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### Transcript of 6. Laminar and turbulent boundary layers .net heat conductionrate out of R + Z R qdR_ | {z } rate

• 6. Laminar and turbulent boundary layers

John Richard Thome

8 avril 2008

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 1 / 34

• 6.1 Some introductory ideas

Figure 6.1 A boundary layer of thickness Boundary layer thickness on a flat surface = fn(u, , , x)

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 2 / 34

• 6.1 Some introductory ideas

The dimensional functional equation for the boundary layer thickness on a flatsurface.

x = fn(Rex ) Rex ux

=ux

(6.1)

= :kinematic viscosity.Rex : Reynolds number.

For a flat surface, where u remains constant.

x =4.92Rex

(6.2)

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 3 / 34

• 6.1 Some introductory ideas

Figure 6.4 Boundary layer on a long, flat surface with a sharp leading edge.

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 4 / 34

• 6.1 Some introductory ideas

(uav )crit : Transitional value of the average velocity.

Recritical D(uav )crit

(6.3)

Rexcritical =uxcrit

(6.4)

Transition from laminar to turbulent flow.

Rex ,c = 5 105

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 5 / 34

• Thermal boundary layer

The wall is at temperature Tw

kf (Ty |y=0) = (Tw T)h (6.5)

Where kf is the conductivity of the fluid. The following condition defined h withinthe fluid instead of specifying it as known information on the boundary.

(

TwTTwT

)( y

L) | y

L =0 =hLkf

= NuL (6.5a)

The physical significance of Nu is given by

NuL =hxkf

=Lt

(6.6)

The nusselt number is inversely proportional to the thickness of the thermalboundary layer t .

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 6 / 34

• Thermal boundary layer

Figure 6.5 The thermal boundary layer during the flow of cool fluid over a warmplate.John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 7 / 34

• 6.2 Laminar incompressible boundary layer on a flat surface

Figure 6.7 A steady, incompressible, two-dimensional flow field represented bystreamlines, or lines of constant .For an incompressible flow, continuity becomes

ux +

vy = 0 (6.11)

5 ~u = ux +

vy +

wz = 0

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 8 / 34

• Conservation of momentum

Figure 6.9 Forces acting in a two-dimensional incompressible boundary layer.John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 9 / 34

• Conservation of momentum

The external forces are :(yx +

yxy dy

)dx yxdx + pdy

(p p

x dx)dy =

(yxy

px

)dxdy

The rate at which A loses x-directed momentum to its surroundings is :(u2 + u

2

x dx)dy u2dy +

[u(v) + uv

y dy]dx uvdx

=

(u2x +

uvy

)dxdy

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 10 / 34

• Conservation of momentum

We obtain one form of the steady, two-dimensional, incompressible boundary layermomentum equation.

u2x +

uvy =

1

dpdx + v

2uy2 (6.12)

A second form of the momentum equation.

u ux + v

uy =

1

dpdx + v

2uy2 (6.13)

If there is no pressure gradient in the flow, if p and u are constant as they wouldbe for flow past a flat plate, so we obtain,

u2x +

(uv)y = u

ux + v

uy = v

2uy2 (6.15)

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 11 / 34

• The skin friction coefficient

The shear stress can be obtained by using Newtons law of viscous shear.

w = uy |y=0 = 0.332

uxRex

The skin local friction coefficient is defined as :

Cf w

u2/2=

0.664Rex

(6.33)

The overall skin local friction coefficient is based on the average of the shearstress :

Cf =1.328ReL

(6.34)

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 12 / 34

• 6.3 The energy equation

Using Fouriers law

h = qTw T= kTw T

Ty |y=0 (6.35)

Pressure variations in the flow are not large enough to affect thermodynamicproperties.Density changes result only from temperature changes and will also be small(incompressible behaviour).Temperature varaitions in the flow are not large enough to change ksignificantly.Viscous stresses do not dissipate enough energy to warm the fluidsignificantly.

