Which one of the following non-dimensional numbers is mostly used for determining the transition from laminar to turbulent flow in free convection?

Explanation:

Non-Dimensional Numbers:

• Non-dimensional numbers are crucial in fluid dynamics and heat transfer as they allow the analysis of complex systems by reducing the number of variables. In free convection, these numbers help determine the transition between laminar and turbulent flow regimes and provide insights into the behavior of the system under different conditions.

Rayleigh Number:

• The Rayleigh number (Ra) is a dimensionless number that plays a critical role in the study of free convection. It is used to predict the transition from laminar to turbulent flow in fluid systems driven by buoyancy forces. The Rayleigh number combines the effects of thermal diffusivity, kinematic viscosity, and buoyancy, and is expressed as:

Ra = Gr × Pr

Where:

• Gr is the Grashof number, which represents the ratio of buoyancy forces to viscous forces.
• Pr is the Prandtl number, which relates momentum diffusivity to thermal diffusivity.

The Rayleigh number signifies the dominance of buoyancy forces compared to viscous and thermal effects. In free convection, the critical Rayleigh number (typically Ra_crit ≈ 10⁶) indicates the onset of turbulence. When Ra is below this threshold, the flow remains laminar. Above this value, the flow transitions to turbulence, characterized by chaotic and irregular motion.

Additional InformationPeclet Number

• The Peclet number (Pe) is a dimensionless parameter that relates the rate of advection of thermal energy to the rate of thermal diffusion. It is expressed as:

Pe = Re × Pr

Where:

• Re is the Reynolds number, representing the ratio of inertial forces to viscous forces.
• Pr is the Prandtl number.

Grashof Number

• The Grashof number (Gr) is a critical parameter in free convection, representing the ratio of buoyancy forces to viscous forces. It is given by:

Gr = (g × β × ΔT × L³) / ν²

Where:

• g is the acceleration due to gravity.
• β is the coefficient of thermal expansion.
• ΔT is the temperature difference.
• L is the characteristic length.
• ν is the kinematic viscosity.

Reynolds Number

• The Reynolds number (Re) is a widely used dimensionless parameter that predicts the transition from laminar to turbulent flow in forced convection. It is defined as:

Re = (ρ × V × L) / μ

Where:

• ρ is the fluid density.
• V is the velocity of the fluid.
• L is the characteristic length.
• μ is the dynamic viscosity