Coanda Effect

The intuition — a jet that refuses to fly straight

Turn on a tap to a thin smooth stream and touch the convex back of a spoon to its edge. Instead of pushing the water away, the spoon yanks itself into the stream, and the water sheet wraps around the curve and shoots off sideways. That stubborn clinging — a free jet bending toward and following a nearby surface instead of going straight — is the Coanda effect.

It feels like the jet is "sticking" to the wall, but nothing glues it there. The real story is a feedback loop:

1. A jet is faster than the still fluid around it, so by viscosity it drags neighboring fluid along — this is entrainment.
2. With a wall on one side, the jet can't replenish the fluid it entrains there as easily as on the open side. Fluid gets evacuated, and the pressure in that gap falls below ambient.
3. The higher ambient pressure on the open side now pushes the jet toward the wall until it touches.
4. Once attached, the jet runs along the surface as a wall jet, and the curved wall keeps supplying the low pressure needed to bend the flow around it.

So the Coanda effect is not one law but a balance of three things: entrainment (viscosity), the pressure asymmetry attachment creates, and the centripetal force needed to turn the streamlines along a curve.

How it works — entrainment and the pressure balance

Take it one piece at a time.

Entrainment. A submerged turbulent jet doesn't keep all its momentum in the original stream. Shear at its edges spins up vortices that fling jet fluid outward and pull ambient fluid inward. A free round jet entrains so much that its total mass flow grows roughly linearly with distance — by several jet-diameters downstream it can be moving many times the fluid it started with. That hunger for surrounding fluid is the engine of attachment.

The pressure asymmetry. Put a surface a small distance from the jet. On the open side the jet entrains freely; on the wall side it cannot, so it draws down the pressure in the gap. The transverse pressure difference Δp across the jet width w produces a net sideways force that curves the jet toward the wall:

F_side ≈ Δp · (jet area facing the wall)

Staying on the curve. Once attached to a convex surface of radius R, the streamlines are bent. Bending flow of density ρ moving at speed U around radius R requires a centripetal pressure gradient pointing toward the center of curvature — i.e. the pressure right at the wall must be lower than the pressure away from it:

dp/dr = ρ·U² / R        (pressure rises outward across a curved jet)

The wall conveniently sits at the low-pressure side, so this is self-consistent: attachment creates the low pressure, and the low pressure is exactly what's needed to keep bending the jet around the wall. Tighter radius R (or faster U) demands a bigger pressure drop — and there's a limit to how big a drop the flow can sustain before it gives up and separates.

Why viscosity matters. In an idealized inviscid fluid there's no entrainment and no boundary layer, so the textbook Coanda feedback wouldn't start. Real attachment is a viscous, usually turbulent phenomenon — which is why it's strongest at moderate-to-high Reynolds numbers and weak or absent in very slow, syrupy flow.

The governing relations

The clean takeaways are the curved-jet pressure relation and the reaction force from turning the jet.

Centripetal pressure across the bend. For a thin jet of thickness w hugging a cylinder of radius R, the pressure difference between the wall and the free side is approximately:

Δp ≈ ρ · U² · w / R

This is the suction the wall must provide. It grows with speed squared and with how tightly you ask the jet to turn (small R) — the practical reason sharp corners break attachment.

Reaction force from deflected momentum. If the jet enters horizontally and leaves at angle θ after wrapping the surface, it has changed its momentum direction. By Newton's second and third laws, the surface feels a reaction. For a jet of mass flow ṁ = ρ·U·A and speed U:

F = ṁ · ΔU = ρ · U · A · U · (change in flow direction)
  → turning the jet by angle θ gives a transverse reaction ≈ ṁ·U·sin θ

This is the force that drags the spoon in, pulls the blown flap's air down to make extra lift, and is the basis of fluidic thrust vectoring.

Bernoulli, locally. Where the attached jet speeds up over the curve, Bernoulli's relation along a streamline says the static pressure falls:

p + ½·ρ·U² = constant   (along a streamline, low-loss region)

Bernoulli is a correct accounting of the pressure once the jet is attached and accelerating — but it does not by itself explain why the jet attached. Entrainment and viscosity do.

Regimes and conditions for attachment

Whether a jet attaches and how far it wraps depends on a handful of knobs.

Knob	Favors attachment	Favors detachment / separation
Surface curvature radius R	Gentle curve (large R)	Sharp curve or sudden corner (small R)
Jet speed / Reynolds number	Fast, turbulent jet (high Re)	Slow, low-Re jet
Offset between jet and wall	Small initial gap	Large gap — jet never finds the wall
Jet thickness w relative to R	Thin jet (small w/R)	Thick jet relative to curve
Pressure gradient along surface	Favorable or mild	Strong adverse gradient → boundary-layer separation
Surface roughness / steps	Smooth, continuous surface	Steps and gaps trip separation

For a turbulent wall jet blown tangentially around a cylinder, the jet can wrap a remarkable amount before letting go — separation typically occurs only after the surface has turned somewhere around 200–245°, far past the 90° you might naively expect. Reduce the speed or sharpen the curve and that wrap angle collapses.

