What is the von K�rm�n vortex street?

The von K�rm�n vortex street (named after Theodore von K�rm�n, 1911) is a repeating pattern of swirling vortices shed alternately from the top and bottom of a bluff body (such as a cylinder) in a flowing fluid. Above a critical Reynolds number (~47 for a circular cylinder), the symmetric wake becomes unstable and vortices are shed periodically, creating two parallel rows of counter-rotating vortices downstream. The shedding frequency is characterised by the Strouhal number St = f�D/U, which equals approximately 0.2 for a circular cylinder over a wide range of Reynolds numbers (200�200,000).

What is the Lattice Boltzmann Method (LBM)?

The Lattice Boltzmann Method (LBM) is a computational fluid dynamics method that simulates fluid flow by tracking the evolution of particle distribution functions on a discrete lattice rather than directly solving the Navier-Stokes equations. In the D2Q9 variant (2D, 9 velocities), each grid cell stores 9 distribution functions f_i corresponding to particles resting or moving in 8 directions. The BGK collision operator relaxes f toward equilibrium: f_i ? f_i + (f_i_eq - f_i)/t, where t is the relaxation time related to kinematic viscosity by ? = cs�(t - �). LBM naturally parallelises, handles complex geometries via simple bounce-back boundary conditions, and efficiently visualises vorticity and flow features.

Why does a splitter plate suppress vortex shedding?

Vortex shedding arises because the wake behind a bluff body is unstable: vorticity from the top and bottom shear layers interacts and triggers alternating roll-up. A splitter plate � a thin plate aligned with the flow and attached to the rear of the cylinder � separates the two shear layers and prevents the interaction that triggers roll-up. This delays or completely suppresses vortex shedding, reducing drag by 30�50% and eliminating the alternating lift force (vortex-induced vibration). The same principle is used in bridge deck design, chimney stabilisers, and marine riser control.

How does Reynolds number control vortex shedding?

Re = ?UD/� = UD/? (ratio of inertial to viscous forces). For a circular cylinder: Re 1000: turbulent wake. The LBM simulator models the 2D incompressible regime Re 50�400, covering the transition from steady wake to fully developed vortex street.

Vortex Shedding Simulator � von K�rm�n Vortex Street & Fluid Instabilities

The von K�rm�n Vortex Street

When a uniform flow encounters a bluff body (a cylinder, bridge pier, chimney, or cable), the wake behind it undergoes a dramatic instability above a critical Reynolds number. The symmetric wake breaks down into an alternating pattern of counter-rotating vortices � the von K�rm�n vortex street, first analysed mathematically by Theodore von K�rm�n in 1911.

The shedding frequency f is remarkably universal, governed by the Strouhal number:

St = f � D / U � 0.2 (Re 200 � 200,000, circular cylinder) where D = cylinder diameter, U = freestream velocity, f = shedding frequency

The key regimes for a circular cylinder:

Re	Flow Regime	Wake Character
< 5	Creeping / Stokes flow	No separation, symmetric streamlines fore/aft
5 � 48	Steady separated	Symmetric recirculation bubbles; no shedding
~47	Hopf bifurcation	Onset of oscillatory instability (symmetry breaking)
48 � 180	Laminar shedding	Periodic vortex street; St � 0.12�0.20
180 � 400	Transitional	3D instabilities (Mode A/B); St � 0.19�0.21
400 � 3�105	Subcritical turbulent	Laminar b.l., turbulent wake; St � 0.20
> 3�105	Supercritical turbulent	Turbulent b.l.; drag crisis; St � 0.27

Engineering Consequences

Vortex-induced vibration (VIV) occurs when the shedding frequency locks onto a structural natural frequency. Famous examples:

Tacoma Narrows Bridge (1940) � catastrophic torsional flutter driven by vortex-induced resonance at Re � 1.1 � 107 (wind speed 19 m/s, D = 8 m deck width).
Chimney stacks � industrial chimneys vibrate at Strouhal frequencies; helical strakes (fin-like spirals) break the coherence of shedding to suppress VIV.
Offshore risers � flexible pipes connecting seabed pipelines to surface platforms can vibrate destructively at VIV frequencies; active flow control or fairings are used.
Overhead power cables � galloping and vortex-induced singing; dampers are added to transmission lines.

Lattice Boltzmann Method (D2Q9)

Rather than solving the incompressible Navier-Stokes equations directly, this simulator uses the Lattice Boltzmann Method (LBM) � a mesoscopic approach that tracks particle distribution functions f_i(x, t) on a lattice. Each cell stores 9 populations corresponding to particles resting or moving in 8 compass directions.

