Introduction - Turbulence in the Atmosphere - page 11

cited as the first LES. DNS and LES, which now use as many as 4096 3 10 10 6 ×

grid points, are leading research tools in turbulence today.

1.5 The equations of turbulence
In the Eulerian description one expresses the fluid velocity u , for example, as
a function of position x in a fixed (relative to the earth) coordinate system, and
time t . One then seeks u ( x ,t) in the flow domain of interest. In the Lagrangian
description, one labels each fluid parcel (with its initial position a , for example)
and seeks its velocity history v ( a ,t) . We will use the Eulerian description almost
exclusively; the one exception is Taylor's solution for dispersion of effluent from a
continuous source, Chapter 4 .
In Part I we shall use the equations for fluids of time-independent, uniform
density (which we shall call simply constant density ), because they contain the
essence of the physics of turbulence. These equations are derived and discussed in
graduate-level fluid mechanics texts (e.g., Kundu , 1990 ). Buoyancy effects stem-
ming from density variations due to heat transfer or phase change can strongly
influence turbulence, particularly in the atmosphere; this is the focus of Part II .
In cartesian tensor notation the fluid continuity or mass-conservation equa-
tion is

∂ρ
∂t +

∂ρu i
∂x i =

0 , (1.17)

where ρ is fluid density, x i = (u 1 ,u 2 ,u 3 )

is velocity. We use the convention that repeated Roman indices are to be summed
over 1, 2, and 3; if we do not wish to sum we use Greek indices. Equation (1.17)
says that at any point in space the time rate of change of fluid density plus the
divergence of the fluid mass flux ρu i is zero. We call ρu i an advective flux, a vector
that represents the amount of fluid mass flowing through unit area per unit time.
In general there is also a molecular flux that represents the diffusive effect of the
random molecular motion, but there is no molecular diffusion of fluid density.
When the fluid density is constant Eq. (1.17) reduces to

(x 1 ,x 2 ,x 3 ) is spatial position, and u i =

∂u i
∂x i =

∂u 1
∂x 1 +

∂u 2
∂x 2 +

∂u 3
∂x 3 =

0 , (1.18)

meaning that the velocity divergence is zero. A fluid that satisfies Eq. (1.18) is
called incompressible - its density does not change with pressure. Gases at low
speeds and liquids are usually treated as incompressible.
Newton's Second Law for a fluid is
ρ Du i

ρ ∂u i u j ∂u i

∂x j

∂p
∂x i +

∂σ ij
∂x j Dt = ∂t +

=− ρg i + , (1.19)