A ring-vortex free-wake model for uniformly loaded propellers. Part I-Model description.

Available online at www.sciencedirect.com Available online at www.sciencedirect.com

Availableonline onlineatatwww.sciencedirect.com www.sciencedirect.com Available

ScienceDirect ScienceDirect

Energy Procedia 00 (2018) 000–000 Energy Procedia (2018) 000–000 Energy Procedia 148 (2018) 360–367 Energy Procedia 0000 (2017) 000–000

www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

73rd Conference of the Italian Thermal Machines Engineering Association (ATI 2018), 12-14 73rd Conference of the Italian Thermal Machines September 2018, Engineering Pisa, Italy Association (ATI 2018), 12-14 September 2018, Pisa, Italy

A free-wake model for uniformly loaded propellers. A ring-vortex ring-vortex free-wake model for uniformly loaded propellers. The 15th International Symposium on District Heating and Cooling Part I – Model description. Part I – Model description. a,∗ a Bontempo , Marcello Mannademand-outdoor Assessing theRodolfo feasibility of using the heat Rodolfo Bontempoa,∗, Marcello Mannaa Dipartimento di Ingegneria Industriale, Universit`a degli Studi di Napoli Federico II, Via Claudio 21, 80125 Naples, Italy temperature function for a long-term district heat Dipartimento di Ingegneria Industriale, Universit` a degli Studi di Napoli Federico II, Via Claudiodemand 21, 80125 Naples,forecast Italy a a

I. Andrića,b,c*, A. Pinaa, P. Ferrãoa, J. Fournierb., B. Lacarrièrec, O. Le Correc Abstract Abstract a IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal The paper presents a free-wake actuator disk model as applied to uniformly loaded propellers without slipstream rotation. The wake b Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, France The paper presents a free-wake actuator model as applied uniformly loaded without slipstream rotation.cylindrical The wake vorticity, originating from the disk edge,disk is represented by thetosuperposition of a4 propellers ringAlfred vortex sheet 44300 and a Nantes, semi-infinite c Département Systèmes Énergétiques et Environnement IMT Atlantique, rue Kastler, Francecylindrical vorticity, originating from the disk edge, is represented by the superposition of a ring vortex sheet and a semi-infinite vortex tube modelling the far wake. The shape and the density strength of the sheet are iteratively evaluated using two conditions. vortex modelling thestability far wake. The shape and the density strength thestatic sheetpressures are iteratively evaluated using twotheconditions. Firstly,tube the vortex sheet condition is strongly enforced, that isofthe just above and beneath sheet are Firstly, the vortex sheet stability condition is strongly enforced, that is the static pressures just above and beneath sheet are required to be equal. Secondly, the vortex sheet is constrained to align with the overall induced flow field, namely it the is compelled required to be equal. Secondly, the vortex sheet is constrained to align with the overall induced flow field, namely it is compelled toAbstract be a streamsurface. By doing so, the contraction of the wake is naturally accounted for in the model and no assumptions on to a streamsurface. doing the contraction of thethe wake is naturally accounted for model in the model andthenodescription assumptions its be shape are made. TheBy first part so, of this work deals with theoretical foundations of the and with of on its its shapecounterpart. are made. The first part of thisof work dealssingularities with the theoretical foundations ofinduced the model and with the description of its discrete The characteristics the flow used to model the flow by the wake are also presented. District heating networks are commonly addressed in the literature as one of the most effective solutions for decreasing the discrete counterpart. The characteristics of the flow singularities used to model the flow induced by the wake are also presented. greenhouse gas emissions from the building sector. These systems require high investments which are returned through the heat csales.  2018 Due The Authors. Authors. Published by Elsevier Elsevier Ltd. and building renovation policies, heat demand in the future could decrease, to the changed climate conditions © 2018 The Published by Ltd. c 2018  The Authors. Published by Elsevier Ltd. This is article under the This is an an open open access articlereturn underperiod. the CC CC BY-NC-ND BY-NC-ND license license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) prolonging theaccess investment This is an and openpeer-review access article under the CC BY-NC-ND licensecommittee (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of the 73rd 73rd Conference of the the Italian Italian Thermal Machines of the scientific the Conference of Thermal Machines The main scope of this paper is to assess the feasibility of using the heatof demand – outdoor temperature function for heat demand Selection andAssociation peer-review under responsibility of the scientific committee of the 73rd Conference of the Italian Thermal Machines Engineering 2018). forecast. The district of(ATI Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665 Engineering Association (ATI 2018). buildings that vary in momentum both construction period and typology. Three weather scenarios (low, medium, high) and three district Keywords: actuator disk; theory; propeller. renovationactuator scenarios developed disk;were momentum theory;(shallow, propeller. intermediate, deep). To estimate the error, obtained heat demand values were Keywords: compared with results from a dynamic heat demand model, previously developed and validated by the authors. The results showed that when only weather change is considered, the margin of error could be acceptable for some applications (the error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered). 1.scenarios, Introduction value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the 1.The Introduction decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and Despite the analysis of the flow propeller can intercept be carried out by for several advanced models, such as three renovation scenarios considered). On through the otheraa hand, function increased 7.8-12.7% per decade (depending on the Despite the analysis of the flow through propeller can be carried out by several advanced models, such dimensional panel methods [32, 44], lifting line [33, 45], lifting surface [31, 50], and CFD based methods [25, as 35,three 39], coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and dimensional panel methodsoriginating [32, 44], lifting line [33, 45], lifting surface [31, 50], [46] and CFD based methods [25, 35, 39], the actuator disk method, from the momentum theory of Rankine and Froude [23, 24], is still the improve the accuracy of heat demand estimations. the diskanalysis method,tool originating from This the momentum theory when of Rankine [46]with and the Froude [23, 24], still the mostactuator employed for propellers. is true especially combined so-called Bladeis Element most employed analysis tool for propellers. This is true especially when combined with the so-called Blade Element Theory 27] to give rise tobythe BladeLtd. Element/Momentum Theory [30, 42, 47]. Besides the simple momentum © 2017 [22, The Authors. Published Elsevier Theory [22,27], 27] more toresponsibility give rise toofthe Element/Momentum Theory [30, 42,Symposium 47].the Besides the simple momentum Peer-review under theBlade Scientific Committee of The 15th International on District Heating and[11], theory [12, advanced actuator disk methods also exist. Not disregarding CFD-based approaches theory [12, 27], more advanced actuator disk methods also exist. Not disregarding the CFD-based approaches [11], Cooling. ∗ Corresponding Keywords: Heat demand; Forecast; Climate changefax: +39-081-2394165. author. Tel.: +39-081-7683287; ∗ Corresponding author. Tel.: +39-081-7683287; fax: +39-081-2394165. E-mail address: [email protected] E-mail address: [email protected]

