PHYS 426 Lab Report¶

Title: Observation of Vortex Shedding from a Cylinder and Smoke Ring Formation
Names: Polina Erofeeva, Anuska Shrestha
Course: PHYS426

Abstract¶

This experiment involved two fluid dynamics demonstrations: (1) observing vortex shedding from a stationary cylinder placed in a flume, and (2) generating and analyzing smoke rings using a smoke ring generator. The objective was to qualitatively observe and describe vortex formation, flow patterns, and ring evolution. Both experiments highlight the natural emergence of organized vortical structures in fluids under certain flow conditions. Results were recorded using flow visualization techniques.

Introduction¶

Vortex structures are a fundamental feature of many fluid flow systems. This lab explored two classic examples:

  1. Vortex shedding from a cylinder: When fluid flows past a bluff body (like a cylinder), alternating vortices are shed from either side, forming a repeating pattern known as a Kármán vortex street.
  2. Smoke rings: Vortex rings are self-propelling doughnut-shaped vortices generated by pushing fluid (in this case, smoke‐filled air) through an orifice.

The aim of the lab was to observe the formation, behavior, and symmetry of these vortices under controlled conditions.

Theory¶

1. Vortex Shedding from a Cylinder¶

Video Reference

Vortex Shedding Mechanism in Bluff-Body Flows
When a fluid flows past a bluff body (a circular cylinder), the boundary layer separates and forms shear layers that roll up into vortices. Initial instabilities cause these vortices to detach alternately from each side in a process called vortex shedding (Vortex shedding – Wikipedia), producing a repeating von Kármán vortex street (Flow separation – Wikipedia). Each shed vortex creates a low-pressure region, resulting in an alternating side-to-side force on the cylinder.

Reynolds Number and Flow Regimes
The dimensionless Reynolds number is
$$ \mathrm{Re} = \frac{U\,D}{\nu} \quad(1) $$ where U is the free-stream flow velocity (m/s), D is the cylinder diameter (m), and ν is the fluid’s kinematic viscosity (the ratio of dynamic viscosity μ to density ρ) (m^2/s).

Physically, Re represents the ratio of inertial forces to viscous forces in the flow. At low Reynolds numbers, viscous forces dominate and the flow around the cylinder remains laminar and steady. For instance, when Re is very small (on the order of a few units), the flow is attached and symmetrical with only a gentle separation; no unsteady vortices appear. In the range of approximately Re ≈ 5 to 40, a pair of steady symmetric eddies forms directly behind the cylinder (a laminar recirculating wake), but the flow is still steady in time. As the Reynolds number increases further, inertial forces begin to overwhelm the viscous effects, and beyond a critical Re the wake loses symmetry and becomes unsteady. Around a critical Reynolds number on the order of Re ≈ 40–50, the previously steady wake undergoes an instability and the cylinder begins to shed vortices periodically. This is when vortex shedding starts. For moderate Re numbers (roughly 40 < Re < 90), the wake flow oscillates and vortices are shed alternately from each side, though the pattern may not yet be perfectly regular. Once the Reynolds number exceeds about 90, the classic von Kármán vortex street appears: a staggered, alternating pattern of vortices forms behind the cylinder. This unsteady vortex shedding continues over a wide range of higher Reynolds numbers. Even at very high Reynolds numbers, when the flow becomes turbulent and the boundary layer transitions to turbulence before separation, a form of vortex shedding usually continues, though the vortices and shedding frequency may become less regular. Thus, we can use Reynolds number to determine whether the flow past a cylinder is steady or unsteady and whether vortex shedding will occur.

Kármán Vortex Street and Strouhal Number

In the vortex shedding regime, the wake behind the cylinder exhibits an alternating vortex pattern known as the von Kármán vortex street. As one vortex sheds from one side of the cylinder, the next vortex sheds from the opposite side, and so on, creating a chain of vortices that are arranged roughly in two rows, staggered downstream. This pattern is very regular under steady flow conditions and is a direct consequence of the alternating nature of the flow separation. Each vortex in the street has a rotation (circulation) opposite to its neighbor, and the street typically forms an angle relative to the flow direction as the vortices propagate downstream. The presence of this alternating street is accompanied by periodic fluctuations in velocity and pressure in the wake, resulting in an oscillatory lift force on the cylinder. A useful way to characterize the vortex shedding is by its frequency, f (cycles per second), which depends on the flow speed and the cylinder size.

