3 Introduction to Computational Fluid Dynamics and Ocean Modelling
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in fluid mechanics. The time derivative
DF
Dt
=
∂F
∂t
+ u · ∇F,
(3.21)
where u = (u, v, w) is a velocity vector in three dimensions, is called the material
derivative.
The motion of Newtonian fluid in a fixed frame of reference is governed by equations expressing the conservation of mass
Dρ
Dt
+ ρ∇ · u = 0
(3.22)
and the conservation of momentum given by the Navier–Stokes equations
ρ
Du
Dt
= −∇p + μμu + ρg N ,
(3.23)
where μ is the molecular viscosity and g N is the Newtonian gravity acceleration.
The acceleration on a fixed (F) and rotating (R) frame of reference are related by
a F = a R − Ω
2 R + 2Ω × u R ,
where Ω is the angular velocity of the rotating reference frame, R is the radius of the
rotation and u R is the velocity (Fig. 3.15). The two additional terms on the right hand
side of the equation represent the centripetal acceleration −Ω 2 R, and the Coriolis
acceleration 2Ω × u R . The conservation of mass (3.22) remains unchanged when
we move to the rotating frame of reference, but the Navier–Stokes equations (3.23)
become
ρ
Du
dt
= −∇p + μμu + ρg − 2ρΩ × u,
(3.24)
where we have dropped the subscripts R for convenience, and defined the effective
gravity acceleration as
g = g N + Ω
2 R.
The angular velocity for the rotation of the Earth is
Ω = 2π rad/day ≈ 0.73 × 10
−4 1/s.
A local Cartesian system on a tangent plane can be applied if the length scale of the
model domain is small compared to the Earth’s radius (6371 km) (Fig. 3.15). The
angular velocity components in the Cartesian system of reference are
Ω x = 0,
Ω y = Ω cos θ,
Ω z = Ω sin θ,