Page 32 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

18 2 Simulating Steady-State Natural Convective Problems
∗
x
∗
T = θ(y ) cos q + (1 − y ) (q = mπ, m = 1, 2, 3, ......). (2.56)
∗
∗
L ∗
Substituting Eqs. (2.55) and (2.56) into Eqs. (2.53) and (2.54) yields the follow-
ing equations:
q q
2
∗ ∗ ∗
f (y ) − f (y ) = Raθ(y ), (2.57)
L ∗ L ∗
q q 2

∗
∗
∗
f (y ) =− θ(y ) + θ (y ). (2.58)
L ∗ L ∗
Combining Eqs. (2.57) and (2.58) leads to an equation containing f (y ) only:
∗
q q q
2 2 2
IV
∗
∗

∗
f (y ) − 2 f (y ) − Ra − f (y ) = 0. (2.59)
L ∗ L ∗ L ∗
It is immediately noted that Equation (2.59) is a linear, homogeneous ordinary
differential equation so that it has a zero trivial solution. For the purpose of find-
ing out a non-zero solution, it is noted that the non-zero solution satisfying both
Equation (2.59) and the boundary conditions in Eqs. (2.46), (2.47) and (2.48) can
be expressed as
∗
∗
f (y ) = sin(ry ) (r = nπ, n = 1, 2, 3, ......). (2.60)
Using this equation, the condition under which the non-zero solution exists for
Eq. (2.59) is derived and expressed as
L ∗ 2 q n m 2
2 2 2
∗
Ra = r + = L + π
q L ∗ m L ∗ (2.61)
(m = 1, 2, 3, ......, n = 1, 2, 3, ......).
∗
It can be observed from Eq. (2.61) that in the case of L being an integer, the
2
∗
minimum Rayleigh number is 4π , which occurs when n = 1 and m = L .How-
2
∗ 2
∗
ever, if L is not an integer, the minimum Rayleigh number is (L + 1 / L ) π ,
∗
which occurs when m = 1 and n = 1. Since the minimum Rayleigh number deter-
mines the onset of natural convection in a fluid-saturated porous medium for the
Horton-Rogers-Lapwood problem, it is often labelled as the critical Rayleigh num-
ber, Ra critical .
For this benchmark problem, the mode shapes for the stream function and
related dimensionless variables corresponding to the critical Rayleigh number can
be derived and expressed as follows:
mπ

Ψ = C 1 sin x ∗ sin(nπy ), (2.62)
∗
L ∗