Pharos University MECH 253 FLUID MECHANICS II

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Pharos UniversityMECH 253 FLUID MECHANICS
II

• Lecture 5
• INVISCID FLOWS

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1 Inviscid Flow

• Inviscid flow implies that the viscous effect is
  negligible. This occurs in the flow domain away
  from a solid boundary outside the boundary layer
  at Re??.
• The flows are governed by Euler
  Equationswhere ?, v, and p can be functions
  of r and t .

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7.1 Inviscid Flow

• On the other hand, if flows are steady but
  compressible, the governing equation
  becomeswhere ?? can be a function of r
• For compressible flows, the state equation is
  needed then, we will require the equation for
  temperature T also.

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7.1 Inviscid Flow

• Compressible inviscid flows usually belong to the
  scope of aerodynamics of high speed flight of
  aircraft. Here we consider only incompressible
  inviscid flows.
• For incompressible flow, the governing equations
  reduce to where ?? constant.

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7.1 Inviscid Flow

• For steady incompressible flow, the governing eqt
  reduce further to where ?
  constant.
• The equation of motion can be rewrited into
• Take the scalar products with dr and integrate
  from a reference at ? along an arbitrary
  streamline ?C , leads to since

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7.1 Inviscid Flow

• If the constant (total energy per unit mass) is
  the same for all streamlines, the path of the
  integral can be arbitrary, and in the flow
  domain except inside boundary layers.
• Finally, the governing equations for inviscid,
  irrotational steady flow are
• Since is the vorticity , flows
  with are called irrotational
  flows.

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7.1 Inviscid Flow

• Note that the velocity and pressure fields are
  decoupled. Hence, we can solve the velocity field
  from the continuity and vorticity equations. Then
  the pressure field is determined by Bernoulli
  equation.
• A velocity potential ? exists for irrotational
  flow, such that, and irrotationality
  is automatically
  satisfied.

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7.1 Inviscid Flow

• The continuity equation becomeswhich is also
  known as the Laplace equation.
• Every potential satisfy this equation. Flows with
  the existence of potential functions satisfying
  the Laplace equation are called potential flow.

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7.1 Inviscid Flow

• The linearity of the governing equation for the
  flow fields implies that different potential
  flows can be superposed.
• If ?1 and ?2 are two potential flows, the sum
  ?(?1?2) also constitutes a potential flow. We
  have
• However, the pressure cannot be superposed due to
  the nonlinearity in the Bernoulli equation, i.e.

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Potential FlowsIntegral Equations

• Irrotational Flow
• Flow Potential
• Conservation of Mass
• Laplace Equation

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7.2 2D Potential Flows

• If restricted to steady two dimensional potential
  flow, then the governing equations become
• E.g. potential flow past a circular cylinder with
  D/L ltlt1 is a 2D potential flow near the middle of
  the cylinder, where both w component and
  ?/?z?0.

U
L
y
x
z
D
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7.2 2D Potential Flows

• The 2-D velocity potential function givesand
  then the continuity equation becomes
• The pressure distribution can be determined by
  the Bernoulli equation,where p is the
  dynamic pressure

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7.2 2D Potential Flows

• For 2D potential flows, a stream function ?(x,y)
  can also be defined together with ?(x,y). In
  Cartisian coordinates, where continuity
  equation is automatically satisfied, and
  irrotationality leads to the Laplace equation,
• Both Laplace equations are satisfied for a 2D
  potential flow

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7.2 Two-Dimensional Potential Flows

• For two-dimensional flows, become
• In a Cartesian coordinate system
• In a Cylindrical coordinate system

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Taking into account
Continuity equation
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Irrotational Flow Approximation

• For 2D flows, we can also use the stream function
• Recall the definition of stream function for
  planar (x-y) flows
• Since vorticity is zero,
• This proves that the Laplace equation holds for
  the stream function and the velocity potential

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Cylindrical coordinate system
In cylindrical coordinates (r , q ,z ) with
-axisymmetric case
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Taking into account
Continuity equation
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7.2 Two-Dimensional Potential Flows

• Therefore, there exists a stream function
  such that
• in the Cartesian
  coordinate system and
• in the cylindrical
  coordinate system.
• The transformation between the two coordinate
  systems

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7.2 Two-Dimensional Potential Flows

• The potential function and the stream function
  are conjugate pair of an analytical function in
  complex variable analysis. The conditions
• These are the Cauchy-Riemann conditions. The
  analytical property implies that the constant
  potential line and the constant streamline are
  orthogonal, i.e.,
• and to imply
  that .

