The start of rotor aerodynamics dates back more than 100 years, when the concept of the actuator disc to represent the action of a propeller was formulated by . In this concept the disc carries only thrust, no torque. Based on this published the first performance prediction that still holds today, for a hovering helicopter rotor or a propeller without forward speed. About 2 years later and published the optimal performance of discs representing wind turbines, for which reason it is called the Betz–Joukowsky maximum (). The names of Betz and Joukowsky are also connected with the two concepts for actuator discs with thrust and torque. The model of was similar to the vortex model of Prandtl for an elliptically loaded wing. This gives an induced velocity which is constant over the wing span, resulting in minimum induced drag. In Betz's model each rotor blade is represented by a lifting line such that the vortex sheet released by the blade has a constant axial velocity. developed the vortex model of a propeller based on a horseshoe vortex of a wing. In his model each blade is modelled by a lifting line with constant circulation.

The constant circulation model of Joukowsky as well as the constant velocity model of Betz represented the ideal rotor. It was not yet possible to compare the models and to conclude which was best. Both models were valid only for lightly loaded rotors as wake expansion or contraction was neglected. A solution for the wake of Betz's rotor, still restricted to lightly loaded propellers, was presented by . The non-linear solution, so including wake deformation, was published by and . A comparison of the models of Betz and Joukowsky for rotors was presented by chap. 4 showing that Joukowsky rotors perform somewhat better than Betz rotors when both operate at the same tip speed ratio. The same conclusion was drawn for actuator discs by : at low tip speed ratio the Joukowsky disc performs somewhat better than the Betz disc. For increasing tip speed ratios, both models become the same as they converge to Froude's actuator disc.

The Joukowsky and Froude discs are still subjects for research as many modern design and performance prediction codes are based on them; see e.g. . Over the last decades the disc received the most attention from the wind energy research community, but recently new propeller research on the actuator disc concept has been published; see . The performance aspects are known by many studies using momentum theory, vorticity or computational fluid dynamics (CFD) methods. Experimental verification is shown by e.g. and . Recent research aims for deriving efficient tip corrections (see e.g. ) or for configurations including a hub or duct .

The present paper addresses the topic which received the least attention: the velocity distribution at the disc. The paper is part of a sequence of papers, starting with concerning flows through wind turbine Froude discs calculated by a potential flow method, followed by concerning the momentum theory and potential flow calculations for wind turbine Joukowsky discs, and the conference paper where the extension to propeller discs was presented. The latter paper was not yet conclusive in the explanation of the difference between wind turbine and propeller discs regarding the velocity distribution at the disc: for wind turbine discs the velocity vector in the plane containing the disc axis, the meridian plane, seems to be uniform, while it seems non-uniform for propeller discs. Compared to all calculations have been redone at equal, highest possible accuracy, leading to slightly different quantitative conclusions and a consistent explanation for the (non-)uniformity of the velocities at the actuator discs. Some of the content of regarding the average velocity at the disc is repeated, in order to make the paper readable independently of the previous papers. The open-access book contains the content of all papers mentioned in this paragraph.

Section  presents the equations of motion and the coordinate system. Section  discusses the average velocity at the disc and some remarkable disc flows, followed by Sect.  treating the velocity distribution for both actuator disc modes. Section  analyses the differences observed between wind turbine and propeller discs, followed by the concluding Sect. .

Figure  shows the coordinate systems. The disc is placed perpendicular to the undisturbed velocity U0, rotating with angular velocity Ω. All vectors are in the positive direction, apart from Γaxis, the vortex at the axis, and γφ, the azimuthal component of the wake boundary vortex sheet. The steady Euler equation is valid: 1ρv⋅∇v=-∇p+f, with f the force density, in this case distributed at the disc with thickness ϵ. The velocity is presented in the cylindrical coordinate system with x pointing downstream: v={vx,vr,vφ}. ρ is the flow density and p the pressure. In some of the equations dimensionless variables for the axial velocity will be used: ud=vx,d/U0 and u1=vx,1/U0, with the subscripts 0,d,1 denoting values far upstream, at the disc and far downstream as indicated in Fig. . Furthermore, v={vs,vn,vφ} is used, where vs=vx2+vr2 is the velocity component along a streamline at the surface with constant Ψ, with Ψ denoting the Stokes stream function.

