ACME 2011 Proceedings of the 19 UK National Conference of the ...

where i Σ is the boundary for phase boundary i Γ , and it is assumed that h hs
⌢
⌢
ℓ = , which is a continuity
'
h ⌢ satisfying equation (10). The jump terms in equation (10) are
condition invoked by
+ + +
] J ⋅ n[ = Js ⋅( − ns ) + Jℓ ⋅( −n ℓ ) and ⎤ρψ ( v − v ) ⋅ n⎡ = ρsψs ( vs − v ) ⋅( − ns ) + ρ ψ ( vℓ − v ) ⋅( −nℓ
⎦ ⎣ )
where subscripts s and ℓ denote different phases.
5 WEIGHTED TRANSPORT EQUATIONS
Consider the following weighted-transport equation,
* *
( ) ( )
ℓ ℓ ,
*
D
Wρψ dV+ Wρψ v− v ⋅ndΓ− ρψ v− v ⋅∇ WdV = − WJ⋅ ndΓ+ ∇W⋅ JdV + ρWbdV
*
D t Ω Γ Ω Γ Ω Ω
∫ ∫ ∫ ∫ ∫ ∫ (11)
* *
where W is transported invariantly with Ω , i.e. D W D t = 0 .
The equation is arrived at by the introduction of W into the integrals in equation (1) but also by
subtracting associated domain integrals involving a derivative of the weighing function for each boundary
integral appearing in (1). The form the additional terms take is as a consequence of the divergence
theorem applied to the weighted boundary terms. Note also that spatial and temporal derivatives of ψ is
avoided, making (11) applicable when a discontinuity is in Ω . Moreover note that on setting W = 1,
equation (2) is returned, which is in an appropriate form for the finite volume method (FVM). Note
however that on applying (11) to an element domain Ω e and adopting a standard Galerkin weighting
gives a finite element formulation in transport form. The full system of FE transport equations with
multiple discontinuities removed are:
D
*
⌢
N ψ dV = ∇N ⋅JdV − N J ⋅ ndΓ + ρN bdV −
∫ ∫ ∫ ∫
*
i i i i
D t Ωe { Γk :k∈Ke }
Ωe Γe Ωe
D
×
k
' '
∑ × i i
k K D e e
e kt
∫ ∑ ∫
∈ Γ k∈K k
e Σk
* × ( k )
⌢ ⌢
N ψ dV − N ψ v − v ⋅ tndΓ
where e / { Γk
: k ∈ K e}
K { k : Ω ∩ Γ ≠ ∅}
which is a subset of { : k 1:
K}
Ω signifies that integration is in the sense of Lebesgue, and where
e = e k
(12)
k = . The velocity fields k
v× track
discontinuities in elements but for convenience match movement of the element boundary. Note that the
LHS of equations (12) is continuous and consequently so is the RHS, so discontinuities have been
annihilated. The evolution of the source term is determined with the following source transport equation:
D
D t
⌢ ⌢
* × ×
( ) ⎤
k ( k ) ⎡ ] [
×
k
×
k
' '
∫ ψ dΓ + ∫ ψ
e e
Γk ∑k v − v ⋅ tn dΣ = ∫ ρψ
⎦ e
Γk v − v ⋅ n dΓ = −
⎣ ∫e
Γk
J ⋅n dΓ
CONCLUSION
The temporal derivatives of ψ ⌢ and ψ are avoided, making the governing equations applicable when a
discontinuity is in Ω . The continuous and source like behaviour of ψ ⌢ facilitates the precise removal of
discontinuities from the governing system of transport FE equations. It is early days for the theories
presented but the transport equations possess conservative properties and capture discontinuous
behaviour.
References
[1] Davey, K and Mondragon, R, A Non-Physical Enthalpy Method for the Numerical Solution of
Isothermal Solidification, Int. Journal for Numerical Methods in Engineering, 2010. 84, pp. 214-
252.
[2] Mondragon, R and Davey, K, Weak Discontinuity Modelling in Solution Modelling, Computers
and Structures, 2011. 89, pp. 681-701.
(13)
12