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Irrotational Flow Approximation

• Irrotational approximation vorticity is
  negligibly small
• In general, inviscid regions are also
  irrotational, but there are situations where
  inviscid flow are rotational, e.g., solid body
  rotation

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7.3 Simple 2-D Potential Flows

• Uniform Flow
• Stagnation Flow
• Source (Sink)
• Free Vortex

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7.3.1 Uniform Flow

• For a uniform flow given by , we have
  
• Therefore,
• Where the arbitrary integration constants are
  taken to be zero at the origin.

and
and
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7.3.1 Uniform Flow

• This is a simple uniform flow along a single
  direction.

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Elementary Planar Irrotational FlowsUniform
Stream

• In Cartesian coordinates
• Conversion to cylindrical coordinates can be
  achieved using the transformation

Proof with Mathematica
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7.3.2 Stagnation Flow

• For a stagnation flow, . Hence,
• Therefore,

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7.3.2 Stagnation Flow

• The flow an incoming far field flow which is
  perpendicular to the wall, and then turn its
  direction near the wall
• The origin is the stagnation point of the flow.
  The velocity is zero there.

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7.3.3 Source (Sink)

• Consider a line source at the origin along the
  z-direction. The fluid flows radially outward
  from (or inward toward) the origin. If m denotes
  the flowrate per unit length, we have
  (source if m is positive and sink if
  negative).
• Therefore,

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7.3.3 Source (Sink)

• The integration leads to
• Where again the arbitrary integration constants
  are taken to be zero at .

and
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7.3.3 Source (Sink)

• A pure radial flow either away from source or
  into a sink
• A ve m indicates a source, and ve m indicates a
  sink
• The magnitude of the flow decrease as 1/r
• z direction into the paper. (change graphics)

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Elementary Planar Irrotational FlowsLine
Source/Sink

• Potential and streamfunction are derived by
  observing that volume flow rate across any circle
  is
• This gives velocity components

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Elementary Planar Irrotational FlowsLine
Source/Sink

• Using definition of (Ur, U?)
• These can be integrated to give ? and ?

Equations are for a source/sink at the origin
Proof with Mathematica
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7.3.4 Free Vortex

• Consider the flow circulating around the origin
  with a constant circulation . We have
  where fluid moves counter clockwise if
  is positive and clockwise if negative.
• Therefore,

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7.3.4 Free Vortex

• The integration leads to
• where again the arbitrary integration constants
  are taken to be zero at

and
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7.3.4 Free Vortex

• The potential represents a flow swirling around
  origin with a constant circulation ?.
• The magnitude of the flow decrease as 1/r.

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7.4. Superposition of 2-D Potential Flows

• Because the potential and stream functions
  satisfy the linear Laplace equation, the
  superposition of two potential flow is also a
  potential flow.
• From this, it is possible to construct potential
  flows of more complex geometry.
• Source and Sink
• Doublet
• Source in Uniform Stream
• 2-D Rankine Ovals
• Flows Around a Circular Cylinder

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7.4.1 Source and Sink

• Consider a source of m at (-a, 0) and a sink of m
  at (a, 0)
• For a point P with polar coordinate of (r, ).
  If the polar coordinate from (-a,0) to P is
  and from (a, 0) to P is
• Then the stream function and potential function
  obtained by superposition are given by

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7.4.1 Source and Sink
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7.4.1 Source and Sink

• Hence,
• Since
• We have

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7.4.1 Source and Sink

• We have
• By
• Therefore,

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7.4.1 Source and Sink

• The velocity component are

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Elementary Planar Irrotational FlowsDoublet

• A doublet is a combination of a line sink and
  source of equal magnitude
• Source
• Sink

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7.4.1 Source and Sink
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7.4.2 Doublet

• The doublet occurs when a source and a sink of
  the same strength are collocated the same
  location, say at the origin.
• This can be obtained by placing a source at
  (-a,0) and a sink of equal strength at (a,0) and
  then letting a ? 0, and m? , with ma keeping
  constant, say 2amM

