2D invicsid incompressible flows

Question 1

What is a potential flow?

Answer 1

A potential flow is a flow which is incompressible and irrotational.

Question 2

What is a stream function?

Answer 2

A streamfunction can only be defined for a planar flow(2D).

The continuity eqn for a 2D flow imposes a relation between the gradients of the flow field.

u,x+v,y=0

We define a scalar function (psi) that satisfies this by construction.

u=psi,y , v=-psi,x

Thus the problem has simplified since we only need to find a scalar function which can give us the velocity field.

Lines of constant psi are also streamlines.

Question 3

What is a velocity potential?

Answer 3

The velocity potential (phi) is a scalar function which by construction satisfies the fact that the flow is irrotational.

V=grad(phi)

Plugging this into the continuity eqn for an incompressible flow gives:

grad dot (V)=0 => laplacian(V)=0

So by solving the laplace eqn for phi the velocity of the flow field can be found.

Question 4

What is a complex potential?

Answer 4

The complex potential is a complex number defined as follows:

w=psi-iphi (velocity potential-istreamfunction)

Question 5

What is Bernoulli’s equation for potential flows?

Answer 5

Reducing the NS eqns for an incompressible flow and using the fact that flow is irrotational, and negelcting the gravity term gives the following relation:

1/2rhou^2+p=constant

This relation is applicable for the whole potential flow field.

Question 6

What are point vortex?

Answer 6

A point vortex is considered to be a string of rotating particles surrounded by fluid moving irrotationally under its influence.

Or a more general definition is that a vortex is a flow system in which a finite area, in a normal section plane contains vorticity.

Question 7

What is the induced velocity of point vortices?

Answer 7

u(x)=gamma/(2pir)

where x is the coordinate of the point of interest.
r= is the perpendicular distance.
Gamma is the circulation strength

Question 8

What are stagnation points?

Answer 8

Stagnation points are any point in a flow field where the velocity is zero.

Question 9

What is the Milne-Thomson circle theorem?

Answer 9

Consider a flow field represented by the complex potential w(z). If a circle with radius abs(z)=a is placed into this flow field, the new complex potential g(z) of the flow field equals:

g(z)=w(z)+w(a^2/conj(z))

Question 10

What is the difference between real and potential flow surface pressure distributions?

Answer 10

Two main things are different between a potential flow and a real flow.
The potential flow does not consider:
- A boundary layer
- Wake/ flow seperation

Question 11

How is the flow around a cylinder, with lift, constructed using potential flow?

Answer 11

Source-sink dipole + free stream + free vortex.

The free vortex provides circulation.

Question 12

How is the flow around a cylinder, without lift, constructed using potential flow?

Answer 12

Source-sink dipole + free stream

Question 13

What is Kutta-Joukowski lift?

Answer 13

L=-gammarhoU

Question 14

What is the Kutta condition?

Answer 14

The Kutta conditon imposes that the trailing edge has to be a stagnation point.

This condition is used to select an appropriate value of the circulation to insure that the TE is a stagnation point.

Basically a way to connect the value of the mathmatical parameter to a phenomen seen in the real world.

Question 15

What is D’Alemberts paradox?

Answer 15

D’Alembert proved that the drag for any body moving in a potential flow is zero.

This is in contradiction to what can be seen in reality.

Question 16

What is the Magnus Effect?

Answer 16

The Magnus effect is a phenomena observered when a rotating body moves through a fluid. An additional forces acts on the object due to the rotational veloctiy. This can be explained by Kutta-Jourkowski lift,
L=-rhoU_infGamma. The rotation introduces circulation.

Question 17

Why do we use conformal maping?

Answer 17

A conformal maping is an analytical function that preserves the local angles adn whose derivative is non-zero everywhere.

It lets us define a complex flow around a cylinder and then transform it into what looks like an airfoil.

Question 18

What is thin airfoil theory?

Answer 18

asd

Question 19

What is the advantage of potential flow?

Answer 19

First of all the problem of finding the velocity field is reduced from finding a vector field to finding a scalar field.

Secondly the scalar function can be found by solving the laplace eqn with BC. This is a linear PDE which allows for superpositioning. Therefore we can find multiple simple solutions and combine them to obtain a more complex solution.

Question 20

What are the elemntary potential flows?

Answer 20

• Uniform flow
• Potential vortex
• Point source or sink
• Source/sink doublett

Question 21

What are the combination of elementary flows around a cylinder?

Answer 21

Source-sink dipole + free stream

Question 22

Given the complex potential how do we find the velocity field?

Answer 22

dw/dz=u-iv

Question 23

What is the unit of circulation gamma?

Answer 23

[gamma]=[m^2]/[T]

Question 24

What is the formula for calculating circulation?

Answer 24

gamma=int_A(omega dot normal)dA

Question 25

What are the thin airfoil assumptions?

Answer 25

-Airfoil thickness i small compared to chord.
-Camber-line shape deviates only slightly from chord line.
Corollary: Theory should be restricted to small alpha.

Question 26

What is a vortex line?

Answer 26

The vortex line is a line everywhere tangent to the local vorticity vector (similary to a stream line.)