MONASH UNIVERSITY
Department of Mechanical and Aerospace Engineering
MEC4447 — Computers in Fluids and Energy —
Assignment 3
Due Monday 8th June 5 pm through Moodle. (Do it much earlier if possible...)
It contributes 10% to the overall subject mark.
Submit modified Matlab code and writeup.
Solving the Navier-Stokes Equations in Streamfunction-Vorticity Form
to Predict Vortex Breakdown in a Cylindrical Container
The unsteady incompressible Navier-Stokes and continuity equations are
∂u
∂t
+ u · ∇u = −∇P + ν∇2u,
and
∇ · u = 0.
Note, that there is no explicit equation for the pressure. This creates some difficulty in
solving the system numerically. This will be discussed further in lectures.
For some problems, it is possible to circumvent this difficulty by writing the equations in
terms of the streamfunction and the vorticity. Because using the streamfunction automati-
cally satisfies the continuity equation by design, this reduces the equation set. The pressure
does not occur in the vorticity equation.
An interesting case is for flows in axisymmetric geometries using cylindrical polar coordinates
when the velocity components do not depend on the azimuthal angle. In this case the
governing equations reduce to
∂ωθ
∂t
+ uz
∂ωθ
∂z
+ ur
∂ωθ
∂r
= ν(
∂2ωθ
∂z2
+
1
r
∂
∂r
(r
∂ωθ
∂r
)−
ωθ
r2
) +
2uθ
r
∂uθ
∂z
+
urωθ
r
, (1)
∂uθ
∂t
+ uz
∂uθ
∂z
+ ur
∂uθ
∂r
+
uruθ
r
= ν(
∂2uθ
∂z2
+
1
r
∂
∂r
(r
∂uθ
∂r
)−
uθ
r2
), (2)
and
−
∂
∂z
(
1
r
∂Ψ
∂z
)−
∂
∂r
(
1
r
∂Ψ
∂r
) = −
1
r
∂2Ψ
∂z2
−
1
r
∂2Ψ
∂r2
+
1
r2
∂Ψ
∂r
= ωθ. (3)
Here, u = (uz, ur, uθ) is the three-dimensional velocity field, with each component only a
function of the axial and radial coordinates z and r, respectively. Also, ωθ is the θ component
of the vorticity and Ψ is the axisymmetric streamfunction, defined below.
1
The vorticity components in cylindrical polar coordinates, and noting there is no variation
with θ, are given by
ωz =
1
r
∂(ruθ)
∂r
−
1
r
∂ur
∂θ
=
1
r
∂(ruθ)
∂r
, ωr =
1
r
∂uz
∂θ
−
∂uθ
∂z
= −
∂uθ
∂z
, ωθ =
∂ur
∂z
−
∂uz
∂r
,
and the relationships between the axial and radial velocity components and streamfunction
are
uz =
1
r
∂Ψ
∂r
ur = −
1
r
∂Ψ
∂z
. (4)
Note that the continuity equation in cylindrical polar coordinates is
∂uz
∂z
+
1
r
∂(rur)
∂r
+
1
r
∂uθ
∂θ
= 0.
Substituting the definitions of ur and uz in terms of the streamfunction into this expression
shows that it is satisfied identically.
So, the first two equations, (1) and( 2) are (Parabolic) time-stepping equations, whilst the
last equation, (3), is an elliptic equation. Each equation can be discretised using central
difference approximations for both first and second spatial derivatives.
For this project we are going to predict the flow inside a cylindrical container driven by a
spinning lid rotating with a constant angular velocity, as shown in the figure below. The
spinning lid centrifuges the fluid near the lid towards the outer radial wall. It then moves
left along the wall to the other end, then back towards the axis. At this stage it travels back
up towards the spinning lid as a strongly vortical (spinning) flow. Under certain conditions,
this vortex can burst or break down, meaning that it expands radially very quickly. This
situation can occur in aeronautical flows with the trailing vortices that occur from wing and
winglet tips. The original F/A-18 jets owned by the Australian Defence Force had a problem
that at high angles of attack the trailing vortex coming off the leading edge extension would
burst over the wings, leading to huge unsteady pressure forces.
rotating lid
H
R
Ω
solution plane
rotation axis
r
z
Figure 1: Left: Spinning lid rig. Right: Vortex breakdown over an F18 simulated in a water
channel.
