Aerodynamics 01
-
Upload
aryengenharia1244 -
Category
Documents
-
view
46 -
download
5
description
Transcript of Aerodynamics 01
![Page 1: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/1.jpg)
Aerodynamics
Lecture 1:
Introduction - Equations of
Motion
G. Dimitriadis
![Page 2: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/2.jpg)
Definition
• Aerodynamics is the science that analyses the flow of air around solid bodies
• The basis of aerodynamics is fluid dynamics
• Aerodynamics only came of age after the first aircraft flight by the Wright brothers
• The primary driver of aerodynamics progress is aerospace and more particularly aeronautics
![Page 3: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/3.jpg)
Applications (1)
Basic phenomena:
Flow around a cylinder
Shock wave Flow around an airfoil
![Page 4: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/4.jpg)
Applications (2)
Trailing vortices High lift devices
Low speed aerodynamics
![Page 5: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/5.jpg)
Applications (3)
High speed aerodynamics
F14 shock wave causes
condensation
F14 shock wave visualized on
water’s surface
![Page 6: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/6.jpg)
Applications (4)
New concepts:
Blended wing body
Micro-air vehicles Forward-swept wings
![Page 7: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/7.jpg)
Applications (5)
Space: Rockets, spaceplanes, reentry,
Airship 1
![Page 8: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/8.jpg)
Applications (6)
Non-aerospace applications: cars,
buildings, birds, insects
![Page 9: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/9.jpg)
Categories of aerodynamics
• Aerodynamics is an all-encompassing term
• It is usually sub-divided according to the speed of the flow regime under investigation: – Subsonic aerodynamics: The flow is subsonic over
the entire body
– Transonic aerodynamics: The flow is sonic or supersonic over some parts of the body but subsonic over other parts
– Supersonic aerodynamics: The flow is supersonic over all of the body
– Hypersonic aerodynamics: The flow is faster than four times the speed of sound over all of the body
![Page 10: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/10.jpg)
Flow type applications
• Subsonic aerodynamics: – Low speed aircraft, high-speed aircraft flying at
low speeds, wind turbines, environmental flows etc
• Transonic aerodynamics: – Aircraft flying at nearly the speed of sound,
helicopter rotor blades, turbine engine blades etc
• Supersonic aerodynamics: – Aircraft flying at supersonic speeds, turbine engine
blades etc
• Hypersonic aerodynamics: – Atmospheric re-entry vehicles, experimental
hypersonic aircraft, bullets, ballistic missiles, space launch vehicles etc
![Page 11: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/11.jpg)
Content of this course (1)
• This course will address mostly
subsonic and supersonic aerodynamics
• Transonic aerodynamics is very difficult
and highly nonlinear
– Small perturbation linearized solutions exist
but their accuracy is debatable
• Hypersonic aerodynamics is beyond the
scope of this course
![Page 12: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/12.jpg)
Content of this course (2)
• Subsonic aerodynamics – Incompressible aerodynamics
• Ideal flow – 2D flow
– 3D flow
• Viscous flow – Viscous-inviscid matching
– Compressibility corrections
• Supersonic aerodynamics – 2D flow
– 3D flow
![Page 13: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/13.jpg)
Simplifications
• The different categories of aerodynamics exist because of the different amount of simplifications that can be applied to particular flows
• Air molecules always obey the same laws, irrespective of the size or speed of the object that is passing through them
• However, the way we analyze flows changes with flow regime because we apply simplifications
• Without simplifications very few useful results can be obtained
![Page 14: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/14.jpg)
Full Navier Stokes Equations
• The most complete model we have of the flow of air is the Navier Stokes equations
• These equations are nevertheless a model: they are not the physical truth
• They represent three conservation laws: mass, momentum and energy
• They are not the physical truth because they involve a number of statistical quantities such as viscosity and density
![Page 15: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/15.jpg)
Navier Stokes for
Aerodynamicists
t+
u( )x
+v( )y
+w( )z
= 0
u( )t
+u2( )x
+uv( )y
+uw( )z
= xx
x+
xy
y+
xz
z
v( )t
+uv( )x
+v 2( )y
+vw( )z
=xy
x+
yy
y+
yz
z
w( )t
+uw( )x
+vw( )y
+w2( )z
=xz
x+
yz
y+
zz
z
E( )t
+uE( )x
+vE( )y
+wE( )z
=q( )t
+uq( )x
+vq( )y
+wq( )z
+xu xx + v xy + w xz( ) +
yu xy + v yy + w yz( ) +
zu xz + v yz + w zz( )
![Page 16: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/16.jpg)
Nomenclature
• The lengths x, y, z are used to define position with respect to a global frame of reference, while time is defined by t.
