Given equivalent boundary conditions the solutions take the same form Analogous Equations if two normalized dimensionless equations take the same form the equations are analogou s The momentum and energy ID: 164109
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Slide1
Fluid Dynamics: Boundary Layers
Given equivalent boundary conditions, the solutions take the same form
Analogous Equations
if two normalized (dimensionless) equations take the same form the equations are
analogou
s
The momentum and energy
boundary layer equations
are
analogous if there is a negligible pressure gradient (dp*/dx* ~ 0) and the Pr ~ 1
Reynolds AnalogySlide2
Fluid Dynamics: Boundary Layers
The Reynolds Analogy
is defined as (when Pr =1)
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)
Colburn
j factor
laminar flows: valid for
dp
*/dx*
~ 0
turbulent flows: generally valid without restriction on
dp
*
/
dx
*
Modified Reynolds Analogy: Chilton-Colburn (empirical)
Defining a new non-dimensional number
Reynolds Analogy
Stanton numberSlide3
Determining Heat Transfer Coefficients
determining heat transfer coefficients requires an accurate knowledge of the flow fieldfew (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
The Nusselt number (and heat transfer coefficient) are functions of the fluid properties (
ν
,
ρ,
α, c,
kf) the effect of variable properties may be considered by evaluating all properties at the film temperaturemost accurate solutions often require iteration on the film properties
External Convection: Overview
local Nusselt number
average Nusselt numberSlide4
Fluid Dynamics: Boundary Layers
Transition
external (flat plate) flow
internal (duct) flow
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 flowSlide5
Fluid Dynamics: Boundary Layers
Transition
Transition leads to a significant increase in the local heat transfer coefficientSlide6
External Convection: Overview
External Flowsboundary 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
compute:
compare to
critical
Reynolds
number
external (flat plate) flow
laminar flow along length of flat plate
transition to turbulent flow
at critical length
Determining external flow
conditionsSlide7
Transition to Turbulence
critical Reynolds number affected by free stream turbulence and surface roughness of platenominally
if the boundary layer is “tripped” at the leading edge:
flow is turbulent along entire length of flat plate
External Convection: Overview
laminar free stream & smooth plate
Average parametersSlide8
Thermal
Conditions at the Surface
(idealized)uniform heat fluxuniform surface temperature
unheated starting lengthExternal Convection: Overview
delays thermal boundary layer growth