6.6 the reynolds analogy the reynolds analogy the reynolds

College of Energy and Power Engineering JHH 1

6.6 The Reynolds analogy


The Reynolds analogyIntegral momentum eq. of flat surface flow without

Integral energy eq. for the case of Tw=const

Distribution of u and T

6 . Laminar and turbulent boundary layers


2)]1([

1

0

fCyduu

uu

dxd

−=⎥⎦

⎤⎢⎣

⎡−

∞∞∫ δ

δ

)()()(

1

0∞∞∞

∞

∞ −=⎥

⎦

⎤⎢⎣

⎡−−

∫ TTucqyd

TTTT

uu

dxd

wp

w

tw ρδφδ

3

21

231 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=Θ

tt

yyδδ

3

21

23

⎟⎠⎞

⎜⎝⎛−=

∞ δδyy

uu

p∇


The Reynolds analogySimilarity of temperature and flow boundary layers, Pr=1

Substituting the result in integral energy eq. by notice

By comparing it with the integral momentum eq.

( 1)w

T T uT T u

∞

∞ ∞

−= − −

−

1

0[ ( 1)]

( )w

p w

qd u u yddx u u c u T T

δδ ρ∞ ∞ ∞ ∞

⎡ ⎤− = −⎢ ⎥ −⎣ ⎦

∫

2 ( )f w

p w

C qc u T Tρ ∞ ∞

=−



tδ δ=

= 1tδφδ=

then

p

hc uρ ∞

= St≡ Stanton number

Reynolds analogy

w

w

T T uT T u∞ ∞

−=

−


The Reynolds analogy

For the case of Pr ≠ 1, Reynolds-Colburn analogy

Stanton number

2Pr 3/2 f

p

Cuc

h=

∞ρ

PrReNuSt

x

x

puch

=≡∞ρ

flow fluid theofcapacity flux heat fluid theflux toheat actual

=Δ

Δ

∞ TucTh

pρ



2 /3St Pr2

fC=

0.6 Pr 50≤ ≤


6.7 Turbulent boundary layers


TurbulenceBoundary layer transition from laminar to turbulent

the smoke rise from a lighted cigarette in a draft-free room

flow along a flat plat




TurbulenceFluctuation of u and other quantities in a turbulent pipe flow



udtu ∫≡=T

T 0

1

0u′ =


TurbulenceMixing length l

Average length that a parcel of fluid moves between interactions (similar to that of the molecular mean free path)For the laminar shear stress

In turbulent case

l become to mixing length

By Navier-Stokes eq. Const=-1

xyτ

u

yx yulC

′=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= )((constant) ρτ

(constant)[ ( )]yx v v uτ ρ′ ′ ′= +

uvuvdtuvuvyx ′′−′−=′′+′−=′ ∫ ρρρτ0

0)]([

T

Tuvyx ′′−=′ ρτ



The shear stress, ,in a laminar or turbulent flow.

C v v v′→ = +

yx yxuy

τ μ τ∂ ′= +∂


TurbulenceShear stress of turbulent flow

Introduce eddy diffusivity εm for momentum

uvyu

yx ′′−∂∂

= ρμτ

muv uy

ρ ρε ∂′ ′− =∂ y

umyx ∂∂

+= )( ενρτ

2

(cnstant)

(cnstant)

mu u uv u l ly y y

u uly y

ρε ρ ρ

ρ

⎛ ⎞⎛ ⎞∂ ∂ ∂′ ′= − = − ±⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠∂ ∂

