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Source: Boundary layer

In physics and fluid mechanics, 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, 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 wing, the velocity boundary layer is the part of the flow close to the wing, where viscous forces distort the surrounding non-viscous flow. In the Earth's atmosphere, 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 transfer to or from the surface.

Types of boundary layers

boundary

The Prandtl boundary layer concept

Boundary layer equations

= Prandtl's transposition theorem

= Von Kármán momentum integral

= Energy integral

boundary

layer

= Von Mises transformation

= Crocco's transformation

Turbulent boundary layers

, the x-momentum equation simplifies to:

u
¯

∂

u
¯

∂
x

+

v
¯

∂

u
¯

∂
y

=
−

1
ρ

∂

p
¯

∂
x

−

∂

∂
y

(

u
′

v
′

¯

)
.

{\displaystyle {\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}-{\frac {\partial }{\partial y}}({\overline {u'v'}}).}

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

η
<<
δ

{\displaystyle \eta <<\delta }

as the y-scale, the leading order momentum equation for this "inner boundary layer" is given by:

0
=
−

1
ρ

∂

p
¯

∂
x

+

ν

∂

2

u
¯

∂

y

2

−

∂

∂
y

(

u
′

v
′

¯

)
.

{\displaystyle 0=-{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'}}).}

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"

η

{\displaystyle \eta }

is a viscous length scale, and is of order

ν

u

∗

{\displaystyle {\frac {\nu }{u_{*}}}}

, with

u

∗

{\displaystyle 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

u
′

v
′

¯

{\displaystyle {\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.
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

[

1
−

(
y
−
h

)

2

h

2

]

=
U

y
h

[

2
−

y
h

]

,

{\displaystyle u(y)=U\left[1-{\frac {(y-h)^{2}}{h^{2}}}\right]=U{\frac {y}{h}}\left[2-{\frac {y}{h}}\right]\;,}

when

y
→
0

{\displaystyle y\rightarrow 0}

, then

u
(
y
)
≈
2
U

y
h

=
θ
y
,

{\displaystyle u(y)\approx 2U{\frac {y}{h}}=\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
)
=
θ
(
x
)
y
.

{\displaystyle u(y)=\theta (x)y.}

This results in a very good approximation, even for low

P
r

{\displaystyle Pr}

numbers, so that only liquid metals with

P
r

{\displaystyle 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

{\displaystyle 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. 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. Analytic solutions can be derived with the time-dependent self-similar Ansatz for the incompressible boundary layer equations including heat conduction.
As is well known from several textbooks, heat transfer tends to decrease with the increase in the boundary layer. Recently, it was observed on a practical and large scale that wind flowing through a photovoltaic generator tends to "trap" heat in the PV panels under a turbulent regime due to the decrease in heat transfer. Despite being frequently assumed to be inherently turbulent, this accidental observation demonstrates that natural wind behaves in practice very close to an ideal fluid, at least in an observation resembling the expected behaviour in a flat plate, potentially reducing the difficulty in analysing this kind of phenomenon on a larger scale.

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

δ

{\displaystyle \delta }

is a function of the Reynolds number for laminar flow.

δ
≈
5.0

x

R
e

{\displaystyle \delta \approx 5.0{x \over {\sqrt {Re}}}}

δ

{\displaystyle \delta }

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

v

∞

{\displaystyle v_{\infty }}

;

x

{\displaystyle x}

is position along the semi-infinite plate, and

R
e

{\displaystyle Re}

is the Reynolds Number given by

ρ

v

∞

x

/

μ

{\displaystyle \rho v_{\infty }x/\mu }

(

ρ
=

{\displaystyle \rho =}

density and

μ
=

{\displaystyle \mu =}

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

v

x

−

v

S

v

∞

−

v

S

=

v

x

v

∞

=

v

y

v

∞

=
0

{\displaystyle {v_{x}-v_{S} \over v_{\infty }-v_{S}}={v_{x} \over v_{\infty }}={v_{y} \over v_{\infty }}=0}

at

y
=
0

{\displaystyle y=0}

v

x

−

v

S

v

∞

−

v

S

=

v

x

v

∞

=
1

{\displaystyle {v_{x}-v_{S} \over v_{\infty }-v_{S}}={v_{x} \over v_{\infty }}=1}

at

y
=
∞

{\displaystyle y=\infty }

and

x
=
0

{\displaystyle x=0}

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

v

S

{\displaystyle 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

{\displaystyle 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

∂
T

∂
x

+

v

y

∂
T

∂
y

=

k

ρ

C

p

∂

2

T

∂

y

2

{\displaystyle v_{x}{\partial T \over \partial x}+v_{y}{\partial T \over \partial y}={k \over \rho C_{p}}{\partial ^{2}T \over \partial y^{2}}}

Mass:

v

x

∂

c

A

∂
x

+

v

y

∂

c

A

∂
y

=

D

A
B

∂

2

c

A

∂

y

2

{\displaystyle v_{x}{\partial c_{A} \over \partial x}+v_{y}{\partial c_{A} \over \partial y}=D_{AB}{\partial ^{2}c_{A} \over \partial y^{2}}}

