AME

60634
Int. Heat Trans.
Fluid Dynamics: Boundary Layers
Analogous Equations
if two normalized (dimensionless) equations take the same form the
equations are analogous

Reynolds Analogy
u*
* * u
*
1 2 u*
u * +v =
The momentum and energy boundary x y *
Re y *2
layer equations are analogous if there is L

advection diffusion
a negligible pressure gradient (dp*/dx* ~ 0)
and the Pr ~ 1

terms term

* *
* T * T 1 2T *
u +v =
x *
y *
Re L y * 2
Given equivalent boundary conditions, the
solutions take the same form

u* = T * ! Re $
C f # & = Nu
u* T * " 2 %
= *
y* y* =0
y y* =0

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 AME 60634
Int. Heat Trans.
Fluid Dynamics: Boundary Layers
Reynolds Analogy

Defining a new non- h Nu

St = = Stanton number
dimensional number VC p RePr

The Reynolds Analogy is defined as

(when Pr =1)

Cf
= St
2
The Reynolds Analogy implies that under certain conditions (no
pressure gradient, Pr = 1) if the velocity parameters are known than the
heat transfer parameters can be determined (and vice versa)

Modified Reynolds Analogy: Chilton-Colburn (empirical)
Cf 2
= St Pr 3 j H 0.6 < Pr < 60 Colburn j factor
2
laminar flows: valid for dp*/dx* ~ 0
turbulent flows: generally valid without restriction on dp*/dx*
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AME 60634
Int. Heat Trans.
External Convection: Overview
Determining Heat Transfer Coefficients
determining heat transfer coefficients requires an accurate knowledge of
the flow field
few (pseudo-)analytical solutions exist (especially for turbulent flow)
similarity solutions, etc.
heat transfer coefficient relations are largely empirical and are
presented based on the Nusselt number

dT * hL h L average Nusselt
Nu = * = = f (x * ,Re L ,Pr) local Nusselt Nu =
dy y= 0
kf number k f number

The Nusselt number (and heat transfer coefficient) are functions of the
fluid properties (, , , c, kf)

the effect of variable properties may be considered by evaluating all
properties at the film temperature
Ts + T
Tf =
2
most accurate solutions often require iteration on the film properties

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AME 60634
Int. Heat Trans.
Fluid Dynamics: Boundary Layers
Transition

Laminar and turbulent boundary layers have different heat transfer

characteristics
turbulent mixing typically increases heat transfer

Critical Reynolds number approximates the location where the

flow transitions from laminar to turbulent flow

Re x,c ~ 10 5 external (flat plate) flow

Vx c
Re x,c
Re D,c ~ 10 3 internal (duct) flow
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AME 60634
Int. Heat Trans.
Fluid Dynamics: Boundary Layers
Transition

Transition leads to a significant increase in the local heat transfer coefficient

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AME 60634
Int. Heat Trans.
External Convection: Overview
External Flows
boundary layers develop freely without constraint (compare to a
internal/duct flow)
boundary layer may be laminar, laminar and turbulent, or entirely
turbulent
simplest external flow: flat plate in parallel flow

Determining external flow conditions

u L
compute: Re L =

compare to critical Reynolds number
Re x,c ~ 10 5 external (flat plate) flow

laminar flow along

Re L < Re x,c
length of flat plate
transition to turbulent x c Re x,c
Re L > Re x,c =
flow at critical length L Re L

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 AME 60634
Int. Heat Trans.
External Convection: Overview
Transition to Turbulence
critical Reynolds number affected by free stream turbulence and surface
roughness of plate
nominally
Re x,c ~ 10 5 laminar free stream & smooth plate
if the boundary layer is tripped at the leading edge:
Re x,c = 0 x c = 0
flow is turbulent along entire length of flat plate

Average parameters
1$ c ' 1# c &
x L x L
s,L = && s,x,lam dx + s,x,turb dx )) hL = %% hx, lam dx + hx,turb dx ((
L% 0 xc ( L$ 0 xc '

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 AME 60634
Int. Heat Trans.
External Convection: Overview
Thermal Conditions at the Surface (idealized)
uniform heat flux q"s"
uniform surface temperature Ts
unheated starting length

delays thermal boundary

layer growth
x < Ts = T
x > Ts or q%s%

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