=∂ ∂

∓

yulm ∂∂

= 2ε Eddy diffusivity for momentum, Boussinesq in 1877



y yxτ

0v′ >

0u′ <

u


Turbulence near wallsAverage momentum eq. in the near wall region

means that shear stress is constant in y and equal to the value on the wall

5 variables in 3 dimensions2 pi-groups



neglect very near the wall

( )yx mu u u uu v v ux y y y y y y

μ ρ τ ρ ν ε⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂′ ′+ = − = = +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

yu

myxw ∂∂

+=≅ )( ενρττ

),,,(fn yu w νρτ=

)(fn*

* νyu

uu=

ρτ /*wu ≡

Friction velocity

Near-wall velocity profile dose not depend directly upon x

Where

m

yw

yu

u dyudενρ

τ+

= ∫∫=

0

)(

0

or fn( )u y+ +=

/ 0yx yτ∂ ∂ ≅


Turbulence near wallsThe viscous sublayer

Very near the wall, the eddies must become tiny, l and thus εmwill tend to zero, so that ν >> εm

This region is called as the viscous sublayerThickness ≈10~100μm

Turbulent mixing is ineffective in the region, it is responsible for a major fraction of thermal resistance

0( )

yw wdy yu y τ τρ ν ρ ν

= =∫

7)/( * ≤νyu



0( )

yw

m

dyu y τρ ν ε

=+∫ →0

*

*

( )u y u yu ν

=ρτ /*wu ≡

u y+ +=


Turbulence near walls

The log layer and y/δ ≤ 0.2Farther away from the wall, l is larger and turbulent shear stress is dominant: εm >> ν

Assuming the velocity gradient to be positive

To fix the constant with experimental data



yu

yul

yu

mw ∂∂

∂∂

=∂∂

≅ 2ρρετ

ldyud w ∫∫ =

ρτ CyuC

ydyuyu +=+= ∫ ln)(

**

κκ

Byuu

yu+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

νκ

*

* ln1)( Log law

30)/( * ≥νyu

5.5≅B

for / 0.20.41

l y yκ δκ= ≤=

Where





30)/( * ≥νyu7)/( * ≤νyu

*

( )u yu

*u yν





*

( )u yu

*u yν

*

*

( )u y u yu ν

=

*

*

( ) 2.44 ln 4.9u y u yu ν

⎛ ⎞= +⎜ ⎟

⎝ ⎠

1/ 7*

*

( ) 8.3lnu y u yu ν

⎛ ⎞= ⎜ ⎟

⎝ ⎠


Turbulence near wallsBuffer layer

more complicated equations for l, εm and u are used to connect the viscous sublayer to the log layer

The outer part of the b.l ( y / δ 0.2）



30)/(7 * << νyu

7/1Re16.0)(

xxx

=δ

7/1Re027.0)(

xf xC =

2]Re06.0[ln(455.0)(

）xf xC =

3/ 2l yκ=

0.09l δ=

Where

Lead to 6 9Re 10 ~ 10x =for

More accurate formula valid for all Rex


6.8 Heat transfer in turbulent boundary layers


The Reynolds-Colburn analogy for turbulent flowEddy diffusivity of heat

Turbulent Prandtl number,

Pr is a physical property of the fluid

Prt is a property of the flow field more than of the fluid



yT

yTkq

hpc

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

−=

≡ ερ

mixing e turbulencreflects whichconstant,another

yTcq hp ∂∂

+−= )( εαρ

h

mt ε

ε≡Pr

yTcq

t

mp ∂

∂+−= )

PrPr( ενρ

hε


The Reynolds-Colburn analogy for turbulent flowAverage b.l energy eq.

means that heat flux is constant in y and equal to the wall heat flux

By integrating,

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

=∂∂

−=∂∂

+∂∂

yTc

yq

yyTv

xTu

t

mp )

PrPr(

near wallry neglect ve

ενρ

0/ ≅∂∂ yq

yTcqq

t

mpw ∂

∂+−== )

PrPr( ενρ

*

* *

Pr thermal sublayer( )

/( ) 1 ln (Pr) thermal log layer

w

w p

u yT T y

q c u u y A

νρ

κ ν

⎧ ⎛ ⎞⎪ ⎜ ⎟

− ⎪ ⎝ ⎠= ⎨⎛ ⎞⎪ +⎜ ⎟⎪ ⎝ ⎠⎩



For Prt =1


The Reynolds-Colburn analogy for turbulent flowFor Pr>0.5 (experimental data)

subtract the dimensionless log-law from its thermal counterpart

In the outer part of the boundary

recall

3.7Pr8.12(Pr) 68.0 −=A

BAu

yuucqyTT

pw

w −=−− (Pr))(

)/()(

**ρ

BAuu

ucqTT

pw

w −=−− ∞∞ (Pr)