2
Boundary Conditions
At solid walls a streamline has to follow the wall, unless there is flow through the wall. In
addition, the same streamline must also pass along the axis due to cylindrical symmetry.
Thus, at all boundaries the streamfunction can be taken as Ψ = 0. The azimuthal vorticity
is not zero - except on the axis. In general, the relationship between the streamfunction and
the vorticity can be obtained by expanding out the streamfunction at a point next to a wall.
Expanding the streamfunction at the point p about the wall (boundary) point w using
Taylor-series gives
Ψp = Ψw +∆z
∂Ψ
∂z
+
(∆z)2
2
∂2Ψ
∂z2
+
(∆z)3
6
∂3Ψ
∂z3
+O((∆z)4).
Here ∂Ψ/∂z = −rur,w, where ur,w is equal to tangential velocity at the wall. In addition,
at the wall, ωθ = ∂ur/∂z (the other component is zero at the wall). Note that terms up to
third-order are required to maintain second-order spatial accuracy, since the vorticity comes
from the second-order term in this expansion, as is seen below. Hence,
(Ψp −Ψw) + ∆z ur,w ≃ −
(∆z)2
2
∂(r ur,w)
∂z
−
(∆z)3
6
∂
∂z
∂(r uz,w)
∂z
,
(Ψp −Ψw) + ∆z ur,w = −
(∆z)2r
2
∂ur,w
∂z
−
(∆r)3r
6
∂2ur,w
∂z2
,
(Ψp −Ψw) + ∆z ur,w = −
(∆z)2r ωθ,w
2
−
(∆z)3r
6
∂ωθ,w
∂z
,
(Ψp −Ψw) + ∆z ur,w = −
(∆z)2r ωθ,w
2
−
(∆z)3r
6
(ωθ,p − ωθ,w)
∆z
,
(Ψp −Ψw) + ∆z ur,w = −
(∆z)2r ωθ,w
3
−
(∆z)2r ωθ,p
6
.
Here, the last term, involving the time derivative of the vorticity, has been replaced by a
first-order finite-difference approximation. This maintains the second-order overall accuracy
because the error in this last term is then O(∆z3 × ∆z) = O(∆z4), consistent with terms
that we are omitting. (To get the expression for the vorticity, we divide by ∆z2, hence the
accuracy is O(∆z4/∆z2) = O(∆z2)). Thus, we can write
ωθ,w = −3((Ψp −Ψw) + ∆r ur,w)/(∆z
2r)−
1
2
ωθ,p +O(∆z
2).
Since ur,w = 0 on the wall, and Ψw = 0, this reduces to
ωθ,w = −3
Ψp
∆z2r
−
1
2
ωθ,p +O(∆z
2).
A similar expression applies at the radial wall (with a few added terms), and at the righthand
wall.
3
wz∆
computational
boundary
p
Figure 2: Evaluation of boundary conditions for vorticity at lefthand wall.
On the central axis, the radial derivative of the axial velocity must vanish by symmetry.
Hence ∂uz/∂r = 0. We need a second-order approximation at the boundary to give overall
second-order accuracy of the solution. An appropriate approximation is
∂uz
∂r
≃
3uz,r=0 − 4uz,r=∆r + uz,r=2∆r
2∆r
= 0,
⇒ uz,r=0 = (4uz,r=∆r − uz,r=2∆r)/3.
The azimuthal velocity is zero on all boundaries except the spinning lid, where uθ = Ωr.
In addition, the azimuthal vorticity is given by ωθ =
∂ur
∂z
−
∂uz
∂r
. Given the constraint on
the axial velocity and that ur = 0 on the boundary because of symmetry, this implies the
azimuthal vorticity is zero on the axis.
Implementation
The governing equations can be expressed in finite-difference form for each internal point.
Generally, the streamfunction is known at all points on the boundary, while the vorticity
depends on the streamfunction and internal vorticity, as above.
