this user's guide - Lammps

SPH-USER Documentation SPH Theory 6
This identity can be rearranged to provide a starting point for the SPH discretisation of the time
evolution of the particle velocity,
Inserting the above line into Eqn. (2), we obtain
The spatial derivatives, ∇ρ and ∇· P
ρ
of a variable field, Eqn. (5):
− 1 ρ ∇ · P ≡ −∇ · P
ρ + P · ∇ρ. (8)
ρ2 dv
dt = − P ρ 2 · ∇ρ − ∇ · P
ρ . (9)
can be discretised using the SPH expression for the gradient
∇ρ = ∑ j
m j ∇ j W ij (10)
∇ · P
ρ
= ∑ j
P j
m j
ρ 2 ∇ j W ij (11)
j
The equation of motion for particle i now reads
dv i
dt = −P i
ρ 2 i
· ∑
m j ∇ j W ij − ∑
j
j
P j
m j
ρ 2 ∇ j W ij , (12)
j
and is immediately written as an expression for pair-wise forces, suitable for implementation in
an Molecular Dynamics code:
dv i
f i = m i
dt = − ∑ ( )
P i
m i m j
ρ 2 + P j
j
i ρ 2 ∇ j W ij . (13)
j
It is evident that this expression for the force is antisymmetric due to the antisymmetry property
of the SPH gradient. It therefore follows that this SPH discretisation preserves total linear
momentum.
3.3. SPH approximation of the Navier-Stokes continuity equation
The continuity equation, Eqn. (1), contains the gradient of the velocity field. As above, we begin
the SPH discretisation by using the identity
which enables us to rewrite the continuity equation as
∇(vρ) = ρ∇v + v∇ρ, (14)
dρ
dt
= ∇(vρ) − v∇ρ. (15)
Applying the SPH discretisation of the gradient of a vector field, Eqn. (5), we obtain:
dρ i
dt = ∑ j
∑
m j v j ∇ j W ij − v i m j ∇ j W ij = − ∑
j
j
3.4. SPH approximation of the Navier-Stokes energy equation
m j v ij ∇ j W ij (16)
In order to derive an SPH expression for the time-evolution of the energy per unit mass, one
can, proceed in analogy to the above steps by evaluating the divergence of the RHS of Eqn. (3).
Here, we only quote the final result:
m i
de i
dt = −1 2
∑
j
(
P i
m i m j
ρ 2 + P j
i ρ 2 j
)
: v ij ∇ j W ij − ∑ j
m i m j (κ i + κ j )(T i − T j )
r ij · ∇ j W ij (17)
ρ i ρ j
r 2 ij