Page 26 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

12 2 Simulating Steady-State Natural Convective Problems

∂Ψ
T
T e ∗ e ∗ T e
ϕϕ V dA + ϕK P dA + ϕK Raϕ T e 2 dA = 0, (2.23)
y y
A A ∂y ∗ A
T T 2 T 2 T
∂ϕ ∂ϕ ∂ ϕ ∂ ϕ
e
e
e
e
ϕu ∗ T dA+ ϕv ∗ T dA− ϕ λ ∗ ex T dA− ϕ λ ∗ ey T dA = 0.
A ∂x ∗ A ∂y ∗ A ∂x ∗2 A ∂y ∗2
(2.24)
Using the Green-Gauss theorem and the technique of integration by parts, the
terms involving the second derivatives in Eq. (2.24) can be rewritten as
∂ ϕ e ∂ϕ ∂ϕ e
2 T T
∗
ϕ λ ∗ ex ∗2 T dA =− λ ∗ ex T dA + ϕq n x dS = 0, (2.25)
x
A ∂x A ∂x ∗ ∂x ∗ S
∂ ϕ e ∂ϕ ∂ϕ e
2 T T
∗
ϕ λ ∗ ey ∗2 T dA =− λ ∗ ey T dA + ϕq n y dS = 0, (2.26)
y
A ∂y A ∂y ∗ ∂y ∗ S
∗
where q and q are the dimensionless heat fluxes on the element boundary of a unit
∗
x y
normal vector, n; A and S are the area and boundary length of the element.
Note that Eqs. (2.21), (2.22), (2.23) and (2.24) can be expressed in a matrix form
as follows:
⎡ e e e ⎤ ⎧ e ⎫ ⎧ e ⎫
M 0 −B x −A x ⎪ U ⎪ ⎪ F ⎪
⎪
0 M e −B e e ⎪ e ⎬ ⎪ x ⎪
⎨
⎨
e ⎬
y y
⎢ −A ⎥ V F
⎢ e y ⎥ e = e , (2.27)
0 0 E
⎣ 0 ⎦
⎪ T ⎪ ⎪ G ⎪
⎪
⎪
⎪
⎪
C e C e 0 0 ⎩ P e ⎭ ⎩ 0 ⎭
x y
e
e
where U and V are the nodal dimensionless velocity vectors of the element in the
e
e
x and y directions respectively; T and P are the nodal dimensionless temperature
e
e
e
e
e
e
e
e
and pressure vectors of the element; A , A , B , B C , C , E and M are the
x y x y x y
e
e
e
property matrices of the element; F , F and G are the dimensionless nodal load
x y
vectors due to the dimensionless stress and heat flux on the boundary of the element.
These matrices and vectors can be derived and expressed as follows:
T
∂ϕ ∂ϕ
e ∗ T e ∗ T e
A = K Ψ dA, B = ϕK Raϕ e 1 dA, C = Ψ dA,
x
x
x
x
x
A ∂x ∗ A A ∂x ∗
(2.28)
∂ϕ ∂ϕ
T
e T e T e
∗
∗
A = K Ψ dA, B = ϕK Raϕ e 2 dA, C = Ψ dA,
y y y y y
A ∂y ∗ A A ∂y ∗
(2.29)
∂ϕ ∂ϕ ∂ϕ
T
e ∗ ∗ e ∗ e ∗
D (u ) = ϕu dA, L = λ dA, F = σ ϕdS,
x ∗ x ∗ ex ∗ x x
A ∂x A ∂x ∂x S
(2.30)