5.2. Threshold Reynolds Number

One of the important parameters that you have read in the fluid flow is Reynolds number. So, there is going to be a Reynolds number associated with this type of phenomenon. That is the occurrence of or the development of the boundary layer which is given by, as given here, R_e at a distance x, is given by, Ux / nu. Where u is the free stream velocity here, x is the distance from the plate, a leading edge and nu is the kinematics viscosity of fluid.

In fluid dynamics, the Reynolds number (R_e) is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is calculated as the ratio of inertial forces to viscous forces, and is given by the formula R_e = Ux / nu. Here, 'U' represents the velocity of the fluid, 'x' is the distance from a reference point (like a leading edge of a plate), and 'nu' is the kinematic viscosity of the fluid, which measures a fluid's resistance to flow. A low Reynolds number indicates that viscous forces are dominant, while a high Reynolds number indicates that inertial forces dominate, influencing the type of flow that occurs (laminar or turbulent).

Imagine trying to swim in both thick syrup and in water. In syrup (high viscosity), you move slowly because the sticky syrup hinders your movement (analogous to a high Reynolds number situation where viscosity dominates). In water (lower viscosity), you move faster and more freely (analogous to low Reynolds number where inertia dominates). The Reynolds number helps determine the conditions and behavior of fluid flow just like your swimming performance depends on the type of fluid you are in.

Near the leading edge of the plate the flow in the boundary layer is laminar. This is important to note. And the length of the plate from the leading edge to the point upto which laminar boundary layer exists is called laminar zone. So, this is the leading edge here. So, the length of the plate from the leading edge, to the point, where the laminar boundary layers, so here, the laminar boundary layer is existing until this point, as indicated in this diagram.

A laminar boundary layer is a smooth and orderly flow of fluid that develops close to the surface of the object (like a plate). This region is characterized by parallel layers of fluid that slide past one another without mixing. The 'laminar zone' refers to the section along the plate where this type of flow occurs, starting from the leading edge (where the fluid first encounters the plate) until the flow begins to transition to turbulent. Typically, this laminar flow can exist up to a Reynolds number of approximately 5 x 10^5, beyond which the flow becomes unstable.

Think of a smooth, straight line of cars on a highway moving at a constant speed. This represents laminar flow where vehicles maintain their paths and speeds without disruptions. However, if the speed limit increases suddenly and some drivers begin to race ahead or slow down for no reason, the orderly flow becomes chaotic, akin to a turbulent flow. Just as cars will behave differently based on the driving conditions, fluid layers behave according to the flow characteristics defined by the Reynolds number.

For a flat plate it has been found out that the laminar boundary layer occurs up to Reynolds number of 5 into 10 to the power 5. Lot of people have done research, experimental results confirmed by numerical analysis and this value has come up. So, if the Reynolds number is less than 5 into 10 to the power 5 for a flow over a plate the laminar boundary layer will occur up to that particular point. And then there is a turbulent boundary layer here, a little bit information on that. Actually with increasing x as we have seen, Reynolds number as a function of x for ux / nu and x is in this directions. So, as you keep on moving in this direction, x is going to increase and therefore, the Reynolds number will increase. Now, when the Reynolds number increases to more than 5 into 10 to the power 5 the laminar boundary layer becomes unstable.

As the Reynolds number increases (due to increased flow velocity or increased distance from the leading edge), the flow may transition from laminar to turbulent. This transition indicates that the orderly flow of the fluid has become disrupted, resulting in eddies, swirls, and more chaotic fluid motion. After the Reynolds number exceeds 5 x 10^5, the laminar boundary layer becomes unstable, leading to fluctuations in velocity and the eventual onset of turbulent flow. This change is significant because turbulent flow has different properties and effects compared to laminar flow.

Imagine a calm lake on a still day; the water flows smoothly and reflects the sky, which is like laminar flow. But when a storm brews and the wind stirs the water, causing waves and turbulence, the calmness is lost, similar to the transition from laminar to turbulent flow. Just like the lake changes character significantly with the onset of wind, the fluid’s behavior changes dramatically when the Reynolds number reaches a certain threshold.

--