AerodynamicsB, AE2115 I, Chapter II

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Aerodynamics Some Fundamental Principles and
Equations

• Dr.ir. M.I. Gerritsma Dr.ir. B.W. van
  Oudheusden
• Delft University of Technology
• Department of Aerospace Engineering
• Section Aerodynamics

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Continuity Equation, Differential Form

• We already found that

Remember that V and V are fixed in space,
therefore
Apply divergence theorem
So the continuity equation becomes
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Continuity Equation, Differential Form

• This equation holds for any arbitrary volume V

If this holds for every imaginable volume V, then
the integrand must be zero, so
Validity 3D, time-dependent, viscous-inviscid,
(in)compressible, allows for discontinuous
solutions (!?)
Stationary flows
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Continuity Equation, Differential Form

• Conservative formulation
• Non-conservative formulation

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Physical Meaning of the Divergence of Velocity

• Moving control volume
• Constant mass
• Shape changes

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Physical Meaning of the Divergence of Velocity

• Assume that the volume VV is so small that
  is constant in V, then

is physically the time rate of change of
the volume of a moving fluid element, per unit
volume
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Momentum Equation

• Newtons second Law

Physical Principle Force time rate of change
of momentum.
Note and are vectors, so the above
identity has to hold for each component.

• 1. Body Forces
• Gravity
• Electromagnetic forces
• Coriolis forces

Which types of forces act on a fluid?

• 2. Surface forces
• normal stresses (perpendicular to plane)
• tangential forces (along the plane)

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Body Forces
Fixed control volume V
Assume is the body force per unit mass
exerted on fluid inside V
The body force acting on a small volume dV
Total force on volume
Example, gravity
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Pressure contribution

• Consider a small elementary surface
• On this surface the pressure p acts, therefore
• in the opposite direction of the outward unit
  normal
• Therefore the total pressure force on V equals

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Viscous forces

• The viscous forces are simply denoted by
  . In Aerodynamics-D AE3-130 explicit formulae for
  will be given.

We want to find an expression for

• Consists of
• Body forces
• Pressure forces
• Viscous forces

Therefore
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• We want to find an expression for

Question How do we evaluate the rate of change
of momentum in the volume V?

• Answer 2 contributions G and H
• G Net flow of momentum out of control volume
  across surface S.
• H Time rate of change of momentum due to
  unsteady fluctuations of flow properties inside V

G Recall that the mass flow across the elemental
area ds is given by Hence, the flow of momentum
per second across ds is Integrating over the
whole surface yields Note that when Ggt0 momentum
is flowing out of the control volume.
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• H Time rate of change of momentum due to
  unstaedy phenomena in V.

Question What is the total amount of momentum in
V at time t.
Answer Within a small volume element dV with
velocity and mass
Therefore the time rate of change of momentum in
the volume V is given by
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Integral form of the momentum equation

• Newton

Therefore
Body forces
Pressure forces
Viscous forces
Momentum transport across surface
Rate of change of momentum in V
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Some Remarks concerning the Momentum Equation

• The momentum equation is a vector equation. In
  fact we have three separate equations in the
  x-direction, y-direction and z-direction.
• The integral equation is valid for unsteady,
  viscous, (in)compressible, three dimensional
  flows.
• will be treated separately
  (Aerodynamics-D AE3-130)
• The integral formulation only provides
  information over integrated quantities, so no
  local values of density, pressure, temperature,
  etc, can be extracted.

Conversion to differential form

• Recipe
• Convert all surface integrals to volume
  integrals using the integral relations given in
  section 2.2.11.
• Interchange and which is allowed
  since V is fixed in space.
• Collect all integrands underneath one integral
  and require that the integrand must vanish, since
  the integral holds for arbitrary volumes.

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• The momentum equation contains the surface
  integral
• Apply the gradient theorem

In order to rewrite the convective surface
integral, we look at the x-component only
Take the x-component of the momentum equation
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• Apply divergence theorem to remaining surface
  integral (Convective term)

Writing the viscous force as
.
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Additional Remarks

• These equations are called the Navier-Stokes
  equations. They consist of a system of non-linear
  partial differential equations (PDEs).
• The PDE connect the variables and the change in
  the variables locally.
• If the flow is stationary,
  inviscid and there are no body
  forces we obtain the so-called Stationary Euler
  equations

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Energy Equation

• We have introduced now 5 unknowns,
  and we have only 4 equations, namely
  the three components of the momentum equation and
  the continuity equation.
• We either have to add another equation, or remove
  one of the unknowns.
• If the flow is incompressible, then the density
  is constant, so we can remove one unknown which
  closes the system
• For compressible flows, one cannot remove any
  unknowns, therefore one has to add another
  equation. This will be the energy equation.
• Physical Principle Energy cannot be created, nor
  destroyed.

