# Reynolds Number

## Article Summary & FAQs

### What is the Reynolds number?

The Reynolds number represents 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 is defined as a characteristic length multiplied by a characteristic velocity and divided by the kinematic viscosity.

### Key Facts

• Osborn Reynolds discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid.
• 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.
• When the inertial forces dominate over the viscous forces (when the fluid is flowing faster and Re is larger) then the flow is turbulent.
• It is defined as: in which V is the mean flow velocity, D a characteristic linear dimension, ρ fluid density, μ dynamic viscosity, and ν kinematic viscosity.
• The Reynolds number can be used to compare a real situation (e.g. air flow around an airfoil and water flow in a pipe) with a small-scale model.
How is Reynolds number calculated?
How is the Reynolds number calculated?

1. Determine the type of flow (internal, external, steady-state etc.).
2. Choose or calculate the characteristic linear dimension
3. From tables, determine the viscosity and the density of fluid.
4. For given flow velocity calculate the Reynolds number.

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

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.

At what Reynolds number is turbulent flow?
At what Reynolds number is turbulent flow?

For external flow, if the Reynolds number is greater than 3500, the flow is turbulent.

What is the critical Reynolds number?
What is the critical Reynolds number?

The critical Reynolds number is associated with the laminar-turbulent transition, in which a laminar flow becomes turbulent. This is an extraordinarily complicated process, which at present is not fully understood.

What are applications of the Reynolds number?
What are applications of the Reynolds number?

Reynolds number plays an important part in calculations in fluid dynamics and heat transfer problems.

1. It is essential for calculation of the friction factor in a few of the equations of fluid mechanics, including the Darcy-Weisbach equation.
2. It is essential for heat transfer calculations, since many other characteristic numbers (e.g. Nusselt number) depend on the flow regime.
3. It is used when modeling the movement of organisms swimming through water.
4. Atmospheric air is considered to be a fluid.

## Reynolds Number

The Reynolds number represents 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 is defined as a characteristic length multiplied by a characteristic velocity and divided by the kinematic viscosity. 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 is flowing faster and Re is larger) then the flow is turbulent. The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, free-stream velocity, surface temperature, and type of fluid, among other things.

It must be noted, the Reynolds number is one of characteristic numbers (standardized in ISO 80000-11:2019), which can be used to compare a real situation (e.g. air flow around an airfoil and water flow in a pipe) with a small-scale model.

The Reynolds number 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.

### Application of Reynolds Number

The Reynolds number plays an important part in calculations in fluid dynamics and heat transfer problems.

• It is essential for calculation of the friction factor in a few of the equations of fluid mechanics, including the Darcy-Weisbach equation.
• It is essential for heat transfer calculations, since many other characteristic numbers (e.g. Nusselt number) depend on the flow regime.
• It is used when modeling the movement of organisms swimming through water.
• It is used to predict the transition from laminar to turbulent flow, and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version.
• The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.

## History of Reynolds Number

The concept was introduced by George Stokes in 1851, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds (1842–1912), who performed exhaustive experiments in the 1880s. Osborn Reynolds discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid. Osborne Reynolds discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid.

This study is closely associated with the boundary layer concept. The study of flows developping along a solid boundary is of primary importance for many engineering problems, such as drag reduction of airplane wings in the aeronautics field. In the boundary layer concept, introduced by Prandtl (1904), the field of wall-bounded flows can be divided into two regions:

• a thin region near the wall called the boundary layer, where strong velocity gradients occur, inducing large viscous shearing forces that must be taken into account and
• a region outside the boundary layer, where the frictions forces can be neglected, and where, therefore, inviscid fluid theory offers a good approximation.

The major contribution of the boundary layer concept was to overcome D’Alembert paradox and to reunify theoretical hydrodynamics (derived in the framework of the perfect fluid hypothesis) with empirical laws from hydraulics. Fundamental research on boundary layers is focused on the study of canonical configurations such as pipe flow, channel flow or flow over a flat plate.

For the flow over a flat plate, it can be seen, near the leading edge of the plate, the régime of the boundary layer is laminar: the flow is two-dimensional and steady. As the boundary layer develops, the flow becomes critical and undergoes transition from the laminar to the turbulent regime: instabilities appear and propagate, giving birth to turbulent spots, irregularly distributed in space and time. As the distance x from the leading edge increases (i.e. as the Reynolds Number increases), the flow becomes unstable and finally for higher Reynolds numbers, the boundary layer is turbulent and the streamwise velocity is characterized by unsteady (changing with time) swirling flows inside the boundary layer. Turbulence always occurs at large Reynolds numbers.

