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Boundary Layer visualization, showing transition from laminar to turbulent condition
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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Fluid mechanics is the study of how Fluids move and the Forces on them FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code 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. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five The planetary boundary layer ( PBL) also known as the atmospheric boundary layer ( ABL) or peplosphere, is the lowest part of the Atmosphere On an aircraft wing the boundary layer is the part of the flow close to the wing. WING "ESPN 1410" is a commercial AM radio station in Dayton Ohio operating with 5000 watts at 1410 kHz with studios offices and transmitter located on David The boundary layer effect occurs at the field region in which all changes occur in the flow pattern. A pattern, from the French patron, is a theme of recurring events or objects sometimes referred to as elements of a set The boundary layer distorts surrounding nonviscous flow. It is a phenomenon of viscous forces. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. In Physics, a force is whatever can cause an object with Mass to Accelerate. This effect is related to the Reynolds number. In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio

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. In Fluid dynamics, the Stokes boundary layer, or oscillatory boundary layer, refers to the Boundary layer close to a solid wall in oscillatory A Blasius boundary layer, in Physics and Fluid mechanics, describes the steady two-dimensional Boundary layer that forms on a semi-infinite plate which 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. In physics the Coriolis effect is an apparent deflection of moving objects when they are viewed from a Rotating frame of reference. In standard boundary-layer theory the effects of viscous Diffusion are usually balanced by convective Inertia. Thermal boundary layers also exist in heat transfer. Multiple types of boundary layers can coexist near a surface simultaneously.

Contents

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. Ludwig Prandtl ( 4 February 1875 &ndash 15 August 1953) was a German Physicist. The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community Heidelberg is a city in Baden-Württemberg, Germany. As of 2006 over 140000 people live within the city's area 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. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a 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 Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such 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. In thermal physics, heat transfer is the passage of Thermal energy from a hot to a colder body

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. In Fluid dynamics, the no-slip condition for viscous fluid states that at a solid boundary the fluid will have zero velocity relative to the boundary 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. The Prandtl number \mathrm{Pr} is a Dimensionless number approximating the ratio of momentum diffusivity ( kinematic viscosity) and Thermal diffusivity 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. Terminology A "glider" is an unpowered Aircraft. The most common types of glider are today used for sporting purposes 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. Displacement thickness is the distance by which a surface would have to be moved parallel to itself towards the reference plane in an ideal fluid stream of velocity u_0 to Secondly, the shear forces at the surface of the wing create skin friction drag. Simple Shear is a special case of Deformation of a fluid where only one component of Velocity vectors has a non-zero value \ V_x=f(xy Parasitic drag (also called parasite drag) is drag caused by moving a solid object through a fluid

At high Reynolds numbers, typical of full-sized aircraft, it is desirable to have a laminar boundary layer. In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio Laminar flow, sometimes known as streamline flow occurs when a fluid flows in parallel layers with no disruption between the layers 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. In Fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic Stochastic property changes The process of a Laminar Boundary layer becoming Turbulent is 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). Porosity is a measure of the void spaces in a material and is measured as a fraction between 0–1 or as a Percentage between 0–100% Boundary layer suction is technique in which an Air pump is used to extract the Boundary layer at the Wing or the inlet of an Aircraft 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. An airfoil (in American English) or aerofoil (in British English) is the shape of a Wing or blade (of a Propeller, rotor The fuselage (from the French fuselé "spindle-shaped" is an Aircraft 's main body section that holds crew and passengers or Cargo 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. In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio 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. An adverse Pressure gradient occurs when the Static pressure increases in the direction of the Flow. 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. All solid objects travelling through a Fluid (or alternatively a stationary object exposed to a moving fluid acquire a Boundary layer of fluid around them where viscous Parasitic drag (also called parasite drag) is drag caused by moving a solid object through a fluid 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 for improving the flow of air over a wing 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 light aircraft. A vortex generator is an Aerodynamic surface consisting of a small Vane that creates a Vortex. 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

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

One 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 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 Navier–Stokes equations of viscous fluid flow can be greatly simplified within the boundary layer. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code Notably, the characteristic of the partial differential equations (PDE) becomes parabolic, rather than the elliptical form of the full Navier–Stokes equations. In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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}\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. In Thermodynamics and Fluid mechanics, compressibility is a measure of the relative volume change of a Fluid or Solid as a response Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress.

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). In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio 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 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. In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing

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. In Fluid dynamics, Bernoulli's principle states that for an Inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in Let u0 be the fluid velocity outside the boundary layer, where u and u0 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 u0 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. Laminar flow, sometimes known as streamline flow occurs when a fluid flows in parallel layers with no disruption between the layers In Fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic Stochastic property changes In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean

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 the theory of Turbulence, Reynolds decomposition is a mathematicaltechnique to separate the average and fluctuating parts of a quantity 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. "A priori" redirects here For other uses see 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. Turbulence modeling is the area of physical modeling where a simpler Mathematical model than the full time dependent Navier-Stokes Equations is used to predict the 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. The Tesla turbine is a bladeless centrifugal flow Turbine expander patented by Nikola Tesla in 1913 There have already been discussions about Tesla's ethnicity on the talk page 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. A turbine is a rotary Engine that extracts Energy from a Fluid flow Boundary layer turbines are also known as cohesion-type turbine, bladeless turbine, and Prandtl layer turbine (after Ludwig Prandtl). Ludwig Prandtl ( 4 February 1875 &ndash 15 August 1953) was a German Physicist.

See also

References

External links

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