Question

What is the physical significance of the Reynolds number? How is it defined for external flow over a plate of length \(L\) ?

Short answer

Answer: The Reynolds number's physical significance lies in its ability to predict the flow behavior of fluids, such as laminar or turbulent flow. It provides a measure of the relative importance of inertial forces (due to fluid velocity) and viscous forces (due to fluid friction) in the flow. The Reynolds number for an external flow over a flat plate of length L can be defined using the equation: \(Re_L = \frac{\rho U L}{\mu}\), where \(Re_L\) is the Reynolds number, \(\rho\) is the fluid density, \(U\) is the free-stream fluid velocity, \(L\) is the plate length, and \(\mu\) is the dynamic viscosity of the fluid.

Step-by-step solution

Physical significance of the Reynolds number

The Reynolds number (Re) is a dimensionless quantity used in fluid dynamics to predict the flow behavior of fluids, such as laminar or turbulent flow. It provides a measure of the relative importance of inertial forces (due to fluid velocity) and viscous forces (due to fluid friction) in the flow. A low Reynolds number indicates that viscous forces dominate the flow and, therefore, the flow pattern is laminar. Conversely, a high Reynolds number indicates that inertial forces are more significant, and the flow is likely to be turbulent.

Definition of the Reynolds number

To define the Reynolds number for an external flow over a flat plate of length \(L\), the following equation is used: \[Re_L = \frac{\rho U L}{\mu}\] Here, \(Re_L\) is the Reynolds number for flow over a plate of length \(L\) \(\rho\) is the fluid density (\(kg/m^3\)), \(U\) is the free-stream fluid velocity (\(m/s\)), \(L\) is the length of the plate (\(m\)), and \(\mu\) is the dynamic viscosity of the fluid (\(N \cdot s/m^2\) or \(Pa \cdot s\)). By knowing the Reynolds number, we can predict the type of flow (laminar, transitional, or turbulent) over the plate, which is essential for understanding fluid-particle interactions, heat transfer, and fluid forces in various engineering applications.