Two-Dimensional Motion Flashcards
Helmholtz’s Equation
Dω/Dt = (ω.∇)u
u and ω are coupled together.
The streamfunction
(u, v, 0) = (∂ψ/∂y , − ∂ψ/∂x , 0)
the incompressibility condition is automatically satisfied.
Steamlines and the stream function
ψ= const
The streamfunction and curl
u = (u, v, 0) = (∂ψ/∂y , − ∂ψ/∂x , 0) = ∇ × (0, 0, ψ)
Cylindrical polar coordinates independent of z
u has a radial component ur and an azimuthal component uθ
The streamfunction in cylindrical polars independent of z.
u = 1/r ∂ψ/∂θ ˆr − ∂ψ/∂r θˆ = ∇ × (0, 0, ψ).
The Cauchy-Riemann Equations
u = ∂ϕ/∂x = ∂ψ/∂y
v = ∂ϕ/∂y = −∂ψ/∂x
What do the Cauchy-Riemann Equations mean?
If both ϕ and ψ are smooth
functions, the combination ϕ+iψ is a differentiable, complex function of z = x + iy, w(z).
What is the complex potential and complex position?
The function w(z) is referred to as the ‘complex potential’, and we sometimes refer to z = x + iy as the ‘complex position’.
What is ∇ϕ.∇ψ
0, Hence ∇ϕ is perpendicular to ∇ψ, streamlines (lines of constant ψ) are orthogonal to lines of constant ϕ (equipotentials); the two sets of lines form an system of orthogonal coordinates for the plane.
What is the complex velocity?
dw/dz = u − iv
How can we work out u and v?
a) Re[w(z)] = velocity potential ϕ, ∇ϕ = (u, v)
b) Im[w(z)] = streamfunction ψ, ∇ψ = (−v, u)
c) Re[dw/dz] = u, Im[dw/dz] = −v
ur − iuθ and the complex potential
ur − iuθ = exp[iθ] dw/dz
What are the 5 special 2-D flows?
a) the Uniform Stream
b) the Source/Sink
c) the Vortex
d) the Dipole
e) the Corner Flow
The uniform stream
Flow is of speed U at an angle α to the positive x-axis (measured anti-clockwise)
(u, v) = (U cos α, U sin α)
dw/dz = u − iv = U cos α − iU sin α = U exp[−iα]
w(z) = Uz exp[−iα] + a constant.
What is a source? What is a sink?
A source is a point where fluid is inserted into a flow.
A sink is a negative source, and is thus a point where fluid is extracted from the flow.
Complex potential for a source at the origin
w(z) = m/2π ln(z)
What can a source or sink be referred to in 3-D?
“Line Source” or “Line Sink”
Fluid velocity of a Source
u = f(r)ˆr
Fluid velocity of a Vortex
u = f(r)θˆ
Complex potential for a vortex at the origin
w(z) = −iK/2π ln(z)
Complex potential of a dipole at the origin
w(z) = −µ exp[iα] / 2πz
µ is dipole strength
What is a dipole?
Combination of a source m and sink -m, imagine the sink at the origin and the source a small distance δexp[iα] away.
What happens if we replace any streamline by a fixed boundary?
The flow is unchanged.
Complex potential for flow in a corner
w(z) = A z^α
α > 1/2
θ = nπ/α
What is the complex velocity in cartesian and polars?
dw/dz = u − iv = exp[−iθ(ur − iuθ)]
What is the method of images?
We can guarantee that u = 0 on x = 0 if we have a flow which is symmetric about x = 0.
Method of images (vortex by a wayy)
If there is a vortex of strength K at (a, 0), in the
presence of the rigid wall at x = 0; we put an image vortex at (−a, 0), we need to reverse the flow) the image must have strength −K.
Method of images (dipole by a wall)
In the presence of the rigid wall at x = 0, there is a Dipole of strength µ, placed at (a, b) and with direction given by exp[iα].
Symmetry requires the image to be another Dipole, placed at (−a, b), strength µ, direction −e
−iα. (The imaginary part of this,
corresponding to the part in the y direction and so parallel to the wall,
is the same as in the real Dipole; however, the real part, corresponding
to the part in the x direction and so perpendicular to the wall, is the
opposite of what is in the real Dipole.)
Method of images (source near a right angle corner)
a) an image Source of strength m at (a, −b),
b) an image Source of strength m at (−a, b), and
c) an image Source of strength m at (−a, −b)
Method of images (vortex near a right angle corner)
For a Vortex of strength K at (a, b) in the
right-angled corner, we require Vortices −K at (a, −b) and (−a, b) and a Vortex of strength +K at (−a, −b).
Complex potential of a source in a uniform stream
w(z) = Uz + m/2π ln(z).
Complex potential of a dipole in a uniform stream
w(z) = Uzexp[−iα] − µexp[i(α+π)] / 2πz
What can the flow of a uniform stream past a dipole be regarded as?
The flow of a Uniform Stream past a circle of radius a.
What is the circle theorem?
w(z) = f(z) + (f(a^2 / z))
