Chapter 6: Differential Analysis of Fluid Flow

Question 1

What is volumetric dilation rate?

Answer 1

The rate of change of the volume per unit volume is called the volumetric dilation rate.
For an incompressible fluid the volumeteric dilation rate is zero.

Question 2

What is vorticity?

Answer 2

The vorticity vector is defined as twice the rotation vector of a fluid element.

Question 3

What is a irrotational flow, and how is it determined weather a flow is irrotational or not?

Answer 3

if grad cross V =0

Then the rotation and vorticity are zero.

Question 4

What is the Eulerian continuity equation in differential form?

Answer 4

nabla(rho)/nabla(t)+grad dot (rho*V))=0
if rho=constant
==> grad dot V =0

Question 5

What is the Lagrangian continuity equation in differential form?

Answer 5

D(rho)/D(t)+rho*grad dot V=0
if rho=constant
==> grad dot V =0

Question 6

What is a stream function?

Answer 6

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 7

What are Euler’s equations of motion?

Answer 7

The Euler equations are the inviscid limit of the Navier stokes equation. (Re&raquo_space;1 => ignore viscosity)
Basically kill the viscous term.

rho(nabla(V)/nabla(t)+V dot grad(V))= grad(p)+rhoG+mu*laplacian(V)

Question 8

What is an ideal fluid?

Answer 8

An ideal fluid is a fluid which is inviscid and incompressible.

Question 9

What is “Bernoulli’s eqn”?

Answer 9

Reducing the NS eqns for an incompressible and steady 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 10

What is the velocity potential?

Answer 10

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 11

What is potential flow? And what are the conditions?

Answer 11

Invisicid, incompressible, irrotational flow fields are governed by the Laplace equation and are called potential flows.

grad dot V=0
grad cross V=0

Question 12

what are equipotential lines?

Answer 12

Lines of constant velocity potential, they are perpendicular to stream lines.

Question 13

What is a flow net?

Answer 13

The combination of all lines of constant stream function and constant velocity potential in a potential flow.
The velocity is inversely proportional to the streamline spacing.

Question 14

What is a uniform flow?

Answer 14

The simplest elementary potential flow, where all the streamlines are straight and parallel.

Question 15

What are sources and sinks?

Answer 15

asd

Question 16

What is a point vortex?

Answer 16

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 17

What is the advantage of potential flow?

Answer 17

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 18

What is a potential vortex?

Answer 18

A potential vortex is a elementary flow in which the streamlines are concentric circles. Swap the streamfunction and the veloctiy potential from the source/sink

Question 19

What is circulation and how is it calculated?

Answer 19

Circulation is defined as the line integral of the tangential component of the velocity field taken around a closed curve in the flow field.

gamma=int_A(omega dot normal)dA=int_CW(V dot ds)

Question 20

What is a doublet?

Answer 20

A doublet is made by combining a source and sink term in a special way. The so-called doublet is formed by letting the source and the sink term approach one another while there strenghts go to infinity.

Question 21

What elementary potential flows is a half body made up of?

Answer 21

• Uniform flow

- source

Question 22

What elementary potential flows is a rankine oval made up of?

Answer 22

• Uniform flow
• Source
• Sink

Question 23

What are the Navier-Stokes equations?

Answer 23

For a incompressible fluid with constant viscosity, the N-S equation can be written as:

rho(nabla(V)/nabla(t)+V dot grad(V))=-grad(p)+rhoG+mu*laplacian(V)

Question 24

What are Stokes equations for creeping flow?

Answer 24

Creeping flow is when Re«1 we can then ignore the inertial term in the Navier-Stokes equation.
rhonabla(V)/nabla(t)=-grad(p)+rhoG+mulaplacian(V)

Question 25

What are the different terms of the Navier Stokes?

Answer 25

• The local derivative of the velocity ( Local acceleration)
• The inertial term
• The pressure term (Pressure forces/unit volume)
• Body force term (Body forces/unit volume)
• The viscous term ( Viscous forces/ unit volume)

Question 26

What causes deformation of a fluid element?

Answer 26

Deformation is caused by different velocities at the corners of the fluid element. => Deformation is linked to velocity gradients.

Question 27

What elementary potential flows make up a cylinder?

Answer 27

• Uniform flow

- A doublet