In flat-plate boundary-layer experiments, these three thicknesses describe different effects of the same velocity deficit. The 99% boundary-layer thickness, (\delta_{99}), is a geometric measure based on where the local velocity reaches 99% of the free-stream value. Displacement thickness, (\delta_1), measures the missing volume flow, while momentum thickness, (\delta_2), measures the missing momentum flow.

The thicknesses are not interchangeable: (\delta_{99}) describes the visible extent of the boundary layer, (\delta_1) describes its mass-flow displacement effect, and (\delta_2) describes its momentum-loss effect.

What the Three Thicknesses Measure

99% Boundary-Layer Thickness

The 99% boundary-layer thickness, commonly written (\delta_{99}), is the distance from the wall at which the local streamwise velocity reaches 99% of the free-stream velocity:

[ u(x,\delta_{99}) = 0.99U_\infty ]

Here, (u(x,y)) is the local velocity at streamwise position (x) and wall-normal position (y), while (U_\infty) is the velocity outside the boundary layer.

For laminar Blasius flow over a flat plate:

[ \delta_{99} \approx 5.0\sqrt{\frac{\nu x}{U_\infty}} ]

where (\nu) is the kinematic viscosity.

Displacement Thickness

The displacement thickness, (\delta_1), represents the reduction in volume flow caused by the slower fluid near the wall. It is the equivalent thickness of ideal, free-stream fluid that would produce the same volume-flow deficit:

[ \delta_1

\int_0^\infty \left(1-\frac{u}{U_\infty}\right),dy ]

Although the actual boundary-layer velocity gradually approaches (U_\infty), the integral is conventionally written to infinity because the velocity deficit becomes negligible outside the boundary layer.

For laminar Blasius flow:

[ \delta_1

1.7208\sqrt{\frac{\nu x}{U_\infty}} ]

This quantity is particularly useful when evaluating effective flow-channel area and the outward displacement of external streamlines.

Momentum Thickness

The momentum thickness, (\delta_2), measures the momentum-flux deficit produced by the boundary layer:

[ \delta_2

\int_0^\infty \frac{u}{U_\infty} \left(1-\frac{u}{U_\infty}\right),dy ]

The factor (u/U_\infty) accounts for the fact that momentum flux depends on velocity as well as mass flow. Therefore, momentum thickness is not simply another geometric measurement of the layer.

For laminar Blasius flow:

[ \delta_2

0.664\sqrt{\frac{\nu x}{U_\infty}} ]

Momentum thickness is commonly used in drag calculations, boundary-layer integral methods, and comparisons between theoretical and measured velocity profiles.

Mathematical Relationships for Laminar Flat-Plate Flow

Common Streamwise Scaling

All three quantities grow with the same characteristic length scale:

[ \sqrt{\frac{\nu x}{U_\infty}} ]

Thus, for fixed fluid properties and free-stream velocity, each thickness increases approximately with (\sqrt{x}) downstream of the leading edge.

The boundary layer becomes thicker farther from the leading edge because viscous effects have had more distance over which to diffuse momentum away from the wall.

Ratios to the 99% Thickness

Using the Blasius coefficients:

[ \frac{\delta_1}{\delta_{99}}

\frac{1.7208}{5.0} \approx 0.344 ]

Therefore:

[ \delta_1 \approx 0.34\delta_{99} ]

Similarly:

[ \frac{\delta_2}{\delta_{99}}

\frac{0.664}{5.0} \approx 0.133 ]

Therefore:

[ \delta_2 \approx 0.13\delta_{99} ]

These ratios explain why the displacement and momentum thicknesses are numerically smaller than the measured or calculated 99% thickness.

Relationship Between Displacement and Momentum Thickness

For laminar Blasius flow:

[ \frac{\delta_1}{\delta_2}

\frac{1.7208}{0.664} \approx 2.59 ]

This ratio is called the shape factor:

[ H

\frac{\delta_1}{\delta_2} ]

For a laminar Blasius boundary layer:

[ H \approx 2.59 ]

The shape factor summarizes the form of the velocity profile. A larger value generally indicates a fuller momentum deficit relative to the displacement effect and can be useful when assessing boundary-layer condition and separation risk.

Why the Definitions Matter in Laboratory Equipment

Interpreting Velocity-Profile Measurements

In a fluid dynamics laboratory rig, a velocity probe measures (u(x,y)) at multiple wall-normal locations. The measured profile can then be used to determine where (u/U_\infty=0.99), providing an experimental estimate of (\delta_{99}).

