Boundary-layer thickness

This page describes some parameters used to measure the properties of boundary layers. Consider a stationary body with a turbulent flow moving around it, like the semi-infinite flat plate with air flowing over the top of the plate. At the solid walls of the body the fluid satisfies a no-slip boundary condition and has zero velocity, but as you move away from the wall, the velocity of the flow asymptotically approaches the free stream mean velocity. Therefore it is impossible to define a clear point at which the boundary layer becomes the free stream. The parameters below overcome this and allow the boundary layer to be measured.

99% Velocity thickness

The boundary layer thickness, δ, is the distance across a boundary layer from the wall to a point where the flow velocity has essentially reached the 'free stream' velocity, \(u_0\). This distance is defined normal to the wall, and the point where the flow velocity is essentially that of the free stream is customarily defined as the point where: \[u(y) = 0.99u_o\]

For laminar boundary layers over a flat plate, the Blasius solution gives: \[ \delta \approx 4.91 \sqrt{ {\nu x}\over u_0} \] \[ \delta \approx 4.91x/ \sqrt{R e_x} \]

For turbulent boundary layers over a flat plate, the boundary layer thickness is given by: \[ \delta \approx 0.382x/ {R e_x}^{1/5} \]

where \[ R e_x = \rho u_0 x/\mu\]

\[\delta\] is the overall thickness (or height) of the boundary layer \[\nu\] is the kinematic viscosity \[x\] is the distance downstream from the start of the boundary layer \[Re_x\] is the Reynolds Number \[\rho\] is the density \[u_0\] is the freestream velocity \[\mu\] is the dynamic viscosity

Displacement thickness

The displacement thickness, δ^* or δ₁ is the distance by which a surface would have to be moved in the direction parallel to its normal vector towards the reference plane in an inviscid fluid stream of velocity \(u_0\) to give the same flow rate as occurs between the surface and the reference plane in a real fluid.

In practical aerodynamics, the displacement thickness essentially modifies the shape of a body immersed in a fluid to allow an inviscid solution. It is commonly used in aerodynamics to overcome the difficulty inherent in the fact that the fluid velocity in the boundary layer approaches asymptotically to the free stream value as distance from the wall increases at any given location.

The definition of the displacement thickness for compressible flow is based on mass flow rate:

\[ {\delta^*}= \int_0^\infty {\left(1-{\rho(y) u(y)\over \rho_0 u_0}\right)dy}\]

The definition for incompressible flow can be based on volumetric flow rate, as the density is constant:

\[ {\delta^*}= \int_0^\infty {\left(1-{u(y)\over u_0}\right)dy}\]

Where \(\rho_0\) and \(u_0\) are the density and velocity in the 'free stream' outside the boundary layer, and \(y\) is the coordinate normal to the wall.

For boundary layer calculations, the density and velocity at the edge of the boundary layer must be used, as there is no free stream. In the equations above, \(\rho_0\) and \(u_0\) are therefore replaced with \(\rho_e\) and \(u_e\).

The displacement thickness is used to calculate the boundary layer's shape factor.

Momentum thickness

The momentum thickness, θ or δ₂, is the distance by which a surface would have to be moved parallel to itself towards the reference plane in an inviscid fluid stream of velocity \(u_0\) to give the same total momentum as exists between the surface and the reference plane in a real fluid.

The definition of the momentum thickness for compressible flow is based on mass flow rate:

\[ \theta = \int_0^\infty {{\rho(y) u(y)\over \rho_0 u_o} {\left(1 - {u(y)\over u_o}\right)}} dy \]

The definition for incompressible flow can be based on volumetric flow rate, as the density is constant:

\[ \theta = \int_0^\infty {{u(y)\over u_o} {\left(1 - {u(y)\over u_o}\right)}} dy \]

Where \(\rho_0\) and \(u_0\) are the density and velocity in the 'free stream' outside the boundary layer, and \(y\) is the coordinate normal to the wall.

For boundary layer calculations, the density and velocity at the edge of the boundary layer must be used, as there is no free stream. In the equations above, \(\rho_0\) and \(u_0\) are therefore replaced with \(\rho_e\) and \(u_e\).

For a flat plate at no angle of attack with a laminar boundary layer, the Blasius solution gives

\[ \theta \approx 0.664 \sqrt{{\nu x}\over u_o}\]

The influence of fluid viscosity creates a wall shear stress, \( \tau_w \), which extracts energy from the mean flow. The boundary layer can be considered to possess a total momentum flux deficit, due to the frictional dissipation.

\[ \rho \int_0^\infty {u(y) \left(u_o - u(y)\right)} dy \]

Other length scales describing viscous boundary layers include the energy thickness, \(\delta_3\).

Shape factor

A shape factor is used in boundary layer flow to determine the nature of the flow.

\[H= \frac {\delta^*}{\theta}\]

where H is the shape factor, \(\delta^*\) is the displacement thickness and θ is the momentum thickness. The higher the value of H, the stronger the adverse pressure gradient. A high adverse pressure gradient can greatly reduce the Reynolds number at which transition into turbulence may occur.

Conventionally, H=2.59 is typical of laminar flows, while H=1.3-1.4 is typical of turbulent flows.

Further Reading

• F.M. White, "Fluid Mechanics", McGraw-Hill, 5th Edition, 2003.
• B.R. Munson, D.F. Young and T.H. Okiishi, "Fundamentals of Fluid Mechanics", John Wiley, 4th Edition, 2002.
• B. Massey and J. Ward-Smith, "Mechanics of Fluids", Taylor and Francis, 8th Edition, 2006.