In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, a boundary layer is the thin layer of
fluid in the immediate vicinity of a
bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a
no-slip boundary condition (zero velocity at the wall). The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.
The air next to a human is heated resulting in gravity-induced convective airflow, airflow which results in both a velocity and thermal boundary layer. A breeze disrupts the boundary layer, and hair and clothing protect it, making the human feel cooler or warmer. On an
aircraft
An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...
wing, the velocity boundary layer is the part of the flow close to the wing, where
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the in ...
forces distort the surrounding non-viscous flow. In the
Earth's atmosphere
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing fo ...
, the
atmospheric boundary layer
In meteorology, the planetary boundary layer (PBL), also known as the atmospheric boundary layer (ABL) or peplosphere, is the lowest part of the atmosphere and its behaviour is directly influenced by its contact with a planetary surface. On Ear ...
is the air layer (~ 1 km) near the ground. It is affected by the surface;
day-night heat flows caused by the sun heating the ground, moisture, or
momentum transfer to or from the surface.
Types of boundary layer
Laminar boundary layers can be loosely classified according to their structure and the circumstances under which they are created. The thin shear layer which develops on an oscillating body is an example of a
Stokes boundary layer, while the
Blasius boundary layer In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional ...
refers to the well-known
similarity solution near an attached flat plate held in an oncoming unidirectional flow and
Falkner–Skan boundary layer, a generalization of Blasius profile. When a fluid rotates and viscous forces are balanced by the
Coriolis effect (rather than convective inertia), an
Ekman layer forms. In the theory of heat transfer, a thermal boundary layer occurs. A surface can have multiple types of boundary layer simultaneously.
The viscous nature of airflow reduces the local velocities on a surface and is responsible for skin friction. The layer of air over the wing's surface that is slowed down or stopped by viscosity, is the boundary layer. There are two different types of boundary layer flow: laminar and turbulent.
Laminar boundary layer flow
The laminar boundary is a very smooth flow, while the turbulent boundary layer contains swirls or "eddies." The laminar flow creates less skin friction drag than the turbulent flow, but is less stable. Boundary layer flow over a wing surface begins as a smooth laminar flow. As the flow continues back from the leading edge, the laminar boundary layer increases in thickness.
Turbulent boundary layer flow
At some distance back from the leading edge, the smooth laminar flow breaks down and transitions to a turbulent flow. From a drag standpoint, it is advisable to have the transition from laminar to turbulent flow as far aft on the wing as possible, or have a large amount of the wing surface within the laminar portion of the boundary layer. The low energy laminar flow, however, tends to break down more suddenly than the turbulent layer.
The Prandtl Boundary Layer Concept
The
aerodynamic boundary layer was first hypothesized by
Ludwig Prandtl in a paper presented on August 12, 1904 at the third
International Congress of Mathematicians in
Heidelberg, Germany. It simplifies the equations of fluid flow by dividing the flow field into two areas: one inside the boundary layer, dominated by
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
and creating the majority of
drag experienced by the boundary body; and one outside the boundary layer, where viscosity can be neglected without significant effects on the solution. This allows a
closed-form solution for the flow in both areas by making significant simplifications of the full
Navier–Stokes equations. The same hypothesis is applicable to other fluids (besides air) with moderate to low viscosity such as water. For the case where there is a temperature difference between the surface and the bulk fluid, it is found that the majority of the
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy ( heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conducti ...
to and from a body takes place in the vicinity of the velocity boundary layer. This again allows the equations to be simplified in the flow field outside the boundary layer. The pressure distribution throughout the boundary layer in the direction normal to the surface (such as an
airfoil) remains relatively constant throughout the boundary layer, and is the same as on the surface itself.
The
thickness of the velocity boundary layer is normally defined as the distance from the solid body to the point at which the viscous flow velocity is 99% of the freestream velocity (the surface velocity of an inviscid flow).
Displacement thickness is an alternative definition stating that the boundary layer represents a deficit in mass flow compared to inviscid flow with slip at the wall. It is the distance by which the wall would have to be displaced in the inviscid case to give the same total mass flow as the viscous case. The
no-slip condition requires the flow velocity at the surface of a solid object be zero and the fluid temperature be equal to the temperature of the surface. The flow velocity will then increase rapidly within the boundary layer, governed by the boundary layer equations, below.