1876-6102 2017The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. c©2018 1876-6102 1876-6102  © 2018 The TheAuthors. Authors. Published by Elsevier Ltd. c under 1876-6102 2018 Published by Elsevier Ltd. Peer-review responsibility ofthe theCC Scientific Committee of The 15th International Symposium on District Heating and Cooling. This is an open access article under BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection responsibility of the scientific committee of theof73rd of the Italian MachinesMachines Engineering Selection and andpeer-review peer-reviewunder under responsibility of the scientific committee the Conference 73rd Conference of the Thermal Italian Thermal Selection and peer-review of the scientific committee of the 73rd Conference of the Italian Thermal Machines Engineering Association (ATI 2018). under Engineering Association (ATI responsibility 2018). Association (ATI 2018). 10.1016/j.egypro.2018.08.089

2

Rodolfo Bontempo et al. / Energy Procedia 148 (2018) 360–367 R. Bontempo and M. Manna / Energy Procedia 00 (2018) 000–000

361

the advanced version of the actuator disk can naturally take into account the effects of the wake contraction and rotation without the typical simplifying assumptions of the momentum theory. For example, Wu [53] developed a nonlinear actuator disk theory which was numerically implemented in [28, 29] and [21]. This method, which has been successfully used to evaluate the accuracy of the momentum theories [7, 13–15], has also been extended to ducted rotors in [1–6, 8–10, 16]. Øye [43] developed a ring-vortex wake method for the energy extracting disk. A similar approach has been proposed by van Kuik and Lignarolo [52] and used to verify the axial momentum theory when applied to a wind turbine. Rosen and Gur [48] modelled the flow through the disk by a distribution of sinks whose strength is directly proportional to the axial velocity along the disk surface. The method has also been applied to skewed configurations [37]. In this paper the theoretical foundations of a free-wake uniformly-loaded actuator disk method without wake rotation are presented. In this method the flow induced by the wake is modelled superimposing a semi-infinite straight vortex cylinder for the far wake, and N ring vortices located at the pivotal points of straight panels. The self-induced velocity of each panel is evaluated taking into account the wake curvature both in the meridional and cross plane. The density strengths of the vortices and the shape of the wake are evaluated by enforcing the wake edge to be force-free and aligned with the overall flow field. The theoretical foundations of the method are described in section 2. The discrete counterpart of the model and the characteristics of the flow singularities used to represent the flow induced by the wake are also illustrated in section 3. Finally, in the companion paper [17], the numerical implementation of the method and the iterative solution procedure are presented. Moreover, the method is used to evaluate some performance parameters, such as the power coefficient and the ideal propulsive efficiency, and to inspect the local characteristics of the flow through a uniformly loaded disk. 2. Description of the model In the actuator disk method the propeller is modelled by axisymmetrically distributing the rotor load over a permeable disk-shaped surface. For this reason, the model can be regarded as a rotor with infinite blades. In fact, as analytically shown by van Kuik [51], the actuator disk represents the limiting case of the Joukowsky vortex model [34] when the number of blades tends to infinity. In the classical Joukowsky rotor, the blades are modelled through a line vortex, while the flow is supposed to be inviscid, incompressible, uniform at infinity and with zero yaw (see Figure 1 (left)). Each blade section experiences a lift, so that, from the Kutta-Joukowsky theorem, a non-zero bound circu-

