The flow is governed by the Euler equation: 1ρf-∇p=v⋅∇v, in which ρ is the fluid density (kg m-3), f the force density (N m-3), p the static pressure (N m-2), v the velocity vector (m s-1) and H=p+12ρv×v the total pressure (N m-2). A cylindrical reference system (x,r,φ) is applied, with the positive x coinciding with the downwind wake axis, and with r and φ the radial and azimuthal coordinate (see Fig. ). For the special case of a disc flow with constant circulation induced by a free potential flow vortex Γ at the axis of the wake with a vortex core having radius δ(x), the azimuthal velocity in the wake is vφ=Γ2πrfor≥δ(x)=Γ2πδ(x)Frδ(x)forr<δ(x). The functions δ(x) and F(r/δ(x)) remain unspecified. Figure  shows (half of) the cross section through the stream tube in the meridian plane, with the disc and fully developed wake indicated. The shaded area is the vortex core with an increasing radius towards the far wake due to the flow deceleration. The fully developed far wake is indicated by the index 1. If there is no index, the variables are taken at the position of the actuator disc. The index 0 is used for flow variables in the undisturbed, upstream flow. The disc has radius R and area A, while A1 is the area of the far wake with radius R1. The analysis starts with δ being non-zero, after which the limit of δ→0 is taken. The only assumption made is that lim⁡δ→0δ1=0whileδ1δ>1.

Only the pressure and the azimuthal velocity will be discontinuous across the infinitely thin disc, so integration of the axial and azimuthal component of Eq. () gives 1ρF=1ρ∫thicknessfdx=exΔpρ+eφvxΔvφ=exΔHρ-12vφ2+eφvxΔvφ, where F denotes a surface load (N m-2), Δ the difference between the down- and upwind side of the disc and e the unit vector. As vφ=0 at the upwind side of the disc, Δvφ=vφ. In Eq. () the Bernoulli equation integrated across the disc thickness has been used: Δp=ΔH-12ρvφ2. The local power converted by the force field f is f×v, which has to be equal to the local contribution to the torque, rfφ, times the rotational speed Ω. The converted power f×v becomes f⋅v=Ωrfφ=v⋅∇H. Integration of Eq. () across the disc combined with the azimuthal component of Eq. () gives the general expression ΔH=ΩrvxFφ=ρΩrvφ. This shows that the work done by the force field is expressed in a change in the total pressure or Bernoulli constant H. With Eq. (), for the Joukowsky disc, 1ρΔH=ΩΓ2πforr≥δ(x)=ΩΓ2πrδ(x)Frδ(x)forr<δ(x). It follows that outside the core, ΔH is constant, by which Eq. () shows that any non-uniformity in the pressure jump is due to the creation of swirl across the disc. The swirl–pressure jump does not change H, so does not contribute to the conversion of power, by which Eq. () may be interpreted as Δp=Δpconverting-H+Δpconserving-H. The sign conventions are that the rotational speed Ω>0 and Γ<0  so ΔH<0, implying that energy is extracted from the flow.

The thrust T is obtained by integration of Eq. () on the disc area. In dimensionless form the thrust coefficient is CT=T/(12ρU02πR2), containing both terms on the right-hand side of Eq. (), here denoted as ΔH and Δφ. For δ→0, CT=CT,ΔH+CT,Δφ=2λΓ2πRU0+Γ2πRU02ln⁡Rδ2, where λ is the non-dimensional tip speed ratio λ=ΩRUo. In the same way the power coefficient is defined as Cp=P/(12ρU03πR2), where P denotes the absorbed power. Evaluation of Cp is done in Sect. .

With the conservation of circulation, rvφ=r1vφ,1, the Bernoulli Eq. () in the wake is written as 1ρp0-p1=12vx,12-U02+vφ,12-ΩΓ2π. Differentiating with respect to r and combining it with the radial pressure equilibrium in the far wake, ∂p1∂r1=-ρvφ,12r1, it is clear that vx,1 is constant. By this Eq. () can be written as p1-p0=-12ρvφ,12+p∗. At the wake boundary the pressure has to be undisturbed (p0), so p∗=12ρvφ,R12 and, with Eq. (), p1-p0=-12ρvφ,12+12ρΓ2πR12. This shows that the pressure variation in the far wake is caused only by the swirl. By merging Eq. () with Eqs. () and (), the second term on the right-hand side appears as a loss in H due to swirl: ΔH=12vx12-U02-12ρΓ2πR12. This is consistent with the optimization of rotors according to Glauert's theory which involves minimization of the swirl (see, e.g., ).