• u, v, w are the local airspeeds. They are functions of position and time.
• p, , are the pressure, density and viscosity of the fluid and they are functions of position and time
• E is the total energy in the flow.
• q is the external heat flux
![Page 17: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/17.jpg)
The stress tensor • Consider a small fluid element.
• In a general flow, each face of the element experiences normal stresses and shear stresses
• The three normal and six shear stress components make up the stress tensor
![Page 18: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/18.jpg)
More nomenclature
• The components of the stress tensor:
• The total energy E is given by:
• where e is the internal energy of the flow
and depends on the temperature and
volume.
xx = p + 2μu
x, yy = p + 2μ
v
y, zz = p + 2μ
w
z
xy = yx = μv
x+
u
y
, yz = zy = μ
w
y+
v
z
, zx = xz = μ
u
z+
w
x
E = e +1
2u2 + v 2 + w2( )
![Page 19: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/19.jpg)
Gas properties
• Do not forget that gases are also
governed by the state equation:
• Where T is the temperature and R is
Blotzmann’s constant.
• For a calorically perfect gas: e=cvT,
where cv is the specific heat at constant
volume.
p = RT
![Page 20: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/20.jpg)
Comments on Navier-Stokes
equations • Notice that aerodynamicists always include the
mass and energy equations in the Navier-Stokes equations
• Notice also that compressibility is always allowed for, unless specifically ignored
• This is the most complete form of the airflow equations, although turbulence has not been explicitly defined
• Explicit definition of turbulence further complicates the equations by introducing new unknowns, the Reynolds stresses.
![Page 21: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/21.jpg)
Constant viscosity
• Under the assumption that the fluid has
constant viscosity, the momentum
equations can be written as
u( )t
+u2( )x
+uv( )y
+uw( )z
=p
x+ μ
2u
x 2+
2u
y 2+
2u
z2
v( )t
+uv( )x
+v 2( )y
+vw( )z
=p
y+ μ
2v
x 2+
2v
y 2+
2v
z2
w( )t
+uw( )x
+vw( )y
+w2( )z
=p
z+ μ
2w
x 2+
2w
y 2+
2w
z2
![Page 22: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/22.jpg)
Compact expressions
• There are several compact expressions
for the Navier-Stokes equations:
DuiDt
=p
xi+ μ
2uixi2Tensor notation:
Vector notation: ut
+1
2u u + u( ) u
= p + μ 2u
Matrix notation: ut
+TuuT
= p + μ 2Tu
![Page 23: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/23.jpg)
Non-dimensional form
• The momentum equations can also be
written in non-dimensional form as
• where
u( )t
+u2( )x
+uv( )y
+uw( )z
=p
x+1Re
2u
x 2+
2u
y 2+
2u
z2
v( )t
+uv( )x
+v 2( )y
+vw( )z
=p
y+1
Re
2v
x 2+
2v
y 2+
2v
z2
w( )t
+uw( )x
+vw( )y
+w2( )z
=p
z+1Re
2w
x 2+
2w
y 2+
2w
z2
= , u =u
U, v =
v
U, w =
w
U, x =
x
L, y =
y
L, z =
z
L, t =
tL
U, p =
p
U 2
![Page 24: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/24.jpg)
Solvability of the Navier-
Stokes equations
• There exist no solutions of the complete
Navier-Stokes equations
• The equations are:
– Unsteady
– Nonlinear
– Viscous
– Compressible
• The major problem is the nonlinearity
![Page 25: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/25.jpg)
Flow unsteadiness
• Flow unsteadiness in the real world arises from two possible phenomena:
– The solid body accelerates
– There are areas of separated flows
• This course will only consider solid bodies that do not accelerate
• Attached flows will generally be considered
• Therefore, unsteady terms will be neglected
– All time derivatives in the Navier-Stokes equations are equal to zero
![Page 26: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/26.jpg)
Unsteadiness Examples
Flow past a circular cylinder
visualized in a water tunnel. The
airspeed is accelerating. The flow is
always separated and unsteady. It
becomes steadier at high airspeeds
Flow past an airfoil visualized in a
water tunnel. The angle of attack is
increasing. The flow attached and
steady at low angles of attack and
vice versa.
![Page 27: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/27.jpg)
Viscosity
• Viscosity is a property of fluids
• All fluids are viscous to different
degrees
• However, there are some aerodynamic
flow cases where viscosity can be
modeled in a simplified manner
• In those cases, all viscous terms are
neglected.
