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Marine Renewables

Tidal Stream Turbine Cp & TSR Simulator

Compare the heart of every tidal turbine — the power coefficient Cp and the tip-speed ratio TSR — across four families: horizontal-axis HATT, vertical-axis Darrieus and Savonius, and ducted rotors. See how close you get to the Lanchester-Betz limit of 16/27 ≈ 59.3% in real time, alongside extracted power, annual energy and the cavitation number σ.

Parameters
Turbine type
HATT / Darrieus / Savonius / ducted change Cp_max and optimum TSR
Rotor diameter D
m
Tidal velocity V
m/s
Spring-tide peak velocity; commercial sites are typically 2-4 m/s
Rotor speed N
RPM
Number of blades B
Pitch angle θ
°
Negative = feathering; above 15° flow stalls and power drops
Depth below surface h
m
Hub depth below mean sea level; feeds σ
Results
—
Tip-speed ratio TSR
—
Power coefficient Cp
—
% of Betz limit
—
Extracted power (kW)
—
Annual energy (MWh/y)
—
Cavitation σ
—
Subsea turbine view — blade rotation and tidal flow

Tidal flow approaches the rotor installed below the sea surface and drives the blades. The Cp gauge on the right shows the current power coefficient against the Lanchester-Betz limit.

Cp – TSR curve (current turbine type)
Cp_max comparison by turbine type
Theory & Key Formulas

$$P = \tfrac{1}{2}\rho A V^{3} C_p,\qquad TSR = \frac{\Omega R}{V},\qquad C_{p,\max} = \frac{16}{27} \approx 0.593$$

Extracted power P and tip-speed ratio TSR. ρ = 1025 kg/m³ (seawater), A = πR² is the swept rotor area, Ω is rotor angular velocity, V is the free-stream speed. The Lanchester-Betz limit 16/27 is the theoretical maximum for an open-flow rotor.

$$C_p(\lambda) \approx C_{p,\max}\,\exp\!\left[-\left(\frac{\lambda-\lambda_{opt}}{3}\right)^{2}\right]$$

A simple bell-shaped Cp(λ) model. λ = TSR, λ_opt depends on rotor family (HATT 6, Darrieus 4, Savonius 1, ducted 5).

$$\sigma = \frac{p_{atm}+\rho g h - p_{vap}}{\tfrac{1}{2}\rho U_{tip}^{2}}$$

Cavitation number σ. h is hub depth, U_tip = ΩR is the tip speed, p_vap ≈ 2500 Pa (15°C). σ < 0.3 is high risk.

Tidal Stream Turbine Power Coefficient Cp — Lanchester-Betz Limit

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A tidal stream turbine is basically an underwater wind turbine, right? What is the actual difference from a wind machine?
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The principle is the same — extract kinetic energy from a moving fluid with a rotor. The only change is that the fluid goes from "air" to "seawater" and gets ~800× denser. At the same diameter and flow speed the theoretical power is 800× larger. Tidal currents are slower than typical winds, so in practice the output is still 10-20× the power density of wind. MeyGen in Scotland already runs four 1.5 MW units for 6 MW grid-connected output.
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Got it! I keep hearing the word "Cp" — what is it, and what's the magic 16/27 number?
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Cp is the "power coefficient", the fraction of incoming fluid energy that the rotor actually captures: Cp = P / (½ρAV³), where A is the swept area and V is the free-stream speed. The famous 16/27 ≈ 0.593 is the Lanchester-Betz limit — Lanchester (1919) and Betz (1920) proved independently that a single open-flow rotor cannot exceed it. Commercial HATT runs at Cp = 0.40-0.45, and ducted designs can apparently exceed it (more on that below).
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Wait — ducted turbines beat Betz? Is that some "limit-breaking" technology?
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Great question. Betz's formula uses the rotor swept area as the reference. A diffuser collects extra flow through the rotor face, so the Cp computed on rotor area reaches 0.6-0.7. If you re-define Cp on the diffuser exit area, 16/27 still holds. It is a definition choice, not a physics violation. Sabella D10 in France and OpenHydro are real-world examples; they work well even where currents are weak.
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Moving the TSR slider draws a bell-shaped Cp curve. What is TSR doing here?
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TSR is the "tip-speed ratio" = ΩR / V (blade tip speed divided by flow speed). Too low, and wake swirl wastes energy; too high, and drag plus turbulent stall take over. So there is an optimum: HATT around 5-7, Darrieus around 4, drag-based Savonius around 1. Real machines vary the blade pitch and rotor speed so they always sit near TSR_opt as the tide rises and falls. That is how Cp_max is held in practice.
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Last thing — what is the cavitation number σ checking for?
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Blade tips move fastest, and if the local pressure there drops below the saturation vapour pressure of water (~2500 Pa at 15°C), vapour bubbles form. When those bubbles collapse, microjets pit the blade surface — that is cavitation damage. σ measures the margin: < 0.3 is high risk, ≥ 1.0 is safe. Fixes are (1) install deeper to raise static pressure, (2) reduce RPM to lower the tip speed, (3) use cavitation-tuned sections (NACA 6-series). Ship propellers have fought this for a century, and tidal turbines inherit the same problem.