The momentum theory results are very sensitive to the choice of δ and δ1 because of the logarithmic singularity resulting from the integration of the pressure due to the azimuthal velocity: at the disc -ρπ∫δRvφ2rdr=-ρΓ24πln⁡Rδ, and similarly in the far wake -ρΓ24πln⁡R1δ1. Previous solutions have dealt with the singularity in different ways. have adopted δδ1=RR1, assuming that the vortex core grows with the stream-tube radius. This removes the singularity as -ρΓ24πln⁡Rδ+ρΓ24πln⁡R1δ1=0, leading to the result that Cp→∞ for λ→0. Van Kuik (2016) assumes δ=δ1 leading to the power coefficient Cp→0 for λ→0. However, as discussed in , both core models do not comply with the inviscid flow equations, so the impact of the vortex core model to the momentum balance merits an additional investigation.

Both analyses used the vortex core boundary as a lower limit in the integration of momentum and energy on the control volume used in momentum theory. This implies that the vortex core is excluded, motivated by its vanishing dimension in the limit δ,δ1→0. Here, it will be included, while the same limit is taken.

With δ(x) denoting the local core radius, with δ≤δ(x)≤δ1, the Bernoulli Eq. () in the vortex core region becomes 1ρp0-p=12vs2-U02+Γ2πδ(x)Frδ(x)2-ΩΓ2πrδ(x)Frδ(x)withrδ(x)≤1, where vs is the velocity in the meridian plane. As vs, U0 and the last term on the right-hand side remain finite for δ(x)→0, they do not contribute in this limit to the axial momentum balance drawn on the core volume. Consequently, this balance reduces to a balance of pressures acting on the control volume boundaries, integrated as a load in x direction: ∫0δp-p02πrdr-∫0δ1p1-p02πrdr+∫δδ1px,δ(x)-p02πδ(x)dδ(x)=0, where the path of integration of the third integral is the core boundary δ(x) with 0≤x≤x1. This integral is evaluated with Eq. (), of which the finite terms do not contribute as the area of the core boundary projected in x direction is π(δ12-δ2), which vanishes for δ(x)→0. Only the pressure terms originating from vφ contribute as these behave like a Delta function: lim⁡δ(x)→0∫δδ1px,δ(x)-p02πδ(x)dδ(x)=-ρπ∫δδ1vφ22πδ(x)dδ(x)=-ρΓ24πln⁡δ1δ. The combination of Eqs. () and () gives ∫0δp-p02πrdr-∫0δ1p1-p02πrdr=ρΓ24πln⁡δ1δ, irrespective of the choice of core model δ(x), Fr/δ(x).

This result will be used in Sect. , where the momentum balance for the entire stream tube is studied. Unless specifically indicated all equations in the forthcoming sections are derived for the flow region outside of the vortex core.

The momentum equation drawn on the stream tube as control volume (see Fig. ) is written as T-∫A1p1-p02πrdr=ρ∫A1vx,1vx,1-U0dA1, where T is the thrust (N), being the integrated pressure jump across the disc. The boundaries of the momentum balance volume are the stream-tube boundary and those of the cross sections A0 and A1, far up- and downstream. As discussed in many references, amongst others in , the pressure at the stream-tube boundary does not contribute to the momentum balance and so is not included in Eq. ().

Figure  shows the pressure distributions appearing on the left-hand side of Eq. (), including the thrust:

constant pressure jump across the disc giving the jump in Bernoulli parameter H according to the first term on the right-hand side of Eq. ().

