Chapter 10 : Turbulent Flow
Lecture 33 :


Boundary Conditions

  • All the components of apparent stresses vanish at the solid walls and only stresses which act near the wall are the viscous stresses of laminar flow. The boundary conditions, to be satisfied by the mean velocity components, are similar to laminar flow.
  • A very thin layer next to the wall behaves like a near wall region of the laminar flow. This layer is known as laminar sublayer and its velocities are such that the viscous forces dominate over the inertia forces. No turbulence exists in it (see Fig. 33.3).
  • For a developed turbulent flow over a flat plate, in the near wall region, inertial effects are insignificant, and we can write from Eq.33.10,

 


Fig 33.3 Different zones of a turbulent flow past a wall


which can be integrated as , =constant

  • We know that the fluctuating components, do not exist near the wall, the shear stress on the wall is purely viscous and it follows

However, the wall shear stress in the vicinity ofthe laminar sublayer is estimated as

(33.11a)

where Us is the fluid velocity at the edge of the sublayer. The flow in the sublayer is specified by a velocity scale (characteristic of this region).

  • We define the friction velocity,
(33.11b)

as our velocity scale. Once is specified, the structure of the sub layer is specified. It has been confirmed experimentally that the turbulent intensity distributions are scaled with . For example, maximum value of the is always about . The relationship between and the can be determined from Eqs (33.11a) and (33.11b) as

Let us assume . Now we can write

       where $\displaystyle \bar{C}\,\,$   is a proportionality constant

(33.12a)

or

(33.12b)

Hence, a non-dimensional coordinate may be defined as, which will help us estimating different zones in a turbulent flow. The thickness of laminar sublayer or viscous sublayer is considered to be .

Turbulent effect starts in the zone of and in a zone of , laminar and turbulent motions coexist. This domain is termed as buffer zone. Turbulent effects far outweight the laminar effect in the zone beyond and this regime is termed as turbulent core .

 
  • For flow over a flat plate, the turbulent shear stress ( ) is constant throughout in the y direction and this becomes equal to $ \tau_w$ at the wall. In the event of flow through a channel, the turbulent shear stress ( ) varies with y and it is possible to write
(33.12c)

where the channel is assumed to have a height 2h and $ \zeta$ is the distance measured from the centreline of the channel . Figure 33.1 explains such variation of turbulent stress.