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 rel ...

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 fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines. ...

wing A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is exp ...

, the velocity boundary layer is the part of the flow close to the wing, where viscous

force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...

s 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 f ...

, the atmospheric boundary layer 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 In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...

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 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 Heidelberg (; Palatine German language, Palatine German: ''Heidlberg'') is a city in the States of Germany, German state of Baden-Württemberg, situated on the river Neckar in south-west Germany. As of the 2016 census, its population was 159,914 ...

. 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 inte ...

and creating the majority of

drag Drag or The Drag may refer to: Places * Drag, Norway, a village in Tysfjord municipality, Nordland, Norway * ''Drág'', the Hungarian name for Dragu Commune in Sălaj County, Romania * Drag (Austin, Texas), the portion of Guadalupe Street adj ...

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 In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...

for the flow in both areas by making significant simplifications of the full

Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician G ...

. 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 conduction ...

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 An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or tur ...

) 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 This page describes some of the parameters used to characterize the thickness and shape of boundary layers formed by fluid flowing along a solid surface. The defining characteristic of boundary layer flow is that at the solid walls, the fluid's ve ...

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 gliders 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 This page describes some of the parameters used to characterize the thickness and shape of boundary layers formed by fluid flowing along a solid surface. The defining characteristic of boundary layer flow is that at the solid walls, the fluid's ve ...

, 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 Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...

. 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 In fluid dynamics, flow separation or boundary layer separation is the detachment of a boundary layer from a surface into a wake. A boundary layer exists whenever there is relative movement between a fluid and a solid surface with viscous ...

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 generator A vortex generator (VG) is an aerodynamic device, consisting of a small vane usually attached to a lifting surface (or airfoil, such as an aircraft wing) or a rotor blade of a wind turbine.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, the well-known governing

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 :

+=0

u+\upsilon=- +\left(+\right)

u+\upsilon=- +\left(+\right)

where

u

and

\upsilon

are the velocity components,

\rho

is the density,

p

is the pressure, and

\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

\upsilon

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 :

u+\upsilon=- +

=0

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

+=0

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

\upsilon

, the wall normal velocity, is small compared with

u

the streamwise velocity. 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

be the fluid velocity outside the boundary layer, where

u

and

U

are both parallel. This gives upon substituting for

p

the following result :

u+\upsilon=U\frac+

For a flow in which the static pressure

p

also does not change in the direction of the flow :

\frac=0

U

remains constant. Therefore, the equation of motion simplifies to become :

u+\upsilon=

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 '' ari ...

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.

Prandtl's transposition theorem

Prandtl observed that from any solution

u(x,y,t),\ v(x,y,t)

which satisfies the boundary layer equations, further solution

u^*(x,y,t),\ v^*(x,y,t)

, which is also satisfying the boundary layer equations, can be constructed by writing :

u^*(x,y,t) = u(x,y+f(x),t), \quad v^*(x,y,t) = v(x,y+f(x),t) - f'(x) u(x,y+f(x),t)

where

f(x)

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.

Von Kármán momentum integral

Von Kármán The term ''von'' () is used in German language surnames either as a nobiliary particle indicating a noble patrilineality, or as a simple preposition used by commoners that means ''of'' or ''from''. Nobility directories like the ''Almanach de Go ...

derived the integral equation by integrating the boundary layer equation across the boundary layer in 1921. The equation is :

\frac = \frac\frac(U\delta_1) + \frac +\frac \frac + \frac

where :

\tau_w = \mu \left( \frac\right)_, \quad v_w = v(x,0,t), \quad \delta_1 = \int_0^\infty \left(1- \frac \right) \, dy, \quad \delta_2 = \int_0^\infty \frac \left(1- \frac\right) \, dy

\tau_w

is the wall shear stress,

v_w

is the suction/injection velocity at the wall,

\delta_1

is the displacement thickness and

\delta_2

is the momentum thickness. Kármán–Pohlhausen Approximation is derived from this equation.

Energy integral

The energy integral was derived by Wieghardt. :

\frac = \frac\frac(\delta_1 + \delta_2) + \frac\frac  +\frac \frac(U^3\delta_3) + \frac

where :

\varepsilon = \int_0^\infty \mu \left( \frac\right)^2 dy, \quad \delta_3 = \int_0^\infty \frac\left(1- \frac\right) \, dy

\varepsilon

is the energy dissipation rate due to viscosity across the boundary layer and

\delta_3

is the energy thickness.

Von Mises transformation

For steady two-dimensional boundary layers, von Mises introduced a transformation which takes

x

and

\psi

(

stream function The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. ...

