Fundamentals of Fluid Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 115

Fig. 5.13 Instantaneous
velocity fluctuating with time
at a specific point in a
turbulent flow

Turbulence Intensity Turbulence is associated with the intense random velocity
fluctuations in the fluid. This behaviour can be exemplified by a typical velocity
measurement as a function of time at a point in the turbulent flow as shown in
Fig. 5.13 . The random nature of flow produces strong velocity fluctuations which
are transient and their instantaneous values vary stochastically with time. Typically,
the velocity is decomposed into a steady mean value u and a fluctuating component ¯

u ( t ) that varies with time. That is u ( t ) u ( t ). In application, a turbulent flow

is characterized by the mean values of the flow properties (

=¯ u +

p , etc.) and the
corresponding statistics of fluctuating property ( u , v , w , p , etc.). This gives rise to
the root mean square (RMS) fluctuation velocities and turbulence intensity which is
a parameter that describes the turbulence level of the flow. The RMS and turbulence
intensity are defined as the following:
u x , u y , u z

u , ¯ v , ¯ w , ¯ ¯

(RMS velocities)

1
3

u x +

u z and

u
¯

where u = u y +

I = u ¯ = u x +¯ ¯ u y +¯ u z (5.22)

u

Here, a bar on the top denotes an ensemble averaging.

Turbulence Shear Stress At wall boundaries, the fluid velocity slows to zero, lead-
ing to a shear stress (i.e. resistance to fluid flow due to the fluid viscosity). Momentum
exchange occurs through fluid particles that are travelling at different velocities within
the boundary layer and moving from one region to another region. A faster moving
particle moving into a slower moving region will increase the momentum at the new
location. If this motion of the fluid particle is caused by molecular diffusion, then the
momentum exchange is due to the fluid viscosity. If we now consider turbulent flow
there is additional mixing of fluid particles, which is caused by turbulence dispersion
due to the instantaneous velocity fluctuations. The instantaneous rate of momentum
exchange due to the velocity fluctuations in the x - and y -direction is given by ρ u x
u y per unit volume of fluid. The turbulence shear stress ( τ T ) is then given as

τ T =− ρu x u y (5.23)