Chapter 3: Fundamentals of Inviscid, Incompressible Flow

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Chapter 3 Fundamentals of Inviscid,
Incompressible Flow

• SONG, Jianyu
• Mar. 8th.2009

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Part 2, Inviscid, Incompressible Flow

• Inviscid frictionless or perfect fluid
• Incompressible the density is constant

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What will we learn from this chapter?

• Bernoullis equation
• Laplaces equation
• Some elementary flows

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Bernoullis equation
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Bernoullis equation

• Recall the x component of the momentum equation
• The form of substantial derivative
  
  
• Translate the equation into substantial
  derivative form
• -------------------------------------------
• For inviscid flow
• with no body forces
• For steady flow

L.H.S
The Math Fact
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• Eulers equation
• Inviscid flow with no body force
• It relate the change in velocity along a
  streamline dV to the change in pressure dp along
  the same steam line

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Bernoullis equation

• For incompressible flow ?constant
• Integrated between any two point along a
  streamline
• Bernoullis equation holds along a streamline in
  either rotational or irrotational case.
• However, for rotational flow it will change from
  different streamline.
• If the flow is irrotational, it is constant for
  the flow
• (Problem 1)

Bernoullis equation
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Laplaces equation
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Stream Function

• Consider 2D steady flow
• Equation for a streamline is
• It can be integrated to yield the algebraic
  equation for a streamline
• Now use symbol instead of f
• The function is called stream function. To get a
  streamline we only have to set c be some
  constant. Figure 2.40
• Let us define the numerical value of such
  that the difference between two streamlines is
  equal to the mass flow between the two
  streamlines.
• Since it is 2D flow, the mass flow between two
  streamlines is defined per unit depth
  perpendicular to the page

Thus choose one streamline of the flow, and give
it an arbitrary value of the stream function. The
value of the stream function for any other
streamline in the flow is fixed.
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Stream Function

• Let be the normal distance between ab
  and cd. Choose it be small enough so that V is
  constant across

Figure 2.41 Due to conservation of mass
According to the chain rule
For incompressible flow
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Velocity Potential

• For an irrotational flow
• Consider the following vector identify
• If f is a scalar function then
• (Due to the trivial Math fact The curl of the
  gradient of a scalar function is identically
  zero. Comparing equations and we see that)
• For an irrotational flow, there exists a scalar
  function f such that the velocity is given by the
  gradient of f. We denote f as velocity potential.

Since the definition of gradient in Cartesian
coordinate is Thus By the way, in
cylindrical coordinate, it is
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CMP

• Stream function

• Velocity potential f

• By differentiating normal to the velocity
  direction
• Either irrotational or rotational flow
• Only Define to 2D flow

• Flow field velocities by differentiating in the
  same direction
• Only for irrotational flow
• Apply to 3D

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CMP

• We see that a line of constant f is an isoline of
  f
• Since f is the velocity potential we give this
  isoline a specific name equipotential line.
• since
• This gradient line is a streamline
• See P171172 for the detail proof of the
  streamline and the equipotential line are
  perperndicular to each other.

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Laplaces equation

• Recall the continuity equation
• For incompressible flow ?constant
• For irrotational flow, it has velocity potential
  function
• Solutions of Laplaces equation are called
  harmonic functions.
• ------------------------------------------
• For 2D incompressible flow, stream function can
  be defined
• In fact, it automatically satisfied the
  condition.
• For irrotational flow

Laplaces equation
Laplaces equation
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Laplaces equation

• Any irrotational, incompressible flow has a
  velocity potential and stream function(for 2D)
  that both satisfy Laplaces equation.
• Conversely, any solution of Laplaces equation
  represents the velocity potential or stream
  function (2D) for an irrotational incompressible
  flow
• Note that Laplaces equation is linear , the sum
  of any particular solution of a linear
  differential equation is also a solution of the
  equation, so a complicated flow pattern for an
  irrotational incompressible flow can be
  synthesize by adding together a number of
  elemental flows that are also irrotational and
  incompressible.

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Boundary Conditions

• Different flows for the different bodies are all
  governed by the Laplace equation, but the
  boundary conditions are different.
• Infinity Boundary Conditions
• Wall Boundary Conditions
• For inviscid flows the velocity at the surface
  can be finite, but because the flow cannot
  penetrate the surface, the velocity vector must
  be tangent to the surface.

Express in velocity potential
Express in streamline function Where s is the
distance measured along the body surface
The body shape function is given as
The shape of the body surface can be expressed as
Express directly in u and v
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Some elementary flows
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Uniform Flow

• It is easy to show that the uniform flow is
  incompressible and irrotational since it
  satisfies
• Hence we can use velocity potential
• For x axils and y axils
• Integrating them w.r.t x and y respectively
• Compare them we get
• The actual value of the velocity potential is not
  important. So without loss of rigor, we get

For incompressible flow, we can express it in the
form of streamline function In a similar
way, we get
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Source Flow

• Let the velocity along each of the streamlines
  vary inversely with distance for O.
• The opposite case is the sink flow which is
  simply a negative source flow
• It is easy to show that
• At every pint except the origin where it becomes
  infinite so the origin is a singular point.
• And it is irrotational every where since
• Hence

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Source Flow

• The total mass flow across the surface of the
  cylinder is
• The rate of volume is
• Define the volume flow rate per unit length along
  the cylinder as source strength
• For which a positive one represents the source
  while the negative one represents the sink.

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Source Flow

• Express the flow in velocity potential
• Integrating them w.r.t. r and ? respectively
• Compare and get ride of the constant
• For the stream function
• Integrating them w.r.t. r and ? respectively
• Compare and get ride of the constant

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Doublet Flow

• This is a special, degenerate case of a
  source-sink pair that leads to a singularity
  called a doublet.
• At any point P, the stream function is
• Let L-gt0 while l? remains constant
• Denote the strength of the doublet is denoted by

The streamline function
In a similar way, we can get the velocity
potential
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Vortex Flow

• Streamlines are concentric circles about a given
  point. The velocity along any given circular
  streamline be constant, but let it vary from one
  streamline to another inversely with distance
  from the common center.
• (1)Vertex flow is physically possible
  incompressible flow, that is at every point
• (2)Vortex flow is irrotational , that is at every
  point except the origin.

The velocity potential can be get in the
following
The stream function can be get in a similar way
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Vortex Flow

• What is the value at r0?

Since And dS has the same direction Let r-gt0
,and the circulation will still remain But r-gt0
so ds-gt0
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Thank you!

• QA