Lecture # 27
The boundary-layer concept:
The concept of a boundary layer was first introduced by Ludwig Prandtl, a German
aerodynamicist, in 1904.
Prior to Prandtl's historic breakthrough, the science of fluid mechanics had been developing
in two rather different directions. Theoretical hydrodynamics evolved from Euler's equation of
motion for a non-viscous fluid (The Euler’s equation of motion). Since the results of
hydrodynamics contradicted many experimental observations (especially, under the
assumption of inviscid flow no bodies experience drag), practicing engineers developed their
own empirical art of hydraulics. This was based on experimental data and differed
significantly from the purelymathematical approach of theoretical hydrodynamics.
Although the complete equations describing the motion of a viscous fluid (the Navier-Stokes
equations) were known prior to Prandtl, the mathematical difficulties in solving these
equations (except for a few simple cases) prohibited a theoretical treatment of viscous flows.
Prandtl showed that many viscous flows can be analyzed by dividing the flow into two
regions, one close to solid boundaries, the other covering the rest of the flow. Only in the thin
region adjacent to a solid boundary (the boundary layer) is the effect of viscosity important.
In the region outside of the boundary layer the effect of viscosity is negligible and the fluid
may be treated as inviscid.
The boundary-layer concept provided the link that had been missing between theory and
practice (for one thing, it introduced the theoretical possibility of drag!). Furthermore, the
boundary-layer concept permitted the solution of viscous flow problems that would have
been impossible through application of the Navier-Stokes equations to the complete flow
field.1 Thus the introduction of the boundary-layer concept marked the beginning of the
modern era of fluid mechanics.
In the boundary layer both viscous and inertia forces are important. Consequently, it is not
surprising that the Reynolds number (which represents the ratio of inertia to viscous forces) is
significant in characterizing boundary-layer flows.
The characteristic length used in the Reynolds number is either the length in the flow direction
over which the boundary layer has developed or some measure of the boundary layer
thickness.
As is true for flow in a duct, flow in a boundary layer may be laminar or turbulent. There is no
unique value of Reynolds number at which transition from laminar to turbulent flow occurs in
a boundary layer. Among the factors that affect boundary layer transition are pressure
gradient, surface roughness, heat transfer, body forces, and freestream disturbances. Detailed
consideration of these effects is beyond the scope of this course.
In many real flow situations, a boundary layer develops over a long, essentially flat surface.
Examples include flow over ship and submarinehulls, aircraft wings, and atmospheric motions
over flat terrain. Since the basic features of all these flows are illustrated in the simpler case
of flow over a flat plate, we consider this first. The simplicity of the flow over an infinite flat
plate is that the velocity U outside the boundary layer is constant, and therefore, because this
region is steady, inviscid, and incompressible, the pressure will also be constant. This constant
pressure is the pressure felt by the boundary layer—obviously the simplest pressure field
possible. This is a zero pressure gradient flow.
A qualitative picture of the boundary-layer growth over a flat plate is shown in Fig. The
boundary layer is laminar for a short distance downstream from the leading edge; transition
occurs over a region of theplate rather than at a single line across the plate. The transition
region extends downstream to the location where the boundary-layer flow becomes
completely turbulent.
For incompressible flow over a smooth flat plate (zero pressure gradient), in the absence of
heat transfer, transition from laminar to turbulent flow in the boundary layer can be delayed
to a Reynolds number, 𝑹𝒆 = 𝝆𝑼𝒙/𝝁, greater than one million if external disturbances are
minimized. (The length 𝒙 is measured from the leading edge.) For calculation purposes, under
typical flow conditions, transition usually is considered to occur at a length Reynolds number
of 500,000. For air at standard conditions, with freestream velocity 𝑼 = 𝟑𝟎 𝒎/𝒔, this
corresponds to 𝒙 ≈ 𝟎. 𝟐𝟒 𝒎. In the qualitative picture of Fig., the turbulent boundary layer is
shown growing faster than the laminar layer, which is indeed true.
Boundary-Layer thickness:
The boundary layer is the region adjacent to a solid surface in which viscous stresses are
present. These stresses are present because we have shearing of the fluid layers, i.e., velocity
gradient, in the boundary layer. As indicated in Fig., both laminar and turbulent layers have
such gradients, but the difficulty is that the gradients only asymptotically approach zero as we
reach the edge of the boundary layer. Hence, the location of the edge, i.e., of the boundarylayer thickness, is not very obvious—we cannot simply define it as where the boundary-layer
velocity u equals the freestream velocity U. Because of this, several boundary-layer definitions
have been developed: the disturbance thickness 𝜹, the displacement thickness 𝜹∗ , and the
momentum thickness 𝜽. (Each of these increases as we move down the plate, in a manner we
have yet to determine.)