The
thermal boundary layer thickness is similarly the distance from the body at which the temperature is 99% of the freestream temperature. The ratio of the two thicknesses is governed by the
Prandtl number. If the Prandtl number is 1, the two boundary layers are the same thickness. If the Prandtl number is greater than 1, the thermal boundary layer is thinner than the velocity boundary layer. 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.
In high-performance designs, such as
glider
Glider may refer to:
Aircraft and transport Aircraft
* Glider (aircraft), heavier-than-air aircraft primarily intended for unpowered flight
** Glider (sailplane), a rigid-winged glider aircraft with an undercarriage, used in the sport of gliding
...
s and commercial aircraft, much attention is paid to controlling the behavior of the boundary layer to minimize drag. Two effects have to be considered. First, the boundary layer adds to the effective thickness of the body, through the
displacement thickness, hence increasing the pressure drag. Secondly, the
shear forces at the surface of the wing create
skin friction drag.
At high
Reynolds numbers, typical of full-sized aircraft, it is desirable to have a
laminar boundary layer. This results in a lower skin friction due to the characteristic velocity profile of laminar flow. However, the boundary layer inevitably thickens and becomes less stable as the flow develops along the body, and eventually becomes
turbulent, the process known as
boundary layer transition. One way of dealing with this problem is to suck the boundary layer away through a
porous surface (see
Boundary layer suction). This can reduce drag, but is usually impractical due to its mechanical complexity and the power required to move the air and dispose of it.
Natural laminar flow (NLF) techniques push the boundary layer transition aft by reshaping the airfoil or
fuselage
The fuselage (; from the French ''fuselé'' "spindle-shaped") is an aircraft's main body section. It holds crew, passengers, or cargo. In single-engine aircraft, it will usually contain an engine as well, although in some amphibious aircraft t ...
so that its thickest point is more aft and less thick. This reduces the velocities in the leading part and the same Reynolds number is achieved with a greater length.
At lower
Reynolds numbers, such as those seen with model aircraft, it is relatively easy to maintain laminar flow. This gives low skin friction, which is desirable. However, the same velocity profile which gives the laminar boundary layer its low skin friction also causes it to be badly affected by
adverse pressure gradients. As the pressure begins to recover over the rear part of the wing chord, a laminar boundary layer will tend to separate from the surface. Such
flow separation causes a large increase in the
pressure drag
Parasitic drag, also known as profile drag, is a type of aerodynamic drag that acts on any object when the object is moving through a fluid. Parasitic drag is a combination of form drag and skin friction drag. It affects all objects regardless of ...
, since it greatly increases the effective size of the wing section. In these cases, it can be advantageous to deliberately trip the boundary layer into turbulence at a point prior to the location of laminar separation, using a
turbulator
A turbulator is a device that turns a laminar boundary layer into a turbulent boundary layer.
Device
Turbulent flow can be desired on parts of the surface of an aircraft wing (airfoil) or in industrial applications such as heat exchangers and t ...
. The fuller velocity profile of the turbulent boundary layer allows it to sustain the adverse pressure gradient without separating. Thus, although the skin friction is increased, overall drag is decreased. This is the principle behind the dimpling on golf balls, as well as
vortex generators on aircraft. Special wing sections have also been designed which tailor the pressure recovery so laminar separation is reduced or even eliminated. This represents an optimum compromise between the pressure drag from flow separation and skin friction from induced turbulence.
When using half-models in wind tunnels, a
peniche is sometimes used to reduce or eliminate the effect of the boundary layer.
Boundary layer equations
The deduction of the boundary layer equations was one of the most important advances in fluid dynamics. Using an
order of magnitude analysis
Scale analysis (or order-of-magnitude analysis) is a powerful tool used in the mathematical sciences for the simplification of equations with many terms. First the approximate magnitude of individual terms in the equations is determined. Then some ...
, the well-known governing
Navier–Stokes equations of
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the in ...
fluid flow can be greatly simplified within the boundary layer. Notably, the
characteristic of the
partial differential equations (PDE) becomes parabolic, rather than the elliptical form of the full Navier–Stokes equations. This greatly simplifies the solution of the equations. By making the boundary layer approximation, the flow is divided into an inviscid portion (which is easy to solve by a number of methods) and the boundary layer, which is governed by an easier to solve
PDE. The continuity and Navier–Stokes equations for a two-dimensional steady
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An ...
in
Cartesian coordinates are given by
:
:
:
where
and
are the velocity components,
is the density,
is the pressure, and
is the
kinematic viscosity of the fluid at a point.
