Ch5-Boundary Layer Flow-Forced Convection - · PDF fileThermal Boundary Layer: ... theory of turbulent flow remains largely undeveloped. ... Boundary Layer Flow-Forced Convection)

Chapter 5: Boundary Layer Flow-Forced Convection

Advanced Heat TransferY.C. Shih Spring 2009


5-1 Introduction5-2 Boundary Layer Equations5-3 Similarity Solution5-4 Integral Method Approximation



All experimental observations indicate that a fluid in motion comes to a complete stop at the surface and assumes a zero velocity relative to the surface (no-slip).The no-slip condition is responsible for the development of the velocity profile.The flow region adjacent to the wall in which the viscous effects (and thus the velocity gradients) are significant is called the boundary layer.

5-1 Introduction (1)

5-1



An implication of the no-slip condition is that heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, and can be expressed as

Heat transfer coefficient

The convection heat transfer coefficient, in general, varies along the flow direction.

2

0

(W/m )conv cond fluidy

Tq q ky =

∂= = −

∂& &

( ) 0 2 (W/m C)fluid y

s

k T yh

T T=

∞

− ∂ ∂= ⋅

−o


)( ∞−= TThq sconv&

5-2



The Nusselt NumberIt is common practice to nondimensionalize the heat transfer coefficient h with the Nusselt number

Heat flux through the fluid layer by convection and by conduction can be expressed as, respectively:

Taking their ratio gives

The Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer.Nu=1 pure conduction.

chLNuk

=

convq h T= Δ&cond

Tq kLΔ

=&

/conv

cond

q h T hL Nuq k T L k

Δ= = =

Δ&

&


5-3



Viscous versus inviscid regions of flowInternal versus external flowCompressible versus incompressible flowLaminar versus turbulent flowNatural (or unforced) versus forced flowSteady versus unsteady flowOne-, two-, and three-dimensional flows


5-4



Velocity Boundary Layer:Consider the parallel flow of a fluid over a flat plate.x-coordinate: along the plate surface y-coordinate: from the surface in the normal direction.The fluid approaches the plate in the x-direction with a uniform velocity V.Because of the no-slip condition V(y=0)=0.The presence of the plate is felt up to d.Beyond d the free-stream velocity remains essentially unchanged.The fluid velocity, u, varies from 0 at y=0 to nearly V at y=d.


5-5



Velocity Boundary Layer:

The region of the flow above the plate bounded by d is called the velocity boundary layer.d is typically defined as the distance y from the surface at which

u=0.99V.The hypothetical line of u=0.99V divides the flow over a plate into two regions:

the boundary layer region, andthe irrotational flow region.


5-6



Surface Shear Stress:Consider the flow of a fluid over the surface of a plate.The fluid layer in contact with the surface tries to drag the plate along via friction, exerting a friction force on it.Friction force per unit area is called shear stress, and is denoted by t.Experimental studies indicate that the shear stress for most fluids is proportional to the velocity gradient.The shear stress at the wall surface for these fluids is expressed as

The fluids that that obey the linear relationship above are called Newtonian fluids.The viscosity of a fluid is a measure of its resistance to deformation.

2

0

(N/m )sy

uy

τ μ=

∂=

∂


5-7



The viscosities of liquids decrease with temperature, whereas the viscosities of gases increase with temperature.In many cases the flow velocity profile is unknown and the surface shear stress ts

from Eq. 6–9 can not be obtained. A more practical approach in external flow is to relate ts to the upstream velocity V as

Cf is the dimensionless friction coefficient (most cases is determined experimentally).The friction force over the entire surface is determined from

22 (N/m )

2s fVC ρτ =

2

(N)2f f sVF C A ρ

=


5-8



Thermal Boundary Layer:Like the velocity a thermal boundary layer develops when a fluid at a specified temperature flows over a surface that is at a different temperature.Consider the flow of a fluidat a uniform temperature of T∞ over an isothermal flat plate at temperature Ts. The fluid particles in the layer adjacent assume the surface temperature Ts. A temperature profile develops that ranges from Ts at the surface to T∞

sufficiently far from the surface.The thermal boundary layer ─ the flow region over the surface in which the temperature variation in the direction normal to the surface is significant.


5-9



The thickness of the thermal boundary layer dt at any location along the surface is defined as the distance from the surface at which the temperature difference T(y=dt)-Ts= 0.99(T∞-Ts).The thickness of the thermal boundary layer increases in the flow direction.The convection heat transfer rate anywhere along the surface is directly related to the temperature gradientat that location.


5-10



Prandtl Number:The relative thickness of the velocity and the thermal boundary layers is best described by the dimensionless parameter Prandtl number, defined as

Heat diffuses very quickly in liquid metals (Pr«1) and very slowly in oils (Pr»1) relative to momentum. Consequently the thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocityboundary layer.

Molecular diffusivity of momentumPrMolecular diffusivity of heat

pckμν

α= = =


5-11



Laminar and Turbulent Flows:Laminar flow ─ the flow is characterized by smooth streamlines and highly-ordered motion.Turbulent flow ─ the flow is characterized by velocity fluctuations and highly-disordered motion.The transition from laminarto turbulent flow does not occur suddenly.


5-12



The velocity profile in turbulent flow is much fuller than that in laminar flow, with a sharp drop near the surface.The turbulent boundary layer can be considered to consist of four regions:

Viscous sublayerBuffer layerOverlap layerTurbulent layer

The intense mixing in turbulent flow enhances heat and momentum transfer, which increases the friction force on the surface and the convection heat transfer rate.


