The fins

How to determine the best shape and size for your rocket?. For this you can use the Barrowman equations. James S Barrowman wrote in 1966 working for the NARAM his findings on the practical calculation of the aerodynamic characteristics of slender finned vehicles. It is a relative simple algebraic method to find the center of pressure (CP) of a rocket flying at sub sonic speeds under small angles of attack (longest size of rocket in line with vector of speed).
Here you find Barrowmans report on calculation of the CP
A handy program for doing these calculations, I find, is VCP from Gary A. Crowell sr. VCP is freeware.
VCP can be downloaded here

If you hear the fins make a rattling sound in the end of the air blowdown phase, this is when the rocket is at maximum speed, then you know that they are not stiff enough and /or the fin size is to large. This phenomenon is known as fin flutter or wing bending torsion flutter. Your rocket will loose speed very quickly and the apogee height will be disappointing. There are several other modes for fins to start to oscillate under the air speed. But at low angles of attack of the air (the common case with a stable rocket) the most common type of flutter is the bending torsion flutter.
On the picture below you find the motion the fins make in this mode.

The fin swings around the longest axis of the rocket and is S-shaped (high amount of drag) in the transition form "up" to "down".

Now lets see if we can estimate the maximum speed of our fins before the flutter kicks in.

As a reference I used the work of Dennis J. Martin from feb 1958 Mr. Martin worked for the NACA (which later became the NASA).
I found in his work the following equations

Where: A is the aspect ratio
t is the fin thickness
c is the fin root chord

Then use the calculated X factor in the second formula

Where
labda is the taper ratio (ratio of tip chord to root chord)
p is the air pressure
p0 is the air pressure at 0 m
So normally for water rocket practice you can use the factor 1 for p/po.

This results in a flutter index FI, the smaller the better. It is a number that represent the sensitivity of your wing dimensions for flutter.
Next you need the shear modulus for your fins.
I don't know the shear modulus by measurement for my sandwich PET type fins.
Bending it, and comparing it with the stiffness of a similar fin of aluminum I think it is about 25% as stiff.
In the work of Mr. Martin (see ref.) fig 3 you find that alu has a GE of about 4e^6. So I use a factor of 1e^6 for my fins, seems reasonable.

With the calculated FI and estimated GE you can calculate the expected max. flutter free max. speed with the formula:

Where
Vf = the expected max flutter free speed
a = speed of sound

With torsion stiff fins it is of course important that you mount the fins torsion-stiff to the rocket body, see the construction section for tips.

As an example here a calculation on the fins I use on the 9L rocket
See the picture for the dimensions.
The fin thickness is 2mm

The aspect ration is 110/85 = 1.294
. t/c^3 -> 2/110^3 -> 6.01e-6
x=4.3^6.
And because p/po = 1 and the taper ratio is also 1 X equals the FI
FI~4e6.
For GE (see above) we assume 1E^6, 25% the shear modules of comparable fins of solid aluminum. The speed of sound = 344 m/sec (@ 20 deg Centigrade)
So received by estimation and calculation I expect my fins to be flutter free to about half the speed of sound or about 170 m/sec
In the simulator the maximum speed for the 9L is just above 80 m/sec. @ 10 bar launch pressure
So in conclusion I expect the fins to be flutter free and to have margin for speed increase.