) as independent variables instead of

x

and

y

and uses a dependent variable

\chi = U^2-u^2

instead of

u

. The boundary layer equation then become :

\frac = \nu \sqrt \, \frac

The original variables are recovered from :

y = \int \sqrt \, d\psi, \quad u = \sqrt, \quad v = u\int \frac \left(\frac\right) \, d\psi.

This transformation is later extended to compressible boundary layer by

von Kármán The term ''von'' () is used in German language surnames either as a nobiliary particle indicating a noble patrilineality, or as a simple preposition used by commoners that means ''of'' or ''from''. Nobility directories like the ''Almanach de Go ...

and HS Tsien.

Crocco's transformation

For steady two-dimensional compressible boundary layer, Luigi Crocco introduced a transformation which takes

x

and

u

as independent variables instead of

x

and

y

and uses a dependent variable

\tau=\mu\partial u/\partial y

(shear stress) instead of

u

. The boundary layer equation then becomes :

& \text \frac=0, \text \frac \frac = \frac\frac. \end

The original coordinate is recovered from :

y = \mu \int \frac \tau .

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 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: :

+=0

\overline+\overline=- + \nu \left(+\right)-\frac(\overline)-\frac(\overline)

\overline+\overline=- +\nu \left(+\right)-\frac(\overline)-\frac(\overline)

Using a similar order-of-magnitude analysis, the above equations can be reduced to leading order terms. By choosing length scales

\delta

for changes in the transverse-direction, and

L

for changes in the streamwise-direction, with

\delta<, the x-momentum equation simplifies to:

: \overline+\overline=- -\frac(\overline). This equation does not satisfy the no-slip condition at the wall. Like Prandtl did for his boundary layer equations, a new, smaller length scale must be used to allow the viscous term to become leading order in the momentum equation. By choosing \eta<<\delta as the ''y''-scale, the leading order momentum equation for this "inner boundary layer" is given by:

: 0=- +-\frac(\overline). In the limit of infinite Reynolds number, the pressure gradient term can be shown to have no effect on the inner region of the turbulent boundary layer. The new "inner length scale" \eta is a viscous length scale, and is of order \frac, with u_* being the velocity scale of the turbulent fluctuations, in this case a friction velocity .

Unlike the laminar boundary layer equations, the presence of two regimes governed by different sets of flow scales (i.e. the inner and outer scaling) has made finding a universal similarity solution for the turbulent boundary layer difficult and controversial. To find a similarity solution that spans both regions of the flow, it is necessary to asymptotically match the solutions from both regions of the flow. Such analysis will yield either the so-called log-law or power-law .

Similar approaches to the above analysis has also been applied for thermal boundary layers, using the energy equation in compressible flows.

The additional term \overline 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 Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real life scenarios, including the flow of blood through the cardiovascular system, the airflow ...

, 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. A constant stress layer exists in the near wall region. Due to the damping of the vertical velocity fluctuations near the wall, the Reynolds stress term will become negligible and we find that a linear velocity profile exists. This is only true for the very near wall region.

Heat and mass transfer

In 1928, the French engineer André Lévêque observed that convective heat transfer in a flowing fluid is affected only by the velocity values very close to the surface. For flows of large Prandtl number, the temperature/mass transition from surface to freestream temperature takes place across a very thin region close to the surface. Therefore, the most important fluid velocities are those inside this very thin region in which the change in velocity can be considered linear with normal distance from the surface. In this way, for :

u(y) = U \left 1 - \frac \right = U \frac \left 2 - \frac \right \;,

when

y \rightarrow 0

, then :

u(y) \approx 2 U \frac = \theta y,

where ''θ'' is the tangent of the Poiseuille parabola intersecting the wall. Although Lévêque's solution was specific to heat transfer into a Poiseuille flow, his insight helped lead other scientists to an exact solution of the thermal boundary-layer problem. Schuh observed that in a boundary-layer, ''u'' is again a linear function of ''y'', but that in this case, the wall tangent is a function of ''x''. He expressed this with a modified version of Lévêque's profile, :

u(y) = \theta(x) y.

This results in a very good approximation, even for low

Pr

numbers, so that only liquid metals with

Pr

much less than 1 cannot be treated this way. In 1962, Kestin and Persen published a paper describing solutions for heat transfer when the thermal boundary layer is contained entirely within the momentum layer and for various wall temperature distributions. For the problem of a flat plate with a temperature jump at

x = x_0

, they propose a substitution that reduces the parabolic thermal boundary-layer equation to an ordinary differential equation. The solution to this equation, the temperature at any point in the fluid, can be expressed as an incomplete

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

. Schlichting proposed an equivalent substitution that reduces the thermal boundary-layer equation to an ordinary differential equation whose solution is the same incomplete gamma function.