D2Q9 Lattice Velocities

Speeds (cx, cy): e0=(0,0) e1=(1,0) e2=(0,1) e3=(-1,0) e4=(0,-1) e5=(1,1) e6=(-1,1) e7=(-1,-1) e8=(1,-1) Weights: w0=4/9 w1?4=1/9 w5?8=1/36 (cs� = 1/3)

BGK Collision (Single Relaxation Time)

f_i(x, t+1) = f_i(x, t) + (f_i^eq - f_i) / t f_i^eq = w_i � ? � [1 + (e_i�u)/cs� + (e_i�u)�/(2cs4) - u�/(2cs�)] Macroscopic: ? = S f_i u = S f_i � e_i / ? Viscosity: ? = cs��(t - �) = (t - 0.5)/3 [lattice units] Reynolds: Re = U�D / ? ? t = U�D/(3�Re) + 0.5

Boundary Conditions

Boundary	Type	Implementation
Cylinder surface / walls	No-slip (bounce-back)	f_i ? f_OPP(i) at obstacle nodes � reverses momentum
Left inlet	Constant velocity (Dirichlet)	f_i set to f_i^eq(?=1, U0, 0) each step
Right outlet	Zero-gradient (extrapolation)	f_i at x=N-1 copied from x=N-2 each step
Top/bottom	No-slip walls	Bounce-back (included in obstacle mask)

Vorticity Visualisation

? = ?uy/?x - ?ux/?y � (uy[x+1,y] - uy[x-1,y] - ux[x,y+1] + ux[x,y-1]) / 2 Colour map: cyan (#22d3ee) = negative ? (clockwise rotation) dark (#001015) = zero vorticity orange (#f97316) = positive ? (counter-clockwise)

Preset Scenarios Explained

🌀 Low Re (Re = 50)

Below the critical Reynolds number (~47), the wake behind the cylinder is steady and symmetric. Two recirculation bubbles (F�ppl vortices) sit in the near wake but no shedding occurs. The drag is dominated by pressure (form) drag. Observe the stable symmetric vorticity pattern without any periodic behaviour.

🔄 Transition (Re = 120)

Just above the Hopf bifurcation (~Re 47), the symmetric wake becomes unstable and periodic shedding begins. The vortex street is laminar and very regular � nearly sinusoidal lift oscillation on the cylinder. The shedding frequency matches St � 0.16�0.18 at this Re.

🌀 Vortex Street (Re = 200)

The classic von K�rm�n vortex street at Re = 200: strong alternating vortices are shed and convect downstream, forming two staggered rows. The street persists far downstream. St � 0.20. The alternating lift force oscillates at f_shed; drag oscillates at 2�f_shed.

🔗 Tandem Cylinders (Re = 160)

Two cylinders placed in series, centre-to-centre distance ~4.5D. The downstream cylinder interacts with the wake of the upstream one � both modifying and being modified by the vortex street. Gap ratio, Re, and cylinder spacing determine whether the wakes merge immediately or whether the downstream cylinder experiences galloping or bistable gap flows.

🏛️ Near Wall (Re = 200)

A cylinder near a flat wall with gap ratio G/D � 0.7. The wall modifies the vortex street asymmetrically: wall-side vortices are suppressed and shed at lower frequency. Below G/D � 0.3, shedding ceases entirely and the cylinder is deflected toward the wall (Magnus-Bernoulli effect). Used in modelling pipelines on the seabed.

🔇 Suppressed (Re = 220, splitter plate)

A rigid splitter plate extending 3.5D downstream of the cylinder separates the shear layers and prevents the roll-up interaction that triggers shedding. This eliminates the alternating lift force and reduces drag by ~40%. The wake remains a thin, steady central jet flanked by two symmetric vorticity layers that diffuse without coalescing.

Key Equations and Analysis

Reynolds Number and Length Scale

Re = ? � U � D / � = U � D / ? ? = kinematic viscosity (m�/s) D = cylinder diameter (m) U = freestream speed (m/s)

Strouhal Number and Shedding Frequency

St = f_s � D / U � 0.198 (1 - 19.7/Re) [Williamson-Brown correlation, Re 50�200] St � 0.2 (Re 1000�200,000, nearly constant) ? f_s = St � U / D Period: T_s = 1/f_s = D/(St�U)

Mean Drag Coefficient vs Re (circular cylinder)

Re	C_D (mean)	C_L (amplitude)	St
50	~1.55 (steady)	0	� (steady)
100	~1.35	~0.30	~0.165
200	~1.30	~0.60	~0.195
400	~1.25	~0.75	~0.206
104�105	~1.0�1.2	~0.5�1.0	~0.20
>3�105	~0.35 (drag crisis)	�	~0.27

Vortex-Induced Vibration Lock-in Criterion

Lock-in occurs when: f_s � f_n (shedding � structural natural frequency) or equivalently: U_r = U / (f_n � D) � 1/St � 5 (reduced velocity at lock-in) During lock-in the structure controls the shedding frequency (synchronisation); amplitudes can grow to D/2 or more without additional damping.

Curriculum Connections

Topic	Qualification	Concepts
Fluid mechanics fundamentals	A-Level Physics / IB HL	Viscosity, Reynolds number, laminar vs turbulent flow
Aerodynamics and drag	A-Level / Engineering	Pressure drag, skin friction, boundary layer separation
Oscillations and resonance	A-Level / Undergraduate	Strouhal frequency, vortex-induced vibration, lock-in
Computational fluid dynamics	Engineering / Masters	LBM, finite difference/volume, boundary conditions
Structural engineering	Civil / Mechanical Engineering	Wind loading, VIV, fatigue, passive flow control

Vortex Shedding Simulator