The shedding frequency (f) relates to velocity and diameter via the Strouhal number:
$$ \mathrm{St} = \frac{f\,D}{U} \quad(2) $$ Here, D is the cylinder diameter and U is the free-stream flow velocity. The Strouhal number measures how many vortex shedding cycles occur per second relative to the convective timescale D/U. Rearranging, the shedding frequency is given by: $$ f = \mathrm{St}\,\frac{U}{D} \quad(3) $$ This relationship shows that, for a given Strouhal number, the vortex shedding frequency is directly proportional to the flow velocity and inversely proportional to the cylinder diameter. In other words, higher fluid velocity results in faster vortex shedding (higher f), while a larger cylinder results in slower shedding (lower f), all else being equal. The Strouhal number for a circular cylinder in cross-flow is nearly constant over a broad range of Reynolds numbers. In the laminar shedding regime (roughly Re ~ 10^2 to 10^5), St typically ranges between about 0.18 and 0.22. A commonly cited representative value is St ≈ 0.20, meaning the shedding frequency is about one-fifth of U/D. For example, if air flows at U = 5 m/s past a cylinder of diameter D = 0.1 m (with Re in the hundreds for air), the shedding frequency would be on the order of f ≈ 0.2 × (5 / 0.1) = 10 Hz. In practice, the Strouhal number varies slightly with Reynolds number and flow regime. An empirical relation valid for moderate to high Re (around 250 < Re < 200,000) is sometimes given as:

$$ \mathrm{St} \approx 0.198\Bigl(1 - \frac{19.7}{\mathrm{Re}}\Bigr) \quad(4) $$ which asymptotically approaches St ≈ 0.198 at large Re. This illustrates that in the typical vortex shedding regime, changes in fluid speed or cylinder size will linearly affect the shedding frequency. In our experiment, the structure's natural frequency is the rate at which the cylinder would naturally oscillate if it were set into motion by a disturbance. If the shedding frequency matches the structure's natural frequency, the resulting resonance can cause large, amplified vibrations.

Reynolds–Strouhal Relation
Combining (2) and (4) gives
$$ \frac{f\,D}{U} = 0.198\Bigl(1 - \frac{19.7}{\mathrm{Re}}\Bigr) \quad(5) $$ Since $$f = \frac{N}{\Delta t}$$ and $$U = \frac{S}{\Delta t},$$ simplifies to
$$ \frac{N\,D}{S} = 0.198\Bigl(1 - \frac{19.7}{\mathrm{Re}}\Bigr) $$ showing that Reynolds number depends on the distance D and count N, not directly on time.

2. Smoke Ring Formation¶

Video Reference

Smoke rings are a great visual demonstration of vortex rings, these rings are generated when a fluid; liquid or gas; are forced through an orifice. When forced through the hole of a smoke ring generator, the fluid is made to curl into itself creating a ring like shape, we can call this a torus shape. Smoke on the outside of the hole is being slowed down by the friction between it and the edges of the generator hole. As the smoke moves from the inside of the generator to rapidly move into the still air of the room, this creates a shear layer at the boundary between the moving and still fluid, as there is friction at the interface between the smoke and still air, at the edges of the generator hole this shear causes a separation of flow; in which the boundary layer layer detaches and rolls up due to the Kelvin-Helmholtz Instability [2], making it so that there is smoke at the center that is moving faster than at the edges creating what we can visualize to be a mushroom cloud; better described to have turbulent vortices curling downward around its edges. This separation and flow pattern is vital for initiating the rotational motion which generates a ring of concentrated vorticity that closes itself into a loop, creating a vortex ring.

We define the local rotation (vorticity) within the ring as:
$$ \omega = \nabla \times \mathbf{u} \quad(6) $$ In which we can define u as the velocity field. As the smoke ring moves out it will circulate on its own loop, if we integrate this rotation around the closed loop we can find circulation to be: $$ \Gamma = \oint \mathbf{u}\cdot d\mathbf{l} \quad(7) $$ Which we can define as a conserved quantity in situations of no negligible viscosity and is integral to understanding the stability of the ring as it moves. The ring will move out while self propagating due to its own induced velocity until the rotation dies and the smoke dissipates. The fluid at the core circulates and simultaneously moves forward creating a stable toroidal motion governed by the conservation of impulse. Impulse is described by the change of momentum of an object, in terms of vortex we can define it to be: $$ \mathbf{I} = \int \mathbf{x}\times \omega \,dV \quad(8) $$ This equation is taking the cross product of position and vorticity, this equation is used to explain why the ring moves forward even though there is nothing pushing the ring forward after the initial push. As we say in equation 1, that discusses the Reynolds number. In our situation U is described to be ring’s velocity; we can define an idealist expression to a thin core ring to be:
$$ U = \frac{\Gamma}{4\pi r}\left(\ln\frac{8R}{\alpha} - \frac{1}{4}\right) \quad (9) $$ This uses circulation and the radius of the diameter of the ring and thickness to estimate the speed of the ring, to predict and better understand how fast the smoke ring will travel based on how it formed.

