Similarly, as a velocity boundary layer develops when fluid flows over a surface, a thermal boundary layer must develop if the bulk temperature and surface temperature differ.
In general, when a fluid flows over a stationary surface, e.g.,, the flat plate, the bed of a river, or the pipe wall, the fluid touching the surface is brought to rest by the shear stress at the wall. The boundary layer is the region in which flow adjusts from zero velocity at the wall to a maximum in the mainstream of the flow. The concept of boundary layers is important in all viscous fluid dynamics and 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 is a characteristic linear dimension, ρ fluid density, μ dynamic viscosity, and ν kinematic viscosity.
The boundary layer is laminar for lower Reynolds numbers, and the streamwise velocity uniformly changes 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. Finally, the boundary layer is turbulent for higher Reynolds numbers, and the streamwise velocity is characterized by unsteady (changing with time) swirling flows inside the boundary layer.
The transition from laminar to turbulent boundary layer occurs when Reynolds number at x exceeds Rex ~ 500,000. The 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 due to 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.
The boundary layer may lift off or “separate” from the body and create an effective shape much different from the physical shape to make things more confusing. 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
Boundary Layer Thickness
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. For laminar boundary layers over a flat plate, the Blasius solution of the flow governing equations gives:
where Rex 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.
Thermal Boundary Layer
Similarly, as a velocity boundary layer develops when fluid flows 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 Twall. At the leading edge, the temperature profile is uniform with Tbulk. 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 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, δt, 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, heat transfer effects penetrate farther into the stream, and the thermal boundary layer grows.
The ratio of these two thicknesses (velocity and thermal boundary layers) is governed by the Prandtl number, 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 do not apply to turbulent boundary layers since turbulent mixing, in this case, may dominate the diffusion processes.