![Page 28: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/28.jpg)
Cases where viscosity is
important
Wake
Shock wave
Boundary layer
![Page 29: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/29.jpg)
Euler equations
• Neglecting the viscous terms, we obtain
the unsteady Euler equations:
t+
u( )x
+v( )y
+w( )z
= 0
u( )t
+u2( )x
+uv( )y
+uw( )z
=p
x
v( )t
+uv( )x
+v 2( )y
+vw( )z
=p
y
w( )t
+uw( )x
+vw( )y
+w2( )z
=p
z
E( )t
+uE( )x
+vE( )y
+wE( )z
=up( )x
vp( )y
wp( )z
![Page 30: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/30.jpg)
Classic form of the Euler
equations
• The Euler equations are usually written
in the form:
• where
Ut+Fx+Gy+Hz= 0
U =
u
v
w
E
, F =
u
p + u2
uv
uw
u E + p( )
, G =
v
uv
p + v 2
uw
v E + p( )
, H =
w
uw
vw
p + w2
w E + p( )
![Page 31: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/31.jpg)
Steady Euler Equations
• Neglecting unsteady terms we obtain
the steady Euler equations:
u( )
x+
v( )
y+
w( )
z= 0
u2( )x
+uv( )
y+
uw( )
z=
p
x
uv( )
x+
v 2( )y
+vw( )
z=
p
y
uw( )
x+
vw( )
y+
w2( )z
=p
z
![Page 32: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/32.jpg)
Example 1
• Notice that in the steady Euler
equations, the energy equation has
disappeared.
• Show that neglecting unsteady and
viscous terms turns the energy equation
into an identity if the air’s internal
energy is constant in space.
![Page 33: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/33.jpg)
Compressibility
• The compressibility of most liquids is negligible for the forces encountered in engineering applications.
• Many fluid dynamicists always write the Navier-Stokes equations in incompressible form.
• This cannot be done for gases, as they are very compressible.
• However, for low enough airspeeds, the compressibility of gases also becomes negligible.
• In this case, compressibility can be ignored.
![Page 34: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/34.jpg)
Compressibility examples
Transonic flow
over airfoil
Supersonic
flow over
sharp wedge
Hypersonic
flow over
blunt wedge
![Page 35: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/35.jpg)
Incompressible, steady Euler
Equations • The incompressible, steady Euler
equations become
u
x+
v
y+
w
z= 0
uu
x+ v
u
y+ w
u
z=
1 p
x
uv
x+ v
v
y+ w
v
z=
1 p
y
uw
x+ v
w
y+ w
w
z=
1 p
z
![Page 36: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/36.jpg)
Comment on the Euler
equations • The Euler equations are much more
solvable than the Navier-Stokes equations
• They are most commonly solved using numerical methods, such as finite differences
• There are very few analytical solutions of the Euler equations and they are not particularly useful
• In order to obtain analytical solutions, the equations must be simplified even further
![Page 37: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/37.jpg)
Flow rotationality
• Rotational flow:
• Irrotational flow:
Fluid particle,
time t1
Fluid rotation
Fluid particle,
time t2
Fluid particle,
time t3
Fluid particle,
time t1
No fluid rotation
Fluid particle,
time t2
Fluid particle,
time t3
![Page 38: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/38.jpg)
Irrotationality (1)
• Some flows can be idealized as
irrotational
• In general, attached, incompressible,
inviscid flows are also irrotational
• Irrotationality requires that the curl of
the local velocity vector vanishes:
• where u=ui+vj+wk and
u = 0
=xi +
yj+
zk
![Page 39: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/39.jpg)
Irrotationality (2)
• This leads to the simultaneous
equations:
• Integrating the momentum equations
using these conditions leads to the well-
known Bernoulli equation
w
y
v
z= 0,
w
x
u
z= 0,
v
x
u
y= 0
1
2u2 + v 2 + w2( ) + P = constant
![Page 40: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/40.jpg)
Example 2
• Integrate the incompressible, steady
momentum equations to obtain
Bernoulli’s equation for irrotational flow
• You can start with the 2D equations
![Page 41: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/41.jpg)
Velocity potential
• Irrotationality allows the definition of the
velocity potential, such that
• It can be seen that all three irrotationality
conditions are satisfied by this function
• Substituting these definitions in the mass
equation leads to
u = -x
, v = -y
, w = -z
2
x 2+
2
y 2+
2
z2= 0
![Page 42: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/42.jpg)
Laplace’s equation
• The irrotational form of the Euler equations is Laplace’s equation.
• This is an equation that has many analytical solutions.
• It is the basis of most subsonic, attached flow aerodynamic assumptions.
• The equation is linear, therefore its solutions can be superimposed
• The complete flow problem has been reduced to a single, linear partial differential equation with a single unknown, the velocity potential.
![Page 43: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/43.jpg)
Potential flow
• Incompressible, inviscid and irrotational flow is also called potential flow because it is fully described by the velocity potential.