For the momentum balance, kinematic viscosity

ν

{\displaystyle \nu }

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

α
=

k

/

ρ

C

P

{\displaystyle \alpha ={k/\rho C_{P}}}

, and by mass diffusivity

D

A
B

{\displaystyle D_{AB}}

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

k

{\displaystyle k}

is its thermal conductivity,

ρ

{\displaystyle \rho }

is its density and

C

P

{\displaystyle C_{P}}

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

α
=

D

A
B

=
ν

{\displaystyle \alpha =D_{AB}=\nu }

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

P
r
=
ν

/

α
=
1

{\displaystyle Pr=\nu /\alpha =1}

and Schmidt number

S
c
=
ν

/

D

A
B

=
1

{\displaystyle Sc=\nu /D_{AB}=1}

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

v

{\displaystyle v}

with

T

{\displaystyle T}

or

c

A

{\displaystyle c_{A}}

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

v

x

−

v

S

v

∞

−

v

S

=

T
−

T

S

T

∞

−

T

S

=

c

A

−

c

A
S

c

A
∞

−

c

A
S

=
0

{\displaystyle {v_{x}-v_{S} \over v_{\infty }-v_{S}}={T-T_{S} \over T_{\infty }-T_{S}}={c_{A}-c_{AS} \over c_{A\infty }-c_{AS}}=0}

at

y
=
0

{\displaystyle y=0}

v

x

−

v

S

v

∞

−

v

S

=

T
−

T

S

T

∞

−

T

S

=

c

A

−

c

A
S

c

A
∞

−

c

A
S

=
1

{\displaystyle {v_{x}-v_{S} \over v_{\infty }-v_{S}}={T-T_{S} \over T_{\infty }-T_{S}}={c_{A}-c_{AS} \over c_{A\infty }-c_{AS}}=1}

at

y
=
∞

{\displaystyle y=\infty }

and

x
=
0

{\displaystyle x=0}

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

τ

0

=

(

∂

v

x

∂
y

)

y
=
0

=
0.332

v

∞

x

R

e

1

/

2

{\displaystyle \tau _{0}=\left({\partial v_{x} \over \partial y}\right)_{y=0}=0.332{v_{\infty } \over x}Re^{1/2}}

And via the boundary conditions, it is known that

v

x

−

v

S

v

∞

−

v

S

=

T
−

T

S

T

∞

−

T

S

=

c

A

−

c

A
S

c

A
∞

−

c

A
S

{\displaystyle {v_{x}-v_{S} \over v_{\infty }-v_{S}}={T-T_{S} \over T_{\infty }-T_{S}}={c_{A}-c_{AS} \over c_{A\infty }-c_{AS}}}

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

(

∂
T

∂
y

)

y
=
0

=
0.332

T

∞

−

T

S

x

R

e

1

/

2

{\displaystyle \left({\partial T \over \partial y}\right)_{y=0}=0.332{T_{\infty }-T_{S} \over x}Re^{1/2}}

(

∂

c

A

∂
y

)

y
=
0

=
0.332

c

A
∞

−

c

A
S

x

R

e

1

/

2

{\displaystyle \left({\partial c_{A} \over \partial y}\right)_{y=0}=0.332{c_{A\infty }-c_{AS} \over x}Re^{1/2}}

So for

P
r
=
S
c
=
1

{\displaystyle Pr=Sc=1}

δ
=

δ

T

=

δ

c

=

5.0
x

R
e

{\displaystyle \delta =\delta _{T}=\delta _{c}={5.0x \over {\sqrt {Re}}}}

where

δ

T

,

δ

c

{\displaystyle \delta _{T},\delta _{c}}

are the regions of flow where

T

{\displaystyle T}

and

c

A

{\displaystyle c_{A}}

are less than 99% of their far field values.
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

P
r
≠
1

{\displaystyle Pr\neq 1}

. His results can be applied to

S
c

{\displaystyle Sc}

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

δ

δ

T

=
P

r

1

/

3

{\displaystyle {\delta \over \delta _{T}}=Pr^{1/3}}

and therefore

δ

δ

c

=
S

c

1

/

3

{\displaystyle {\delta \over \delta _{c}}=Sc^{1/3}}

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.

q
A

=
−
k

(

∂
T

∂
y

)

y
=
0

=

h

x

(

T

S

−

T

∞

)