The same profile can be integrated numerically to calculate (\delta_1) and (\delta_2). These integral quantities use the entire velocity profile rather than one threshold location.

Assessing Effective Flow Area

The boundary layer reduces the volume flow carried by the region near the wall. Displacement thickness expresses this reduction as an equivalent blockage or outward movement of the external streamlines.

In a channel, duct, or pilot-scale flow apparatus, the effective flow area is therefore smaller than the purely geometric area. The impact becomes more important as boundary layers grow from opposing walls.

Connecting Momentum Loss to Drag

Momentum thickness is directly related to the streamwise momentum deficit created by wall friction. In a zero-pressure-gradient boundary layer, its downstream growth is associated with the momentum removed from the external flow.

This makes (\delta_2) more relevant than (\delta_{99}) when analyzing skin friction, drag, and integral momentum balances.

Understanding the Trade-offs

(\delta_{99}) Depends on a Chosen Threshold

The 99% thickness is convenient, but the boundary layer does not end at a sharply defined physical surface. Choosing 95%, 99%, or 99.9% produces different numerical thicknesses.

Consequently, (\delta_{99}) is useful for reporting the visible extent of the layer, but it is less fundamental than the integral thicknesses.

Integral Thicknesses Depend on Profile Quality

Calculating (\delta_1) and (\delta_2) requires accurate velocity measurements over the full region containing the velocity deficit. Sparse measurements, probe positioning errors, wall interference, and uncertainty in (U_\infty) can significantly affect the integrations.

Experimental data are often interpolated before numerical integration, but the chosen interpolation and the treatment of the outer measurement limit should be documented.

Blasius Relations Have Specific Assumptions

The formulas

[ \delta_{99}=5.0\sqrt{\frac{\nu x}{U_\infty}}, \qquad \delta_1=1.7208\sqrt{\frac{\nu x}{U_\infty}}, \qquad \delta_2=0.664\sqrt{\frac{\nu x}{U_\infty}} ]

apply to a steady, incompressible, two-dimensional, laminar boundary layer over a flat plate with approximately zero pressure gradient.

They should not be applied without modification to turbulent flow, strong pressure gradients, curved surfaces, three-dimensional cross-flow, compressible flow, or separated flow. In those cases, the definitions remain useful, but the numerical relationships change.

How to Apply This to Your Project

Use the three measures according to the question your laboratory experiment is intended to answer:

If your primary focus is boundary-layer growth: Use (\delta_{99}) to report the geometric thickness and compare its measured variation with (5.0\sqrt{\nu x/U_\infty}).
If your primary focus is flow blockage or effective channel area: Use (\delta_1) because it quantifies the volume-flow deficit and streamline displacement.
If your primary focus is drag or momentum loss: Use (\delta_2) and evaluate the shape factor (H=\delta_1/\delta_2).
If your primary focus is validating a laminar flat-plate experiment: Compare all three quantities and check whether (\delta_1/\delta_{99}\approx0.34), (\delta_2/\delta_{99}\approx0.13), and (H\approx2.59).

Selecting the thickness definition that matches the physical quantity of interest turns a measured velocity profile into a meaningful engineering result.

Summary Table:

Thickness	What it measures	Key formula	Blasius result
99% boundary-layer thickness, δ₉₉	Distance where velocity reaches 99% of free-stream velocity	u(x, δ₉₉) = 0.99U∞	δ₉₉ ≈ 5.0√(νx/U∞)
Displacement thickness, δ₁	Volume-flow deficit and outward displacement of external streamlines	δ₁ = ∫₀∞(1 − u/U∞)dy	δ₁ = 1.7208√(νx/U∞)
Momentum thickness, δ₂	Momentum-flux deficit associated with drag and momentum loss	δ₂ = ∫₀∞(u/U∞)(1 − u/U∞)dy	δ₂ = 0.664√(νx/U∞)
Shape factor, H	Velocity-profile shape and separation tendency indicator	H = δ₁/δ₂	H ≈ 2.59

Laminar flat-plate relationships: δ₁ ≈ 0.34δ₉₉, δ₂ ≈ 0.13δ₉₉, and all three quantities scale with √(νx/U∞).

Apply Boundary-Layer Theory to Your Laboratory

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LABPARK provides Educational and Vocational Unit Operations Pilot Plants for universities, research institutes, and enterprises, including fluid dynamics laboratory equipment for flat-plate boundary-layer experiments. Our systems help users measure velocity distributions, calculate δ₉₉, δ₁, and δ₂, evaluate shape factor, and compare experimental results with Blasius theory. Contact our team to identify the right configuration for teaching, research, process development, or industrial training.