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24 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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Lecture 7

Flow over immersed bodies.

Boundary layer.

Analysis of inviscid flow.

Flow over immersed bodies

flow classification: 2D,
axisymmetric, 3D

bodies: streamlined
and blunt

Shuttle landing: examples of various body types

Lift and Drag

shear stress and pressure
integrated over the surface of a
body create force


force component in the
direction of upstream velocity


force normal to upstream
velocity (might have 2
components in general case)

Flow past an object

Dimensionless numbers involved

for external flow: Re>100 dominated by inertia, Re<1

by viscosity

Flow past an object

Character of the steady, viscous
flow past a circular cylinder:

(a) low Reynolds number flow, (b)
moderate Reynolds number flow,
(c) large Reynolds number flow.

Boundary layer characteristics

for large enough Reynolds number flow can be divided into
boundary region where viscous effect are important and
outside region where liquid can be treated as inviscid

Laminar/Turbulent transition

Near the leading edge of a flat
plate, the boundary layer flow is

f the plate is long enough, the flow
becomes turbulent, with random,
irregular mixing. A similar
phenomenon occurs at the
interface of two fluids moving with
different speeds.

Boundary layer characteristics

Boundary layer thickness

Boundary layer displacement thickness:

Boundary layer momentum thickness (defined in terms of momentum flux):

Prandtl/Blasius boundary layer solution



boundary conditions:

Let’s consider flow over large thin plate:

Prandtl/Blasius boundary layer solution

as dimensionless velocity profile should
be similar regardless of location:

dimensionless similarity variable

stream function

Drag on a flat plate

Drag on a flat plate is related to the momentum deficit within
the boundary layer

Drag and shear stress can be calculated just by assuming
some velocity profile in the boundary layer

Transition from Laminar to Turbulent flow

The boundary layer will become turbulent if a plate is long enough

turbulent profiles are flatter and produce
larger boundary layer

Inviscid flow

no shearing stress in inviscid flow, so

equation of motion is reduced to Euler equations

Bernoulli equation

let’s write Euler equation for a steady flow along a streamline

now we multiply it by

along the streamline

Irrotational Flow

Analysis of inviscide flow can be further simplified if we
assume if the flow is irrotational:

Example: uniform flow in x

Bernoulli equation for irrotational flow

Thus, Bernoulli equation can be applied between any two
points in the flow field

always =0, not only along a stream line

Velocity potential

equations for irrotational flow will be satisfied automatically if
we introduce a scalar function called velocity potential such

As for incompressible flow conservation of mass leads to:

Laplace equation

Some basic potential flows

As Laplace equation is a linear one,
the solutions can be added to each
other producing another solution;

stream lines (
=const) and
equipotential lines (
=const) are
mutually perpendicular



satisfy Laplace’s equation

Uniform flow

constant velocity, all stream lines are straight and

Source and Sink

Let’s consider fluid flowing radially outward from a line through
the origin perpendicular to x
y plane

from mass conservation:


now we consider situation when ther
stream lines are concentric circles i.e.
we interchange potential and stream


in case of vortex the circulation is zero
along any contour except ones
enclosing origin

Shape of a free vortex

at the free surface p=0:


let’s consider the equal strength, source
sink pair:

if the source and sink are close to
each other:


strength of a doublet


Superposition of basic flows

basic potential flows can be combined to form new
potentials and stream functions. This technique is
called the
method of superpositions

superposition of source and uniform flow

Superposition of basic flows

Streamlines created by injecting
dye in steadily flowing water show
a uniform flow. Source flow is
created by injecting water through
a small hole. It is observed that for
this combination the streamline
passing through the stagnation
point could be replaced by a solid
boundary which resembles a
streamlined body in a uniform flow.
The body is open at the
downstream end and is thus called
a halfbody.

Rankine Ovals

a closed body can be modeled as a combination of a
uniform flow and source and a sink of equal strength

Flow around circular cylinder

when the distance between source and sink approaches 0,
shape of Rankine oval approaches a circular shape

Potential flows

Flow fields for which an incompressible
fluid is assumed to be frictionless and
the motion to be irrotational are
commonly referred to as potential flows.

Paradoxically, potential flows can be
simulated by a slowly moving, viscous
flow between closely spaced parallel
plates. For such a system, dye injected
upstream reveals an approximate
potential flow pattern around a
streamlined airfoil shape. Similarly, the
potential flow pattern around a bluff
body is shown. Even at the rear of the
bluff body the streamlines closely follow
the body shape. Generally, however, the
flow would separate at the rear of the
body, an important phenomenon not
accounted for with potential theory.

Effect of pressure gradient

d’Alemberts paradox: drug on an object in inviscid liquid is
zero, but not zero in any viscous liquid even with vanishingly
small viscosity

inviscid flow

viscous flow

Effect of pressure gradient

At high Reynolds numbers, non
streamlined (blunt) objects have wide,
low speed wake regions behind them.

As shown in a computational fluid
dynamics simulation, the streamlines for
flow past a rectangular block cannot
follow the contour of the block. The flow
separates at the corners and forms a
wide wake. A similar phenomenon
occurs for flow past other blunt objects,
including bushes. The low velocity wind
in the wake region behind the bushes
allows the snow to settle out of the air.
The result is a large snowdrift behind the
object. This is the principle upon which
snow fences are designed

Drag on a flat plate

Drag coefficient diagram

Drag dependence

Low Reynolds numbers Re<1:

Drag dependence

Moderate Reynolds numbers. Drag coefficient on flat plate ~Re
; on blunt
bodies relatively constant (and decreases as turbulent layer can travel
further along the surface resulting in a thinner wake



The drag coefficient for an
object can be strongly
dependent on the shape of
the object. A slight change
in shape may produce a
considerable change in


6.60. An ideal fluid flows past an
infinitely long semicircular hump. Far
from hump flow is uniform and pressure
is p
. Find maximum and minimum
pressure along the hump. If the solid
surface is
=0 streamline, find the
equation for the streamline passing
/2, r=2a.

9.2. The average pressure and
shear stress acting on the surface of
1 m
square plate are as indicated.
Determine the lift and drag

9.12. Water flows past a flat plate with an upstream velocity of
U=0.02 m/s. Determine the water velocity 10mm from a plate at
distances 1.5mm and 15m from the leading edge.