First Law of Thermodynamics
Increment in internal energy
Work done on the system
Added heat
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Energy Equation.

• Apply to a control
  volume

• Call
• B1 the amount of heat added to the volume from
  the environment
• B2 the amount of work exerted on the volume
• B3 rate of change of energy as it flows
  through the control volume

B1 B2 B3
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Determination of added heat B1

• A small volume element dV receives from its
  environment an amount of heat per unit time and
  unit mass

So the total amount of heat added to dV per unit
time is
Sources are radiation, emission, combustion,
condensation, etc.

• Through the surface V heat is added to the
  volume by conduction and mass diffusion. These
  are so-called viscous processes, denoted by

So the total heat added to the system, B1, will
be equal to
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Rate of work done on the volume, B2

• Work

• On surface element a pressure force
  acts, so the amount of work per unit
  time due to the pressure will be
  So the
  total pressure contribution to the rate of work
  over the entire volume is given by
• On dV a body force acts, therefore
  the rate of work due to the body force on dV is
  The total rate of work on V is
  given by

• Viscous rate of work

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Determination of the rate of change of energy, B3

• Energy in V consists of internal energy, e per
  unit mass and kinetic energy per
  unit mass, so the total energy per unit mass is
  given by

At time t the amount of energy in dV is given by
So the instationary rate of change of energy

• The rate of mass flowing through the surface
  carries a net amount of energy given by

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Energy Equation
Thermal conduction, mass diffusion
Work done by body forces
Work done by pressure forces
Volumetric heating
Work done by viscous forces
Net rate of energy outflow
Transient fluctuations

• This is the integral form of conservation of
  energy, valid for (in)compressible,
  viscous/inviscid fluids, stationary and time
  dependent problems.
• Introduction of the enthalpie
• Introduction of the total enthalpie

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Energy Equation in terms of H

• Total energy
• total enthalpy

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Differential Formulation of Energy Equation

• Divergence Theorem

• Results

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Special Cases Energy Equation

• If the flow is
• stationary
• inviscid
• adiabatic
• no body forces

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Conclusion The change in H in the direction of
the flow is zero (remember directional
derivative), therefore H does not change in the
direction of the flow for a stationary,
inviscid, adiabatic flow without body forces the
total enthalpy is constant along streamlines!
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Important Concepts in Aerodynamics

• Substantial, Convective, Local Derivative,
• Pathlines, Streamlines and Streaklines,
• Angular Velocity, Vorticity and Strain
• Circulation
• Stream Function
• Velocity Potential
• Relationship between Stream Function and
  Velocity Potential

section 2.9 2.10 section 2.11 section
2.12 section 2.13 section 2.14 section
2.15 section 2.16
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Substantial Derivative

• When we consider a scalar quantity q we can
  either write this quantity with respect to fixed
  points in space, X, or with repect with a
  coordinate system that is moving with the flow,
  x.
• Obviously, x, will be a function of time t, so
  x(t).
• In the first case, we have the following time
  derivative
• In the second case we have

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Substantial Derivative

• So we have the new time derivative
• This can be written more succinctly as
• The new time derivative is called the substantial
  or material time derivative

Convective Derivative
Substantial Derivative
Local Derivative
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Fundamental Equations in terms of the Substantial
Derivative

• Conservation of Mass
• Conservation of Momentum
• Conservation of Energy

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Pathlines Streamlines

• A pathline is the curve in space traced out by a
  particular particle in time
• A streamline is a curve in space which is
  locally tangent to the velocity vector

For steady flows pathlines and streamline
coincide, for unsteady flows these lines are
generally different!
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Direction of a Streamline

• Since the curve is locally tangent to the
  velocity vector we have

Writing
So the equations for a streamline are
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Vorticity Strain

• Different aspects of the movement of a particle
• The trajectory (already discussed)
• Orientation (translation and rotation)
• Deformation.

