Boundary layer

In physics and Reynolds number.

Laminar boundary layers come in various forms and 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 layer, whilst the Ekman layer. Thermal boundary layers also exist in heat transfer. Multiple types of boundary layers can coexist near a surface simultaneously.

Aerodynamics

The aerodynamic boundary layer was first defined by Ludwig Prandtl in a paper presented on August 12, 1904 at the third International Congress of Mathematicians in Heidelberg, Germany. It allows aerodynamicists to simplify the equations of fluid flow by dividing the flow field into two areas: one inside the boundary layer, where heat transfer to and from a body also takes place within the boundary layer, again allowing the equations to be simplified in the flow field outside the boundary layer.

The thickness of the velocity boundary layer is normally defined as the distance from the solid body at which the flow velocity is 99% of the freestream velocity, that is, the velocity that is calculated at the surface of the body in an inviscid flow solution. The no-slip condition requires that the flow velocity at the surface of a solid object is zero and that the fluid temperature is 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 temperature found from an inviscid solution. 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 sailplanes and commercial transport aircraft, much attention is paid to controlling the behavior of the boundary layer to minimize drag. Two effects must to be considered. First, the boundary layer adds to the effective thickness of the body, through the skin friction drag.

At high Boundary layer suction). This can result in a reduction in drag, but is usually impractical due to the mechanical complexity involved and the power required to move the air and dispose of it.

At lower turbulator. 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 the drag is decreased. This is the principle behind the dimpling on golf balls, as well as vortex generators on light aircraft. Special wing sections have also been designed which tailor the pressure recovery so that 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.

Naval Architecture

Many of the principles that apply to aircraft also apply to ships and offshore platforms, however there are a few key differences.

One key difference is the mass of the boundary layer. Since a good portion of the boundary layer travels at or near the speed of the ship, the energy required to accelerate and decelerate this additional mass must be taken into account. When calculating the power required by the engine, this mass is added to the mass of the ship. In aircraft, this additional mass is not usually taken into account because the weight of the air is so small. However, in ship design, this mass can easily reach 1/4 or 1/3 of the weight of the actual ship and therefore represents a significant drag in addition to frictional drag.

Boundary layer equations

The deduction of the boundary layer equations was perhaps one of the most important advances in fluid dynamics. Using an order of magnitude analysis, the well-known governing incompressible flow in cartesian coordinates are given by

${\partial u\over\partial x}+{\partial v\over\partial y}=0$

$u{\partial u \over \partial x}+v{\partial u \over \partial y}=-{1\over \rho} {\partial p \over \partial x}+{\nu}({\partial^2 u\over \partial x^2}+{\partial^2 u\over \partial y^2})$

$u{\partial v \over \partial x}+v{\partial v \over \partial y}=-{1\over \rho} {\partial p \over \partial y}+{\nu}({\partial^2 v\over \partial x^2}+{\partial^2 v\over \partial y^2})$

where u and v are the velocity components, $ρ$ is the density, p is the pressure, and $ν$ 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 $u$ and $v$ be streamwise and transverse (wall normal) velocities respectively inside the boundary layer. Using asymptotic analysis, it can be shown that the above equations of motion reduce within the boundary layer to become

${\partial u\over\partial x}+{\partial v\over\partial y}=0$

$u{\partial u \over \partial x}+v{\partial u \over \partial y}=-{1\over \rho} {\partial p \over \partial x}+{\nu}{\partial^2 u\over \partial y^2}$

and the remarkable result that

${1\over \rho} {\partial p \over \partial y}=0$

The asymptotic analysis also shows that $v$ , the wall normal velocity, is small compared with $u$ the streamwise velocity, and that variations in properties in the streamwise direction are generally much lower than those in the wall normal direction.

Since the static pressure $p$ is independent of $y$ , 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 $u 0$ be the fluid velocity outside the boundary layer, where $u$ and $u 0$ are both parallel. This gives upon substituting for $p$ the following result

$u{\partial u \over \partial x}+v{\partial u \over \partial y}=u_0{\partial u_0 \over \partial x}+{\nu}{\partial^2 u\over \partial y^2}$

with the boundary condition

${\partial u\over\partial x}+{\partial v\over\partial y}=0$

For a flow in which the static pressure $p$ also does not change in the direction of the flow then

${\partial p\over\partial x}=0$

so $u 0$ remains constant.

