Physique du vol stationnaire et conception des rotors

1. Actuator-Disk (Momentum) Theory and Disk Loading

For a hovering rotor modeled as an actuator disk of area $A$ in air of density $\rho$, the mass flow rate through the disk is $\dot m = \rho A v_i$, where $v_i$ is the induced velocity at the disk. The far wake velocity is approximately $2 v_i$ in hover, so momentum theory gives thrust[^1]

$$T = 2 \rho A v_i^2. \tag{1}$$

Rearranging, the induced velocity is

$$v_i = \sqrt{\frac{T}{2 \rho A}}. \tag{2}$$

The disk loading is defined as $T/A$. Combining with equation (2) shows that higher disk loading implies higher induced velocity. The ideal induced power is $P_i = T v_i$, so[^2][^1]

$$P_i = T \sqrt{\frac{T}{2 \rho A}}. \tag{3}$$

For fixed thrust, increasing $A$ (larger disk) reduces $v_i$ and hence induced power, which is why large rotors are fundamentally more efficient in hover and produce gentler downwash. Helicopters typically operate at disk loading around 5–10 lb/ft² (≈24–48 N/m²), whereas high-disk-loading VTOL fans can exceed 500 lb/ft², with much higher induced power and downwash.[^3][^1][^2]

Example: 5 kg vehicle within 55 cm footprint

Take a 5 kg AUW multirotor at sea level. Weight is $T = mg ≈ 49\,N$. Suppose a single circular rotor nearly fills the 55 cm diameter envelope: $D = 0.55\,m$, radius $R=0.275\,m$, disk area $A = \pi R^2 ≈ 0.238\,m²$.

• Disk loading: $T/A ≈ 49 / 0.238 ≈ 206\,N/m²$.
• Induced velocity: with $\rho = 1.225\,kg/m³$ and equation (2), $v_i ≈ \sqrt{49/(2·1.225·0.238)} ≈ 9.5\,m/s$.
• Ideal induced power: $P_i = T v_i ≈ 49 · 9.5 ≈ 465\,W$.

This disk loading is higher than full-scale helicopter values but in the range studied for UAM-scale multirotors (e.g., 287–861 N/m²), which show the expected trend of increasing power with disk loading.[^4][^3]

If the same thrust is shared by multiple rotors whose total disk area sums to $A_{tot}$, equations (2)–(3) hold with $A = A_{tot}$ (neglecting interference). Thus, maximizing total rotor disk area within a 55 cm footprint directly minimizes ideal hover power.

2. Figure of Merit and Power Loading

The hover figure of merit (FM) is defined as

$$\text{FM} = \frac{P_{ideal}}{P_{actual}}, \tag{4}$$

where $P_{ideal}$ comes from momentum theory (equation 3) and $P_{actual}$ is measured shaft power including profile drag, tip losses, non-uniform inflow, and mechanical losses. An ideal actuator disk has FM = 1, but practical rotors achieve FM ≈ 0.75–0.8 for good full‑scale helicopter rotors and ≈0.5 or less for poor designs.[^5][^6][^7]

Small UAV rotors tend to have lower FM due to higher relative profile drag, Reynolds-number effects, and manufacturing tolerances. Experiments on a micro coaxial UAV achieved a best single-rotor FM ≈ 0.42, with BEMT analysis indicating that profile drag accounted for about 45% of the losses versus around 30% in full-scale helicopters.[^8][^9]

Power loading (thrust per power) can be written as $T/P_{actual} = \text{FM}·T/P_{ideal}$. Higher FM directly improves power loading. Helicopters with disk loading around 10 lb/ft² typically achieve power loadings of about 4 kg/kW (≈39 N/kW), while higher disk loading VTOL concepts have worse power loading.[^2][^3]

Realistic efficiency for a 5 kg multirotor

For the 5 kg, 0.55 m rotor example, ideal induced power was $≈465\,W$. If a well-designed rotor system achieves FM ≈ 0.65–0.7 (reasonable for small optimized rotors, lower than large helicopters but higher than early micro UAVs), the actual hover power per rotor system becomes $P_{actual} ≈ P_{ideal} / \text{FM} ≈ 465/0.7 ≈ 660\,W$.[^7][^9][^5]

The corresponding power loading is $T/P_{actual} ≈ 49/660 ≈ 0.074\,N/W$, or ≈13.5 g/W. This is broadly consistent with empirical multicopter design tools and literature values for small multirotors hovering near optimal operating points.[^10][^11]

