The roozen paper refers to:
P. Merkli and H. Thomann, “Thermoacoustic effects in a resonance tube,” (1975), published at ETH zurich.
The access to the paper seems to be restricted.
I'll try to find a way!
Thank you @Dmitrij_S ,
the strouhal number seems to relate a "characteristic dimension" (not further defined, can be diameter, radius ...) to the particle displacement.
In addition your reference seems to give an inverse value (displacement/characteristic dimension).
So in future whenever I refer to strouhal number I should probably include the dimension and definition it refers to, which is the port radius (effective port exit radius in my specific case), following roozen and merkli/thomann.
the strouhal number seems to relate a "characteristic dimension" (not further defined, can be diameter, radius ...) to the particle displacement.
In addition your reference seems to give an inverse value (displacement/characteristic dimension).
So in future whenever I refer to strouhal number I should probably include the dimension and definition it refers to, which is the port radius (effective port exit radius in my specific case), following roozen and merkli/thomann.
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I think I may have found the reason for different definitions of Sr.
The "classic" strouhal number relates to a solid cylinder in a fluid flow, see wikipedia. Strouhal himself apparently studied cables oscillating in the wind, caused by vortex shedding.
With a loudspeaker port the number relates to a fluid flow inside a (hollow) cylinder.
Anyone has more knowledge about this?
V.M.Garcia-Alcaide et al. studied turbulence in ports using the CFD modeling package OpenFOAM. They also used the definition of the Strouhal number given by Roozen, and used the radius of the port rounding as a characteristic size. It seems that Roozen's definition of the Strouhal number is quite well-established in the ports turbulence studies.
@Dmitrij_S
thank you very much for pointing me to this paper and confirming the definition of strouhal number by roozen!
It's encouraging to see how similar the flow simulation/high speed camera images are to my slow motion videos!
thank you very much for pointing me to this paper and confirming the definition of strouhal number by roozen!
It's encouraging to see how similar the flow simulation/high speed camera images are to my slow motion videos!
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Excellent read! Reminds me of how race cars create downforce with ground effects...under the front of the car takes incoming air pressure (inner flare of the port), speeds up the air in the middle of the car (middle tube of the port), and then slow down the air with rear diffusers for higher cornering speeds (outer flare of the port).
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It is interesting that the Roozen Strouhal number definition is in effect the same as the one given by the general formula if the characteristic length is assumed to be the port tube circumference, rather than the diameter.
General formula:
St = f * L / V
Where:
St = Strouhal number
V = Velocity
L = Characteristic length
f = Frequency
If characteristic length = circumference then L = 2 * pi * port radius
St = f * (2 * pi * port radius) / V
St = port radius / (V / (2 * pi * f))
Particle displacement = V / (2 * pi * f)
St = port radius / particle displacement
I gather that you would prefer to use the Strouhal 'circumference' definition rather than the 'diameter' one, and that the design aim is to ensure that the Strouhal Number is greater than one at all frequencies of interest. Is this correct?
(NOTE - To be consistent with Roozen you should be using the mean-to-peak rather than peak-to-peak displacement value).
Perfectly correct, yes!
Thank you very much once more!
While embarrasing, fortunately it does not completely prove my tests wrong because the transition to turbulent flow is rather gradual.
It may even be more consistent with my H3 measurement in post #556.
In addition using the correct mean-peak value results more reasonable port sizes.
UPDATED: how to find a suitable port exit diameter using the Strouhal number
UPDATE regarding corrected strouhal number calculation: mean-peak value instead of peak-peak value, see post #591 by David McBean above!For my 2-way speaker the goal was to use the old drivers I already had, even if they are quite outdated spec-wise. These drivers and speakers have a sentimental value, and I challenged myself to get the best out of them.
The speakers will mostly be used for low level music reproduction in a small room. Also, the sensitivity is not bad for such a small driver. There is no shorting ring but as far as I understand the low voice coil overhang keeps inductance low.
I first define the maximum sensible input voltage for the small 4” midwoofers Xmax of 2 mm in the usable frequency range (above tuning freq).
I use a hornresp simulation tuned accordingly using a simple tube port and adjust the input voltage to keep displacement near Xmax:
of course the excursion will exceed Xmax below tuning frequency, but that’s to be expected with a ported box and such a low Xmax and Sd woofer.
This loudspeaker will not play loud and not very low.
3,3 V equals not even 3 W at 4 Ohm!
I would like to get (mostly) uncompressed and non-chuffing port output at the determined max. sensible input voltage, thus the Strouhal number should never go below 1 (one) with 3,3 V input.