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7.4.2 Doublet

• For source of m at (-a,0) and sink of m at (a,0)
• Under these limiting conditions of a?0, m? ,
  we have

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7.4.2 Doublet

• Therefore, as a?0 and m? with 2amM
• The corresponding velocity components are

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7.4.2 Doublet
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7.4.3 Source in Uniform Stream

• Assuming the uniform flow U is in x-direction and
  the source of m at(0,0), the velocity potential
  and stream function of the superposed potential
  flow become

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7.4.3 Source in Uniform Stream
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7.4.3 Source in Uniform Stream

• The velocity components are
• A stagnation point occurs at
• Therefore, the streamline passing through the
• stagnation point when .
• The maximum height of the curve
  is

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7.4.3 Source in Uniform Stream

• For underground flows in an aquifer of constant
  thickness, the flow through porous media are
  potential flows.
• An injection well at the origin than act as a
  point source and the underground flow can be
  regarded as a uniform flow.

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7.4.4 2-D Rankine Ovals

• The 2D Rankine ovals are the results of the
  superposition of equal strength sink and source
  at xa and a with a uniform flow in x-direction.
• Hence,

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7.4.4 2-D Rankine Ovals

• Equivalently,

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7.4.4 2-D Rankine Ovals

• The stagnation points occur at
• where with corresponding .

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7.4.4 2-D Rankine Ovals

• The maximum height of the Rankine oval is
• located at when ,i.e.,
• which can only be solved numerically.

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7.4.4 2-D Rankine Ovals
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7.4.5 Flows Around a Circular Cylinder

• Steady Cylinder
• Rotating Cylinder
• Lift Force

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7.4.5.1 Steady Cylinder

• Flow around a steady circular cylinder is the
  limiting case of a Rankine oval when a?0.
• This becomes the superposition of a uniform
  parallel flow with a doublet in x-direction.
• Under this limit and with M2a. mconstant,
• is the radius of the
  cylinder.

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7.4.5.1 Steady Cylinder

• The stream function and velocity potential
  become
• The radial and circumferential velocities are

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7.4.5.1 Steady Cylinder
ro
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Examples of Irrotational Flows Formed by
Superposition

• Flow over a circular cylinder Free stream
  doublet
• Assume body is ? 0 (r a) ? K Va2

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Examples of Irrotational Flows Formed by
Superposition

• Velocity field can be found by differentiating
  streamfunction
• On the cylinder surface (ra)

Normal velocity (Ur) is zero, Tangential velocity
(U?) is non-zero ?slip condition.
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Rotating Cylinder

• The potential flows for a rotating cylinder is
  the free vortex flow. Therefore, the potential
  flow of a uniform parallel flow past a rotating
  cylinder at high Reynolds number is the
  superposition of a uniform parallel flow, a
  doublet and free vortex.
• Hence, the stream function and the velocity
  potential are given by

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Rotating Cylinder

• The radial and circumferential velocities are
  given by

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Rotating Cylinder

• The stagnation points occur at
• From

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Rotating Cylinder
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Rotating Cylinder
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Rotating Cylinder

• The stagnation points occur at
• Case 1
• Case 2
• Case 3

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Rotating Cylinder

• Case 1

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Rotating Cylinder

• Case 2
• The two stagnation points merge to one at
  cylinder surface where .

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Rotating Cylinder

• Case 3
• The stagnation point occurs outside the cylinder
• when where . The condition
  of
• leads to
• Therefore, as , we have
  

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Rotating Cylinder

• Case 3

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Lift Force

• The force per unit length of cylinder due to
  pressure on the cylinder surface can be obtained
  by integrating the surface pressure around the
  cylinder.
• The tangential velocity along the cylinder
  surface is obtained by letting rro,

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Lift Force

• The surface pressure as obtained from
  Bernoulli equation is
• where is the pressure at far away from the
  cylinder.

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Lift Force

• Hence,
• The force due to pressure in x and y directions
  are then obtained by

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Lift Force

• The lift on rotating bodies is called the Magnus
  effect.
• the lift force is due to the circulation around
  the body.
• An airfoil without rotation can develop a
  circulation around the airfoil when Kutta
  condition is satisfied at the rear tip of the
  airfoil.
• Therefore, The tangential velocity along the
  cylinder surface is obtained by letting rro
• This forms the base of aerodynamic theory of
  airplane.