Fluid Kinematics

Question 1

Define fluid particles

Answer 1

Points within the continuum that are moved by particle motion

Question 2

Define the continuum model of fluids

Answer 2

A model where the fluid is defined as some continuous medium made up of fluid parcels.

Question 3

Define fluid parcels

Answer 3

An infinitesimal portion of the continuum which contains an average of the properties of the fluid inside it.

Question 4

Define the Eularian picture

Answer 4

Where fluid particles are labelled by their current position x.

Question 5

Define the Lagrangian picture

Answer 5

Where fluid particles are labelled by their initial position a.

Question 6

Define particle path (or trajectory). Which picture does this represent?

Answer 6

The path of a particle x(a,t) parametetrised by its initial position a and time t. Eulerian picture.

Question 7

Define the time derivatives for the Eularian and Lagrangian pictures

Answer 7

(partial/partial t) holding x and a constant respectively. Denoted by (partial/partial t) and D/Dt respectively.

Question 8

Define the material (or convective) derivative

Answer 8

The Lagrangian derivative of an Eulerian function, where we consider the rate of change of the function moving with the fluid.

Question 9

Give (and explain) the two criteria for a particle path

Answer 9

Continuous and invertible, such that we can determine the initial position of a passing through x at time t.

Question 9

Define particle velocity

Answer 9

The Lagrangian derivative of particle position x.

Question 10

Define material curve

Answer 10

A line which threads through a subset of fluid elements x at time t and moves with them.

Question 11

Define a streamline

Answer 11

The integral curve of Eulerian velocity u(x,t) at a fixed time. Streamlines satisfy the differential equation:
dx/ds = u(x(s),t)

Question 12

Given a velocity field, compare the methods for finding streamlines and particle paths.

Answer 12

Streamlines - integrate with respect to some dummy variable s, defining initial particle positions as x_0, y_0, z_0.

Particle paths - integrate with respect to time and , defining initial particle positions as a,b,c.

Question 13

State the case where streamlines and particle paths coincide

Answer 13

Where the flow is steady (not dependent on time).

Question 14

Define a stream function (and the two conditions under which it can be defined)

Answer 14

A scalar function ψ(x,y,t) whose grad describes u when it is two-dimensional and divergence free.

Question 15

Give the volume flow rate (per unit height) between to fluid particles.

Answer 15

The difference in the stream function evaluated at the location of the fluid particles i.e. psi(x_2)-psi(x_1).

Question 16

Is the volume flow rate independent of the curve connecting the fluid particles? Why?

Answer 16

Yes: the flow is conservative and so independent of the curve.

Question 17

Define a streakline

Answer 17

A line formed from all the fluid elements passing through a point (a,b) between some t=0 and t>0.

Question 18

How do we determine streaklines

Answer 18

Find the particles paths, introducing parameter epsilon (between 0,t) such that a fluid particle passes through (a,b) at t=epsilon. Substitute the integration parameters this gives, into the particle paths.

Question 19

Describe how the Jacobian relates Lagrangian and Eulerian pictures

Answer 19

The Jacobian defines how a line segment connecting two adjacent fluid particles at t=0 evolve with time. In three dimensions, it describes how the volume of a fluid parcel evolves with time.

Question 20

Give (and explain) a constraint on the Jacobian

Answer 20

J cannot =0, since we require the map to be invertible.

Question 21

Define an incompressible flow

Answer 21

A flow which preserves the volume of every material subregion in the fluid domain D_t.

Question 22

Define a material scalar (or Lagrange invariant)

Answer 22

A function which has a zero material (or convective) derivative Df/Dt = 0

Question 23

Given a function f(x(a,t), t), give the material derivative and describe.

Answer 23

Df/Dt = ∂f/∂t + u · ∇f

LHS: Lagrangian
RHS: Eulerian
Found by chain rule

Question 24

Give the expression for the Jacobian relating E and L pictures

Answer 24

dx dy dz = J da db dc

Question 25

What are the Jacobian and divergence of an incompressible flow?

Answer 25

J = 1 and ∇ · u = 0.

Question 26

What is the relative change in jacobian J moving with the fluid?

Answer 26

1/J DJ/Dt = ∇ · u

Question 27

Define local analysis

Answer 27

Considering two neighboring points of the flow x + δx

Question 28

Define velocity gradient tensor

Answer 28

The tensor ∇u(x)_ij = = ∂u_i/∂x_j which defines the local structure of the velocity field

Question 29

What three flow components do we find in local analysis?

Answer 29

Body translation across Dt given by u(x,t)

Stretching/compression of Dt from symm part ∇u(x).

Rotation of Dt from anti-symm part of ∇u(x).

Question 30

Define vorticity

Answer 30

The pseudo-vector ω = ∇ × u

Question 31

Define circulation

Answer 31

The integral of u · dl around a closed curve γ

Question 32

State Stoke’s theorem

Answer 32

The integral of a function around closed loop γ is equivalent to the curl of the function through the surface enclosed by the loop

Question 33

Define a vortex tube

Answer 33

A bundle of vortex lines: the streamline analogue for vorticity; an integral curve of ω(x, t) at fixed time.

Question 34

Give Helmholtz’ rule for vortex tubes

Answer 34

If γ1 and γ2 are two closed curves encircling a vortex tube, their circulation is equal.

Question 35

Briefly describe ‘flow reconstruction’

Answer 35

Recovering an incomp velocity from a vorticity, knowing the velocity is not unique.

I.e. u and u’ := u + ∇φ have the same vorticity field.

Question 36

What are the boundary conditions for flow reconstruction?

Answer 36

∆φ = 0
n · (∇φ + u_v) = 0

Question 37

Define the method of images for flow reconstruction

Answer 37

Adding addition vortices which ensures the boundary condition, since BS is linear in w.

Question 38

What is the trick for differentiating over a changing volume?

Answer 38

Express integral in Lagrangian coords (over fixed volume D0 at t=0), diff and then re-express in Eulerian.

Question 39

What does circulation define for a vortex tube?

Answer 39

The ‘strength’ of the vortex tube; invariant along its length.

Question 40

State the divergence theorem

Answer 40

The volume integral of the divergence of a function is equivalent to the function flowing through a surface.