3. Tip Speed, Mach Number, and Prop Size

The blade tip speed is $V_{tip} = \Omega R$, where $\Omega$ is angular speed and $R$ rotor radius. The corresponding tip Mach number is $M_{tip} = V_{tip}/a$, with $a$ the speed of sound (≈340 m/s at sea level).[^12][^13]

Propeller designers generally keep $M_{tip}$ below ≈0.85–0.9 for efficient operation and to avoid sharp drag and noise increases associated with compressibility and shock formation. For low-noise applications, even lower $M_{tip}$ (≈0.5–0.7) are desirable, while too low a tip speed can also reduce efficiency by increasing induced and profile losses per unit thrust.[^14][^15][^13][^12]

For a 0.55 m rotor, keeping $M_{tip} ≤ 0.6$ implies $V_{tip} ≤ 0.6·340 ≈ 204 m/s$, so the RPM limit is

$$\text{RPM} = \frac{60 V_{tip}}{2 \pi R} \approx \frac{60·204}{2 \pi·0.275} \approx 7100\,\text{rpm}. \tag{5}$$

In practice, a quiet, efficient rotor would likely operate significantly below this upper bound (e.g., 2000–4000 rpm) while still providing required thrust, provided the disk area and blade geometry are adequate.[^16][^17]

The key intuition is that for a given thrust, a larger rotor can operate at a lower induced velocity and lower tip speed (for the same thrust coefficient), reducing induced power and compressibility-related profile losses. A small rotor must run at higher $\Omega$ and/or higher lift coefficient, increasing profile drag and often operating at less favorable Reynolds numbers, which degrades efficiency.[^7][^16]

4. Blade Geometry: Blades, Solidity, Chord, Pitch, Aspect Ratio, Taper, Twist

Solidity and chord

Blade solidity $\sigma$ is the ratio of blade planform area to disk area: $\sigma = B c_{avg} / (\pi R)$ for a rotor with $B$ blades and average chord $c_{avg}$. Higher solidity allows the rotor to generate the same thrust at lower blade lift coefficient (and potentially lower angle of attack), which can reduce profile drag and delay stall, but increases blade area and drag if overdone.[^18][^17][^5]

For a given thrust coefficient, there is an optimal range of $C_T/\sigma$ (mean lift coefficient) that maximizes FM; too low solidity forces high $C_l$ and high profile drag, while too high solidity adds unnecessary wetted area. Small UAV blade optimization studies using BEMT and testing have shown FM improvements from ≈0.31 to ≈0.68 by optimizing rotational speed, chord, taper, twist, airfoil profile, and blade number.[^17][^5][^18]

Pitch, twist, and aspect ratio

• Pitch/collective: Higher pitch at a given $\Omega$ increases thrust but also increases power and risk of stall. Optimal hover design sets collective such that most of the blade operates near the airfoil’s maximum $L/D$ at the design thrust.
• Twist: Ideal twist distributes angle of attack so each radial station operates near optimal $L/D$, compensating for the fact that local inflow angle varies with radius due to changing tangential velocity and induced velocity. Untwisted blades are easier to manufacture but less efficient.[^19][^20][^18]
• Aspect ratio: High aspect ratio blades (large span, small chord) reduce induced drag per unit lift but can be structurally challenging. For small rotors, moderate aspect ratios are common; span is capped by the 0.55 m envelope.

Blade count

Increasing blade count at fixed disk area and thrust increases solidity, allowing lower $C_l$ per blade and lower tip loading, which can reduce noise and redistribute induced velocity, but adds profile drag and weight. There is a trade-off: for small UAV rotors, 2–3 blades are typical; more than 3 blades per rotor for small coaxial configurations is often not recommended due to mechanical complexity and diminishing returns.[^21][^22][^5][^7]

Practical impact on RPM and efficiency

Given a required thrust and disk area, changing blade geometry mainly shifts how much $C_l$ and drag each section must generate at a given RPM. More blade area (higher solidity) allows reducing RPM or collective to maintain thrust, which can help stay in a favorable tip Mach and Reynolds regime. Optimized geometry (airfoil choice, twist, taper) can significantly raise FM and reduce hover power at the same thrust and disk loading.[^9][^18][^17]

5. Maximizing Total Disk Area Within a 55 cm Footprint

Single large rotor

A single rotor that nearly fills the 55 cm circle has area $A = 0.238\,m²$, as above. This is the maximum disk area obtainable from one circular rotor in that envelope. Hovering at 5 kg, disk loading is ≈206 N/m² and ideal induced power ≈465 W. With FM ≈0.7, actual hover power might be ≈660 W. This is close to the lower bound achievable without going to exotic configurations.

