditions on
(I)
V
and U = or velocity components ; p =pressure ; p = density (mass/volume) ; f • volume forces ; v = cinematic viscosity ; A= penalty parameter
(106 '1'2, '1'3> 'l'p (that is'¥, 'l'p and U,p belong to identical functionnal spaces). We perform as well necessary integration by parta in order to reduce order of derivatives on U and p (no pressure derivative appear in the integral).
w..
Jvr[i=li
• p
dV +
('¥.
r 1
)v
'¥
u. p
grad
u.
1
+
v
grad'¥ . • grad 1
(div • U- p/A) dV +
r
Jav
u. 1.
'¥. f.)]- 1/p div • '¥ l.
1
('¥ • p - IJ'. llU./lln)ds 1 n l.
(2)
The solution of (2) approaches that of problem (I) as A+ oo, the variational formulation satisfying LBB condition in the continuum space. 1.1. Finite element discretisation: The finite elements-discretisation leads to W .. l: we • 0 where we is evaluated on each element. For the mixed formulation (2) ., the velocity-pressure finite element approximations should not only satisfy the necessary continuity requirement but the discretised equivalence of LBB condition as well for numerically stable solutions. We choose cO approximation for velocity and c-1 for pressure (discontinuous field). The approximations are '¥1 • {~}, ul • {u~} and similar approximation for ('¥ 2 ,
(3-a)
Uz)
and ('1'3, u 3 )
K. Morgan et al. (eds.), Analysis of Laminar Flow over a Backward Facing Step © Springer Fachmedien Wiesbaden 1984
163
'I' p
= {V'}, p p
=
p
{pn}
(3-b)
The discretised representation of We
[k] {If} with
(4-b) [k]
= fk*]
-1 ] [cT]
- A [c] [m
{5)
One may remark that for constant pressure approximation, [m] is a scalar (area or volume of element) and for linear pressure approximation, [m] is diagonal if three pressure nodes are located at three integration points (T7L element). In order to obtain correct solution, velocitypressure approximation should lead to discretised formulation satisfying LBB condition : dv Sup """'"'---"'-~~- > a II Ph II 0
lit
(6)
where a is a positive constant independant of element size ph are finite element approximations of U, p.
h
and
Uh'
1.2. Triangular element T7L : Among a family of triangular elements presented in [1], we retain a rather expensive but precise element T7L for this study [2].
n~ ~-nt 'tt~
ou,v
The discontinuous pressure field is represented by a linear variation over each element (p = a + b~ + en) and the continuous field for velocity components is given by an incomplete 7 terms cubic approximation (1, ~. n, ~2 ,-n2, ~n. {I-~- n) ~n). The element has 14 df representing velocity components, three pressure terms are eliminated at element level (4-b, 5). Only necessary velocity components need be specified at the boundary, with no explicit restriction on pressure terms. The evaluation of pressure field is obtained from the velocity components via eqs (3-b, 5-b). That is, at a point (~,n) of an element : -1 T _..n (7) p{~,n) ,.._ (A [m] [c] {u }) p
The nodal press
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