Convective transfer constants from boundary layer analysis

Paul Richard Heinrich Blasius derived an exact solution to the above laminar boundary layer equations. The thickness of the boundary layer

\delta

is a function of the Reynolds number for laminar flow. :

\delta \approx 5.0

\delta

= the thickness of the boundary layer: the region of flow where the velocity is less than 99% of the far field velocity

v_\infty

;

x

is position along the semi-infinite plate, and

Re

is the Reynolds Number given by

\rho v_\infty x / \mu

(

\rho =

density and

\mu =

dynamic viscosity). The Blasius solution uses boundary conditions in a dimensionless form: :

=  = = 0

y=0

=  = 1

y=\infty

and

x=0

Velocity and Temperature boundary layer similarity

Note that in many cases, the no-slip boundary condition holds that

v_S

, the fluid velocity at the surface of the plate equals the velocity of the plate at all locations. If the plate is not moving, then

v_S = 0

. A much more complicated derivation is required if fluid slip is allowed. In fact, the Blasius solution for laminar velocity profile in the boundary layer above a semi-infinite plate can be easily extended to describe Thermal and Concentration boundary layers for heat and mass transfer respectively. Rather than the differential x-momentum balance (equation of motion), this uses a similarly derived Energy and Mass balance: Energy:

v_x  + v_y  =

Mass:

v_x  + v_y  = D_

For the momentum balance, kinematic viscosity

\nu

can be considered to be the ''momentum diffusivity''. In the energy balance this is replaced by thermal diffusivity

\alpha =

, and by mass diffusivity

D_

in the mass balance. In thermal diffusivity of a substance,

k

is its thermal conductivity,

\rho

is its density and

C_P

is its heat capacity. Subscript AB denotes diffusivity of species A diffusing into species B. Under the assumption that

\alpha = D_ = \nu

, these equations become equivalent to the momentum balance. Thus, for Prandtl number

Pr = \nu/\alpha = 1

and Schmidt number

Sc = \nu/D_ = 1

the Blasius solution applies directly. Accordingly, this derivation uses a related form of the boundary conditions, replacing

v

with

T

c_A

(absolute temperature or concentration of species A). The subscript S denotes a surface condition. :

=  = = 0

y=0

=  =  = 1

y=\infty

and

x=0

Using the streamline function Blasius obtained the following solution for the shear stress at the surface of the plate. :

\tau_0 = \left(  \right) _=0.332  Re^

And via the boundary conditions, it is known that :

=  =

We are given the following relations for heat/mass flux out of the surface of the plate :

\left(  \right) _=0.332  Re^

\left(  \right) _=0.332  Re^

So for

Pr=Sc=1

\delta =\delta _T= \delta _c=

where

\delta_T,\delta_c

are the regions of flow where

T

and

c_A

are less than 99% of their far field values.Geankoplis, Christie J. Transport Processes and Separation Process Principles: (includes Unit Operations). Fourth ed. Upper Saddle River, NJ: Prentice Hall Professional Technical Reference, 2003. Print. Because the Prandtl number of a particular fluid is not often unity, German engineer E. Polhausen who worked with Ludwig Prandtl attempted to empirically extend these equations to apply for

Pr\ne 1

. His results can be applied to

Sc

as well. He found that for Prandtl number greater than 0.6, the thermal boundary layer thickness was approximately given by:

= Pr^

and therefore

= Sc^

From this solution, it is possible to characterize the convective heat/mass transfer constants based on the region of boundary layer flow. Fourier's law of conduction and Newton's Law of Cooling are combined with the flux term derived above and the boundary layer thickness. :

= -k \left( \right)_ = h_x(T_S-T_\infty)

h_x = 0.332 Re^_x Pr^

This gives the local convective constant

h_x

at one point on the semi-infinite plane. Integrating over the length of the plate gives an average :

h_L = 0.664 Re^_L Pr^

Following the derivation with mass transfer terms (

k

= convective mass transfer constant,

D_

= diffusivity of species A into species B,

Sc = \nu / D_

), the following solutions are obtained: :

k'_x = 0.332 Re^_x Sc^

k'_L = 0.664 Re^_L Sc^

These solutions apply for laminar flow with a Prandtl/Schmidt number greater than 0.6.

Naval architecture

Many of the principles that apply to aircraft also apply to ships, submarines, and offshore platforms. For ships, unlike aircraft, one deals with incompressible flows, where change in water density is negligible (a pressure rise close to 1000kPa leads to a change of only 2–3 kg/m³). This field of fluid dynamics is called hydrodynamics. A ship engineer designs for hydrodynamics first, and for strength only later. The boundary layer development, breakdown, and separation become critical because the high viscosity of water produces high shear stresses.

Boundary layer turbine

This effect was exploited in the

Tesla turbine Tesla turbine at Nikola Tesla Museum The Tesla turbine is a bladeless centripetal flow turbine patented by Nikola Tesla in 1913. It is referred to as a ''bladeless turbine''. The Tesla turbine also known as the ''boundary-layer turbine'', ' ...