However, a single rotor requires some way to cancel torque (tail rotor or other counter-torque mechanism) and complicates the airframe; it also creates a central hub where the payload must fit below or above the disk.

Multiple separate rotors within the same footprint

For multirotors, individual rotor disks must fit inside the 55 cm overall diameter. The sum of their areas can, in principle, approach or even exceed the single-disk area if disks extend outside the body but remain inside the overall envelope. However, practical layout typically leaves gaps between disks, so the total disk area is often somewhat lower than $\pi (0.275)^2$.

A common arrangement is four rotors (quad) placed at the corners of a square or X-frame. If each rotor has radius $r$ and the diagonal span between opposite motor centers is less than 0.55 m, the per-rotor diameter must satisfy geometric constraints. For example, if motor-to-motor diagonal is $d$ and prop diameter is $D_p$, then $d + D_p ≤ 0.55\,m$. To maximize disk area, push $D_p$ close to this limit and minimize hub spacing, while respecting structural and control needs.

In terms of physics, for a given total thrust $T$ and total disk area $A_{tot} = n·\pi r^2$, ideal induced power is fully determined by $A_{tot}$ (equation 3), independent of how thrust is split among the rotors, assuming flows do not interfere. Realistically, interaction between closely spaced rotors and the fuselage will slightly degrade FM.[^23][^1][^3][^2]

Coaxial stacks

Coaxial rotors (two rotors on the same axis) are attractive when footprint is constrained because they double disk loading for a given planform area, increasing thrust capacity in a compact area. However, the wake interaction between upper and lower rotors reduces hover efficiency relative to two isolated rotors.

Studies of coaxial mini-UAVs and experimental propeller data show that coaxial pairs typically suffer 15–30% thrust or efficiency penalties for the same total power, depending on spacing and rotor matching. Optimal axial spacing is often quoted as a fraction of diameter ($z/D ≈ 0.1\text{–}0.4$); too close and the lower rotor sees highly non-uniform inflow; too far and the system behaves more like two separate disks but with more structural height.[^24][^22][^21]

Unequal propeller combinations with the smaller rotor upstream can reduce losses somewhat, but equal-diameter pairs generally maximize total thrust for a given RPM and power. Overall, for endurance-focused hover, coaxial stacking is usually less efficient than separate non-overlapping disks of the same total area.[^24]

Overlap and rotor placement

Allowing partial overlap between different rotors within the 55 cm envelope can increase total blade-swept area beyond $\pi R^2$ in a purely geometric sense, but overlapping regions of flow suffer strong interference. Where wakes overlap, induced velocities add, increasing local $v_i$ and induced power for a given thrust. From a momentum-theory perspective, the effective area is closer to the union of disk projections than the sum when significant overlap exists.

Wind-tunnel and computational studies of multirotor interference show that tightly clustered rotors suffer reductions in FM and power loading compared to isolated rotors, especially in hover. Thus, to maximize practical total disk area and FM within 55 cm, it is better to:[^16][^7]

• Use as few rotors as possible consistent with control requirements (e.g., a well-designed quad or tri-rotor rather than many small rotors).
• Maximize individual rotor diameter within geometric constraints, minimizing overlap and avoiding excessive coaxial stacking.

Few large slow rotors vs many small fast rotors

Given a fixed 55 cm vehicle size, many small rotors must each have substantially smaller radius than a few large ones, dramatically increasing disk loading on each rotor and total ideal induced power (since $P_i ∝ T^{3/2}/\sqrt{A}$). To compensate, they operate at higher $\Omega$ and often higher tip Mach, increasing profile power and noise.[^1][^2][^7][^16]

In contrast, fewer, larger rotors maximize total disk area and allow lower RPM and tip Mach, reducing both induced and profile losses for a given thrust. This is why large, slow props are preferred for endurance multirotors and why eVTOL design guidance emphasizes low disk loading (<20 lb/ft²) for good hover efficiency.[^3][^16]

Within a 55 cm diameter constraint, the physics strongly favor configurations with:

• Rotor diameters as close as possible to 55 cm (or the largest feasible subset) with minimal overlap.
• The minimum rotor count that still gives acceptable controllability and redundancy.

