Speed 0 BL/sTail beat 0 HzSt 0School saving 0 %Fish 0FPS 0

Rings: green fish save swimming energy in the wake of fish ahead, red fish pay extra. Blue streaks are the induced flow. Drag to orbit, wheel or pinch to zoom. Slow motion is on so the tail beat is readable.

Fish School Hydrodynamics

Every tail beat sheds a pair of vortices. Where should a neighbour swim, and when should its tail beat, to use less energy?

Swimming

School geometry

Live readouts

Saving of every fish (front to back)

Follower map behind one leader

Energy saving of one tuned follower for every position behind a single leader (green = saves, red = pays). The dot is your current spacing.

Experiment: saving vs streamwise spacing

Goals

Independent numerical checks

Controls

Drag or one finger: orbit; wheel or pinch: zoom. Presets set the formation (line, diamond, rectangle, V-echelon, abreast phalanx).
Tail phase: sync = every fish beats in phase; tuned = each fish, from the front to the back, picks the beat phase that costs least in the wakes of the fish ahead.
Slow motion slows the animation and the flow but not the numbers. Top view shows only the analysis layer. Quality Low = fewer layers and tracers (also ?q=low).

The science

Tail beat. A fish of length L swimming at U beats its tail with amplitude A = 0.2 L at frequency f; efficient swimmers sit near the Strouhal number St = f A / U ≈ 0.3, so f = St U / A.

Wake. Each half period the tail sheds a vortex of alternating sign (a reverse Kármán street, which carries a backward jet and so makes thrust). Each vortex is a Lamb-Oseen vortex, v_θ = Γ/(2πr) (1 − e^−r²/r_c²), with Γ = c_Γ U A, a core that grows as r_c² = r_c0² + 4ν_et and a circulation that decays over about 2.5 beats. The street convects at 0.85 U relative to the water, so its wavelength is λ = 0.85 U/f and the mean jet speed between the rows is Γ/λ.

Cost model. Relative to a solo fish: P/P_solo = (1 − q) ⟨(1 − u_x/U)³⟩ + q ⟨(ḣ − v_w)²⟩ / ⟨ḣ²⟩. The first term is body drag in the induced streamwise flow u_x (averaged along the body); the second is the lateral work of the tail against the wake's lateral flow v_w at the tail, with ḣ = 2πfA cos(ωt + φ) and q = 0.5 the share of lateral work in the solo cost. A tail that beats with the incoming wake flow does less work. Saving = 1 − P/P_solo.

What the model shows. A fish directly behind another swims into its backward jet and pays a lot (line, rectangle). Fish staggered between the wakes of the fish ahead (echelon, diamond) can save a few percent when their tail phase is tuned and lose when it is not. The numbers are model values: c_Γ, q, the 0.85 convection factor, the decay time and the eddy viscosity are illustrative assumptions [перевірити за джерелом].

Simplifications. 2-D wake in one horizontal layer (the other layers repeat it without interaction); vortices do not move each other (frozen wake); each fish feels only the fish ahead; no body-vortex collision, no 3-D vortex rings, no turbulence, constant speed, rigid lattice. Real school savings depend on all of these. Virtual animals only; no real fish are involved.

Blender-modelled fish and seabed, Three.js r160. Educational model values.