## Boundary Layer

**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**. The concept of boundary layers is of importance in all of viscous fluid dynamics and also in the theory of heat transfer.

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**.

**The Reynolds number** is 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.

**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**

## Boundary Layer Thickness

**thickness**of the boundary layer as the distance from the wall to the point where the velocity is 99% of the “free stream” velocity. For

**laminar boundary layers**over a flat plate, the

**Blasius solution**of the flow governing equations gives:

where **Re _{x}** is the Reynolds number based on the length of the plate.

For a **turbulent flow** the boundary layer thickness is given by:

This equation was derived with several assumptions. The turbulent boundary layer thickness formula assumes that the flow is turbulent right from the start of the boundary layer.

**0.1 m/s**past a flat plate 1 m long. Compute the boundary layer thickness in the middle of the plate. Assume that kinematic viscosity of water at 20°C is equal to

**1×10**.

^{-6}m^{2}/s**The Reynolds number** for the middle of the plate is equal to:

Re_{L/2} = 0.1 [m/s] x 0.5 [m] / 1×10^{-6} [m^{2}/s] = **50 000**

This satisfies the laminar conditions. The boundary layer thickness is therefore equal to:

δ ≈ 5.0 x 0.5 / (50 000)^{½} = **0.011 m**

## Thermal Boundary Layer

**velocity boundary layer**develops when there is fluid flow over a surface, a

**thermal boundary layer**must develop if the bulk temperature and surface temperature differ. Consider flow over an isothermal flat plate at a constant temperature of

**T**. At the leading edge the temperature profile is uniform with

_{wall}**T**. Fluid particles that come into contact with the plate achieve thermal equilibrium at the plate’s surface temperature. At this point, energy flow occurs at the surface

_{bulk}**purely by conduction**. These particles exchange energy with those in the adjoining fluid layer (by conduction and diffusion), and temperature gradients develop in the fluid. The region of the fluid in which these temperature gradients exist is the

**thermal boundary layer**. Its

**thickness**,

**δ**, is typically defined as the distance from the body at which the temperature is 99% of the temperature found from an inviscid solution. With increasing distance from the leading edge, the effects of heat transfer penetrate farther into the stream and the thermal boundary layer grows.

_{t}The ratio of these two thicknesses (velocity and thermal boundary layers) is governed by the Prandtl number, which is defined as the** ratio** of **momentum diffusivity** to **thermal diffusivity**. A Prandtl number of unity indicates that momentum and thermal diffusivity are comparable, and velocity and thermal boundary layers almost coincide with each other. If the Prandtl number is less than 1, which is the case for air at standard conditions, the thermal boundary layer is thicker than the velocity boundary layer. If the Prandtl number is greater than 1, the thermal boundary layer is thinner than the velocity boundary layer. Air at room temperature has a **Prandtl number** of **0.71** and for water at 18°C it is around **7.56**, which means that the thermal diffusivity is more dominant for air than for water.

Similarly as for **Prandtl Number**, the **Lewis number** physically relates the relative thickness of the thermal layer and mass-transfer (concentration) boundary layer. The **Schmidt number** physically relates the relative thickness of the velocity boundary layer and mass-transfer (concentration) boundary layer.

where n = 1/3 for most applications in all three relations. These relations, in general, are applicable only for laminar flow and are not applicable to turbulent boundary layers since turbulent mixing in this case may dominate the diffusion processes.

**Reactor Physics and Thermal Hydraulics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
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- 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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