6. Blade Element Momentum Theory (BEMT) for Design and Optimization

BEMT basics

Blade Element Momentum Theory combines 1D momentum theory (actuator disk) with blade element theory. The blade is divided into many radial elements (annuli). For each annulus, the local flow is assumed quasi-2D, and lift and drag are computed using airfoil polars and local relative velocity.[^20][^25][^26][^19]

Momentum theory provides relationships for thrust and torque as functions of induced velocity (axial and tangential induction factors) in each annulus, while blade element theory provides thrust and torque from the aerodynamic forces on the blade sections. BEMT solves for induction factors that satisfy both sets of equations and then integrates along the span to obtain total thrust and torque.[^19][^20]

Assumptions include steady, axisymmetric inflow, inviscid flow, and negligible radial interactions. Corrections such as tip-loss factors, high-induction corrections, and unsteady inflow models are often added to improve accuracy.[^20][^19]

Accuracy for small UAV rotors

For steady hover and moderate advance ratios, BEMT has been shown to predict rotor performance (thrust, torque, power) reasonably well for both helicopter-scale and small UAV rotors, provided airfoil data are appropriate and tip/hub loss corrections are used. Micro rotary-wing MAV studies have used BEMT to match hover test data and to attribute efficiency losses to profile drag and other non-idealities.[^8][^9][^19][^7][^16]

Recent research has refined BEMT formulations to avoid singularities and to reinterpret figure of merit as the product of component efficiencies (e.g., induced and profile components). These improvements make BEMT robust and suitable for optimization workflows.[^18]

Suitability for large design sweeps and genetic algorithms

BEMT is computationally inexpensive because each evaluation involves solving a 1D problem per radial station and integrating over the span, with no full CFD mesh. It is widely used in wind-turbine and propeller design tools that routinely run thousands of design evaluations in optimization loops on desktop PCs.[^25][^19]

Studies on rotor optimization for UAVs using BEMT have successfully explored large design spaces including rotational speed, chord distribution, taper, twist, airfoil selection, and blade count, demonstrating large gains in FM and hover endurance. This indicates that using BEMT inside a genetic-algorithm-based simulator to evaluate thousands of rotor designs on a standard PC is not only feasible but common practice in similar domains.[^17][^18]

For this use case, the main inputs to a BEMT model would be:

• Rotor radius and hub geometry.
• Number of blades.
• Chord and twist distributions (possibly parameterized by a small set of design variables).
• Airfoil polars ($C_l(\alpha, Re)$, $C_d(\alpha, Re)$) across relevant Reynolds and Mach numbers.
• Operating RPM or collective (depending on control variable).

Outputs include thrust, torque, power, FM, and detailed spanwise load distributions, which can be fed into structural and noise models.

7. Key Levers for Maximizing Endurance in a 5 kg, ≤55 cm Multirotor

Pulling the previous sections together, the main physical levers for maximizing hover endurance within this constraint are:

• Maximize total disk area: Use as few, as large rotors as a 55 cm footprint and control requirements allow, minimizing rotor overlap and avoiding coaxial stacks unless footprint constraints absolutely demand them.[^1][^2][^3]
• Target low disk loading: Within the geometric constraints, aim to minimize $T/A_{tot}$. Lower disk loading reduces induced velocity and induced power, improving power loading and endurance.[^2][^1]
• Optimize blade geometry for FM: Use BEMT plus experiments to tune solidity, chord, twist, taper, and airfoil selection to operate blades near maximum $L/D$ at the hover thrust, aiming for FM in the 0.6–0.7 range for small rotors.[^5][^9][^17]
• Set tip speed in a moderate Mach regime: Choose RPM so that $M_{tip}$ is well below transonic (e.g., 0.4–0.6) but not extremely low, balancing induced and profile losses while helping with noise.[^15][^13][^12]
• Avoid unnecessary interference: Place rotors to reduce mutual inflow interference and keep a reasonable distance from the fuselage and payload structures where possible, as fuselage proximity and coaxial interactions can reduce FM.[^22][^23][^3]
• Use BEMT-driven optimization: Implement a BEMT model to explore thousands of designs, using a genetic algorithm or similar to search over rotor radius (within the 55 cm constraint), blade count, solidity, twist, and RPM, with FM or specific hover power (W/N or W/kg) as the objective.[^25][^19][^18]

For a 5 kg urban delivery multirotor constrained to 55 cm overall, the physics favor a configuration with one or a few large, slow rotors with high FM and low disk loading rather than many small, fast rotors, provided control and integration issues can be solved.

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