, patented by

Nikola Tesla Nikola Tesla ( ; ,"Tesla"
'' 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).

Predicting transient boundary layer thickness in a cylinder using dimensional analysis

By using the transient and viscous force equations for a cylindrical flow you can predict the transient boundary layer thickness by finding the Womersley Number (

N_w

). Transient Force =

\rho v w

Viscous Force =

Setting them equal to each other gives: :

\rho v w=

Solving for delta gives: :

\delta_1=\sqrt=\sqrt

In dimensionless form: :

N_w

where

N_w

= Womersley Number;

\rho

= density;

v

= velocity;

w=

\delta_1

= length of transient boundary layer;

\mu

= viscosity;

L

= characteristic length.

Predicting convective flow conditions at the boundary layer in a cylinder using dimensional analysis

By using the convective and viscous force equations at the boundary layer for a cylindrical flow you can predict the convective flow conditions at the boundary layer by finding the dimensionless Reynolds Number (

Re

). Convective force:

\rho v^2\over\ L

Viscous force:

Setting them equal to each other gives: :

=

Solving for delta gives: :

\delta_2=\sqrt

In dimensionless form: :

\sqrt

where

Re

= Reynolds Number;

\rho

= density;

v

= velocity;

\delta_2

= length of convective boundary layer;

\mu

= viscosity;

L

= characteristic length.

Boundary layer ingestion

Boundary layer ingestion promises an increase in aircraft fuel efficiency with an aft-mounted

propulsor {{short description, Mechanical device to propel a vessel A propulsor is a mechanical device that gives propulsion. The word is commonly used in the marine vernacular, and implies a mechanical assembly that is more complicated than a propeller. The ...

ingesting the slow

boundary layer and re-energising the

wake Wake or The Wake may refer to: Culture *Wake (ceremony), a ritual which takes place during some funeral ceremonies *Wakes week, an English holiday tradition * Parish Wake, another name of the Welsh ', the fairs held on the local parish's patron s ...

to reduce drag and improve

propulsive efficiency In aerospace engineering, concerning aircraft, rocket and spacecraft design, overall propulsion system efficiency \eta is the efficiency with which the energy contained in a vehicle's fuel is converted into kinetic energy of the vehicle, to accel ...

. To operate in distorted airflow, the fan is heavier and its efficiency is reduced, and its integration is challenging. It is used in concepts like the Aurora D8 or the French research agency Onera’s Nova, saving 5% in cruise by ingesting 40% of the fuselage boundary layer.

Airbus Airbus SE (; ; ; ) is a European multinational aerospace corporation. Airbus designs, manufactures and sells civil and military aerospace products worldwide and manufactures aircraft throughout the world. The company has three divisions: '' ...

presented the Nautilius concept at the ICAS congress in September 2018: to ingest all the fuselage boundary layer, while minimizing the

azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...

al flow distortion, the fuselage splits into two spindles with 13-18:1 bypass ratio fans. Propulsive efficiencies are up to 90% like counter-rotating open rotors with smaller, lighter, less complex and noisy engines. It could lower fuel burn by over 10% compared to a usual underwing 15:1 bypass ratio engine.

References

* * A.D. Polyanin and V.F. Zaitsev, ''Handbook of Nonlinear Partial Differential Equations'', Chapman & Hall/CRC Press, Boca Raton – London, 2004. * 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. * Hermann Schlichting, Klaus Gersten, E. Krause, H. Jr. Oertel, C. Mayes "Boundary-Layer Theory" 8th edition Springer 2004 * John D. Anderson, Jr.
"Ludwig Prandtl's Boundary Layer"
''Physics Today'', December 2005 * * H. Tennekes and J. L. Lumley, "A First Course in Turbulence", The MIT Press, (1972).
Lectures in Turbulence for the 21st Century by William K. George

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, "

'". NASA Glenn Learning Technologies.

''Boundary layer equations: Exact Solutions''
– from EqWorld * Jones, T.V
''BOUNDARY LAYER HEAT TRANSFER''
* {{DEFAULTSORT:Boundary Layer Aircraft wing design Heat transfer

Types of boundary layer

The Prandtl Boundary Layer Concept

Boundary layer equations

Prandtl's transposition theorem

Von Kármán momentum integral

Energy integral

Von Mises transformation

Crocco's transformation

Turbulent boundary layers

Heat and mass transfer

Convective transfer constants from boundary layer analysis

Naval architecture

Boundary layer turbine

Predicting transient boundary layer thickness in a cylinder using dimensional analysis

Predicting convective flow conditions at the boundary layer in a cylinder using dimensional analysis

Boundary layer ingestion

See also

References

External links