When all contributions are expressed in Γ by Eqs. () and (), integrated, subjected to lim⁡δ→0, substituted in Eq. () and divided by the disc surface πR2, the result is ΩΓ2π-12Γ2πR2-Γ2πR2ln⁡Rδ-ln⁡R1δ1-ln⁡δ1δadbce=vx,1vx,1-U0R1R2, where the terms on the left-hand side have been named in accordance with Fig. . The term between square brackets simplifies to ln⁡(R/R1). In other words: only the wake expansion area c>R contributes to this term. The mass balance is vx‾vx,1=R1R2, with the bar above vx indicating that it is the average value. The energy balance follows from Eq. (): ΩΓ2π-12Γ2πR12=12vx,12-U02. Mixing Eqs. () and () simplifies the momentum balance, yielding ΩΓ2π-12Γ2πR2-Γ2πR2ln⁡RR1=vx‾vx,1-U0. The non-dimensional vortex q=-Γ2πRUo is introduced. As Γ<0 q>0. Furthermore, from here on vx‾ and vx,1 indicate the dimensionless value vx‾U0 and vx,1U0. Herewith Eq. () becomes 1ρΔHU02=-λq, and the momentum balance becomes 2λq+q21+ln⁡RR12=2vx‾1-vx,1, and the energy balance becomes 2λq+q2RR12=1-vx,12. An analytical solution of Eqs. () and () is not found. An implicit expression of vx,1 in the independent variables λ, q is obtained by writing Eq. () as an expression for vx‾ with the help of Eq. () and substituting this in Eq. (): 1-vx,1vx,1q21+2λq-vx,12=qλ-12q21-ln⁡q21+2λq-vx,12. This can be solved numerically for vx,1. The wake expansion follows from Eq. () and the velocity at the disc from Eq. (). The power coefficient follows by integration of Eq. () on the disc area: Cp=2λqvx‾.

By mixing Eqs. () and (), the velocity at the disc can be written as vx‾=12vx,1+12λq+q21+ln⁡RR122λq+q2RR12. As (1+ln⁡R/R12)<R/R12 for R<R1 the ratio is <1. Consequently vx‾<0.5vx,1+1. The ratio in Eq. () is the ratio between the left-hand sides of the momentum balance Eq. () and energy balance Eq. () or, in other words, between the total load exerted on the flow in the stream-tube control volume and the non-conservative load which is the load performing work. By this, Eq. () is equivalent to Eq. (6) of where the distinction between conservative and non-conservative loads is used to explain the results of the momentum theory applied to an annulus of the stream tube. This analysis, without swirl, is done again with swirl in Sect. .

For large values of λ, the wake angular momentum should go to 0, and the momentum theory should become the one-dimensional theory yielding the well-known Betz–Joukowsky maximum value for Cp. According to Eq. () q is inversely proportional to λ for constant ΔH or λq. In the balances Eqs. () and (), the q2 terms vanish for λ→∞, with which the momentum theory without wake swirl is indeed recovered.

For the limit λ→0, flow states with λq being constant are studied. The energy balance Eq. () shows that the highest value for q2(R/R1)2 is obtained for vx,1=0: 2λq+q2RR12=1. With vx,1=0, the right-hand side of the momentum balance is 0 as is clear from Eq. (), by which it becomes 2λq+q21-ln⁡R1R2=0. Elimination of q2 from Eqs. () and () gives the wake expansion for the highest q-lowest λ: R1R21-ln⁡R1R2=2λq2λq-1. As an example, 2λq=8/9 results in R1R=2.77, q=0.924 from Eq. () and λ=0.48. Both vx‾ and vx,1 are 0, but the ratio of vx‾vx,1→7.69. This flow state is characterized by a full blockage by the disc, creating a wake with azimuthal flow only, so there is no change in axial momentum. The associated pressure distributions in the wake and at the disc balance each other. A lower value of λ is not possible for this value of λq. For λq=0 with λ=0, Eq. () gives ln⁡R1R2=1; Eq. () gives R1R=q=e=1.648 although vx=vx,1=0. In the wake only the azimuthal velocity is non-zero, reaching qRR1=1 at the far wake boundary r=R1. The wake expansion is close to the experimental value ≈1.6 of the wake expansion behind a solid disc reported in .

Cp,max(λ) is obtained by optimizing the solutions for fixed λ varying q.

The computer code described in has been adapted to include wakes with swirl. Axial and radial velocities are calculated by summation of the induction by each of the several thousand vortex rings which constitute the wake boundary. The azimuthal velocities are calculated from Eq. (). The shape and strength of the vortex rings are adapted in the convergence scheme to satisfy the two boundary conditions: zero pressure jump across the wake boundary, and zero cross flow. The first boundary condition Δpwake-boundary=0 is expressed in v and input parameter ΔH: Δ(12ρv2)-ΔH=0. In , v only had an axial and radial component, but now the azimuthal component also enters this boundary condition. The strength of the vortex at the axis follows from Eq. () expressed in H and the second input parameter λ:q=-ΔH/(ρU02λ). Apart from these changes, the code and the numerical parameters are unmodified. The results satisfy the same accuracy requirements as described in . Figures  and show the streamlines, expressed in the stream-function Ψ and isobars of the disc flow with ΔH/(12ρU02)=-0.8888 and λ=0.731 and 1.018. The isobars in the wake show the pressure gradient due to the swirl.