• The first part of this course will look at potential flow solutions:
– First in two dimensions
– Then in three dimensions
• Potential flow solutions have provided us with the most useful and trustworthy aerodynamic results we have to date.
• Their limitations must be kept in mind at all times.
![Page 44: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/44.jpg)
Potential flow solutions
• We now have a basis for modelling the
flow over 2D or 3D bodies. All we need
to do is:
– Solve Laplace’s equation
– With two boundary conditions (2nd order
problem):
• Impermeability: Flow cannot enter or exit a
solid body
• Far field: The flow far from the body is
undisturbed.
![Page 45: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/45.jpg)
Boundary conditions (1)
qn
qt
Impermeability:
The normal flow
velocity component must be equal to
zero.
qn = n surface
= 0
n
n: unit vector normal to the surface
qn: normal flow velocity component
qt: tangential flow velocity component
Neumann boundary condition
![Page 46: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/46.jpg)
Boundary conditions (1bis)
(x,y,z)
i(x,y,z)
An alternative form of
the impermeability
condition states that the potential inside
the body must be a
constant:
i(x,y,z)=constant
Dirichlet boundary condition
![Page 47: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/47.jpg)
Boundary conditions (2)
Far field: Flow far from
the body is undisturbed.
This usually is expressed as:
* 0, as r
r
r
r2=x2+y2+z2
![Page 48: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/48.jpg)
2D Potential Flow
• Two-dimensional flows don’t exist in reality but they are a useful simplification
• Two-dimensionality implies that the body being investigated: – Has an infinite span
– Does not vary geometrically with spanwise position
• As examples, consider an infinitely long circular cylinder or an infinitely long rectangular wing
![Page 49: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/49.jpg)
2D Potential equations
• Laplace’s equation in two dimensions is
simply
• While the irrotationality condition is
• We still need to find solutions to this
equation.
2
x 2+
2
y 2= 0
v
x
u
y= 0
![Page 50: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/50.jpg)
Streamlines
• A streamline is a curve that is
instantaneously tangent to the velocity
vector of the flow
u
xs
x is the position vector of a point on a streamline, u is the velocity vector at that point and s is the distance on the streamline of the point from the origin
![Page 51: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/51.jpg)
Streamline definition
• A streamline is defined mathematically
as:
• Where u has components u, v, w and x has components x, y, z.
• It can be easily seen that the definition
leads to:
dxds
= u
dx
ds= u,
dy
ds= v,
dz
ds= w, and therefore
dx
u=
dy
v=
dz
w
![Page 52: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/52.jpg)
The stream function
• The stream function is defined at right
angles to the flow plane, i.e.
• Where u=[u v 0] and =[0 0 ]. It can
be seen that
• The stream function is only defined for
2D or axisymmetric flows.
u =
u =y
, v = -x
![Page 53: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/53.jpg)
Properties of the stream
function
• The stream function automatically
satisfies the continuity equation.
• The stream is constant on a flow
streamline
• But, on a streamline
• Therefore
u
x+
v
y=
x y
+
y x
=
2
x y
2
x y= 0
d =xdx +
ydy = vdx + udy
dx
u=dy
v
d = udy + udy = 0
![Page 54: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/54.jpg)
Elementary solutions
• There are several elementary solutions
of Laplace’s equation:
– The free stream: rectilinear motion of the
airflow
– The source: a singularity that creates a
radial velocity field around it
– The sink: the opposite of a source
– The doublet: a combined source and sink
– The vortex: a singularity that creates a
circular velocity field around it.
![Page 55: Aerodynamics 01](https://reader036.fdocuments.in/reader036/viewer/2022081721/563dbbaa550346aa9aaf329e/html5/thumbnails/55.jpg)
Historical perspective
• 1738: Daniel Bernoulli developed Bernoulli’s principle, which leads to Bernoulli’s equation.
• 1740: Jean le Rond d'Alembert studied inviscid, incompressible flow and formulated his paradox.
• 1755: Leonhard Euler derived the Euler equations.
• 1743: Alexis Clairaut first suggested the idea of a scalar potential.
• 1783: Pierre-Simon Laplace generalized the idea of the scalar potential and showed that all potential functions satisfy the same equation: Laplace’s equation.
• 1822: Louis Marie Henri Navier first derived the Navier-Stokes equations from a molecular standpoint.
• 1828: Augustin Louis Cauchy also derived the Navier-Stokes equations
• 1829: Siméon Denis Poisson also derived the Navier-Stokes equations
• 1843: Adhémar Jean Claude Barré de Saint-Venant derived the Navier-Stokes equations for both laminar and turbulent flow. He also was the first to realize the importance of the coefficient of viscosity.
• 1845: George Gabriel Stokes published one more derivation of the Navier-Stokes equations.