boundary layer

Dictionary: boundary layer

n.
The layer of reduced velocity in fluids, such as air and water, that is immediately adjacent to the surface of a solid past which the fluid is flowing.

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Britannica Concise Encyclopedia: boundary layer

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In fluid mechanics, a thin layer of flowing gas or liquid in contact with a surface (e.g., of an airplane wing or the inside of a pipe). The fluid in the boundary layer is subjected to shear forces. A range of velocities is established across the boundary layer, from zero (provided the fluid is in contact with the surface) to maximum. Flow in boundary layers is more easily described mathematically than is flow in the free stream. Boundary layers are thinner at the leading edge of an aircraft wing and thicker toward the trailing edge; such boundary layers generally have laminar flow in the leading (upstream) portion and turbulent flow in the trailing (downstream) portion. See also drag.

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Sci-Tech Encyclopedia: Boundary-layer flow

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That portion of a fluid flow, near a solid surface, where shear stresses are significant and the inviscid-flow assumption may not be used. All solid surfaces interact with a viscous fluid flow because of the no-slip condition, a physical requirement that the fluid and solid have equal velocities at their interface. Thus a fluid flow is retarded by a fixed solid surface, and a finite, slow-moving boundary layer is formed. A requirement for the boundary layer to be thin is that the Reynolds number of the body be large, 10³ or more. Under these conditions the flow outside the boundary layer is essentially inviscid and plays the role of a driving mechanism for the layer. See also Reynolds number.

A typical low-speed or laminar boundary layer is shown in the illustration. Such a display of the streamwise flow vector variation near the wall is called a velocity profile. The no-slip condition requires that u(x, 0) = 0, as shown, where u is the velocity of flow in the boundary layer. The velocity rises monotonically with distance y from the wall, finally merging smoothly with the outer (inviscid) stream velocity U(x). At any point in the boundary layer, the fluid shear stress τ is proportional to the local velocity gradient, assuming a newtonian fluid. The value of the shear stress at the wall is most important, since it relates not only to the drag of the body but often also to its heat transfer. At the edge of the boundary layer, τ approaches zero asymptotically. There is no exact spot where τ = 0; therefore the thickness δ of a boundary layer is usually defined arbitrarily as the point where u = 0.99U. See also Laminar flow.

Typical laminar boundary-layer velocity profile.

When a flow enters a duct or confined region, boundary layers immediately begin to grow on the duct walls. An inviscid core accelerates down the duct center, but soon vanishes as the boundary layers meet and fill the duct with viscous flow. Constrained by the duct walls into a no-growth condition, the velocity profile settles into a fully developed shape which is independent of the streamwise coordinate. The pressure drops linearly downstream, balanced by the mean wall-shear stress. This is a classic and simple case of boundary-layer flow which is well documented by both theory and experiment.

A classic incompressible boundary-layer flow is a uniform stream at velocity U, moving past a sharp flat plate parallel to the stream. In the Reynolds number range 1 × 10³ to 5 × 10⁵, the flow is laminar and orderly, with no superimposed fluctuations. The boundary-layer thickness δ grows monotonically with x, and the shape of the velocity profile is independent of x when normalized. The profiles are said to be similar, and they are called Blasius profiles.

The Blasius flat-plate flow results in closed-form algebraic formulas for such parameters as wall-shear stress and boundary-layer thickness as well as for temperature and heat-transfer parameters. These results are useful in estimating viscous effects in flow past thin bodies such as airfoils, turbine blades, and heat-exchanger plates.

The flat plate is very distinctive in that it causes no change in outer-stream velocity U. Most body shapes immersed in a stream flow, such as cylinders, airfoils, or ships, induce a variable outer stream U(x) near the surface. If U increases with x, which means that pressure decreases with x, the boundary layer is said to be in a favorable gradient and remains thin and attached to the surface. If, however, velocity falls and pressure rises with x, the pressure gradient is unfavorable or adverse. The low-velocity fluid near the wall is strongly decelerated by the rising pressure, and the wall-shear stress drops off to zero. Downstream of this zero-shear or separation point, there is backflow and the wall shear is upstream. The boundary layer thickens markedly to conserve mass, and the outer stream separates from the body, leaving a broad, low-pressure wake downstream. Flow separation may be predicted by boundary-layer theory, but the theory is not able to estimate the wake properties accurately.

In most immersed-body flows, the separation and wake occur on the rear or lee side of the body, with higher pressure and no separation on the front. The body thus experiences a large downstream pressure force called pressure drag. This happens to all blunt bodies such as spheres and cylinders and also to airfoils and turbomachinery blades if their angle of attack with respect to the oncoming stream is too large. The airfoil or blade is said to be stalled, and its performance suffers.

All laminar boundary layers, if they grow thick enough and have sufficient velocity, become unstable. Slight disturbances, whether naturally occurring or imposed artificially, tend to grow in amplitude, at least in a certain frequency and wavelength range. The growth begins as a selective group of two-dimensional periodic disturbances, called Tollmien-Schlichting waves, which become three-dimensional and nonlinear downstream and eventually burst into the strong random fluctuations called turbulence. The critical parameter is the Reynolds number. The process of change from laminar to turbulent flow is called transition.

The turbulent flow regime is characterized by random, three-dimensional fluctuations superimposed upon time-mean fluid properties, including velocity, pressure, and temperature. The fluctuations are typically 3–6% of the mean values and range in size over three orders of magnitude, from microscale movements to large eddies of size comparable to the boundary-layer thickness. They are readily measured by modern instruments such as hot wires and laser-Doppler velocimeters. See also Anemometer.

