A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

where the mean quantities are defined by these time averages:
2. Governing Equations and Boundary Conditions 23
t+ T
t + T
t+ T
t+ T
u
i
= ∫ --
1
ũ
T i
t
dt
∫
1
, p = -- p˜ dt
T
t ∫
1
, ρ = -- ρ˜ dt
T
t ∫
1
, s = -- s˜ dt
T
t
. (2.13)
The size of the averaging interval T is difficult to prescribe exactly, but in general it must be long compared to the time scale
of the individual fluctuations but small compared to the time scale at which the unsteady mean flow is varying. In estuaries where
the astronomical tides are a primary source of unsteadiness, the mean flow can vary significantly in just 15 minutes. The minimum
required value for T will vary between flows and can truly be determined only by experiment. Indeed, a criticism of the time aver-
age is that T cannot always be chosen to include all of the time scales of the stochastic (turbulent) motion. An argument also can
be made that an average should be taken in space as well as time to average the instantaneous quantities over the spatial turbulent
eddy structure. The Reynolds time average, however, still is used extensively in practice and has not yet been shown to be
inadequate in a significant way.
Substituting the relations 2.12 into equations 2.4, 2.9, and 2.10 and time averaging each term in the way indicated by 2.13
yields the mean-flow and transport equations (For details, see most texts on fluid mechanics or turbulence such as Sabersky,
Acosta, and Hauptmann, 1971; Hinze, 1975; and Tennekes and Lumley, 1972):
Continuity equation,
Momentum equation,
Salt transport equation,
Equation of state,
∂u
i
------ = 0 ; (2.14)
∂x
i
∂u
i
∂u
i
ρ ------
0
ρ
∂t
0
uj ------
∂p
∂ ⎛ ∂u
i ⎞
+ = – ------ – 2ε Ω
∂xj
∂x
ijk jρ 0
uk + ρg
i
+ ρ
0 ⎜ν------ – u
∂x
i
j ∂x
i
′ uj ′ ⎟ ; (2.15)
⎝ j ⎠
∂s
∂u
i
s ∂ ∂s
---- + --------- = ⎛λ------ – u
∂t
∂x
∂x
i i
∂x
i
′ s′ ⎞ ; (2.16)
⎝ ⎠
i
ρ = f( s, Θ)
. (2.17)
These equations are identical in form to the instantaneous equations except for the extra terms u i ′ u j ′ and u i ′ s′ that have been
introduced by the Reynolds averaging process (the overbar represents the Reynolds time average). These terms originate with the
nonlinear advection terms and are non-zero because correlations exist between the fluctuating velocities and between the velocities
and salt fluctuations. In a physical sense, these terms, when multiplied by density, represent fluxes of momentum and salt
associated with the turbulent fluctuations. The momentum flux acts as an effective stress on the fluid that is analogous to the
viscous stress in gases caused by momentum flux associated with molecular fluctuations. The presence of the heretofore
undetermined turbulent stress and salt transport terms in the mean flow equations make the number of unknowns greater than the
number of equations. Resolving this imbalance between equations and unknowns is the problem of turbulence closure.