Lecture 21. Boundary Layer part1

Transcription

1 Lecture 21 Boundary Layer part1 Mecânica de Fluidos Ambiental 2015/2016

2 Flat Boundary Layer xeuxiom Inside the BL there is equilibrium between inertia and viscous forces. Outside there is no friction and the Bernoulli equation is applicable. u t i u j u x i j p x i x j u x i j g i Termo convectivo Escala velocidade U e Escala comprimento - X 1 Termo difusivo Escala velocidade U e Escala comprimento -

3 Reynolds number and Boundary layer thickness

4 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:

5 BL evolution As the fluid moves along x 1, 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:

6 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 x 1 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. Mecânica de Fluidos Ambiental 2015/2016

7 Flows where the Boundary Layer concept is Mecânica de Fluidos Ambiental 2015/2016 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 )

8 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 u 1 (x 2 ) 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.

9 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.

10 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.

11 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.

12 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.