introduction to the boundary layer concept

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Introduction to the Boundary Layer concept Content:

– Introduction to the Bounday Layer concept;

– Constraint and unconstraint boundary layers

– Free shear flows and wakes

– Laminar and turbulent boundary layers;

– Boundary layer separation;

– Thin boundary layer equations

– Longitudinal pressure gradient effects on the boundary layer growth.


Introduction to the Boundary Layer

Movies (6), 88, 89

MFM: BL, Impulsive Started Flow, Overview

MFM: BL, BL Concepts,Viscous effects near boundaries


Introduction to the Boundary Layer Boundary layer: flow region in the vicinity of a wall

where viscous/diffusive effects and energy dissipation are significative.

U U

x

y Outer invisicid flow

Boundary layer: significantyu

(x)

Boundary layer width: u0,99 U


Introduction to the Boundary Layer Streamlines over a flat plat

1. The streamlines moves away slowly from the wall. Why?

U U

x

y Limit strealine of Boundary layer

Streamlines

2. This separation of streamlines is most intense outside the boundary layer. Why?


Introduction to the Boundary Layer Notes about boundary layer:

1. The boundary layer may be laminar or turbulent

2. Thin boundary layer if (x)<<x

3. Boundary layer confined: cannot grow free (ex: tube or between plates)


Introduction to the Boundary Layer Constrained Boundary Layer:

(x) Fully Developed Velocity ProfielEntrance Region

Boundary Layer

External Flow

1. Entrance region: the velocity increases at the center line (to keep the mass flow rate) and the pressure falls (Bernoulli’s Equation)–> dp/dx<0.

2. After the union of all boundary layers, all the flow is boundary layer flow . In turbulent flow, the eddy dimension is limited by d.

Rx


Introduction to the Boundary Layer Boundary layer in external flows (unconstrained)

1. is not limited, it grows with the distance to the leading edge x (beginning).

2. Nondimensional velocity profile can stabilize


Introduction to the Boundary Layer

Shear flows (longitudinal convective transport of momentum

affected by diffusion):

o Free Shear Flows: ex: free jet

o Wake: flow zone resulting from the joining of the boundary layers on the two faces of the plate


Introduction to the Boundary Layer Transition from laminar to turbulent:

Ux

cesviscousfor

rcesinertialfoRe x – Start of Boundary Layer (BL)

•Start of BL: 0x 0Re Laminar Flow

•Long plate: Re increases

Critical Re (5105)

Transition to turbulent

00 yyu Very high

00 yyu decreases


Introduction to the Boundary Layer Regions of turbulent boundary layer:

– Linear sub-layer or laminar sub-layer;– Transition layer;– Logaritmical profile zone;– Outer zone (turbulent vorticity and non turbulent outer flow).

mfm – BL/ Instability, Transition and Turbulence:Boundary Layer transitionFully turbulent BL flowInstability and transition in pipe and duct flow

Fully turbulent duct flow


Laminar Thin Boundary Layer Equations (<<x) over flat plate

Steady flow, e constants. Streamlines slightly divergents0 yp

dxdpxp e

2

2

2

21

y

u

x

u

x

p

y

uv

x

uu

2D Navier-Stokes Equations at x direction:

Compared with 2

2

y

u

dxdpe


Laminar Thin Boundary Layer Equations (<<x) over flat plate

2

21

y

u

dx

dp

y

uv

x

uu e

Laminar thin boundary layer equations (<<x) to flat plates

pe external pressure, can be calculated bu Bernoulli’s Equation because there is not viscous effects outer the Boundary Layer

Note 1. The plate is considered flat if d is lower then the local curvature radius

Note 2. At the separation point, the BD grows a lot and no longer thin


z

wu

y

vu

x

uu

y

u

dx

dp

y

uv

x

uu e

2

21

Turbulent Thin Boundary Layer Equations (<<x) over flat plate

2D Thin Turbulent Boundary Layer Equation (<<x) to flat plates:

Resulting from Reynolds Tensions (note the w term)

0 0


Boundary Layer Separation Boundary Layer Separation: reversal of the flow by

the action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects

mfm: BL / Separation / Flow over edges and blunt bodies


Boundary Layer Separation Boundary layer separation: reversal of the flow by the

action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects


Boundary Layer Separation Bidimensional (2D) Thin Boundary Layer (<<x)

Equations to flat plates:

2

21

y

u

dx

dp

y

uv

x

uu e

Close to the wall (y=0) u=v=0 :

dx

dp

y

u e

y1

0

2

2

Similar results to turbulent boundary layer - close to the wall there is laminar/linear sub-layer region.


Boundary Layer Separation Outside Boundary layer: 0

2

2

y

u

The external pressure gradient can be:o dpe/dx=0 <–> U0 constant (Paralell outer streamlines):

o dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):

o dpe/dx<0 <–> U0 increases (Convergent outer streamlines):

Close to the wall (y=0) u=v=0 :

dx

dp

y

u e

y1

0

2

2

Same sign


Zero pressure gradient:dpe/dx=0 <–> U0 constant (Paralell outer streamlines):

y

u

Inflection point at the wall

No separation of boundary layer

02

2

yy

u

00

2

2

yy

u

Boundary Layer Separation

Curvature of velocity profile is constant


Favourable pressure gradient:dpe/dx<0 <–> U0 increases (Convergent outer streamlines):

02

2

yy

u y

00

2

2

yy

u

Curvature of velocity profile remains constant

No boundary layer separation



Adverse pressure gradient:dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):

02

2

yy

u

00

2

2

yy

u Curvature of velocity profile can change

Boundary layer Separation can occur

y

P.I.


Separated Boundary Layer


Sum of viscous forces:2

2

y

u

Become zero with velocity

Can not cause by itself the fluid stagnation (and the separation of Boundary Layer)



Effect of longitudinal pressure gradient:

0dx

dpe (Convergent outer streamlines)

0dx

dpe (Divergent outer streamlines)

Viscous effects retarded Viscous effects reinforced

Fuller velocity profiles

Less full velocity profiles

...11

dx

dp

ux

u e

Decreases BL growth Increases BL growths



Effect of longitudinal pressure gradient:

Fuller velocity profiles

Less full velocity profiles

...11

dx

dp

ux

u e

Decreases BL growth Increases BL growths

Fuller velocity profiles – more resistant to adverse pressure gradients

Turbulent flows (fuller profiles)- more resistant to adverse pressure gradients



Boundary Layer Sepaation

Longitudinal and intense adverse pressure gradient does not cause separation

=> there’s not viscous forces


No viscous forces – no separation of Boundary Layer:

(ds displacement over a streamline)

V=0 (stagnation point) 0ds

dp

ds

dp

ds

dVV

1


From pressure forcesNo reversal of the flow


MECÂNICA DOS FLUIDOS II Contents:

– Boundary Layer;

– Boundary Layer thickness;

– Limiting line of Boundary Layer;

– Deviation of streamlines at Boundary Layer;

– Thin Boundary Layer;

– Constrained and unconstrained boundary layer;

– Free Shear flows;

– Wakes;

– Thin boundary layer equations.


Contents:– Boundary Layer Separation: conditions to separation

– Adverse, favourable and zero pressure gradient;

– Effects of pressure gradient on the Boundary layer evolution

Separação da Camada limite


MECÂNICA DOS FLUIDOS II

Bibliography :– Sabersky – Fluid Flow: 8.1, 8.2

– White – Fluid Mechanics: 7.1, 7.3, 7.5

introduction to the boundary layer concept

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