• Rotation and deformation are determined by the
  velocity field.
• Consider a 2 dimensional square

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• Convention counter clockwise rotations are
  positive
• The angle is given by
• and the rotation of AB by

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• Definition of angular velocity
• The angular velocity of a small line element is
  given by
• Therefore the angular velocity of AB is
• And the angular velocity of AC is
• The angular velocity of the fluid element ABCD
  is given by the average angular velocity

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• For general 3 dimensional flows the angular
  velocity is a vector given by

Vorticity is defined as
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Irrotational Flow

• If at every point in the flow,
  the flow is called rotational.
• If at every point in the flow,
  the flow is called irrotational.

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Irrotational Flow

• Irrotational flow is of practical use
• - Subsonic flow around airfoils
• - Supersonic flow around slender bodies (no
  curved shock waves)
• - Subsonic/Supersonic flow in nozzles, jets,
  wind tunnels, etc.

Note, that in many flows the flow outside the
boundary layer may be assumed to be irrotational,
but inside the boundary layer the flow is
rotational, since
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Strain / Deformation

• Strain is the change in k, where positive strain
  corresponds with decrasing strain

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Time rate of strain

• Definition Time rate of strain
• Deformation rates!
• Analogously

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Circulation

• Circulation is strongly connected to the notion
  of lift
• Frederick Lanchaster (1878-1946)
• Wilhelm Kutta (1867-1944)
• Nicolai Joukowski (1847-1921)
• Definition

• Remarks
• Circulation depends on the contour C chosen
• It is customary to use a positive circulation in
  the clockwise direction!!

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Connection between Circulation and Vorticity

• The circulation of a flow around a closed curve C
  is by definition
• Now using Stokes Theorem we get

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Circulation and Vorticity

• Stokes Theorem applied to circulation yields the
  following result
• The circulation around a contour C can also be
  obtained by integrating the vorticity over the
  surface enclosed by this contour
• Remarks
• If everywhere (i.e. an
  irrotational flow) the circulation is zero for
  every contour C.
• Conversely, if for every imaginable
  contour C, then the vorticity must be zero
  everywhere and therefore the flow will be
  irrotational.
• Suppose that the contour shrinks such that it
  encloses a very small surface over which the
  vorticity is more or less constant, then the
  circulation is given by

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Stream function

• Consider a two dimensional flow
• The definition of a streamline is given by

This equation constitutes a differential equation
with solution in which denotes the
integration constant. Each different constant
defines a different streamline. Suppose that
streamline A corresponds to the constant
and streamline B corresponds to
is called a stream function if
denotes the mass flow between the
streamlines A and B.
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Stream function

• Mass flow

Remark Differentiating the stream function in
the direction perpendicular to the streamline
produces the mass flux

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Stream function

• If the streamfunction is given, the above
  formulae give a way to determine the momentum per
  unit volume
• Note that differentiation of the stream function
  in the y-direction produces the mass flux in the
  x-direction and vice versa.
• For an incompressible flow we can redefine the
  streamfunction by setting by dividing by the
  constant density, in which case the above
  formulae produce the velocity components.
• In polar coordinates the above equation become
• The stream function provides a very usefull
  tool given the stream function we can
  immediately determine the streamlines by setting
  yconst and we can determine the velocity field
• Note that the velocity components obtained from
  te stream function identically satisfy the
  stationary continuity equation.

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Velocity Potential

• The introduction of a velocity potential only
  makes sense for irrotational flows.
• Irrotational
• Vector identity
• Therefore set

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Potential Function

• Remarks
• If is given, the velocity
  components follow immediately by differentiation,
  so the potential function replace 3 unknowns u, v
  and w, by 1 unknown
• The velocity component in a particular direction
  is obtained by differentiating the potential
  function in that particular direction.
• By introducing the potential function we
  identically satisfy the condition for
  irrotational flow, i.e.
• By replacing the velocity components by the
  gradient of the potential function in the
  continuity equation and momentum equation we
  obtain the so-called potential equation.
  Irrotational flows satisfying this equation are
  called potential flows.

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Relation between the stream function and the
potential function

• Consider a two dimensional, incompressible flow
• Continuity equation
• Streamlines
• Equipotential lines

We know that is orthogonal to
the equipotential lines and oriented in the
direction of the flow so are the streamlines,
therefore equipotential lines are orthogonal to
the streamlines. A mathematical proof of this
assertion can be found in section 2.16.
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Proof orthogonality of streamlines and
equipotential lines

• Note that is perpendicular to the
  streamlines and is perpendicular to the
  equipotential lines, in which
• so these gradients are perpendicular to
  eachother, and therefore the lines whose normals
  they are, are perpendicular as well.

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