The effect of superimposing a wide spectrum of eddies on a viscous flow is to greatly increase mixing and transport of mass, momentum, and heat across the flow. Turbulent boundary layers are thicker than laminar layers and have higher heat transfer and friction. The turbulent mean-velocity profile is rather flat, with a steep gradient at the wall. The edge of the boundary layer is a ragged, fluctuating interface which separates the nonturbulent outer flow from large turbulent eddies in the layer. The thickness of such a layer is defined only in the time mean, and a probe placed in the outer half of the layer would show intermittently turbulent and nonturbulent flow.

As the stream velocity U becomes larger, its kinetic energy, U²/2, becomes comparable to stream enthalpy, c_pT, where c_p is the specific heat at constant pressure and T is the absolute temperature. Changes in temperature and density begin to be important, and the flow can no longer be considered incompressible. Liquids flow at very small Mach numbers, and compressible flows are primarily gas flows. See also Gas; Mach number.

In a flow with supersonic stream velocity, the no-slip condition is still valid, and much of the boundary-layer flow near the wall is at low speed or subsonic. The fluid enters the boundary layer and loses much of its kinetic energy, of which a small part is conducted away although most is converted into thermal energy. Thus the near-wall region of a highly compressible boundary layer is very hot, even if the wall is cold and is drawing heat away. The basic difference between low and high speed is the conversion of kinetic energy into higher temperatures across the entire boundary layer.

In a low-speed (incompressible) boundary layer, a cold wall simply means that the wall temperature is less than the free-stream temperature. The heat flow is from high toward lower temperature, that is, into the wall. For a low-speed insulated wall, the boundary-layer temperature is uniform. For a high-speed flow, however, an insulated wall has a high surface temperature because of the viscous dissipation energy exchange in the layer.

Except for the added complexity of having to consider fluid pressure, temperature, and density as coupled variables, compressible boundary layers have similar characteristics to their low-speed counterparts. They undergo transition from laminar to turbulent flow but typically at somewhat higher Reynolds numbers. Compressible layers tend to be somewhat thicker than incompressible boundary layers, with proportionally smaller wall-shear stresses. They tend to resist flow separation slightly better than incompressible flows.

In a supersonic outer stream, shock waves can always occur. Shocks may form in the boundary layer because of obstacles in the layer or downstream, or they may be formed elsewhere and impinge upon a boundary. In either case, the pressure rises sharply behind the shock, an adverse gradient, and this tends to cause early transition to turbulence and early flow separation. Special care must be taken to design aerodynamic surfaces to accommodate or avoid shockwave formation in transonic and supersonic flows. See also Compressible flow.

As boundary layers move downstream, they tend to grow naturally and undergo transition to turbulence. Boundary layers encountering rising pressure undergo flow separation. Both phenomena can be controlled at least partially. Airfoils and hydrofoils can be shaped to delay adverse pressure gradients and thus move separation downstream. Proper shaping can also delay transition. Wall suction removes the low-momentum fluid and delays both transition and separation. Wall blowing into the boundary layer, from downward-facing slots, delays separation but not transition. Changing the wall temperature to hotter for liquids and colder for gases delays transition. Practical systems have been designed for boundary-layer control, but they are often expensive and mechanically complex. See also Airfoil; Streamlining; Viscosity.

Geography Dictionary: boundary layer

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In meteorology, any layer of the atmosphere significantly affected by its lower boundary: the earth's surface. The laminar boundary layer is the few millimetres above the surface; the turbulent boundary layer, or surface boundary layer, is an ill-defined layer covering the conspicuously turbulent part of the atmosphere.

Sports Science and Medicine: boundary layer

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The layer of fluid immediately adjacent to a body in a fluid. Adhesion between fluid particles and the body surface create viscous stresses (see viscosity), which increase drag.

Wikipedia: Boundary layer

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Boundary layer visualization, showing transition from laminar to turbulent condition

In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. In the Earth's atmosphere, the planetary 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 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 boundary layer, whilst the Blasius boundary layer refers to the well-known similarity solution for the steady boundary layer attached to a flat plate held in an oncoming unidirectional flow. When a fluid rotates, viscous forces may be balanced by the Coriolis effect, rather than convective inertia, leading to the formation of an Ekman layer. Thermal boundary layers also exist in heat transfer. Multiple types of boundary layers can coexist near a surface simultaneously.

1 Aerodynamics
2 Naval architecture
3 Boundary layer equations
4 Turbulent boundary layers
5 Boundary layer turbine
6 See also
7 References
8 External links

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 viscosity is dominant and the majority of the drag experienced by a body immersed in a fluid is created, 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, which is a significant simplification over the solution of the full Navier–Stokes equations. The majority of the 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. An alternative definition, the displacement thickness, recognises the fact that the boundary layer represents a deficit in mass flow compared to an inviscid case 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 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 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 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. Natural laminar flow is the name for techniques pushing the boundary layer transition aft by shaping of an aerofoil or a fuselage so that their thickest point is 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, 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 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.

Naval architecture

Please help improve this article by expanding it. Further information might be found on the talk page. (April 2009)

Many of the principles that apply to aircraft also apply to ships, submarines, and offshore platforms.

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 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 continuity and Navier–Stokes equations for a two-dimensional steady incompressible flow in Cartesian coordinates are given by

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

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

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

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 scale 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 if the fluid is incompressible (as liquids are under standard conditions):

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

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 laminar or 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:

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

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

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

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 a major obstacle in 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 (1992). Fundamentals of Aerodynamics (2nd edition ed.). Toronto: S.S.CHAND. pp. 711–714. ISBN 0-07-001679-8.

External links

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

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boundary layer

Contents

Aerodynamics

Naval architecture

Boundary layer equations

Turbulent boundary layers

Boundary layer turbine

See also

References

External links