Reynolds averaging and turbulent stress

Turbulent flow is a complex motion that is fundamentally three-dimensional and highly unsteady. Figure 7.34a depicts a typical variation of a flow variable,/, such as velocity or pressure, with time at a fixed point in a turbulent flow. The usual approach in engineering, originating with Reynolds*, is to take a time average. Thus the instantaneous velocity is given by

/=/+/

where the time average is denoted by ( ) and ( )’ denotes the fluctuation (or deviation from the time average). The strict mathematical definition of the time average is

(7.100)

where to is the time at which measurement is notionally begun. For practical meas­urements T is merely taken as suitably large rather than infinite. The basic approach is often known as Reynolds averaging.

f

V

f

(a)

(b)

Fig. 7.34 * Reynolds, O. (1895) ‘ On the dynamical theory of incompressible viscous fluids and the determination of the criterion’, Philosophical Transactions of the Royal Society of London, Series A, 186, 123.

We will now use the Reynolds averaging approach on the continuity equation (2.94) and x-momentum Navier-Stokes equation (2.95a). When Eqn (7.99) with и for / and similar expressions for v and w are substituted into Eqn (2.94) we obtain

Taking a time average of a fluctuation gives zero by definition, so taking a time average of Eqn (7.101) gives

We now take a time average of each term, noting that although the time average of a fluctuation is zero by definition (see Fig. 7.34b), the time average of a product of fluctuations is not, in general, equal to zero (e. g. plainly u’u’ = ua > 0, see Fig. 7.34b). Let us also assume that the turbulent boundary-layer flow is two-dimensional when time-averaged, so that no time-averaged quantities vary with z and w = 0. Thus if we take the time average of each term of Eqn (7.104), it simplifies to

dp (d2u ~d~x + tlM

The term marked with * can be written as

=0 from Eqn (7.103)

<9uV

h

=0 no variation with z

So that Eqn (7.105) becomes

( du _du dp (daxx daxy .

where we have written

– du ~n ’ axy = ^-f^v

This notation makes it evident that when the turbulent flow is time-averaged — pua and – pu’V take on the character of a direct and shear stress respectively. For this reason, the quantities are known as Reynolds stresses or turbulent stresses. In fully turbulent flows, the Reynolds stresses are usually very much greater than the viscous stresses. If the time-averaging procedure is applied to the full three-dimensional Navier-Stokes equations (2.95), a Reynolds stress tensor is generated with the form

(7.107)

It can be seen that, in general, there are nine components of the Reynolds stress comprising six distinct quantities.