{\displaystyle {q \over A}=-k\left({\partial T \over \partial y}\right)_{y=0}=h_{x}(T_{S}-T_{\infty })}

h

x

=
0.332

k
x

R

e

x

1

/

2

P

r

1

/

3

{\displaystyle h_{x}=0.332{k \over x}Re_{x}^{1/2}Pr^{1/3}}

This gives the local convective constant

h

x

{\displaystyle h_{x}}

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

h

L

=
0.664

k
x

R

e

L

1

/

2

P

r

1

/

3

{\displaystyle h_{L}=0.664{k \over x}Re_{L}^{1/2}Pr^{1/3}}

Following the derivation with mass transfer terms (

k

{\displaystyle k}

= convective mass transfer constant,

D

A
B

{\displaystyle D_{AB}}

= diffusivity of species A into species B,

S
c
=
ν

/

D

A
B

{\displaystyle Sc=\nu /D_{AB}}

), the following solutions are obtained:

k

x

′

=
0.332

D

A
B

x

R

e

x

1

/

2

S

c

1

/

3

{\displaystyle k'_{x}=0.332{D_{AB} \over x}Re_{x}^{1/2}Sc^{1/3}}

k

L

′

=
0.664

D

A
B

x

R

e

L

1

/

2

S

c

1

/

3

{\displaystyle k'_{L}=0.664{D_{AB} \over x}Re_{L}^{1/2}Sc^{1/3}}

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, with water as the primary fluid of concern rather than air. As water is not an ideal fluid, ships moving in water experience resistance. The fluid particles cling to the hull of the ship due to the adhesive force between water and the ship, creating a boundary layer where the speed of flow of the fluid forms a small but steep speed gradient, with the fluid in contact with the ship ideally has a relative velocity of 0, and the fluid at the border of the boundary layer being the free-stream speed, or the relative speed of the fluid around the ship.
While the front of the ship faces normal pressure forces due to the fluid surrounding it, the aft portion sees a lower acting component of pressure due to the boundary layer. This leads to higher resistance due to pressure known as 'viscous pressure drag' or 'form drag'.
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, 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).

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

{\displaystyle N_{w}}

).
Transient Force =

ρ
v
w

{\displaystyle \rho vw}

Viscous Force =

μ
v

δ

1

2

{\displaystyle {\mu v \over \delta _{1}^{2}}}

Setting them equal to each other gives:

ρ
v
w
=

μ
v

δ

1

2

{\displaystyle \rho vw={\mu v \over \delta _{1}^{2}}}

Solving for delta gives:

δ

1

=

μ

ρ
w

=

v

w

{\displaystyle \delta _{1}={\sqrt {\mu \over \rho w}}={\sqrt {\ v \over \ w}}}

In dimensionless form:

L

δ

1

=

L

w

v

=

N

w

{\displaystyle {L \over \delta _{1}}={L{\sqrt {w \over \ v}}}=N_{w}}

where

N

w

{\displaystyle N_{w}}

= Womersley Number;

ρ

{\displaystyle \rho }

= density;

v

{\displaystyle v}

= velocity;

w
=

{\displaystyle w=}

?;

δ

1

{\displaystyle \delta _{1}}

= length of transient boundary layer;

μ

{\displaystyle \mu }

= viscosity;

L

{\displaystyle 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 (

R
e

{\displaystyle Re}

).
Convective force:

ρ

v

2

L

{\displaystyle \rho v^{2} \over \ L}

Viscous force:

μ
v

δ

2

2

{\displaystyle {\mu v \over \delta _{2}^{2}}}

Setting them equal to each other gives:

ρ

v

2

L

=

μ
v

δ

2

2

{\displaystyle {\rho v^{2} \over \ L}={\mu v \over \delta _{2}^{2}}}

Solving for delta gives:

δ

2

=

μ
L

ρ
v

{\displaystyle \delta _{2}={\sqrt {\mu L \over \rho v}}}

In dimensionless form:

L

δ

2

=

ρ
v
L

μ

=

R
e

{\displaystyle {L \over \delta _{2}}={\sqrt {\rho vL \over \mu }}={\sqrt {Re}}}

where

R
e

{\displaystyle Re}

= Reynolds Number;

ρ

{\displaystyle \rho }

= density;

v

{\displaystyle v}

= velocity;

δ

2

{\displaystyle \delta _{2}}

= length of convective boundary layer;

μ

{\displaystyle \mu }

= viscosity;

L

{\displaystyle L}

= characteristic length.

Boundary layer ingestion

Boundary layer ingestion promises an increase in aircraft fuel efficiency with an aft-mounted propulsor ingesting the slow fuselage boundary layer and re-energising the wake to reduce drag and improve propulsive efficiency.
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 presented the Nautilius concept at the ICAS congress in September 2018:
to ingest all the fuselage boundary layer, while minimizing the azimuthal 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

Chanson, H. (2009). Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages. ISBN 978-0-415-49271-3.
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
Hermann 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 ed.). Toronto: S.S.CHAND. pp. 711–714. ISBN 0-07-001679-8.
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, "Boundary layer". NASA Glenn Learning Technologies.
Boundary layer separation
Boundary layer equations: Exact Solutions – from EqWorld
Jones, T.V. BOUNDARY LAYER HEAT TRANSFER
"The revolutionary concept of "boundary layer" and its prevalence in aeronautics by Sourabh S. Diwan". YouTube. International Centre for Theoretical Sciences. February 18, 2022.

Types of boundary layers

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

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