Strouhal number = port radius / mean - peak air particle displacement
(refer to post #591 for details on strouhal number calculation!)
Thus the requirement is:
port radius ≥ mean-peak air particle displacement
The maximum air particle displacement (and air speed) in the port happens at tuning frequency (fb), which in my case is around 50 Hz.
particle displacement can be calculated using the max. airspeed value from a simulation (see #391):
m-p air particle displacement [m] = max. airspeed [m/s] / (2 * Pi * f [1/s])
so for the Strouhal requirement:
port radius ≥ m-p air particle displacement
we can postulate:
Max air speed [m/s] = port radius [m] * 2 * Pi * fb [1/s]
or:Max air speed [m/s] = port diameter [m] * Pi * fb [1/s]
It is now either possible to
- approximate the required max. airspeed in the simulation tool by adjusting the port surface in the simulation while considering the updated port radius.
- or it’s possible interpolating the value found with sensible starting parameters:
For my speaker I start with a 5 cm diameter tube port (19,6 cm²) and get the following result:
Max airspeed = 6,57 m/s
m-p particle displacement = 6,57 [m/s] / 3,141 * 2 * 50 [1/s] = 0,021 m = 2,1 cm
The displacement is below port radius, at 3,3 W input and a 5 cm port.
the Strouhal number is above 1 (indicating probable safe non-turbulent flow):
Strouhal Number = 2,5 cm / 2,1 cm = 1,19
I still would like some headroom, so I will increase the desired strouhal number to 2.
(to be honest: I previuosly used the wrong calculation formula and got some headroom "for free"
)Increasing the port diameter will raise the port radius proportionally but also reduce the air speed by division through square of the increase factor (because of squared increase of port cross section surface).
Note that this is just an approximate method, assuming uniform air movement in the port therefore it’s important to use sensible entry data. As far as I know hornresp does simulate realistic air movement in the port!
Thus an increase of port diameter or radius by factor x will increase the Strouhal number by x³.
We have a Strouhal number of 1,19 and want a Strouhal number of 2, thus:
x³ = 2 / 1,19 = 1,68
To get x (port radius increase factor) we need to calculate the cubic root of x³:
x = cubic root of (1,68) = 1,19
The required minimum effective port exit diameter is
1,19 * 5 cm = 5,95 cm
As can be seen increasing the diameter by just 19% raises the strouhal number by 67%.
This relation is not very intuitive, thus I am very grateful that hornresp will provide a strouhal number calculation in the update and facilitate finding a useful port size!
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In terms of the UI would it be possible to include a note to guide the user that the strouhal number should ideally be used to define the port exits size and not the port central / minimum size? I think some users might think it represents the latter, as max air speed is often interpreted this way if they have not read up. Or perhaps @stv would be willing to assist in writing a section for the help file similar to his post here?
Also, thank you very much!
That's a bit tricky, because how do you define exit diameter with some totally random flare, chamfer or round over?
As far as I know these calculations assume linear systems.
Because if it's not linear anymore, there isn't much to calculate and we have to rely on simulations.
Also I still think that this number is also only valid when the port's entrance and exit are symmetrical?
So it's another way of saying and asking how much value this number gives in reality. Especially for the average user.
Don't get me wrong, I still think it's really handy to have!
Speaking of which.
@stv I am mostly still very curious if there is some kind of formula possible for those better performing tubes?
Because they obviously shift the tuning frequency as well.
I think it's enough to provide the strouhal number, just as air speed is provided. The user can decide how to interpret this data.
And as hornresp only offers tube ports I think no further description of the effect regarding port center/ terminations is needed.
Yes, I will suggest a method, based on a formula for length calculation of flared ports in the salvatti, devantier, button paper!
Not sure if you refer to a "corresponding diameter" for simulation or calculation of tuning?
If so:
For a very slightly curved "Roozen-Type" Port (as my first 3d-printed port, scroll down in the post for a drawing) I got quite accurate tuning by using arithmetic mean of cross section surface, taking only two samples (end, center).
This will not work with more curved wall ports, however, because the influence of the narrowest section is much bigger.
So until now for my experiments with elliptic wall ports I just did it by trial and error.
Here is an excerpt of the Salvatti/Devantier/Button Paper showing a simple formula for calculating the equivalent effective port length and surface area:
I verified it with my last "BIG port" and got a very close match.
Note that "end correction" factors are already included in most simulation tools.
That needs to be considered before applying the "L eff" correction factor when simulating using corresponding tube ports.
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