Mecânica de Fluidos Ambiental 2015/2016

Boundary Layer – part1Lecture 21

Mecânica de Fluidos Ambiental 2015/2016

Flat Boundary Layer

• Inside the BL there is equilibrium between inertia and viscous forces. Outside there is no friction and the Bernoulli equation is applicable.

https://www.youtube.com/watch?v=7SkWxEUXIoM

ij

i

jij

ij

i gxu

xxp

xuu

tu

Termo convectivo

Escala velocidade – Ue

Escala comprimento - X1

Termo difusivo Escala velocidade – Ue

Escala comprimento -

Mecânica de Fluidos Ambiental 2015/2016

Reynolds number and Boundary layer thickness

Mecânica de Fluidos Ambiental 2015/2016

Reynolds number and thin Boundary layer

• If the thickness of the BL is very small when compared with the longitudinal scale High Reynolds number flow

• In the BL we have:

Mecânica de Fluidos Ambiental 2015/2016

BL evolution

• As the fluid moves along x1, the viscous forces diminish and flow disturbances have a higher probability of not being dissipated and consequently the flow can become turbulent. When we define the Reynolds number, Re, we have to take that into consideration:

Mecânica de Fluidos Ambiental 2015/2016

Evolution of BL regimes• Infinitely long flat plate along which flows a viscous fluid:

• The fluid will be delayed in very close vicinity of the solid wall.• Suppose that the BL is initially laminar and as it increases the distance x1 from

the leading edge the Reynolds number will increase, reached the critical value, and the flow becomes turbulent.

• The turbulent region correspond to a increase in the boundary layer thickness • Sub laminar layer in the turbulent region near the solid wall.

https://www.youtube.com/watch?v=WEX72jeXTGM

Mecânica de Fluidos Ambiental 2015/2016

Flows where the Boundary Layer concept is applied

• The common characteristic of the boundary layer type flow, regardless the presence or absence of solid boundaries, is the existence of convection in the predominant direction of flow balanced by the diffusion in the transverse direction.

• Examples of boundary layer type flow cases where there is no interference of solid walls, the so-called free flow:• Free jet (“jacto livre”)• Wake (“esteira”)• Mixing layer ( “camada de mistura”)

Mecânica de Fluidos Ambiental 2015/2016

Free jet : Boundary Layer type flow• Ejecting a fluid through a ring contraction to a space filled with a

fluid at rest (e.g. exhaust of a jet engine to the atmosphere at rest)• In the contraction exit the velocity profile u1(x2) is uniform , from section A to B

it happens essentially a pressure transformation into kinetic energy and the viscous effect of the wall does not have time to diffuse into the interior

• After the exit section (B) a strong diffusion in the peripheral zone occurs, i.e. in the zone of large velocity gradients, until section C where the diffusion reaches the jet axis, and the uniform part of the velocity profile no longer exists.

Mecânica de Fluidos Ambiental 2015/2016

Free jet : Boundary Layer type flow• From B to C there is a central conic zone without velocity gradients,

where there is no diffusion – perfect fluid region• In the velocity gradients zone , the transversal momentum diffusion -

responsible for the widening of the jet – is counterbalanced by the longitudinal convection; Boundary Layer region. There is equilibrium between inertia and viscous forces.

Mecânica de Fluidos Ambiental 2015/2016

Wake BL type flow• Wake is a delayed fluid zone formed downstream of a solid body

immersed in a flow.• Consider that the solid body is a cylinder and the current is uniform:

• The velocity distribution in the wake is generally complex in the immediate vicinity of the body.

Mecânica de Fluidos Ambiental 2015/2016

Wake BL type flow• However, from a certain distance downstream the direct effect of the body

into the flow disappear and the streamlines are again approximately straight and parallel. This distant region is a BL zone because the predominant processes are the longitudinal convection and transversal diffusion, the latter being responsible for enlargement of the wake and by the velocity profile uniformization trend.

• The wake characteristics, namely its dimension, are imposed by the shape and dimension of the body that originate the wake and by the Reynolds number of the flow .

Mecânica de Fluidos Ambiental 2015/2016

Mixing layer BL flow• A mixing layer can be obtain through two adjacent uniform flows,

with the same velocity direction and different velocity modulus • Initially the velocity profiles are uniforms. The high velocity gradient in the

contact zone between the two flows will cause a strong tranversal difusion momentum that will reduce this gradient. The involved diffusive process will also enlarge the real fluid zone downstream.