The approximation states that, for a sufficiently high
Reynolds number the flow over a surface can be divided into an outer region of inviscid flow unaffected by viscosity (the majority of the flow), and a region close to the surface where viscosity is important (the boundary layer). Let
and
be streamwise and transverse (wall normal) velocities respectively inside the boundary layer. Using
scale analysis, it can be shown that the above equations of motion reduce within the boundary layer to become
:
:
and if the fluid is incompressible (as liquids are under standard conditions):
:
The order of magnitude analysis assumes the streamwise length scale significantly larger than the transverse length scale inside the boundary layer. It follows that variations in properties in the streamwise direction are generally much lower than those in the wall normal direction. Apply this to the continuity equation shows that
, the wall normal velocity, is small compared with
the streamwise velocity.
Since the static pressure
is independent of
, then pressure at the edge of the boundary layer is the pressure throughout the boundary layer at a given streamwise position. The external pressure may be obtained through an application of
Bernoulli's equation. Let
be the fluid velocity outside the boundary layer, where
and
are both parallel. This gives upon substituting for
the following result
:
For a flow in which the static pressure
also does not change in the direction of the flow
:
so
remains constant.
Therefore, the equation of motion simplifies to become
:
These approximations are used in a variety of practical flow problems of scientific and engineering interest. The above analysis is for any instantaneous
laminar or
turbulent boundary layer, but is used mainly in laminar flow studies since the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
flow is also the instantaneous flow because there are no velocity fluctuations present. This simplified equation is a parabolic PDE and can be solved using a similarity solution often referred to as the
Blasius boundary layer In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional ...
.
Prandtl's transposition theorem
Prandtl observed that from any solution
which satisfies the boundary layer equations, further solution
, which is also satisfying the boundary layer equations, can be constructed by writing
:
where
is arbitrary. Since the solution is not unique from mathematical perspective, to the solution can added any one of an infinite set of eigenfunctions as shown by
Stewartson and
Paul A. Libby
Paul Andrews Libby (September 4, 1921 – November 2, 2021) was a professor of mechanical and aerospace engineering at the University of California, San Diego, a specialist in the field of combustion and aerospace engineering.
Biography
Libby r ...
.
Von Kármán momentum integral
Von Kármán derived the integral equation by integrating the boundary layer equation across the boundary layer in 1921. The equation is
:
where
:
:
is the wall shear stress,
is the suction/injection velocity at the wall,
is the displacement thickness and
is the momentum thickness.
Kármán–Pohlhausen Approximation is derived from this equation.
Energy integral
The energy integral was derived by
Wieghardt.
:
where
:
:
is the energy dissipation rate due to viscosity across the boundary layer and
is the energy thickness.
Von Mises transformation
For steady two-dimensional boundary layers,
von Mises introduced a transformation which takes
and
(
stream function) as independent variables instead of
and
and uses a dependent variable
instead of
. The boundary layer equation then become
:
The original variables are recovered from
:
This transformation is later extended to compressible boundary layer by
von Kármán and
HS Tsien.
Crocco's transformation
For steady two-dimensional compressible boundary layer,
Luigi Crocco
is a fictional character featured in video games and related media released by Nintendo. Created by Japanese video game designer Shigeru Miyamoto, Luigi is portrayed as the younger fraternal twin brother and sidekick of Mario, Nintendo's masc ...
introduced a transformation which takes
and
as independent variables instead of
and
and uses a dependent variable
(shear stress) instead of
. The boundary layer equation then becomes
:
The original coordinate is recovered from
:
Turbulent boundary layers
The treatment of turbulent boundary layers is far more difficult due to the time-dependent variation of the flow properties. One of the most widely used techniques in which turbulent flows are tackled is to apply
Reynolds decomposition
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...
. Here the instantaneous flow properties are decomposed into a mean and fluctuating component with the assumption that the mean of the fluctuating component is always zero. Applying this technique to the boundary layer equations gives the full turbulent boundary layer equations not often given in literature:
:
:
:
Using a similar order-of-magnitude analysis, the above equations can be reduced to leading order terms. By choosing length scales
for changes in the transverse-direction, and
for changes in the streamwise-direction, with