Taking equation (1) and using a forward-time centred-space approximation gives:
ω
(n+1)
θ,i,j − ω
(n)
θ,i,j
∆t
+ u
(n)
z,i,j(
ω
(n)
θ,i+1,j − ω
(n)
θ,i−1,j
2∆z
) + ur,i,j(
ω
(n)
θ,i,j+1 − ω
(n)
θ,i,j−1
2∆r
) =
ν(
ω
(n)
θ,i+1,j − 2ω
(n)
θ,i,j + ω
(n)
θ,i−1,j
∆z2
+
ω
(n)
θ,i,j+1 − 2ω
(n)
θ,i,j + ω
(n)
θ,i,j−1
∆r2
+
1
ri,j
ω
(n)
θ,i,j+1 − ω
(n)
θ,i,j−1
2∆r
−
ω
(n)
θ,i,j
r2i,j
)+
2u
(n)
θ,i,j
ri,j
u
(n)
θ,i+1,j − u
(n)
θ,i−1,j
2∆z
+
u
(n)
r,i,jω
(n)
θ,i,j
ri,j
. (5)
4
This can be rearranged slightly to give an update equation for ωθ, to calculate the values at
the new timestep:
ω
(n+1)
θ,i,j = ω
(n)
θ,i,j +∆t

−u(n)z,i,j(ω
(n)
θ,i+1,j − ω
(n)
θ,i−1,j
2∆z
)− ur,i,j(
ω
(n)
θ,i,j+1 − ω
(n)
θ,i,j−1
2∆r
)+
ν(
ω
(n)
θ,i+1,j − 2ω
(n)
θ,i,j + ω
(n)
θ,i−1,j
∆z2
+
ω
(n)
θ,i,j+1 − 2ω
(n)
θ,i,j + ω
(n)
θ,i,j−1
∆r2
+
1
ri,j
ω
(n)
θ,i,j+1 − ω
(n)
θ,i,j−1
2∆r
−
ω
(n)
θ,i,j
r2i,j
)+
2u
(n)
θ,i,j
ri,j
u
(n)
θ,i+1,j − u
(n)
θ,i−1,j
2∆z
+
u
(n)
r,i,jω
(n)
θ,i,j
ri,j

 . (6)
A similar approximation can be written for the uθ equation (2). You need to do this for this
assignment and insert your expression into the code template.
Equation (3) for the streamfunction Ψ is an elliptic equation that can be solved by relaxation.
Using second-order finite-difference approximations for the spatial derivatives gives:
−
1
ri,j
Ψi+1,j − 2Ψi,j +Ψi−1,j
∆z2
−
1
ri,j
Ψi,j+1 − 2Ψi,j +Ψi,j−1
∆r2
+
1
r2i,j
Ψi,j+1 −Ψi,j−1
2∆r
= ωθ,i,j, (7)
⇒ −
Ψi+1,j − 2Ψi,j +Ψi−1,j
∆z2
−
Ψi,j+1 − 2Ψi,j +Ψi,j−1
∆r2
+
1
ri,j
Ψi,j+1 −Ψi,j−1
2∆r
= ri,jωθ,i,j, (8)
To use successive over-relaxation (SOR), we need to obtain an expression for Ψi,j. Rear-
ranging
⇒ (
2
∆z2i,j
+
2
∆r2i,j
)Ψi,j =
Ψi+1,j +Ψi−1,j
∆z2
+
Ψi,j+1 +Ψi,j−1
∆r2
−
1
ri,j
Ψi,j+1 −Ψi,j−1
2∆r
+ri,jωθ,i,j, (9)
⇒ Ψ∗i,j =
(
Ψi+1,j +Ψi−1,j
∆z2
+
Ψi,j+1 +Ψi,j−1
∆r2
−
1
ri,j
Ψi,j+1 −Ψi,j−1
2∆r
+ ri,jωθ,i,j
)
/(
2
∆z2i,j
+
2
∆r2i,j
),
(10)
This gives a new estimate for the streamfunction at each point on the mesh given current
values. It is basically the Gauss-Seidel update equation. SOR just involves using this new
estimate (Ψ∗i,j) to give ∆Ψi,j = Ψ
∗
i,j−Ψ
old
i,j , then the new value of the streamfunction is given
by Ψnewi,j = Ψ
old
i,j + r∆Ψi,j, with r the relaxation parameter.