Materials and Methods¶

Experiment 1: Vortex Shedding
Equipment: Flume tank; rods (Ø 1 cm & 3 mm); blue + pearlescent dye; phone camera
Procedure: In this experiment, we filled up the tank with water, added some sparkly pearlescent dye with a regular dye for better visualisation and put a ruler next to it to define our observation window. Holding a rod horizontally at the surface, we dragged it smoothly through the water so that it traveled from one side to the other at controlled speeds (e.g. 0.30 m in 15 s, 0.37 m in 15 s, or 0.40 m in 5 s), while the phone camera recorded the wake. After each pass, we replayed the video in slow motion to count the vortices shed within the most well-defined span and noted the duration, then wiped down the tank, refreshed the dye, and repeated each trial three times for both rod sizes before averaging the results for our Reynolds and Strouhal analyses.

Experiment 2: Smoke Rings
Equipment: Smoke chamber; electrician’s tape; sucrose smoke generator; phone camera
Procedure: We started by turning on the smoke generator and waited until the light turned on and the smoke was ready to come out. We then taped our desired shape at the end of the smoke camber and put the smoke release pipe into the created hole and filled the camber with smoke. We then used both the attached lever and hand to hit the end of the camber to create the smoke rings in slow intervals. Due to the nature of the environment within the lab room, we found it best to use a darker poster type panel to be able to see the smoke better and to put the camber on the floor or a stable situation. We then redid the experiment with different shapes and sizes, making sure to collect data as we went.

Results¶

Vortex Shedding¶

For our experiment, we measured:

  • d = 1 cm = 0.01 m – diameter of a medium glass rod
  • l = 3 mm = 0.003 m – diameter of a small metal rod

Low Reynolds number¶

Over a time interval of Δt = 15 s, N = 5 eddies were created over a distance of S = 30 cm with a medium glass rod. We are also using an assumption for kinematic viscosity v = 1.5×10⁻⁶ m²/s.

f = (number of eddies) / Δt = 5 / 15 = 0.333 Hz U = S / Δt = 0.3 m / 15 s = 0.02 m/s Re = U D / v = 0.02 * 0.01 / 1.5×10⁻⁶ => Re ≈ 133

As we can see, the Reynolds number we got is still not nearly as low as we wanted to get. This is because it is very hard to achieve a low Reynolds number in water.

Medium Reynolds number:¶

For N = 7 eddies created over a distance of S = 0.37 m within 12 seconds:
f = N / Δt = 7 / 12 = 0.333 Hz
U = S / Δt = 0.37 m / 12 s = 0.031 m/s
=> Re ≈ 205.6
This Reynolds number is higher as we were moving the rod faster (i.e. higher stream velocity) and, thereby, more inertial forces were dominating, resulting in a higher Reynolds number.

High Reynolds number:¶

For the distance of 0.4 m over the time of 5 seconds:
Re = 533.3
Then we can find the shedding frequency using the Reynolds-Strouhal number relation (since Re > 250) and compare it to our observation:
(f · D) / (S/Δt ) = 0.198 (1 - 19.7 / Re) => (f · 0.01) / (0.4/5) = 0.198 (1 - 19.7 / 533.3) => f = 1.53 Hz or N=fΔt=5*5=7.65 eddies were created. This aligns well what we see in the video - within the time period of 5 seconds 8 eddies were created.

Small rod:¶

Re = U D / v = 0.68 * 0.003 / 1.5×10⁻ =150
f = 5.0 Hz => N = fΔt= 5*8 = 40 eddies created within 8 seconds.
Manually calculated number of eddies on this interval was 44, which lies within the 10% uncertainty.
We can see that for a much narrower rod, it is much easier to get a lower Reynolds number and, thereby, higher shedding frequency.

Results are better represented in the code below:

In [27]:
scenarios = [
    {"name": "Medium rod (slow)", "D": 0.01, "distance": 0.3, "time": 15, "eddies": 6},
    {"name": "Medium rod (moderate)", "D": 0.01, "distance": 0.37, "time": 12, "eddies": 7},
    {"name": "Medium rod (fastest)", "D": 0.01, "distance": 0.4, "time": 5, "eddies": 8},
    {"name": "Small rod",        "D": 0.003, "distance": 0.6, "time": 8,  "eddies": 47}
]
nu = 1.5e-6  # kinematic viscosity (m^2/s)
for case in scenarios:
    D = case["D"]
    U = case["distance"] / case["time"]              # flow speed
    Re = U * D / nu                                  # Reynolds number
    St = 0.2
    if Re > 250:                                     # use Re-dependent St if applicable
        St = 0.198 * (1 - 19.7 / Re)
    f_theor = St * U / D                             # theoretical shedding frequency
    f_exp = case["eddies"] / case["time"]
    print(f"{case['name']}: Re ≈ {Re:.1f}, f_theor ≈ {f_theor:.2f} Hz," +
          (f" f_exp ≈ {f_exp:.2f} Hz"))