## Critical Reynolds Number

The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number. The value of the critical Reynolds number is different for different geometries.

• For flow over a flat plate, the generally accepted value of the critical Reynolds number is Rex ~ 500000.
• For flow in a pipe of diameter D, experimental observations show that for “fully developed” flow, laminar flow occurs when ReD < 2300 and turbulent flow occurs when ReD > 3500.
• For a sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the characteristic velocity is that of the sphere relative to the fluid. Purely laminar flow only exists up to Re = 10 under this definition.

## Reynolds’ law of similarity

In order for two flows to be similar, they must have the same geometry and equal Reynolds and Euler numbers. When comparing fluid behavior at corresponding points in a model and a full-scale flow, the following holds:

Remodel = Re
Eumodel = Eu

For example, let us compare Reynolds numbers of an actual vehicle and a half scale model as shown in the following diagram. The Reynolds numbers of both agree when the velocity of the half scale model is doubled. In this state, the proportions of viscous force and inertia force of both cases are equal; hence, the surrounding flows can be defined as similar.

This allows engineers to perform experiments with reduced scale models in water channels or wind tunnels and correlate the data to the actual flows, saving on costs during experimentation and on lab time. Note that true dynamic similitude may require matching other dimensionless numbers as well, such as the Mach number used in compressible flows, or the Froude number that governs open-channel flows.

## 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 in 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.
• Most common type of flow.
Classification of Flow Regimes
From a practical engineering point of view the flow regime can be categorized according to several criteria.

All fluid flow is classified into one of two broad categories or regimes. These two flow regimes are:

• Single-phase Fluid Flow
• Multi-phase Fluid Flow (or Two-phase Fluid Flow)

This is a basic classification. All of the fluid flow equations (e.g. Bernoulli’s Equation) and relationships that were discussed in this section (Fluid Dynamics) were derived for the flow of a single phase of fluid whether liquid or vapor. Solution of multi-phase fluid flow is very complex and difficult and therefore it is usually in advanced courses of fluid dynamics. Another usually more common classification of flow regimes is according to the shape and type of streamlines. All fluid flow is classified into one of two broad categories. The fluid flow can be either laminar or turbulent and therefore these two categories are:

• Laminar Flow
• Turbulent Flow

Laminar flow is characterized by smooth or in regular paths of particles of the fluid. Therefore the laminar flow is also referred to as streamline or viscous flow. In contrast to laminar flow, turbulent flow is characterized by the irregular movement of particles of the fluid. The turbulent fluid does not flow in parallel layers, the lateral mixing is very high, and there is a disruption between the layers. Most industrial flows, especially those in nuclear engineering are turbulent.

The flow regime can be also classified according to the geometry of a conduit or flow area. From this point of view, we distinguish:

• Internal Flow
• External Flow

Internal flow is a flow for which the fluid is confined by a surface. Detailed knowledge of behaviour of internal flow regimes is of importance in engineering, because circular pipes can withstand high pressures and hence are used to convey liquids. On the other hand, external flow is such a flow in which boundary layers develop freely, without constraints imposed by adjacent surfaces. Detailed knowledge of behaviour of external flow regimes is of importance especially in aeronautics and aerodynamics.

Table from Life in Moving Fluids

## 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 Red,crit = 2300.

Transitional flow. At Reynolds numbers between about 2000 and 4000 the flow is unstable as a result of 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.

## Reynolds Number and Internal Flow

The internal flow (e.g. flow in a pipe) configuration 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 of importance in engineering, because circular pipes can withstand high pressures and hence are used to convey liquids. Non-circular ducts are used to transport low-pressure gases, such as air in cooling and heating systems.

For internal flow regime an entrance region is typical. In this region a nearly inviscid upstream flow converges and enters the tube. To characterize this region the hydrodynamic entrance length is introduced and is approximately equal to: The maximum hydrodynamic entrance length, at ReD,crit = 2300 (laminar flow), is Le = 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 Le is relatively shorter. For any given problem, Le / D has to be checked to see if Le is negligible when compared to the pipe length. At a finite distance from the entrance, the entrance effects may be neglected, because the boundary layers merge and the inviscid core disappears. The tube flow is then fully developed

Flow Velocity Profile - Power-law velocity profile Source: U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2 and 3. June 1992.

## 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 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 at the picture. In short, the value n increases with increasing Reynolds number. The one-seventh power-law velocity profile approximates many industrial flows.

## Hydraulic Diameter

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

For these purposes the Reynolds number is defined as: where Dh is the hydraulic diameter:  The hydraulic diameter, Dh, 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 are in contact with the flow.