The most straight forward definition is the disturbance thickness, 𝜹. This is usually defined as
the distance from the surface at which the velocity is within 1% of the free stream, 𝒖 ≈
𝟎. 𝟗𝟗𝑼 . The other two definitions are based on the notion that the boundary layer retards
the fluid, so that the mass flux and momentum flux are both less than they would be in the
absence of the boundary layer. We imagine that the flow remains at uniform velocity U, but
the surface of the plate is moved upwards to reduce either the mass or momentum flux by the
same amount that the boundary layer actually does. The displacement thickness, 𝜹∗ , is the
distance the plate would be moved so that the loss of mass flux (due to reduction in uniform
flow area) is equivalent to the loss the boundary layer causes. The mass flux if we had no
∞
boundary layer would be ∫𝟎 𝝆𝑼𝒅𝒚 𝒘, where 𝒘 is the width of the plate perpendicular to the
∞
flow. The actual flow mass flux is ∫𝟎 𝝆𝒖𝒅𝒚 𝒘. Hence, the loss due to the boundary layer is
∞
∫𝟎 𝝆(𝑼 − 𝒖)𝒅𝒚 𝒘. If we imagine keeping the velocity at a constant U, and instead move the
plate up a distance 𝜹∗ , the loss of mass flux would be 𝝆𝒖𝜹∗ 𝒘. Setting these losses equal to
one another gives
∞
𝝆𝒖𝜹∗ 𝒘 = ∫ 𝝆(𝑼 − 𝒖)𝒅𝒚 𝒘
𝟎
For Incompressible flow 𝜹 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 and
∞
𝜹
𝒖
𝒖
𝜹∗ = ∫ (𝟏 − )𝒅𝒚 ≈ ∫ (𝟏 − )𝒅𝒚
𝑼
𝑼
𝟎
𝟎
Since 𝒖 ≈ 𝑼 at 𝒚 ≈ 𝜹, the integrand is essentially zero for 𝒚 ≥ 𝜹.
The momentum thickness, 𝜽, is the distance the plate would be moved so that the loss of
momentum flux is equivalent to the loss the boundary layer actually causes. The momentum
flux if we had no boundary layer would be
∞
∞
∫𝟎 𝝆𝑼𝒅𝒚 𝒘, (the actual mass flux is ∫𝟎 𝝆𝒖𝒅𝒚 𝒘, and the momentum per unit mass flux of
∞
the uniform flow is U itself). The actual momentum flux of the boundary layer is ∫𝟎 𝝆𝒖𝟐 𝒅𝒚 𝒘.
∞
Hence, the loss of momentum in the boundary layer is ∫𝟎 𝝆𝒖(𝑼 − 𝒖)𝒅𝒚 𝒘. If we imagine
keeping the velocity at a constant 𝑼, and instead move the plate up a distance 𝜽 (as shown in
𝜽
Fig. 3c), the loss of momentum flux would be ∫𝟎 𝝆𝑼𝑼𝒅𝒚 𝒘 = 𝝆𝑼𝟐 𝜽𝒘. Setting these losses
equal to one another gives
∞
𝟐
𝝆𝑼 𝜽 = ∫ 𝝆𝒖(𝑼 − 𝒖)𝒅𝒚
𝟎
∞
𝜹
𝒖
𝒖
𝒖
𝒖
𝜽 = ∫ 𝝆 (𝟏 − )𝒅𝒚 ≈ ∫ 𝝆 (𝟏 − )𝒅𝒚
𝑼
𝑼
𝑼
𝑼
𝟎
𝟎
the integrand is essentially zero for 𝒚 ≥ 𝜹.
The displacement and momentum thicknesses, 𝜹∗ and 𝜽, are integral thicknesses, because
their definitions, are in terms of integrals across the boundary layer. Because they are defined
in terms of integrals for which the integrand vanishes in the freestream, they are appreciably
easier to evaluate accurately from experimental data than the boundary-layer disturbance
thickness, 𝜹. This fact, coupled with their physical significance, accounts for their common use
in specifying boundary-layer thickness.
We have seen that the velocity profile in the boundary layer merges into the local freestream
velocity asymptotically. Little error is introduced if the slight difference between velocities at
the edge of the boundary layer is ignored for an approximate analysis. Simplifying
assumptions usually made for engineering analyses of boundary layer development are:
1. 𝒖 → 𝑼 𝒂𝒕 𝒚 = 𝜹.
𝝏𝒖
2. 𝝏𝒚 → 𝟎 𝒂𝒕 𝒚 = 𝜹.
3. 𝒖 << 𝑼
Results of the analyses developed in the next two sections show that the boundary layer is
very thin compared with its development length along the surface. Therefore it is also
reasonable to assume:
4. Pressure variation across the thin boundary layer is negligible. The freestream pressure
distribution is impressed on the boundary layer.