Embed Size (px)
Citation preview
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
Transport processes – Part 4
Ron ZevenhovenÅbo Akademi University
Thermal and Flow Engineering / Värme- och strömningstekniktel. 3223 ; [email protected]
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
3/42 4
Chapters 7-8-9 (not part of this course)
Except when Re = ∞: inviscid flow)
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
4/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
5/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
6/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
7/42 4
scoordinate Cartesianin
y
v
x
vx
v
z
vz
v
y
v
vvrot
xy
zx
yz(viscous effects neglected: ”inviscid”)
Steady state: ∂../∂t = 0
- sign because h↑ and g↓
v not 0 rot v = 0 for streamline
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
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.
8/42
kk
y
v
x
vx
v
z
vz
v
y
v
vvrot
xy
zx
yz
0
0
0
00
00
x
yv
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
9/42 4
dyvdxv
dyy
dxx
d
yx
xy
),(
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
10/42 4
v vectorany for 0
scalarany for 0
v
) chargeelectric an for : forceelectric and fieldelectric an for ,(Similarly qvoltageq
FEFE elec
elecelecelec
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
11/42 4
.
),(),(),(
: etcdxyx
yx
dxy
yxvv
y
yxv
y
v
x
vcontinuity y
xyyx
02 ..ei
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
12/42 4
r
θ
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
13/42 4
note: constant A = A(4.23)·C(4.25)
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
14/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
15/42 4
½ρv2+p+ρgh = constant
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
16/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
17/42 4
D’Alembertparadox
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
18/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
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
pv
and y
p
μz
v
y
v
x
v
x
p
μz
v
y
v
x
v yyyxxx
and y
p
μz
v
x
p
μz
v yx
x
z
y z=0
z=h
vx, vy, vz
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
20/42 4
functionflow potential 6
6 and
6
2
,2,2
µ
ph
vh
µ
x
pv
h
µ
x
pmeanymeanx
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
21/42 4
Re << 1
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
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 vectorv(x,y) = (vx,vy) and for the stream functionψ(x,y).
22/42
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
23/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
24/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
25/42 4
@ y ≥ δ(x) : vy = 0
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
26/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
27/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
28/42 4
V
x ~
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
29/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
30/42 4
4.4) (Fig.velocity flow free ) V(or V, y
)()( fxA
d
dfVfVv x
)()('
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
31/42 4
see next 4 slides
2
2
3
3
2 d
fdd
fdf
d
dfVfVv x
)()('
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
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):
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
Boundary layers – Blasius /2• Producing from this the terms for the Prandtl
equations gives resultwhere 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’sequation gives finallywith boundary conditions
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
Boundary layers – Blasius /3• A numerical solution was produced by Howarth (1938)
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
Boundary layers – Blasius /4• Note that the equation (4.63), however, defines
η(x,y) as which differsby 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:
)(vdyvdx
ddy
y
vdy
x
v:Note yx
yx
000
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
36/42 4
0yv
use Leibniz
)(vdyvdx
d
dyy
vdy
x
v:Note
yx
yx
0
00
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
37/42 4
44
33
2210
bb
bbbV
v y
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
38/42 4
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
39/42 4
µVV
V
y
v
y
v xx
220
332
0
32
)(@
)(
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
A classroom exercise - 5• Blasius’ boundary layer
analysis describes the velocity profile (vx,vy) in a laminar boundary layer witha function f(η) where η = ½y√(v∞/xν) = ψ(x,y)/√(v∞xν), with kinematic viscosity ν, position x along the surface on which the boundary layer builds up, position y from the surface, and undisturbedflow (v∞, 0). See course material § 4.3.2 + addedmaterial. (continues)
40/42
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
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
41/42
νxv
x x
x
ReRe
with 5
Tra
nspo
rt p
roce
sses
(TR
P)
VST rz18
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