5-13



Reynolds Number:The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, flow velocity, surface temperature, and type of fluid.The flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid.This ratio is called the Reynolds number, which is expressed for external flow as

At large Reynolds numbers (turbulent flow) the inertia forces are large relative to the viscous forces.At small or moderate Reynolds numbers (laminar flow), the viscous forces are large enough to suppress these fluctuations and to keep the fluid “inline.”Critical Reynolds number ─ the Reynolds number at which the flow becomes turbulent.

Inertia forcesReViscous forces

c cVL VLρν μ

= = =


5-14



Heat and Momentum Transfer in Turbulent Flow:

Turbulent flow is a complex mechanism dominated by fluctuations, and despite tremendous amounts of research the theory of turbulent flow remains largely undeveloped.Knowledge is based primarily on experiments and the empirical or semi-empirical correlations developed for various situations.Turbulent flow is characterized by random and rapid fluctuationsof swirling regions of fluid, called eddies.The velocity can be expressed as the sum of an average value u and a fluctuating component u’

'u u u= +


5-15



It is convenient to think of the turbulent shear stress as consisting of two parts: the laminar component, andthe turbulent component.

The turbulent shear stress can be expressed as

The rate of thermal energy transport by turbulent eddies is

The turbulent wall shear stress and turbulent heat transfer

mt ─ turbulent (or eddy) viscosity.

kt ─ turbulent (or eddy) thermal conductivity.

' 'turb pq c v Tρ=&

' 'turb u vτ ρ= −

' ' ; turb t turb p tu Tu v q c vT ky y

τ ρ μ ρ∂ ∂= − = = = −

∂ ∂&


5-16



The total shear stress and total heat flux can be expressed as

and

In the core region of a turbulent boundary layer ─ eddy motion (and eddy diffusivities) are much larger than their molecular counterparts.Close to the wall ─ the eddy motion loses its intensity. At the wall ─ the eddy motion diminishes because of the no-slip condition.

( ) ( )turb t tu uy y

τ μ μ ρ ν ν∂ ∂= + = +

∂ ∂

( ) ( )turb t p tT Tq k k cy y

ρ α α∂ ∂= − + = − +

∂ ∂&


5-17



In the core region ─ the velocity and temperature profiles are very moderate.In the thin layer adjacent to the wall ─ the velocity and temperature profiles are very steep.

Large velocity and temperature gradients at the wall surface.

The wall shear stress and wall heat flux are much larger in turbulent flow than they are in laminar flow.


5-18



5-2 Boundary Layer Equations (1)

Consider the parallel flow of a fluid over a surface.Assumptions:

steady two-dimensional flow, Newtonian fluid, constant properties, andlaminar flow.

The fluid flows over the surface with a uniform free-stream velocity V, but the velocity within boundary layer is two-dimensional (u=u(x,y), v=v(x,y)).Three fundamental laws:

conservation of mass continuity equationconservation of momentum momentum equationconservation of energy energy equation 5-19



Boundary Layer ApproximationAssumptions:

1) Velocity components:u>>v

2) Velocity gradients:∂v/∂x≈0 and ∂v/∂y≈0∂u/∂y >> ∂u/∂x

3) Temperature gradients:∂T/∂y >> ∂T/∂x

When gravity effects and other body forces are negligible the y-momentum equation

0Py

∂ =∂


5-20



Consider laminar flow of a fluid over a flat plate. Steady, incompressible, laminar flow of a fluid with constant properties

Continuity equation

Momentum equation

Energy equation

2

2

u u uu vx y y

ν∂ ∂ ∂+ =

∂ ∂ ∂

u vx y∂ ∂

+∂ ∂

2

2

T T Tu vx y y

α∂ ∂ ∂+ =

∂ ∂ ∂


5-21



Boundary conditions

At x=0

At y=0

As y ∞

When fluid properties are assumed to be constant, the first two equations can be solved separately for the velocity components u and v.knowing u and v, the temperature becomes the only unknown in the last equation, and it can be solved for temperature distribution.

( ) ( )0, , 0,u y V T y T∞= =

( ) ( ) ( ),0 0, ,0 0, ,0 su x v x T x T= = =

( ) ( ), , ,u x V T x T∞∞ = ∞ =


5-22



The continuity and momentum equations are solved by transforming the two partial differential equations into a single ordinary differential equation by introducing a new independent variable (similarity variable).The argument ─ the nondimensional velocity profile u/Vshould remain unchanged when plotted against the nondimensional distance y/d.d is proportional to (nx/V)1/2, therefore defining dimensionless similarity variable as

might enable a similarity solution.

Vy xη ν=

5-3 Similarity Solution (1)

5-23



Introducing a stream function y(x, y) as

Defining a function f(h) as the dependent variable as

The velocity components become

; u vy xψ ψ∂ ∂

= = −∂ ∂

( )/

fV x V

ψην

=

12 2

x df V dfu V Vy y V d x d

x df V V dfv V f fx V d Vx x d

ψ ψ η νη η ν η

ψ ν ν ν ηη η

∂ ∂ ∂= = = =∂ ∂ ∂

⎛ ⎞∂= − = − − = −⎜ ⎟∂ ⎝ ⎠


5-24



3 2

3 22 0d f d ffd dη η

+ =

( )0

0 0, 0, 1df dffd dη ηη η= →∞

= = =


5-25



5-4 Integral Method Approximation

Mathematical Simplification• Number of independent variables are reduced• Reduction in order of differential equation

5-26

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Ch5-Boundary Layer Flow-Forced Convection - · PDF fileThermal Boundary Layer: ... theory of turbulent flow remains largely undeveloped. ... Boundary Layer Flow-Forced Convection)