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 13 / 34

• 6.3 The energy equation

We write conservation of energy in the form.

ddt

Ru dR

rate of internal energy increase in R

=

S

(h)~u ~n dS

rate of internal energy and work out of R

S(k 5 T ) ~n dS

net heat conduction rate out of R

+

Rq dR

rate of heat generation in R

(6.36)

For a constant pressure flow field.

cp

Ttenergy storage

+ ~u 5T enthalpy convection

= k 52 T heat conduction

+ qheat generation

(6.37)

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 14 / 34

• 6.3 The energy equation

In a steady two-dimensional flow field without heat sources, equation 6.37 takesthe following forme.With this assumption, 2T/x2 2T/y2, so the boundary layer form is

u Tx + v

Ty =

2Ty2

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 15 / 34

• 6.4 The Prandtl number and the boundary layer thickness

To look more closely at the implications of the similarity between the velocity andthe thermal boundary layers.

h = fn(k, x , , cp, , u)

We can find the following number by dimension analysis.Prandtl number

Pr

Relative effectiveness of momentum and energy transport by diffusion in thevelocity and thermal boundary layers where for laminar flow.Relationship with other dimensionless number.

Nux = fn(Rex ,Pr)

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 16 / 34

• 6.4 The Prandtl number and the boundary layer thickness

For simple monatomic gases, Pr = 23 .For diatomic gases in which vibration is unexcited, Pr = 57 .As the complexity of gas molecules increase, Pr approaches an upper value ofunity.For liquids composed of fairly simple molecules, excluding metals, Pr is of theorder of magnitude of 1 of 10.For liquid metals, Pr is of the order of magnitude of 102 or less.

Boundary layer thickness, and t , and the Prandtl numberWhen Pr > 1, > t , and when Pr < 1, < t . This is because high viscosityleads to a thick velocity boundary layer, and a high thermal diffusivity should givea thick thermal boundary layer.

t= fn

( only

).

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 17 / 34

• 6.5 Heat transfer coefficient for laminar, incompressibleflow over a flat surface

The following equation expresses the conservation of thermal energy in integratedform.

ddx

t0

u(T T)dy =qwcp

(6.47)

Predicting the temperature distribution in the laminar thermal boundarylayer

T TTw T

= 1 32yt

+12

(yt

)3(6.50)

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 18 / 34

• Predicting the heat flux in the laminar boundary layer

Figure 6.14 The exact and approximate Prandtl number influence on the ratio ofboundary layer thicknesses.

t

=1

1.025Pr1/3[1

(2t /142

)]1/3 ' 11.025Pr1/3 (6.54)So we can write for 0.6 Pr 50.

t

= Pr1/3

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 19 / 34

• Predicting the heat flux in the laminar boundary layer

The following expression gives very accurate results under the assumptions onwhich it is based : a laminar two-dimensional boundary layer on a flat surface,with Tw constant and 0.6 Pr 50

Nux = 0.332Re1/2Pr1/3 (6.58)

High Pr At high Pr, equation 6.58 is still close correct. The exact solution is

Nux 0.339Re1/2x Pr1/3, Pr

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 20 / 34

• Some other laminar boundary layer heat transfer equations

Figure 6.15 A laminar boundary layer in a low-Pr liquid. The velocity boundarylayer is so thin that u ' u in the thermal boundary layer.Low Pr In this case, t , the influence of the viscosity were removed from theproblem and for all pratical purposes u = u everywhere. With a dimensionanalysis, we can find :

Nux =hxk

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 21 / 34

• Some other laminar boundary layer heat transfer equations

We can define a new dimensionless number.Peclet number

Pex RexPr =ux

(6.61)

Peclet number can be interpreted as the ratio of heat capacity rate of fluid inthe b.l. to axial heat conductance of b.l.The exact solution of the boundary layer equations gives, in this case : ForPex 100, Pr 1100 , Rex 10

4.

Nux = 0.565Pe1/2x (6.62)

NuL = 1.13Pe1/2L

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 22 / 34

• Some other laminar boundary layer heat transfer equations

Churchill-Ozoe cor