Convection in Boundary Layers

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

A tiny layer but very significant………..

Pr,,Re,

*

**

dx

dpxf L

Prandtl Number: The ratio of momentum diffusion to heat diffusion.

T

m

Pr

Other scales of reference:

Length of plate: L & Free stream velocity : uoo

Potential for diffusion of momentum change (Deficit or excess) created by a solid boundary. Potential for Diffusion of thermal changes created by a solid boundary.

Momentum Vs Thermal Effects

0

''

y

s y

TkTThq

0*

0 *scaleLength

scale eTemperatur

yyyy

T

0

**

y

sfluids yL

TTkTTh

Pr,,Re,*

**

0*

* dx

dpxf

k

hL

y Lfluidy

This dimensionless temperature gradient at the wall is named asNusselt Number:

resistance Convection

resistance Conduction1

h

kL

k

hLNu fluid

fluid

Pr,,Re,*

**

0*

* dx

dpxf

k

hx

yNu x

fluidy

Local Nusselt Number

Average Nusselt Number

avgfluid

avgavg k

LhNu

,

Computation of Dimensionless Temperature Profile

First Law of Thermodynamics for A CV

Energy Equation for a CV

How to select A CV for External Flows ?

Relative sizes of Momentum & Thermal Boundary Layers …

T

m

Pr

Liquid Metals: Pr <<< 1

y*

1.0u*(y*) y*)

Gases: Pr ~ 1.0

y

1.0

u*(y*)

y*)

Water :2.0 < Pr < 7.0

y

1.0

u*(y*)

y*)

Oils:Pr >> 1

y

1.0

u*(y*)

y*)

The Boundary Layer : A Control Volume

For pr < 1

xx uT &dxxdxx uT &

CV

CM VdVbt

b

dt

dB

.

Reynolds Transport Theorem

The relation between A CM and CV for conservation of any extensive property B.

• Total rate of change of any extensive property B of a system(C.M.) occupying a control volume C.V. at time t is equal to the sum of

• a) the temporal rate of change of B within the C.V.

• b) the net flux of B through the control surface C.S. that surrounds the C.V.

Conservation of Mass

• Let b=1, the B = mass of the system, m.

The rate of change of mass in a control mass should be zero.

CV

CM VdVtdt

dm

.

0.

CV

VdVt

Above integral is true for any shape and size of the control volume, which implies that the integrand is zero.

0.

Vt

Conservation of Momentum

• Let b=V, the B = momentum of the system, mV.

The rate of change of momentum for a control mass should be equalto resultant external force.

CV

CM VdVVt

V

dt

Vmd

.

FVdVVt

V

CV

.

bodysurface ffVVt

V

.

Momentum equation of per unit volume:

For a boundary layer :

gVVt

Vij ˆ..

For an incompressible flow

gVVt

Vij ˆ..

g

Dt

DVij ˆ.

Conservation of Energy

• Let b=e, the B = Energy of the system, me.

The rate of change of energy of a control mass should be equalto difference of work and heat transfers.

CV

CM VdVet

e

dt

dE

.

WQVdVet

e

CV

.

Energy equation per unit volume:

wqVet

e

.

Tkq .

Using the law of conduction heat transfer:

The net Rate of work done on the element is:

ijVw ..

From Momentum equation: N S Equations

g

Dt

DVij ˆ.

Then

Vg

Dt

DVVV ij .ˆ..

gzV

he 2

2

wqVet

e

.

j

iijij x

uVTkVg

Dt

DVV

Dt

Dh ....

j

iij x

uVg

Dt

DVVTkVg

Dt

DVV

Dt

Dh .ˆ..

Substitute the work done by shear stress:

j

iij x

uTk

Dt

Dh .

This is called the first law of thermodynamics for fluid motion.

For an Incompressible fluid:

Vpx

u

x

u

j

i

j

iij ij

.'

Dt

DpVp .

Invoking conservation of mass:

j

iij x

uTk

Dt

Dh .

First law for a fluid motion:

0. VDt

D

Dt

Dp

x

uTk

Dt

Dh

j

iij

'.

Dt

DpTk

Dt

Dh

.

is called as viscous dissipation.

Dt

DpTk

Dt

DTC p

.