Frequently Asked Questions

It is the theoretical upper bound on the fraction of kinetic energy that a single open-flow rotor can extract from a fluid, derived from ideal actuator-disc (momentum) theory. The value is 16/27 ≈ 0.593 (59.3%), proved independently by Lanchester in 1919 and Betz in 1920. The same limit applies to both wind turbines and tidal stream turbines, but ducted (diffuser-augmented) machines redirect more flow through the rotor area, so a Cp defined on rotor area can reach 0.6-0.7.
If TSR = ΩR/V (blade tip speed divided by free-stream speed) is too low, wake swirl losses dominate; if it is too high, drag and turbulent stall between blades dominate. Horizontal-axis tidal turbines (HATT) peak around TSR = 5-7, Darrieus rotors near TSR = 4, and drag-based Savonius rotors near TSR = 1. Commercial machines run variable-pitch / variable-speed control to track the optimum TSR through the tidal cycle and hold Cp near its peak.
Power scales with fluid density ρ. Seawater is 1025 kg/m³ versus 1.225 kg/m³ for air — about 837× denser. At the same flow speed and rotor diameter the theoretical power is ~800× larger. Real tidal flows are slower than typical winds (2-3 m/s vs 8-12 m/s), and power scales as the cube of speed, but tidal stream still produces 10-20× the power density of wind. As a bonus, tides follow celestial mechanics, so output is predictable on a 12.4 h cycle — great news for the grid.
σ = (p_atm + ρgh − p_vap) / (½ρU²). When local pressure near the blade tip drops below the vapour pressure, vapour bubbles form (cavitation) and pit the blade surface when they collapse. σ < 0.3 is high risk, 0.3-1.0 is marginal, and ≥ 1.0 is generally safe. Mitigation: (1) install deeper to raise the static pressure, (2) increase rotor diameter and reduce RPM to drop the tip speed, (3) use cavitation-tuned blade sections such as NACA 6-series.

Real-World Applications

MeyGen (Pentland Firth, Scotland): The world's largest commercial tidal-stream project. SIMEC Atlantis Energy operates four AR1500 turbines (rotor D = 18 m, rated 1.5 MW) for a total of 6 MW grid-connected. The default settings of this tool (D = 18 m, V = 2.5 m/s) are very close to MeyGen, and field Cp values sit in the 0.40-0.45 band. The project roadmap targets a 86 MW phase expansion and an eventual 398 MW.

Sabella D10 (Brittany, France): A horizontal-axis, un-ducted machine — D = 10 m, rated 1 MW. With six blades and no diffuser, it is tuned for low-flow sites (~2 m/s) and once provided primary power to the Ushant island grid. Reproduce it in this tool with "Horizontal-axis (HATT)" and D = 10 m.