Numbers — how much air a Coanda device actually moves

System	Coanda role	Concrete figure
Dyson-style bladeless fan	Slot jet wraps a ring airfoil, entrains room air	~15× amplification: ~15 L of air out for every 1 L through the slot
Wall jet around a cylinder	Tangential jet stays attached around the curve	Wraps to ≈200–245° before separation (turbulent, high Re)
Blown flap (STOL aircraft)	Engine bleed air blown over flap stays attached, turns down	Lift coefficient C_L raised from ~1.5 to 3–5+
NOTAR helicopter tail	Slot air on the tail boom + downwash makes anti-torque side force	Removes the tail rotor entirely; quieter, safer
Fluidic (no-moving-parts) valve	Control jet flips a power jet between two attached walls	Switches with no mechanical parts; works at high temperature
Spoon under a tap	Water sheet wraps convex back, deflects sideways	Visible sideways tug toward the stream — a kitchen demo

The bladeless-fan figure is the headline number: most of the breeze you feel is room air the small primary jet has entrained, not air pushed through the motor — which is exactly why it feels smooth and "bladeless."

Where the Coanda effect shows up

• Bladeless fans and air multipliers. A thin slot jet wraps a ring-shaped airfoil and drags in roughly an order of magnitude more room air.
• High-lift wings (blown flaps, circulation control). Air blown tangentially over a rounded flap or trailing edge stays attached and turns sharply down, boosting lift for short take-off and landing aircraft.
• NOTAR helicopters. No tail rotor — slotted air around the tail boom plus the main-rotor downwash uses Coanda attachment to make an anti-torque force.
• Fluidic logic and oscillators. Jets that attach to one of two walls form bistable switches and flip-flops with no moving parts — used where electronics can't survive (extreme heat, radiation). Windshield-washer fluidic nozzles fan the spray this way.
• Thrust vectoring and flow control. Small control jets steer a much larger main jet by changing where it attaches.
• Combustion and HVAC. Coanda nozzles spread or steer flames and ventilation air along ceilings and walls without ducting.
• Everyday surprises. The spoon-under-a-tap demo, the way a ping-pong ball can ride an angled air jet, and why smoke from a candle hugs a nearby wall.

Worked example — a wall jet wrapping a cylinder

A thin air jet of speed U = 30 m/s and thickness w = 2 mm blows tangentially around a cylinder of radius R = 25 mm. Air density ρ ≈ 1.2 kg/m³.

Estimate the suction the wall must supply to bend the jet:

Δp ≈ ρ · U² · w / R
   = 1.2 · (30)² · (0.002) / (0.025)
   = 1.2 · 900 · 0.002 / 0.025
   ≈ 86 Pa  (below ambient, against the wall)

That's about 0.09% of atmospheric pressure — small in absolute terms, but it acts over the whole wrap and is enough to hold the jet on the curve. Now halve the radius to R = 12.5 mm: the required Δp doubles to ≈172 Pa. Keep tightening the curve and you eventually demand more suction than the flow can sustain, the boundary layer separates, and the jet flies off straight. That single relation — Δp scaling as U²w/R — explains both why fast jets wrap so far and why a sharp corner kills attachment.

Common misconceptions and edge cases

• "It's just Bernoulli." Bernoulli explains the pressure once the jet is attached and accelerating, not why it attached. Attachment is driven by entrainment and viscosity — kill the viscosity and the effect disappears.
• "Lift is just the Coanda effect." Air does follow a wing's curved top (Coanda-style downwash), but quantitative lift needs circulation and the Kutta condition. Coanda is part of the story, not the whole of it.
• "The jet sticks to any surface." Only while the wall can supply the low pressure to bend it. Too sharp a curve, too slow a jet, or a step in the surface and the boundary layer separates.
• "It works the same in honey as in air." No — it's a high-Reynolds-number, turbulent-entrainment phenomenon. Very viscous, low-speed flow doesn't show classic Coanda attachment.
• "Coanda attaches flow to concave surfaces too." The dramatic free-jet attachment is a convex-surface phenomenon. On a concave wall the flow is already turning into the surface for other reasons.
• "It explains why you can blow a paper strip up." Often cited, but that demo mixes several effects; be careful attributing every blow-and-bend trick purely to Coanda.