At low values of Reynolds number the viscous force is large compared to the inertial
force.

In this range, viscous forcing dampens out disturbances in the flow field that are a result
of surface roughness or pressure gradients. As the Reynolds number increases, the viscous
force becomes relatively smaller and at some point it becomes possible for small
perturbation to grow. The flows become unstable and can transition to turbulence where large
fluctuations in the velocity field continue to develop.

Figure 1 shows two different
flow patterns in pipes for laminar flow and turbulent flow. It is evident in the image that
turbulent flow undergoes irregular flow patterns while laminar flow moves in smooth layers
while maintaining a constant flow direction. The turbulent flows happen at a high Reynolds
number where inertial forces are higher than viscous forces and perturbations can become
amplified, whereas the laminar flows occur at a low Reynolds number in which any induced
perturbations are damped out due to relatively strong viscous forces.

Figure 2 shows the time history
of local velocity variations for laminar flows and turbulent flows, respectively. These
patterns can be obtained from hot wire anemometer measurements or CFD simulations. For the
pipe flow cases, laminar flows have nearly constant velocity, while turbulent flows have
random or chaotic velocity fluctuation patterns. For a cylinder in cross flow, laminar flows
produce a sine wave velocity pattern at a downstream location of the cylinder, while the
turbulent flows have similar wave patterns but with embedded fluctuations.

If you consider a flat plate immersed in a flow field with finite viscosity, a thin
boundary layer will begin to develop as a function of the distance traveled along the plate
(x). The image below shows a comparison of time-averaged velocity profiles for laminar flow
and turbulent flow over a flat plate. The time-averaged velocity profile in a turbulent flow
appears more uniform than in a laminar flow because the eddy motions in turbulent flow
transport momentum more actively from one place to another. This process results in a more
uniform profile outside the boundary layer. The velocity gradient near the wall is higher
than the one seen in a laminar flow resulting in a larger skin friction coefficient (${C}_{f}$) than the laminar flow. The skin friction coefficient can be
defined as

where ${\tau}_{w}=\mu \frac{\partial U}{\partial y}$ is the wall shear stress, $\mu $ is the dynamic viscosity, $\rho $ is the density and $U$ is the mean velocity. The local Reynolds number ($R{e}_{x}$) is calculated by using the distance from the leading edge of
the flat plate as the length scale. $R{e}_{x}$ measures the local ratio of inertial to viscous forcing and
provides an indication of the state of the flow regime as it moves from laminar to
turbulent.

The information below includes equations of boundary layer thickness and skin friction
coefficient for turbulent flow and laminar flow. These equations where developed through
empirical relationships between local Reynolds number and boundary layer characteristics.
The information below also provides a summary of general characteristics that are present in
turbulent flow and laminar flow.

Table 1. Laminar Flow vs Turbulent Flow: Boundary Layer Thickness and Skin Friction
Coefficient