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2.3 Reynolds-Averaged Navier-Stokes Equations 55

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(3 X = ua xx + va xy xyi + G ^ f a ) + Vx^-{a )
P r ( 7 - 1 ) dry
(2.2.48)
Ua xy + VOyy + ^ _

2.3 Reynolds-Averaged Navier-Stokes Equations

The Navier-Stokes equations of the previous section also apply to turbulent
flows if the values of fluid properties and dependent variables are replaced by
their instantaneous values. A direct approach to solving the equations for tur-
bulent flows is to solve them for specific boundary conditions and initial values
that include time-dependent quantities. Mean values are needed in most prac-
tical cases, so an ensemble of solutions of time-dependent equations is required.
Even for the most restricted cases, this approach, referred to as direct numer-
ical simulation (DNS) and discussed in [6], becomes a difficult and extremely
expensive computing problem because the unsteady eddy motions of turbu-
lence appear over a wide range of scales. The usual procedure is to average the
equations rather than their solutions, as discussed in [2,3,7].
In this section and following sections, we shall consider the differential form
of the conservation equations and, for simplicity, restrict the discussion to in-
compressible flows. The treatment of compressible flow equations is similar but
rather lengthy. Whenever appropriate, the governing equations for compressible
flows will be given without derivation.
In order to obtain the conservation equations for turbulent flows, we replace
the instantaneous quantities in the equations by the sum of their mean and
fluctuating parts. For example, the instantaneous values of the u-, v- and w-
velocities are expressed by the sum of their mean iZ, v, w and fluctuating parts
7
u\ v' and w\ and the temperature T by T and T , that is,
u = u-\- u , v — v + v w = w + w T = T + T' (2.3.1)

With the help of the continuity equation, (2.2.1), one can now write the
left-hand sides of the momentum and energy equations in conservation form
and introduce the above relations into the continuity, momentum and energy
equations. After time averaging and making use of the substantial derivatives
given by Eq. (2.2.5), the Reynolds averaged Navier-Stokes (RANS) equations
for three-dimensional incompressible flow can be written in the following form:
du dv dw
(2.3.2)
dx dy dz

Du dp 72,- ,2) d
+ /iV u + 4<" Q—{u'v') U W (2.3.3)
} gf x Q-Q- ZK ' ')
Di dx dy