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## 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**.

## 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**.

## 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 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.

## 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