As shown in Figs.  and the pressure at the upstream side of the disc is constant, which implies, by the Bernoulli equation, that the absolute velocity v upstream of the disc is constant. Figure  shows the values of the axial, radial and azimuthal velocity component at the disc as well as the absolute value |v|meridian=vx2+vr2. Similar distributions of the axial velocity have been calculated by several others, e.g. , , , and . The fact that |v|meridian is constant has been reported by for actuator disc flows without swirl. The explanation given in is now extended to include discs with swirl.

The radial component of Eq. () just upstream of the disc is ρvs∂vr∂s=-∂p∂r, with s being the coordinate along the streamline and r the radial coordinate. The pressure does not depend on r when it is shown that the radial velocity reaches a maximum at the disc when following a streamline. Along any streamline passing the disc, vr increases when the position of observation s0 travels from far upstream to the disc sdisc, due to the decreasing distance to the vorticity γ in the wake boundary, so ∂vr/∂s>0. Following the streamline in the wake, so with s0>sdisc, two regions can be distinguished: the vorticity between sdisc and s0 induces a negative vr and so contributes to ∂vr/∂s<0, while the induction by the vorticity at s>s0 is approximately constant as the wake downstream of s0 remains semi-infinite, with γ depending only slightly on s for s>s0. The result is that ∂vr/∂s=0 at the disc position, so from Eq. () the pressure upstream of the disc is constant and, from the Bernoulli, equation |v|meridian is constant; QED.

Figure  shows the calculated strength of the vortex sheet for the load case of Fig. , confirming the reasoning. With |γ| having a maximum at its leading edge, the non-uniformity of γ contributes to a negative induction of vr at streamline positions s0>sdisc. It should be noted that the distribution in Fig.  does not show the irregular behaviour at the leading edge as shown in Fig. 9 of . The explanation is that the distance between the first vortex rings in this previous paper is smaller than the radius of the vortex ring core, leading to this irregularity. Calculations with a smaller core size, not yet reported, have removed this irregularity, thereby not having any observable impact on the flow pattern and integrated numerical results. In the present calculation the distance between rings is always larger than the radius of the core of the vortex ring.

Now that the pressure at the upstream side of the disc is known to be constant, the radial derivative of Eq. () becomes ∂p-∂r=-ρvφ2r, with p- being the pressure at the downstream side of the disc. This is the radial equilibrium expression Eq. () for the flow in the wake. Apparently the radial distribution of p is only linked to vφ, not to the other velocity components.

In , the non-uniformity of the axial velocity at the disc is explained by applying the momentum theory to annuli instead of the entire stream tube. An annulus is defined as the volume of flow between the stream-tube values Ψn and Ψn-1. With 1<n≤10 and Ψn-Ψn-1=0.1Ψstream tube, Figs.  and shows 10 annuli as the volume between the plotted streamlines passing the disc. For the flow case shown in Fig. , the momentum balance per annulus is evaluated as follows.

The balance for the entire stream tube is defined by Eq. (). In this equation it is implicitly assumed that the pressure acting at the stream-tube boundary does not contribute, as discussed in many publications, amongst which are . This is confirmed by the calculations: the force in x direction resulting from the pressure integrated along the wake boundary for -25<x/R<25 is 0.2 % of the non-conservative disc load ΔHπR2. However, when applying Eq. () per annulus, the pressure acting at the boundaries of the annulus has to be added, so the second term in Eq. () is expanded to become ∮Sp-p02πrdr, where S is the contour in the meridian plane of the annulus control volume:

Tannulus-∮Sannulusp-p02πrdr=ρ∫A1,annulusvx,1vx,1-U0dA1. The pressure integral is calculated with x/R=±25R as up- and downstream limits. The momentum balance Eq. () yields the wake velocity vx,1,annulus which, combined with the mass conservation

vx‾vx,1annulus=A1Aannulus, results in vx,annulus‾ at the disc. Figure  shows the distribution of vx resulting from the flow field calculation, the associated average value per annulus, and the value resulting from Eqs. () and (). Apart from the outer annulus, the calculated average per annulus coincides with the momentum balance average values. This confirms results as published by and , both for disc flows without swirl: the annuli cannot be assumed to be independent, as the pressure field contributes to the axial momentum exchange leading to the non-uniform distribution of vx.