Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 67

Box 3.2 Reynolds' approach, 1895. For constant density, isothermal, steady, uniform flows:
1 There is an instantaneous flux of momentum per unit volume of fluid in a streamwise direction.
2 The instantaneous velocity comprises the sum of the mean and the instantaneous fluctuation (see Figs 3.46 and 3.47).
3 The instantaneous momentum flux (a force) comprises both the mean and fluctuating contributions:

u ( ru ) = ru 2 = r ( u + u ') 2 = r ( u 2 + 2 uu ' + u ' 2 )

4 The mean flux of turbulent momentum involves only the sum of the mean and turbulent contributions (the central
subterm in brackets on the right-hand-side above becomes zero in the mean, since all mean fluctuations are zero by definition).

Hence, ru 2

= r (u 2 + u' 2

)

So, going back to our earlier point concerning accelerations and forces, net force due to turbulence in steady, uniform
turbulent flows cause rate of change of momentum applied. Or, more correctly since we are viewing the flow
from the point of view of accelerations, the turbulent acceleration requires a net force to produce it.

the dynamic viscosity, , for it varies in time and space for

different flows (i.e. it is anisotropic) and must always be
measured experimentally.

3.11.4 Reynolds' accelerations for turbulent flow

Now back to Reynolds': he proposed to take the Second
Law and repla ce the total acceleration term involving
mean vel oc ity, , by a term also involving the turbulent
velocity,

u

u '. After some manipulation (Box 3.2)
although the arithmetic looks complicated, it is not (see
Cookie 8). The total acceleration term for a steady, uni-
form turbulent flow becomes simply the spatial change in
any velocity fluctuation. The result is staggering - despite
the fact that a turbulent flow may be steady and uniform
in the mean there exist time-mean accelerations due to
gradients in space of the turbulent fluctuations. The accel-
eration gradients, when multiplied by mass per unit fluid
volume, are conventionally expressed as Reynolds' stresses .
Net forces produce the gradients because there is change of
momentum due to the turbulence. Or, since we are dis-
cussing accelerations, we say the turbulent acceleration
requires a net force to produce it. We shall return to this
topic in Section 4.5; in fact we constantly think about it.

u

Boussinesq

Fig. 3.49 Eddies provide a variable turbulent friction far greater in
magnitude than viscous friction. Boussinesq added the turbulent
friction as an “eddy viscosity” term, , to Newton's viscous shear

expression: ( ) d u /d y .

additional to those molecular forces created by the action
of the change of velocity gradient on dynamic viscosity
(see Section 3.10). The extra mixing process resulting
from turbulence was given the name eddy viscosity , symbol

, by Boussinesq in 1877 (Fig. 3.49). Although this was a
useful illustrative concept,

is not a material constant like

3.12 Overall forces of fluid motion

We have seen that in stationary fluids the static forces of
hydrostatic pressure and buoyancy are due to gravity.
These forces also exist in moving fluids but with additional
dynamic forces present - viscous and inertial - due to gra-
dients of velocity and accelerations affecting the flow. In
order to understand the dynamics of such flows and to be
able to calculate the resulting forces acting we need to

understand the interactions between the dynamic and
static forces that comprise F , the total force. This will

enable us to eventually solve some dynamic force equa-
tions, the equations of motion, for properties such as
velocity, pressure, and energy. Such a development will
inform Chapter 4 concerning the nature of physical envi-
ronmental flows.