Transport processes Part 4 Ron Zevenhoven bo Akademi University - - PDF document

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Transport processes – Part 4

Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik

• tel. 3223 ; ron.zevenhoven@abo.fi

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Chapters 7-8-9 (not part of this course) Except when Re = ∞: inviscid flow)

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VST rz18 7/42 4 s coordinate Cartesian in                                      y v x v x v z v z v y v v v rot

x y z x y z

(viscous effects neglected: ”inviscid”)

Steady state: ∂../∂t = 0

• sign because h↑ and g↓

v not 0  rot v = 0 for streamline

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Example rot v

• Assume a flow field v = (vx,vy,vz) = (k·y,0,0),

with a constant y.

• For this case
• gives a vector with a non-zero component in

z-direction.

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                                                               k k y v x v x v z v z v y v v v rot

x y z x y z

x y v

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dy v dx v dy y dx x d y x

x y

               ) , ( Transport processes (TRP)

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v vector any for scalar any for          v  

) charge electric an for : force electric and field electric an for , (Similarly q voltage q F E F E

elec elec elec elec

  

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VST rz18 11/42 4 . ) , ( ) , ( ) , ( : etc dx y x y x dx y y x v v y y x v y v x v continuity

y x y y x

 

                    

2

   . .e i

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r θ

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½ρv2+p+ρgh = constant

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D’Alembert paradox

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Creeping flow

• The expression (4.38) gives, with vz << vy and << vx

which with ∂vx/∂y and ∂vx/∂x << ∂vx/∂z, and similarly ∂vy/∂y and ∂vy/∂x << ∂vy/∂z simplifies to (4.39, 4.40)

μ p v   

and y p μ z v y v x v x p μ z v y v x v

y y y x x x

                       

           

and y p μ z v x p μ z v

y x

           

   

x z y z=0 z=h vx, vy, vz

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function flow potential 6 6 and 6

2 , 2 , 2

µ ph v h µ x p v h µ x p

mean y mean x

           

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Re << 1

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A classroom exercise - 4

• An inviscid incompressible fluid flow can be

described by a two-dimensional stream function ψ(x,y) and potential Φ, for –L ≤ x ≤ L, –L ≤ y ≤ L . With velocity v∞ at x = L, y = L, the velocity potential Φ is given by : Φ = v∞· x· y /L.

• Give the expression for the velocity vector

v(x,y) = (vx,vy) and for the stream function ψ(x,y).

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@ y ≥ δ(x) : vy = 0

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V x    ~

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4.4) (Fig. velocity flow free ) V (or V ,  



 y

) ( ) (  f x A  

   d df V f V v x ) ( ) ( '    

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see next 4 slides

2 2 3 3

2   d f d d f d f       d df V f V v x ) ( ) ( '    

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Boundary layers – Blasius /1

• The starting point for Blasius’ analysis are Prandtl’s

boundary layer equations, which with dp/dx ≈ 0 (or at least dp/dx << the other terms) are with boundary conditions v∞=v∞(x) in the undisturbed flow, vx=vy=0 at x=0, vx=v∞ at y=∞.

• Considering a two-dimensional flow (i.e. symmetry

in third dimension) described by stream function ψ(x,y) and introducing a dimensionless variable η(x,y) = y/√(ν·x/v∞) ~ y/δ gives a function f(η):

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Boundary layers – Blasius /2

• Producing from this the terms for the Prandtl

equations gives result where f´ = ∂f/∂η, f´´ = ∂2f/∂η2 and noting that ∂f/∂x = ∂f/∂η·∂η/∂x = f´·∂η/∂x and ∂f/∂y = ∂f/∂η·∂η/∂y = f´·∂η/∂y

• Using this in Prandtl’s

equation gives finally with boundary conditions

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Boundary layers – Blasius /3

• A numerical solution was produced by Howarth (1938)

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Boundary layers – Blasius /4

• Note that the equation (4.63), however, defines

η(x,y) as which differs by a factor 2 from the expression used by Blasius.

• This then gives instead of f´´´+ f·f´´ = 0 the

expression 2·f´´´+ f·f´´ = 0 with boundary conditions f(0) = f´(0) = 0 and f´(∞) = 1.

• The numerical solution for this

is given in the table:

) ( v dy v dx d dy y v dy x v : Note

y x y x

         

  

  

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

y

v

use Leibniz

) ( v dy v dx d dy y v dy x v : Note

y x y x

         

  

  

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4 4 3 3 2 2 1

            



b b b b b V v y

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        

  

                 µV V V y v y v

x x

2 2 3 3 2

3 2

) (@ ) (

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A classroom exercise - 5

• Blasius’ boundary layer

analysis describes the velocity profile (vx,vy) in a laminar boundary layer with a function f(η) where η = ½y√(v∞/xν) = ψ(x,y)/√(v∞xν), with kinematic viscosity ν, position x along the surface

• n which the boundary layer

builds up, position y from the surface, and undisturbed flow (v∞, 0). See course material § 4.3.2 + added

• material. (continues)

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A classroom exercise - 5

• Using the analytical solution

by Howarth, given in the table for η, f´(η)= ∂f/∂η and f´´(η)= ∂²f/∂η², show that the thickness of the boundary layer, defined by vx/v∞ = 0.99, can be approximated by

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ν x v x

x x 

   Re Re with 5

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Sources used

(besides course book Hanjalić et al.)

• Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena”

Wiley, 2nd edition (1999)

• R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley,

New York (1960)

• * C.J. Hoogendoorn ”Fysische Transportverschijnselen II”, TU Delft /

D.U.M., the Netherlands 2nd. ed. (1985)

• * C.J. Hoogendoorn, T.H. van der Meer ”Fysische Transport-

verschijnselen II”, TU Delft /VSSD, the Netherlands 3nd. ed. (1991)

• J.R. Welty, C.E. Wicks, R.E. Wilson. “Fundamentals of momentum,

heat and mass transfer” Wiley New York (1969)

* Earlier versions of Hanjalić et al. book but in Dutch 42/42 4