Fluid Kinematics Flashcards
Define fluid particles
Points within the continuum that are moved by particle motion
Define the continuum model of fluids
A model where the fluid is defined as some continuous medium made up of fluid parcels.
Define fluid parcels
An infinitesimal portion of the continuum which contains an average of the properties of the fluid inside it.
Define the Eularian picture
Where fluid particles are labelled by their current position x.
Define the Lagrangian picture
Where fluid particles are labelled by their initial position a.
Define particle path (or trajectory). Which picture does this represent?
The path of a particle x(a,t) parametetrised by its initial position a and time t. Eulerian picture.
Define the time derivatives for the Eularian and Lagrangian pictures
(partial/partial t) holding x and a constant respectively. Denoted by (partial/partial t) and D/Dt respectively.
Define the material (or convective) derivative
The Lagrangian derivative of an Eulerian function, where we consider the rate of change of the function moving with the fluid.
Give (and explain) the two criteria for a particle path
Continuous and invertible, such that we can determine the initial position of a passing through x at time t.
Define particle velocity
The Lagrangian derivative of particle position x.
Define material curve
A line which threads through a subset of fluid elements x at time t and moves with them.
Define a streamline
The integral curve of Eulerian velocity u(x,t) at a fixed time. Streamlines satisfy the differential equation:
dx/ds = u(x(s),t)
Given a velocity field, compare the methods for finding streamlines and particle paths.
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.
State the case where streamlines and particle paths coincide
Where the flow is steady (not dependent on time).
Define a stream function (and the two conditions under which it can be defined)
A scalar function ψ(x,y,t) whose grad describes u when it is two-dimensional and divergence free.
Give the volume flow rate (per unit height) between to fluid particles.
The difference in the stream function evaluated at the location of the fluid particles i.e. psi(x_2)-psi(x_1).
Is the volume flow rate independent of the curve connecting the fluid particles? Why?
Yes: the flow is conservative and so independent of the curve.
Define a streakline
A line formed from all the fluid elements passing through a point (a,b) between some t=0 and t>0.
How do we determine streaklines
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.
Describe how the Jacobian relates Lagrangian and Eulerian pictures
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.
Give (and explain) a constraint on the Jacobian
J cannot =0, since we require the map to be invertible.
Define an incompressible flow
A flow which preserves the volume of every material subregion in the fluid domain D_t.
Define a material scalar (or Lagrange invariant)
A function which has a zero material (or convective) derivative Df/Dt = 0
Given a function f(x(a,t), t), give the material derivative and describe.
Df/Dt = ∂f/∂t + u · ∇f
LHS: Lagrangian
RHS: Eulerian
Found by chain rule
Give the expression for the Jacobian relating E and L pictures
dx dy dz = J da db dc
What are the Jacobian and divergence of an incompressible flow?
J = 1 and ∇ · u = 0.
What is the relative change in jacobian J moving with the fluid?
1/J DJ/Dt = ∇ · u
Define local analysis
Considering two neighboring points of the flow x + δx
Define velocity gradient tensor
The tensor ∇u(x)_ij = = ∂u_i/∂x_j which defines the local structure of the velocity field
What three flow components do we find in local analysis?
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).
Define vorticity
The pseudo-vector ω = ∇ × u
Define circulation
The integral of u · dl around a closed curve γ
State Stoke’s theorem
The integral of a function around closed loop γ is equivalent to the curl of the function through the surface enclosed by the loop
Define a vortex tube
A bundle of vortex lines: the streamline analogue for vorticity; an integral curve of ω(x, t) at fixed time.
Give Helmholtz’ rule for vortex tubes
If γ1 and γ2 are two closed curves encircling a vortex tube, their circulation is equal.
Briefly describe ‘flow reconstruction’
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.
What are the boundary conditions for flow reconstruction?
∆φ = 0
n · (∇φ + u_v) = 0
Define the method of images for flow reconstruction
Adding addition vortices which ensures the boundary condition, since BS is linear in w.
What is the trick for differentiating over a changing volume?
Express integral in Lagrangian coords (over fixed volume D0 at t=0), diff and then re-express in Eulerian.
What does circulation define for a vortex tube?
The ‘strength’ of the vortex tube; invariant along its length.
State the divergence theorem
The volume integral of the divergence of a function is equivalent to the function flowing through a surface.