The pressure and azimuthal velocity are discontinuous across the disc when ϵ→0. For such an infinitely thin disc, integration of Eq. () yields 2F=lim⁡ϵ→0∫ϵfdx=exΔp+eφρvxΔvφ, where Δ denotes the jump across the disc and F the applied surface load. A Joukowsky disc has a wake with swirl, induced by a vortex Γ at the axis. The vortex core radius δ is assumed to be infinitely thin. The azimuthal velocity is 3vφ=Γ2πr. The Bernoulli equation reads 4p+12ρv⋅v=H. When this is integrated across the disc and combined with Eq. (), the axial component of Eq. () becomes 5Fx=Δp=ΔH-12ρΔvφ2=ΔH-12ρΓ2πr2. The power converted by an annulus dr of the actuator disc equals the torque Q times rotational speed Ω, giving ΩdQ=2πΩfφr2dr, but also the integrated value of f⋅v with the use of Eq. (), giving 2πr(v⋅∇)Hdr. Equating both expressions shows that 6f⋅v=Ωrfφ=(v⋅∇)H. fφ is expressed by the φ component of the Euler Eq. (): fφ=ρ(v⋅∇)vφ. Herewith 7Ωrfφ=ρ(v⋅∇)(Ωrvφ), which gives with Eq. () 81ρ∇H=∇Ωrvφ=∇ΩΓ2π. Consequently, for a Joukowsky disc 9ΔH=ρΩΓ2π=constant. In the wind turbine mode ΔH<0, as energy is taken from the flow. With Ω always taken positive, Γ and vφ are negative in the wind turbine mode and positive in the propeller mode. This explains why Γaxis is shown with a negative sign in Fig. . Furthermore Eq. () shows that for Ω→∞ meanwhile keeping ΔH constant, Γ vanishes and, by Eq. (), also fφ. The result is the Froude disc without torque and wake swirl.

The power P converted by the disc follows by integration of Eq. () on the actuator disc. In dimensionless notation this becomes 10Cp=112ρU03Ad∫Af⋅vdAd=u‾dΔH12ρU02=2u‾dΩRU0Γ2πRU0. With λ=ΩR/U0 and q=Γ/(2πRU0), Eq. () becomes 11ΔH12ρU02=2qλ, and similarly Eq. () becomes 12Cp=2qλu‾d. The thrust T is derived in the same way, based on Eq. (). Dimensionless, the thrust coefficient is CT=T/(12ρU02Ad)=CT,ΔH+CT,Δvφ according to the two terms on the right-hand side of (): 13CT=CT,ΔH+CT,ΔvφCT,ΔH=2λqCT,Δvφ=-q2ln⁡Rδ2. CT,Δvφ does not contribute directly to the conversion of power, as it does not appear in Eq. (). It is a conservative contribution to CT, delivering the radial pressure gradient balancing the swirl immediately behind the disc. For finite q and δ→0, CT,Δvφ→∞. For a non-zero δ combined with high λ and low q, CT,Δvφ becomes small. For typical wind turbine parameters λ=8, CT,ΔH=-8/9 and δ=0.05R with δ representing the blade root cut-out area, CT,Δvφ≈-0.02.

The power and thrust have the same sign as ΔH or q: positive for propeller discs and negative for wind turbine discs. Consequently, the thrust and power (coefficients) are negative for discs extracting energy from the wake and positive for discs adding energy to the wake.

The velocity in the far wake is characterized by vr=0. Herewith the Bernoulli Eq. () becomes in the far wake 141ρp0-p1=12vx,12-U02+vφ,12-ΔH. The radial derivative is ∂p1/∂r1=ρ(vφ,12/r1-vx,1∂vx,1/∂r). When this is compared with the condition for radial pressure equilibrium in the fully developed wake, given by substitution of vr=0 in the radial component of Eq. (), 15∂p1∂r1=ρvφ,12r1, the result is vx,1=constant or, dimensionless, u1=vx,1/U0=constant.

With vs being the velocity in the meridian plane, vs/U0 at the upstream side of the disc equals |v|m=vx,d2+vr,d2/U0. Table  gives the numerical values of u‾d and |v|m for the flow cases considered. The differences between u‾d as calculated numerically and as resulting from the momentum theory are 0.2% or less. The |v|m value in the table is the value for r=0. Figure  shows the distribution of the axial and radial velocity components and the meridional velocity. Most striking is the distribution of this meridional velocity being practically uniform. The explanation of this is presented in Sect. , but first the velocity distributions shown in Fig.  are analysed.

The Euler equation of motion (Eq. ) offers a first-order explanation for the observation that |v|m is practically uniform for λ≥1. The radial component of Eq. () reads 20∂p∂r=-ρvs∂vr∂s+vφ2r. Equation () for vφ combined with Bernoulli's Eq. () gives a second equation for ∂p/∂r: 21∂p∂r=-ρvs∂vs∂r+vφ2r, so the result is ∂vs/∂r=∂vr/∂s, or at the disc 22∂vs,d∂r=∂vr,d∂s. Consequently, the distribution of vs,d is determined by the derivative ∂vr/∂s along the streamline. In case vr has a maximum or minimum at the disc, vs,d/U0=|v|m is uniform.

Qualitative observations regarding the increase or decrease in vr are possible when moving the position along a streamline in the meridian plane. The radial velocity depends only on the vorticity γφ distributed along the wake boundary and the position of observation s*. For a disc with an expanding wake, the following relations hold. a.

At the upwind side of the streamline, when moving towards the disc, the distance to γφ decreases, so vr increases, and ∂vr/∂s>0.

With respect to the average velocity at the actuator disc, the following applies.

For Joukowsky discs in wind turbine and propeller mode, the average velocity has been found, from λ=0 up to λ→∞ or J→∞ to J=0.