Reynolds Number for Turbulent Flow

Reynolds Number Regimes

Laminar flow. For practical purposes, if the Reynolds number is less than 2000, the flow is laminar. The accepted transition Reynolds number for flow in a circular pipe is Re_d,crit = 2300.

Transitional flow. At Reynolds numbers between about 2000 and 4000, the flow is unstable due to the onset of turbulence. These flows are sometimes referred to as transitional flows.

Turbulent flow. If the Reynolds number is greater than 3500, the flow is turbulent. Most fluid systems in nuclear facilities operate with turbulent flow.

Definition of Reynolds Number

The Reynolds number is the ratio of inertial forces to viscous forces and is a convenient parameter for predicting if a flow condition will be laminar or turbulent. It can be interpreted that when the viscous forces are dominant (slow flow, low Re), they are sufficient enough to keep all the fluid particles in line, then the flow is laminar. Even very low Re indicates viscous creeping motion, where inertia effects are negligible. When the inertial forces dominate over the viscous forces (when the fluid flows faster and Re is larger), the flow is turbulent.

It is a dimensionless number comprised of the physical characteristics of the flow. An increasing Reynolds number indicates increasing turbulence of flow.

It is defined as:

where:
V is the flow velocity,
D is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter etc.)
ρ fluid density (kg/m³),
μ dynamic viscosity (Pa.s),
ν kinematic viscosity (m²/s);  ν = μ / ρ.

Laminar vs. Turbulent Flow

Laminar flow:

• Re < 2000
• ‘low’ velocity
• Fluid particles move in straight lines
• Layers of water flow over one another at different speeds with virtually no mixing between layers.
• The flow velocity profile for laminar flow in circular pipes is parabolic in shape, with a maximum flow at the center of the pipe and a minimum flow at the pipe walls.
• The average flow velocity is approximately one-half of the maximum velocity.
• Simple mathematical analysis is possible.
• Rare in practice in water systems.

Turbulent Flow:

• Re > 4000
• ‘high’ velocity
• The flow is characterized by the irregular movement of particles of the fluid.
• Average motion is in the direction of the flow.
• The flow velocity profile for turbulent flow is fairly flat across the center section of a pipe and drops rapidly extremely close to the walls.
• The average flow velocity is approximately equal to the velocity at the center of the pipe.
• Mathematical analysis is very difficult.
• The most common type of flow.

Reynolds Number and Turbulent Flow

Flow Velocity Profile - Power-law velocity profile

Power-law velocity profile – Turbulent velocity profile

The velocity profile in turbulent flow is flatter in the central part of the pipe (i.e., in the turbulent core) than in laminar flow. The flow velocity drops rapidly extremely close to the walls. This is due to the diffusivity of the turbulent flow.

In the case of turbulent pipe flow, there are many empirical velocity profiles. The simplest and the best known is the power-law velocity profile:

where the exponent n is a constant whose value depends on the Reynolds number. This dependency is empirical, and it is shown in the picture. In short, the value n increases with increasing Reynolds number. The one-seventh power-law velocity profile approximates many industrial flows.

The configuration of the internal flow (e.g.,, flow in a pipe) is a convenient geometry for heating and cooling fluids used in energy conversion technologies such as nuclear power plants.

In general, this flow regime is important in engineering because circular pipes can withstand high pressures and hence convey liquids. Non-circular ducts are used to transport low-pressure gases, such as air in cooling and heating systems.

For the internal flow regime, an entrance region is typical. In this region, a nearly inviscid upstream flow converges and enters the tube. The hydrodynamic entrance length is introduced to characterize this region and is approximately equal to:

The maximum hydrodynamic entrance length, at Re_D,crit = 2300 (laminar flow), is L_e = 138d, where D is the diameter of the pipe. This is the longest development length possible. In turbulent flow, the boundary layers grow faster, and L_e is relatively shorter. For any given problem, L_e / D has to be checked to see if L_e is negligible compared to the pipe length. The entrance effects may be neglected at a finite distance from the entrance because the boundary layers merge and the inviscid core disappears. The tube flow is then fully developed.

Hydraulic Diameter

Since the characteristic dimension of a circular pipe is an ordinary diameter D and especially reactors contain non-circular channels, the characteristic dimension must be generalized.

For these purposes, the Reynolds number is defined as:

where D_h is the hydraulic diameter:

The hydraulic diameter, D_h, is a commonly used term when handling flow in non-circular tubes and channels. The hydraulic diameter transforms non-circular ducts into pipes of equivalent diameter. Using this term, one can calculate many things in the same way as for a round tube. In this equation, A is the cross-sectional area, and P is the wetted perimeter of the cross-section. The wetted perimeter for a channel is the total perimeter of all channel walls that contact the flow.

See above:

Reynolds Number