Boundary Layer Equations

Consider the flow over a parallel flat plate.

Assume two-dimensional, incompressible, steady flow with constant properties.

Neglect body forces and viscous dissipation.

The flow is nonreacting and there is no energy generation.

The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation:

Boundary Conditions

Scale Analysis

Define characteristic parameters:

L : length

u ∞ : free stream velocity

T ∞ : free stream temperature

General parameters:

x, y : positions (independent variables)

u, v : velocities (dependent variables)

T : temperature (dependent variable)

also, recall that momentum requires a pressure gradient for the movement of a fluid:

p : pressure (dependent variable)

Define dimensionless variables:

L

xx *

L

yy *

u

uu*

u

vv*

s

s

TT

TT

2*

u

pp

Similarity parameters can be derived that relate one set of flow conditions to geometrically similar surfaces for a different set of flow conditions:

0*

*

*

*

y

v

x

u

2*

*2

*

*

*

**

*

**

Re

1

y

u

x

p

y

vv

x

uu

L

2*

2

**

**

PrRe

1

yyv

xu

L

Boundary Layer Parameters

• Three main parameters (described below) that are used to characterize the size and shape of a boundary layer are:

• The boundary layer thickness,

• The displacement thickness, and

• The momentum thickness.

• Ratios of these thicknesses describe the shape of the boundary layer.

Boundary Layer Thickness

• The boundary layer thickness, signified by , is simply the thickness of the viscous boundary layer region.

• Because the main effect of viscosity is to slow the fluid near a wall, the edge of the viscous region is found at the point where the fluid velocity is essentially equal to the free-stream velocity.

• In a boundary layer, the fluid asymptotically approaches the free-stream velocity as one moves away from the wall, so it never actually equals the free-stream velocity.

• Conventionally (and arbitrarily), we define the edge of the boundary layer to be the point at which the fluid velocity equals 99% of the free-stream velocity:

• Because the boundary layer thickness is defined in terms of the velocity distribution, it is sometimes called the velocity thickness or the velocity boundary layer thickness.

• Figure  illustrates the boundary layer thickness. There are no general equations for boundary layer thickness.

• Specific equations exist for certain types of boundary layer.

• For a general boundary layer satisfying minimum boundary conditions:

0 ;)( ;0)0(

y

y

uuuu

The velocity profile that satisfies above conditions:

2

22

yy

uu

Further analysis shows that:

xx Re

5.5

Where:

xu

xRe

Variation of Reynolds numbers

All Engineering Applications

Laminar Velocity Boundary Layer

The velocity boundary layer thickness for laminar flow over a flat plate:

as u∞ increases, δ decreases (thinner boundary layer)

The local friction coefficient:

and the average friction coefficient over some distance x:

x

xRe

5.5

Laminar Thermal Boundary Layer

022

2

d

df

pr

d

d

Boundary conditions:

1 00

This differential equation can be solved by numerical integration.

One important consequence of this solution is that, for pr >0.6:

3/1

0

332.0 prd

d

Local convection heat transfer coefficient:

0

**

y

fluidx yL

kh

0

**

y

sfluids yL

TTkTTh

Local Nusselt number:

0

x

ukh fluidx

000

Re

xfluid

xx

xu

x

ux

k

xhNu

3/1Re332.0 prk

xhNu x

fluid

xx

Average heat transfer coefficient:

L

xfluid

L

xavg dxprx

k

Ldxh

Lh

0

3/1

0

Re332.011

L

fluidavg

x

dxpr

u

x

k

Lh

0

3/1332.01

xavg hh 2

6.0 Re664.0 3/1 prprk

LhNu L

fluid

avgavg

A single correlation, which applies for all Prandtl numbers,Has been developed by Churchill and Ozoe..

100

0468.01

Re338.0

41

32

3/1

xx

x Pe

pr

prNu

xavg NuNu 2

Turbulent Flow

• For a flat place boundary layer becomes turbulent at Rex ~ 5 X 105.

• The local friction coefficient is well correlated by an expression of the form

7x

51

, 10Re Re059.0

xxfC

Local Nusselt number: 60 0.6 Re029.0 3/154

prprNu xx

Local Sherwood number: 60 0.6 Re029.0 3/154

ScScSh xx