Medium rod (slow): Re ≈ 133.3, f_theor ≈ 0.40 Hz, f_exp ≈ 0.40 Hz
Medium rod (moderate): Re ≈ 205.6, f_theor ≈ 0.62 Hz, f_exp ≈ 0.58 Hz
Medium rod (fastest): Re ≈ 533.3, f_theor ≈ 1.53 Hz, f_exp ≈ 1.60 Hz
Small rod: Re ≈ 150.0, f_theor ≈ 5.00 Hz, f_exp ≈ 5.88 Hz

Smoke Rings¶

In our experiment, we tried different shapes, a large triangle and a small circle.
Due to the nature of the experiment we cannot directly measure the speed of the smoke ring, so instead we use the distance traveled.

Large triangle
Time of ring motion is about t_ring = 0.2 s
Distance travelled L ≈ 0.25 m

U = L / t_ring = 0.25 / 0.2 = 1.25 m/s

Orifice diameter D = 4 cm = 0.04 m
Kinematic viscosity (air) ν = 1.5×10^-5 m²/s

Reynolds number:

R = U * D / ν = (1.25 m/s * 0.04 m) / 1.5×10^-5 m²/s = 4166.667 ≈ 4167

This is a high Reynolds number (Re > 3000), indicating turbulent onset. The rings form a donut shape but dissipate quickly.

Small circle
Time of ring motion is about t_ring = 0.6 s
Distance travelled L ≈ 0.30 m

Using v = d/t:

U = L / t_ring = 0.30 / 0.6 = 0.5 m/s

Orifice diameter D = 1.2 cm = 0.012 m
Kinematic viscosity (air) ν = 1.5×10^-5 m²/s

Reynolds number:

R = U * D / ν = (0.5 m/s * 0.012 m) / 1.5×10^-5 m²/s = 400

This is a low Reynolds number, evident as the ring moves smoothly with little mixing until dissipation.

Discussion¶

Vortex shedding¶

In our experiment we measured the shedding frequency using rods of different diameters and dragging them through water at different speeds. Our theoretical results were consistent with the measured experimental frequencies within uncertainty. We proved that for higher stream velocity, Reynolds number and, thereby, the shedding frequency increase. We can also increase shedding frequency by decreasing the diameter of the rod (or making an experiment in an even lower viscous liquid).

Smoke rings¶

We can understand that theoretically the larger the surface area for air to be pushed out the greater the ring will be and the greater of a chance there is for instability. This is consistent with our results as we see a great example of stability between our two tests which varies in length.

Sources of error¶

Sources of error might include:

Vortex shedding frequency:

  • Inconsistent flow rates in the flume because of the manually dragged rod
  • Boundary layers due to relatively narrow tank
  • Limited visibility due to dye diffusion

Smoke rings:

  • Irregular membrane tapping in the smoke ring experiment
  • Air movement within the room causing smoke rings to travel in unexpected way
  • Camber’s hole being tampered with by smoke pipe causing their to be holes.
  • Limited visibility due to smoke colour in comparison to room settings.

Conclusion¶

Both vortex shedding and smoke‐ring experiments demonstrate how flow instabilities produce organized vortical structures. Vortex shedding behind a moving rod follows the classic Kármán street behavior, with shedding frequency scaling as (f ~ 0.20 U/D), while smoke rings illustrate toroidal vortex formation and self‐propelled motion governed by circulation and impulse conservation.

Appendix: Video References¶

  • Low Reynolds:
    https://youtu.be/qMi_C0Xz75Q

  • Medium Reynolds:
    https://youtu.be/KIP43nbmVd4

  • High Reynolds:
    https://youtu.be/RWj0m-9ercY

  • Small rod:
    https://youtu.be/B41XbK0uJYs

  • Smoke vortex rings – small circle:
    https://youtu.be/51iLJZrrSV8

  • Smoke vortex rings – triangle:
    https://youtu.be/Pe07rW3h2QY

References¶

  1. Wikipedia Contributors. “Vortex Ring.” Wikipedia, The Free Encyclopedia, Wikimedia Foundation, December 5, 2019. https://en.wikipedia.org/wiki/Vortex_ring (accessed April 16, 2025).

  2. Darrigol, Olivier. “Stability and Instability in Nineteenth-Century Fluid Mechanics.” Revue d’Histoire des Mathématiques 8, no. 1 (2025): 5–65. https://www.numdam.org/item/RHM_2002__8_1_5_0/ (accessed April 16, 2025).

  3. Wikipedia Contributors. “Vortex shedding.” Wikipedia, The Free Encyclopedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Vortex_shedding (accessed April 16, 2025).

  4. Wikipedia Contributors. “Flow separation.” Wikipedia, The Free Encyclopedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Flow_separation (accessed April 16, 2025).