## 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 is flowing faster and Re is larger) then the flow is turbulent.

**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/m^{3}),

μ dynamic viscosity (Pa.s),

ν kinematic viscosity (m^{2}/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 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**.

**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**.

## 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 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 **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**has to be

_{e}/ D**checked**to see if L

_{e}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**.

### 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

**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 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 are in contact with the flow.

`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/m^{3}). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m^{3}) 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**,*D*, is equal to 2 cm_{h} - the kinematic viscosity of the water at 290°C is equal to 0.12 x 10
^{-6}m^{2}/s

See also: Example: **Flow rate through a reactor core**

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:

**Re _{D}** = 17 [m/s] x 0.7 [m] / 0.12×10

^{-6}[m

^{2}/s] =

**99 000 000**

This fully satisfies the **turbulent conditions**.

The Reynolds number inside the fuel channel is equal to:

**Re _{DH}** = 5 [m/s] x 0.02 [m] / 0.12×10

^{-6}[m

^{2}/s] =

**833 000**

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 **Re _{x} ~ 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.

See also: Boundary layer thickness

See also: Tube in crossflow – external flow

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

**1 m/s**stream of water at

**20°C**. Assume that kinematic viscosity of water at 20°C is equal to

**1×10**.

^{-6}m^{2}/sAt **what distance x** from the leading edge will be the **transition** from laminar to turbulent boundary layer (i.e. find Re_{x} ~ 500,000).

**Solution:**

In order to locate the transition from laminar to turbulent boundary layer, we have to find x at which **Re _{x} ~ 500,000**.

**x** = 500 000 x 1×10^{-6} [m^{2}/s] / 1 [m/s] = **0.5 m**

**Reactor Physics and Thermal Hydraulics:**

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

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