## Laminar Flow

In fluid dynamics, **laminar flow** is characterized by **smooth or in regular paths** of particles of the fluid, in contrast to turbulent flow, that is characterized by the irregular movement of particles of the fluid. The fluid flows in **parallel layers** (with minimal lateral mixing), with no disruption between the layers. Therefore the laminar flow is also referred to as **streamline or viscous flow**.

The term streamline flow is descriptive of the flow because, in laminar flow, layers of water flowing over one another at different speeds with virtually no mixing between layers, fluid particles move in definite and observable paths or streamlines.

When a fluid is flowing through a closed channel such as a pipe or between two flat plates, either of two types of flow (laminar flow or turbulent flow) may occur depending on the **velocity**, **viscosity** of the fluid and the **size of the pipe**. **Laminar flow** tends to occur at **lower velocities** and **high viscosity**. On the other hand turbulent flow tends to occur at higher velocities and low viscosity.

Since laminar flow is common only in cases in which the flow channel is relatively small, the fluid is moving slowly, and its viscosity is relatively high, laminar flow is not common in industrial processes. Most industrial flows, especially those in nuclear engineering are turbulent. Nevertheless laminar flow **occurs at any Reynolds number** near solid boundaries in a thin layer just adjacent to the surface, this layer is usually referred to as the** laminar sublayer** and it is very important in heat transfer.

Despite the small thickness of the **laminar sublayer** (usually much less than 1 percent of the pipe diameter), since it strongly influences the flow in the rest of the pipe. Any irregularity or roughness on the surface disturbs this layer and significantly affects the flow. Therefore, unlike laminar flow, **the friction factor** in turbulent flow is a strong function of surface roughness.

## 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.

`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.**

## Laminar Flow – Heat Transfer Coefficient

### External Laminar Flow

The average **Nusselt number** over the entire plate is determined by:

This relation gives the average **heat transfer coefficient **for the entire plate when the flow is laminar over the entire plate.

### Internal Laminar Flow

**Constant Surface Temperature**

In** laminar flow** in a tube with constant surface temperature, both the friction factor and the **heat transfer coefficient** remain constant in the fully developed region.

**Constant Surface Heat Flux**

Therefore, for fully developed **laminar flow** in a circular tube subjected to constant surface heat flux, the Nusselt number is a constant. There is no dependence on the Reynolds or the Prandtl numbers.

**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