Understanding the Reynolds Number for Turbulent External Flow

The Reynolds number, denoted by Re, is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is defined as the ratio of inertial forces to viscous forces within a fluid. The formula for the Reynolds number is given by:

Re frac{rho v L}{mu}

Where:

(rho) is the density of the fluid (kg/m3) (v) is the velocity of the fluid relative to the object (m/s) (L) is the characteristic length (m) often taken as the diameter of a pipe or the length of a flat plate (mu) is the dynamic viscosity of the fluid (Pa·s or kg/m·s)

Reynolds Number for Turbulent Flow

In the context of external flow, the flow is considered turbulent when the Reynolds number exceeds a certain threshold. The specific value of this threshold can vary based on the specific situation, such as flow over a flat plate around a cylinder. However, a common rule of thumb is:

Turbulent flow typically occurs when Re > 4000.

For flow over a flat plate or in a pipe, a Reynolds number above approximately 2000 is often considered the transition point from laminar to turbulent flow. However, it's important to note that the transition can be influenced by surface roughness, flow disturbances, and other factors.

Reynolds Number for Transitional Flow

Flow can be considered as transitional when the Reynolds number falls within the range of:

2000 Re 5000.

The condition for turbulent flow requires Re to be greater than 5000.

Determination of Critical Reynolds Number

The critical Reynolds number, above which the laminar time-independent solution to the Navier-Stokes equation becomes unstable, is a significant concept in fluid dynamics. The determination of this value from the stability studies of the Navier-Stokes (NS) equation has historically been known as the Orr-Sommerfeld problem. This problem was first solved by C.C. Lin, who wrote a monograph on the subject.

Attempting to solve for the Reynolds number without the necessary data points, such as fluid density ((rho)), velocity ((v)), characteristic length ((L)), and dynamic viscosity ((mu)), is not possible. Ensure all required data is provided for accurate calculations.

Conclusion

Understanding the Reynolds number is crucial for predicting fluid flow patterns in external situations. The specific values and thresholds can vary based on the application, but the general principles of laminar, transitional, and turbulent flow can be applied to a wide range of scenarios.