OpenHydro / ducted designs: Naval Energies' rim-driven, ring-mounted ducted turbine has an unmistakable look. Select "Ducted (HATT)" here and the Cp_max jumps to 0.55. Pilots in Canada's Bay of Fundy (16 m tidal range) demonstrated the concept but also exposed the harsh reality of turbulent tidal sites — mechanical failures showed how tough this market is.

Pre-sizing for BEM and CFD: Before running a full Blade Element Momentum (BEM) analysis or a CFD case (OpenFOAM interFoam / k-ω SST), use a Cp(TSR) estimator like this to fix turbine family, diameter and RPM. If a CFD Cp differs from the estimate by an order of magnitude, it is a strong hint to recheck mesh quality and turbulence model settings.

Common Misconceptions and Pitfalls

The biggest trap is "ducted turbines break the Betz limit". In reality the diffuser just funnels more flow through the rotor area; if you redefine Cp using the diffuser inlet area, 16/27 still holds. When reading papers, always check whether Cp is defined on rotor area, diffuser inlet area, or a virtual free-stream area. Mix these up and a ducted machine looks 50% better than HATT — or exactly equivalent. This tool uses rotor area as the reference, so the Cp_max = 0.55 for "ducted" is a convenience figure, not a Betz violation.

Next is "annual energy = rated power × 8760 h". Tidal flow varies sinusoidally between zero and a peak on a 12.4 h cycle, so the mean speed is closer to √(2)/2 of the peak and the mean power is 1/8 to 1/4 of the rated value. Add maintenance and slack-water shutdowns and the realistic capacity factor (CF) lands at 30-40%. This tool assumes CF = 0.35; tune it to site-specific data. Any business case quoting CF > 0.5 deserves a second look.

Finally, the cavitation σ trap of looking only at the mean static pressure. Near the blade leading edge the local pressure can drop 2-5× below the static value, so even a mean σ > 1.0 can fail locally. Use BEM or CFD to inspect the blade-surface pressure coefficient distribution and verify that the minimum Cp_min > −(σ − 0.05). Installing deeper helps most, but in sites with large tidal range the low-tide surface clearance must also be checked.

How to Use

  1. Enter rotor diameter (3–20 m typical for commercial tidal turbines) in the first field
  2. Input tidal velocity (0.5–2.5 m/s for deployable sites; neap vs. spring cycles affect this)
  3. Set rotor RPM (15–50 rpm typical; higher RPM increases TSR but reduces torque)
  4. Specify blade count (3–4 blades standard; affects Cp profile and cavitation risk)
  5. Read TSR, Cp, Betz efficiency %, extracted power in kW, and cavitation number σ from outputs
  6. Adjust RPM to optimize Cp near 0.45–0.50 for your velocity regime

Worked Example

Horizontal-axis tidal turbine: rotor diameter 10 m, tidal velocity 1.8 m/s, 25 rpm, 3 blades. Blade tip speed = π × 10 m × (25/60 Hz) = 13.1 m/s. TSR = 13.1 / 1.8 = 7.28. At TSR 7.28, Cp typically reaches 0.48 (95.2% of 0.504 Betz limit). Rotor area = 78.5 m². Available power = 0.5 × 1025 kg/m³ × 78.5 m² × (1.8)³ = 233 kW. Extracted power = 0.48 × 233 = 112 kW. Annual energy at mean 1.8 m/s ≈ 980 MWh/y. Cavitation σ = (patm − pvap) / (0.5ρV²) ≈ 5.2 (safe; threshold typically σ > 2.0 for seawater).

Practical Notes

  1. Spring tide velocity spikes 30–40% above neap conditions; re-run simulator for both scenarios to size power electronics and structural ratings
  2. Cp peaks near TSR 6–8 for most horizontal-axis designs; deviation causes stall or excessive tip-speed losses in seawater (kinematic viscosity = 1.0 × 10⁻⁶ m²/s)
  3. Cavitation risk increases above 3 m/s velocity and shallow depths (<15 m); validate σ > 3.0 margin for blade erosion prevention
  4. Moorings and drive-train fatigue scale with annual MWh; 800+ MWh/y favors larger capital investment in gearbox and generator redundancy