)/( **ρ

22

*fw C

uuu

=≡∞∞ ρ

τ BAC

Cucq

TT

f

f

pw

w −=−−

∞

∞ (Pr)22)/(ρ




The Reynolds-Colburn analogy for turbulent flowRearrangement

Substituting B=5.5 and

Works for either uniform Tw or uniform qw

Cf can be determined by using equation (6.102), p15 in this lecturenote

2/](Pr)[12/

))(( f

f

wp

w

CBAC

TTucq

−+=

− ∞∞ρ

0.68

/ 2St Pr 0.5

1 12.8[Pr 1] / 2f

xp f

Chc u Cρ ∞

≡ = ≥+ −



3.7Pr8.12(Pr) 68.0 −=A


Other equations for heat transfer in turbulent b.l.

For Pr ≈1, Re not too far from transition

For wide-range (Žukauakas)

Flat plate flow with low-Re

More accurate



1 Pr for Pr2

St 3/2 ≈⎟⎟⎠

⎞⎜⎜⎝

⎛= −f

x

C

755/1 10Re105

Re0592.0

≤≤×≅ xx

fC

380Pr 0.7for Pr2

St 57.0 ≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛= −f

x

C

PrRe0296.0Nu 43.08.0xx =

6543.08.0 105Re102 PrRe032.0Nu ×≤≤×= xxx


Other equations for heat transfer in turbulent b.l.Average Nusselt number for uniform Tw

It may be used for either Tw=const or qw=const and for

43.08.0 PrRe0370.0Nu LL =

7103Re ×≤L

x0 0 0

Nu1 1 1 L L L xkh qdx h dx dx

L T L L x= = =

Δ ∫ ∫ ∫



0.43 0.8

0 0

Nu 1Nu 0.0296 Pr ReL Lx

L xhL dx dxk x x

⎡ ⎤= = = ⎢ ⎥⎣ ⎦∫ ∫


Other equations for heat transfer in turbulent b.l.Laminar and turbulent b.l

For liquid, Whitaker suggested

( )4/1

8.043.0 9200RePr0370.0Nu ⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∞

wLL

μμ



⎥⎦⎤

⎢⎣⎡ +=

Δ= ∫∫∫ dxhdxh

Lqdx

TLh

L

x

xL

turbulentlaminar00 trans

trans1 1

[ ]{ }2/1trans

097.08.0trans

8.043.0 )(RePr95.17ReRePr0370.0Nu −−= LL


A correlation for laminar, transitional, and turbulent flow

Churchill suggests

Where

Reu is the Reynolds number at the end of the turbulent transition region It is for uniform TwMay be used for uniform qw, if0.3387 0.4637, 0.0468 0.02052



2/1

5/22/7

5/32/1

])/(1[)2600/(1)3387.0(45.0Nu ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

++=φφ

φφu

x

2/13/23/2

Pr0468.01PrRe

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+≡ xφ

75 10~10≈uφ )Re(Re uxu =≈φφ


A correlation for laminar, transitional, and turbulent flow

Average Nusselt number

Where

Applicable for either uniform Tw or uniform qw

2/1

5/22/7

5/32/1

])/(1[)12500/(1)6774.0(45.0Nu ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

++=φφ

φφum

L

)Re(Re875.1 uxum =≈ φφ




Homework

补充题

水以2kg/s的质量流量流过直径为40mm，长为4m的圆管，管壁温度保持在90℃，水的进口温度为30℃。

求水的出口温度和管子对水的散热量。水的物性参数按40℃的水查取。不考虑由温差引起的修正。

6.306.31



6.6 the reynolds analogy the reynolds analogy the reynolds

Documents