Boundary layer

In physics and fluid mechanics, the boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. In the atmosphere the boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the flow close to the wing. The Boundary layer effect occurs at the field region in which all changes occur in the flow pattern. The boundary layer distorts surrounding nonviscous flow. It is a phenomenon of viscous forces. This effect is related to the Leidenfrost effect and the Reynolds number.

Aerodynamics

The aerodynamic boundary layer was discovered by Ludwig Prandtl at the beginning of the twentieth century and represents one of the greatest discoveries in the history of aerodynamics. It is particularly important in aerodynamics because it is directly responsible for the drag experienced by a body immersed in a fluid. 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 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. This can result in a reduction in drag, but is usually impractical due to the mechanical complexity involved.

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 tends to separate from the surface. Such separation causes a large increase in the pressure drag, 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. 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. 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 the induced turbulence

Boundary layer equations

The deduction of the boundary layer equations was perhaps one of the most important advances in aerodynamics. Using an order of magnitude analysis, the well-known governing Navier-Stokes equations of viscous 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 Navier-Stokes equations for a two-dimensional steady 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

$\nu$ 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 asymtotic 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 througout 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 laminar or turbulent boundary layer, but are 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 give the full turbulent boundary layer equations not often given in literature, viz.

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

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

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

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 necessitate 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 aerodynamics.

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 turbines, bladeless turbine, and Prandtl layer turbine (after Ludwig Prandtl).

External links

• Insert non-formatted text heresary.shtml#boundary_layer National Science Digital Library - Boundary Layer
• Moore, Franklin K., "Displacement effect of a three-dimensional boundary layer". NACA Report 1124, 1953.
• Benson, Tom, "Boundary layer". NASA Glenn Learning Technologies.
• Boundary layer separation
• Boundary layer equations: Exact Solutions - from EqWorld

Bibliography

• 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: 3540662707