### Example: Reynolds number for a primary piping and a fuel bundle

`It is an illustrative example, following data do not correspond to any reactor design.`

Pressurized water reactors are cooled and moderated by high-pressure liquid water (e.g. 16MPa). At this pressure water boils at approximately 350°C (662°F). Inlet temperature of the water is about 290°C (⍴ ~ 720 kg/m3). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m3) as the water flows through the core.

The primary circuit of typical PWRs is divided into 4 independent loops (piping diameter ~ 700mm), each loop comprises a steam generator and one main coolant pump. Inside the reactor pressure vessel (RPV), the coolant first flows down outside the reactor core (through the downcomer). From the bottom of the pressure vessel, the flow is reversed up through the core, where the coolant temperature increases as it passes through the fuel rods and the assemblies formed by them.

Assume:

• the primary piping flow velocity is constant and equal to 17 m/s,
• the core flow velocity is constant and equal to 5 m/s,
• the hydraulic diameter of the fuel channel, Dh, is equal to 1 cm
• the kinematic viscosity of the water at 290°C is equal to 0.12 x 10-6 m2/s

Determine

• the flow regime and the Reynolds number inside the fuel channel
• the flow regime and the Reynolds number inside the primary piping

The Reynolds number inside the primary piping is equal to:

ReD = 17 [m/s] x 0.7 [m] / 0.12×10-6 [m2/s] = 99 000 000

This fully satisfies the turbulent conditions.

The Reynolds number inside the fuel channel is equal to:

ReDH = 5 [m/s] x 0.01 [m] / 0.12×10-6 [m2/s] = 416 600

This also fully satisfies the turbulent conditions.

## Reynolds Number and External Flow

The Reynolds number describes naturally the external flow as well. In general, when a fluid flows over a stationary surface, e.g. the flat plate, the bed of a river, or the wall of a pipe, the fluid touching the surface is brought to rest by the shear stress to at the wall. The region in which flow adjusts from zero velocity at the wall to a maximum in the main stream of the flow is termed the boundary layer.

Basic characteristics of all laminar and turbulent boundary layers are shown in the developing flow over a flat plate. The stages of the formation of the boundary layer are shown in the figure below: Boundary layers may be either laminar, or turbulent depending on the value of the Reynolds number.

Also here the Reynolds number represents the ratio of inertia forces to viscous forces and is a convenient parameter for predicting if a flow condition will be laminar or turbulent. It is defined as: in which V is the mean flow velocity, D a characteristic linear dimension, ρ fluid density, μ dynamic viscosity, and ν kinematic viscosity.

For lower Reynolds numbers, the boundary layer is laminar and the streamwise velocity changes uniformly as one moves away from the wall, as shown on the left side of the figure. As the Reynolds number increases (with x) the flow becomes unstable and finally for higher Reynolds numbers, the boundary layer is turbulent and the streamwise velocity is characterized by unsteady (changing with time) swirling flows inside the boundary layer.

Transition from laminar to turbulent boundary layer occurs when Reynolds number at x exceeds Rex ~ 500,000. Transition may occur earlier, but it is dependent especially on the surface roughness. The turbulent boundary layer thickens more rapidly than the laminar boundary layer as a result of increased shear stress at the body surface.

The external flow reacts to the edge of the boundary layer just as it would to the physical surface of an object. So the boundary layer gives any object an “effective” shape which is usually slightly different from the physical shape. We define the thickness of the boundary layer as the distance from the wall to the point where the velocity is 99% of the “free stream” velocity.

To make things more confusing, the boundary layer may lift off or “separate” from the body and create an effective shape much different from the physical shape. This happens because the flow in the boundary has very low energy (relative to the free stream) and is more easily driven by changes in pressure.

Special reference: Schlichting Herrmann, Gersten Klaus. Boundary-Layer Theory, Springer-Verlag Berlin Heidelberg, 2000, ISBN: 978-3-540-66270-9

Example: Transition Layer
A long thin flat plate is placed parallel to a 1 m/s stream of water at 20°C. Assume that kinematic viscosity of water at 20°C is equal to 1×10-6 m2/s.

At what distance x from the leading edge will be the transition from laminar to turbulent boundary layer (i.e. find Rex ~ 500,000).

Solution:

In order to locate the transition from laminar to turbulent boundary layer, we have to find x at which Rex ~ 500,000.

x = 500 000 x 1×10-6 [m2/s] / 1 [m/s] = 0.5 m

References:
Reactor Physics and Thermal Hydraulics:

1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2 and 3. June 1992.
10. White Frank M., Fluid Mechanics, McGraw-Hill Education, 7th edition, February, 2010, ISBN: 978-0077422417

Fluid Dynamics