This defines a Gauss-Seidel update equation for Ψ, which can be used with SOR to iterate Ψ
given the current estimate of the vorticity field. Typically, this equation should be solved to
some convergence criterion at each timestep. In the code, the minimum number of iterations
is set to 5. That is, it completes at least 5 iterations to update Ψ before stepping forward
ωθ and uθ to the next timestep.
So, in summary a timestep consists of
5
• updating the Ψ field given the current vorticity field using SOR (only internal points
are updated since Ψ = 0 on the boundaries of the computational domain);
• then stepping forward the vorticity and azimuthal velocity to the next timestep using
the updated value of Ψ. Boundary conditions on ωθ (called vort in the code) are also
implemented during this update.
These two steps are repeated until the solution reaches its asymptotic state.
For the assignment you only need to fill in the update equation for uθ, which is similar in form
to equation 6, and then set other parameters such as the Reynolds number: Re = ΩR2/ν,
the physical dimensions: zlen = H and rlen = R, the relaxation parameter: relax , the
timestep: dt = ∆t, etc.
Program
You can download the template code from Moodle. You need to add some lines
defining the update equation for uθ and one of the ωθ boundary conditions. (These bits
are marked by ???? in the template). The rest should be OK. Please have a look at the
structure of the program to understand broadly how it works.
Note that the routine is supplied as a Matlab function rather than a script file (because
often this makes the code run much faster). However, make sure you are running the latest
version of MATLAB. Older versions can be very much slower...
You can call the routine by name (typing cyl cavity time in the command window - as long
as the file is in your matlab path) or just load it and run it.
You should supply the arguments required.
These arguments include:
• the number of cells in z and r: nz, nr;
• the domain size in z and r: zlen, rlen;
• the Reynolds number: re;
• the relaxation parameter for the Ψ equation: relax;
• the convergence criteria: eps psi, eps omega, eps ut;
• the timestep: dt;
To call the program as a function, supply (name, value) pairs as the arguments.
6
For example, typing
cyl cavity time(’re’, 1000, ’nz’, 60, ’nr’, 60, ’relax’, 1.5, ’dt’, 0.05);
calls the function with the values: re=100, nz=60, nr=60, relax=1.5, dt = 0.05.
Alternatively, just change the values directly in the program and rerun it.
Note that there is a primitive restart facility. After running through a case, the fields are
saved and can be used to restart the next run by changing the restart parameter to 1. If you
alter the grid size, Matlab kindly interpolates the old solution onto the new grid for you.
Note that the maximum timestep is controlled by both a Courant condition and a diffusion
condition. For fine grids you may need to use a smaller timestep to get the timestepping to
work.
Other parameters not specified are taken as default values defined at the top of the function.
The defaults are: eps psi=0.000001, eps omega=0.000001, eps ut=0.000001, nz=60, nr=60,
re=1000, zlen=2.5, rlen = 1.0, Ω = 1.
The Project
1. Use the streamfunction vorticity formulation to find the maximum and minimum
streamfunction for Re = 1994. Do a grid resolution study. Explain how you have
ensured that the predictions are accurate to within 2%. Supply images of the con-
verged streamfunction, azimuthal vorticity and azimuthal velocity fields. [2 marks].
2. Run the model at Re = 2765. In this case, the solution is periodic. Determine the
period of oscillation to within 2%. Again explain how you did this. (Note you should
be able to get the period to within sufficient accuracy from the convergence history
plot). [2 marks].
Project Writeup
I want a short report containing the following information.
• Derivation of the finite-difference equation for uθ, finally expressed as an update equa-
tion for uθ at the new timestep. (You can write this by hand and photograph them to
include in your writeup) [2 marks].
• Final Matlab program (submitted through Moodle) [2 marks].
• Brief answers to the questions above.
• Discretionary [2 marks] for the writeup.
7
A standard reference for this problem is
(1) Lopez, J. M., Axisymmetric vortex breakdown. Part I: Confined swirling flow. Journal
of Fluid Mechanics, 221, 533-552, 1990. (Copy on Moodle page).
8