Flow in Pipes - Handout 3

Turbulent flow is prevalent in engineering applications and affects velocity distribution and wall shear stress.
Understanding turbulence is crucial despite its complex behavior and underdeveloped theories.
Analysis of turbulent flow often relies on dimensional considerations and empirical relations from experiments.

Random and rapid fluctuations are a hallmark of turbulent flow, characterized by swirling regions called eddies.
These eddies enhance momentum and energy transfer, contrasting with laminar flow, where diffusion accounts for such transfers.
- Momentum & Energy Transfer:
  - Laminar flow: Transferred by molecular diffusion.
  - Turbulent flow: Transferred through the rapid movement of eddies.

Laminar Flow Velocity Profile:
- Mean velocity (V(mean)) is approximately half of the maximum velocity (V(max)):
  - V(mean) = V(max) / 2
Turbulent Flow Velocity Profile:
- Mean velocity is higher than in laminar flow:
  - V(mean) ~ 0.8 V(max)

Turbulent flow presents significant fluctuations, affecting velocity, temperature, and pressure even if average flow appears steady.
Instantaneous velocity (u) can be represented as:
- u = V(mean) + u', where u' represents the fluctuating component.

The wall shear stress in turbulent flow is significantly greater than in laminar flow due to the velocity profile differences.

Reynolds stress can be calculated considering both laminar and turbulent components:
- ( \tau_{total} = \tau_{lam} + \tau_{turb} )
Boussinesq's hypothesis relates turbulent shear stress to eddy viscosity, which accounts for momentum transport:
- ( \tau_{turb} = - \rho \frac{\partial u}{\partial y} )
Eddy viscosity varies with flow conditions, while molecular viscosity is a fixed property of the fluid.

Turbulent flow near a wall consists of three distinct regions:
- Viscous Sublayer:
  - Dominated by viscous effects, typically <1% of pipe diameter.
  - The wall dampens eddy motion, leading to laminar characteristics and a linear velocity profile.
- Overlap Layer:
  - Turbulent effects are prominent but not dominant.
- Outer Layer:
  - Turbulent effects dominate over viscous diffusion.
In the viscous sublayer, the wall shear stress can also be expressed as:
- ( \tau_{w} = \rho u_{}^{2} ) where ( u_{} ) is the shear velocity.

Surface roughness significantly impacts the viscous sublayer behavior, affecting overall flow characteristics in turbulent flow:
- Hydrodynamically Smooth vs. Rough Surfaces:
  - A surface is smooth if the roughness height (k) is less than the viscous sublayer thickness (d').
  - Conversely, it is rough if k > d'.
Different materials can appear smooth or rough depending on magnification and characteristics in fluid dynamics.

The roughness Reynolds number determines the effect of roughness on flow characteristics:
- Conditions change when roughness is comparable to the viscous sublayer thickness.

Velocity measurements can be plotted on linear and logarithmic scales to illustrate flow characteristics.

The overlap layer features a logarithmic relationship between velocity and distance from the wall:
- Velocity in this layer is described by the logarithmic law:
- ( u = \frac{u_{*}}{\kappa} \ln(y) + B ) where ( \kappa ) (von Karman constant) is approximately 0.4.
For smooth surfaces, velocity scales with the viscous length scale, whereas for rough surfaces, it scales with roughness height.
The normalized velocity profile in the turbulent outer layer is independent of fluid viscosity.