Therefore, the equation of motion simplifies to become

$u{\partial u \over \partial x}+v{\partial u \over \partial y}={\nu}{\partial^2 u\over \partial y^2}$

These approximations are used in a variety of practical flow problems of scientific and engineering interest. The above analysis is for any instantaneous turbulent boundary layer, but is used mainly in laminar flow studies since the mean flow is also the instantaneous flow because there are no velocity fluctuations present.

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. Here the instantaneous flow properties are decomposed into a mean and fluctuating component. Applying this technique to the boundary layer equations gives the full turbulent boundary layer equations not often given in literature:

${\partial \overline{u}\over\partial x}+{\partial \overline{v}\over\partial y}=0$

$\overline{u}{\partial \overline{u} \over \partial x}+\overline{v}{\partial \overline{u} \over \partial y}=-{1\over \rho} {\partial \overline{p} \over \partial x}+{\nu}({\partial^2 \overline{u}\over \partial x^2}+{\partial^2 \overline{u}\over \partial y^2})-\frac{\partial}{\partial y}(\overline{u'v'})-\frac{\partial}{\partial x}(\overline{u'^2})$

$\overline{u}{\partial \overline{v} \over \partial x}+\overline{v}{\partial \overline{v} \over \partial y}=-{1\over \rho} {\partial \overline{p} \over \partial y}+{\nu}({\partial^2 \overline{v}\over \partial x^2}+{\partial^2 \overline{v}\over \partial y^2})-\frac{\partial}{\partial x}(\overline{u'v'})-\frac{\partial}{\partial y}(\overline{v'^2})$

Using the same order-of-magnitude analysis as for the instantaneous equations, these turbulent boundary layer equations generally reduce to become in their classical form:

${\partial \overline{u}\over\partial x}+{\partial \overline{v}\over\partial y}=0$

$\overline{u}{\partial \overline{u} \over \partial x}+\overline{v}{\partial \overline{u} \over \partial y}=-{1\over \rho} {\partial \overline{p} \over \partial x}+{\nu}{\partial^2 \overline{u}\over \partial y^2}-\frac{\partial}{\partial y}(\overline{u'v'})$

${\partial \overline{p} \over \partial y}=0$

The additional term $\overline{u'v'}$ in the turbulent boundary layer equations is known as the Reynolds shear stress and is unknown a priori. The solution of the turbulent boundary layer equations therefore necessitates the use of a turbulence model, which aims to express the Reynolds shear stress in terms of known flow variables or derivatives. The lack of accuracy and generality of such models is the single major obstacle which inhibits the successful prediction of turbulent flow properties in modern fluid dynamics.

Boundary layer turbine

This effect was exploited in the Tesla turbine, patented by Nikola Tesla in 1913. It is referred to as a bladeless turbine because it uses the boundary layer effect and not a fluid impinging upon the blades as in a conventional turbine. Boundary layer turbines are also known as cohesion-type turbine, bladeless turbine, and Prandtl layer turbine (after Ludwig Prandtl).

References

A.D. Polyanin and V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton - London, 2004. ISBN 1-58488-355-3
A.D. Polyanin, A.M. Kutepov, A.V. Vyazmin, and D.A. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, 2002. ISBN 0-415-27237-8
Herrmann Schlichting, Klaus Gersten, E. Krause, H. Jr. Oertel, C. Mayes "Boundary-Layer Theory" 8th edition Springer 2004 ISBN 3-540-66270-7
John D. Anderson, Jr, "Ludwig Prandtl's Boundary Layer", Physics Today, December 2005
Anderson, John (1991). Fundamentals of Aerodynamics, 2nd edition, Toronto: McGraw-Hill, 711-714. ISBN 0-07-001679-8.

Boundary layers

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Boundary_layer". A list of authors is available in Wikipedia.

Boundary layer

Aerodynamics

Naval Architecture

Boundary layer equations

Turbulent boundary layers

Boundary layer turbine

See also

References