Flow over immersed bodies.

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Lecture 7

• Flow over immersed bodies.
• Boundary layer. Analysis of inviscid flow.

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Flow over immersed bodies

• flow classification 2D, axisymmetric, 3D
• bodies streamlined and blunt

Shuttle landing examples of various body types
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Lift and Drag

• shear stress and pressure integrated over the
  surface of a body create force
• drag force component in the direction of
  upstream velocity
• lift force normal to upstream velocity (might
  have 2 components in general case)

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Flow past an object

• Dimensionless numbers involved

• for external flow Regt100 dominated by inertia,
  Relt1 by viscosity

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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.
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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

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Laminar/Turbulent transition

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

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Boundary layer characteristics

• Boundary layer thickness

Boundary layer displacement thickness
Boundary layer momentum thickness (defined in
terms of momentum flux)
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Prandtl/Blasius boundary layer solution
Lets consider flow over large thin plate

• approximations

• than

• boundary conditions

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Prandtl/Blasius boundary layer solution

• as dimensionless velocity profile should be
  similar regardless of location

• dimensionless similarity variable

• stream function

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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

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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
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Inviscid flow

• no shearing stress in inviscid flow, so

• equation of motion is reduced to Euler equations

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Bernoulli equation

• lets write Euler equation for a steady flow
  along a streamline

• now we multiply it by ds along the streamline

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Irrotational Flow

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

• Example uniform flow in x-direction

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Bernoulli equation for irrotational flow
always 0, not only along a stream line

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

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Velocity potential

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

• As for incompressible flow conservation of mass
  leads to

Laplace equation
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Some basic potential flows

• As Laplace equation is a linear one, the
  solutions can be added to each other producing
  another solution
• stream lines (yconst) and equipotential lines
  (fconst) are mutually perpendicular

Both f and y satisfy Laplaces equation
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Uniform flow

• constant velocity, all stream lines are straight
  and parallel

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Source and Sink

• Lets consider fluid flowing radially outward
  from a line through the origin perpendicular to
  x-y planefrom mass conservation

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Vortex

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

• circulation

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

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Shape of a free vortex
at the free surface p0
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Doublet

• lets consider the equal strength, source-sink
  pair

if the source and sink are close to each other
K strength of a doublet
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Summary
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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

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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.

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Rankine Ovals

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

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Flow around circular cylinder

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

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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.

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Effect of pressure gradient

• dAlemberts 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
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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

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Drag on a flat plate
Drag coefficient diagram
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Drag dependence

• Low Reynolds numbers Relt1

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Drag dependence

• Moderate Reynolds numbers. Drag coefficient on
  flat plate Re-½ on blunt bodies relatively
  constant (and decreases as turbulent layer can
  travel further along the surface resulting in a
  thinner wake

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Examples
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Examples

• 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 drag.

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Problems

• 6.60. An ideal fluid flows past an infinitely
  long semicircular hump. Far from hump flow is
  uniform and pressure is p0. Find maximum and
  minimum pressure along the hump. If the solid
  surface is y0 streamline, find the equation for
  the streamline passing through Qp/2, r2a.

• 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 generated.

• 9.12. Water flows past a flat plate with an
  upstream velocity of U0.02 m/s. Determine the
  water velocity 10mm from a plate at distances
  1.5mm and 15m from the leading edge.