Figure 1. Sketch of the rotor vortex model. Left: Joukowsky model. Right: vorticity decomposition.

lation exists. Then, applying the Stokes theorem, it can be shown that the vorticity flux over all surfaces containing a blade section must be different from zero. Having said that, if the bound circulation changes in the spanwise direction, then the vorticity flux changes too. Consequently, since the vorticity is a divergence-free vector field, a vorticity flux has to leave the blade giving rise to the so-called trailing vorticity whose density strength is equal to the derivative of the bound circulation in the spanwise direction. If the rotor is uniformly loaded, i.e., there is not a spanwise variation of the bound circulation, then no trailing vorticity is spread all along the trailing edge of the blades. However, due to the solenoidal behaviour of the vorticity field, two vortices stem from the tip and the root of each blades. Hence, the rotor model is made up of three vortex systems (see Figure 1 (left)). The first one represents the bound vortices of the

Rodolfo Bontempo et al. / Energy Procedia 148 (2018) 360–367 R. Bontempo and M. Manna / Energy Procedia 00 (2018) 000–000

362

3

blades. The second one is the root vortex, that is a straight line vortex whose strength is given by the product between the number of blades and the strength of the single bound vortex. Finally, the last system is the helical vortex filament originating from the tip of the blades. As suggested by Coleman et al. [20], this filament can be decomposed into a tangential wake-vorticity and in an axial wake-vorticity system (see Figure 1 (right)). When the number of blades tends to infinity, the bound vorticity is axisymmetrically spread over the rotor disk surface, whereas the root system is unchanged. The two wake vortex systems, which have to be uniformly distributed over the wake surface, are represented by two sheet vortices. Coming now to the classical actuator disk without wake rotation, note that the tangential vorticity induces both an axial and a radial velocity, whereas a tangential velocity can only be induced by the root, the disk and the axial wake-vorticity. Consequently, the classical actuator disk without wake rotation has to be modelled through the sole tangential sheet vortex. Both the strength and the shape of this sheet, which represents the wake edge, are not known in advance. As shown in the companion paper, the shape can be evaluated by constraining it to align with the overall induced flow field, namely the sheet is compelled to be a streamsurface. The r strength is evaluated enforcing a static pressure stability condition, that is the static pressures just above (pa ) and beneath (pb ) the sheet are required to be equal between each other (∆psheet = 0). To properly formulate this condition, consider Figure 2 which shows a meridional sketch of the tangential sheet vortex. As clearly reported in the figure, a cylindrical coordinate system (z, r, θ) is introduced with r

ds va

ds s

Disk

R

Disk

vb

R

s

vb

va

Wake

Wake z

Figure 2. Meridional view of the wake vortex model.

origin in the centre of the disk. By convention, the sheet strength is considered to be clockwise-positive around the positive z axis. In the same figure, a curvilinear abscissa s is also defined along the sheet, namely along the wake edge. Therefore, the strength of an infinitesimal element of the sheet vortex can be written as γ ds, where γ is the unknown distribution of the density strength to be evaluated through the force-free condition across the sheet, i.e., ∆psheet = 0. To this aim, the Bernoulli principle is applied to the streamtube passing through the disk. By doing so, it is easy to prove that, for a uniformly loaded disk without slipstream rotation, the jump in the static pressure across the disk ∆pdisk must be equal to the difference between the total pressure outside (p0∞ ) and inside (p0w ) the wake, i.e., 2 2 ) = (p0w − p0∞ )/( 12 ρU∞ ), CT = ∆pdisk /( 12 ρU∞

(1)

γ = va − vb .

(2)

vsheet = (va + vb )/2.

(3)

def

where ρ is the fluid density, U∞ is the freestream velocity and CT is the thrust coefficient. Applying the Stokes theorem to the infinitesimal control volume highlighted in Figure 2 through a dotted line, the strength of the sheet vortex γds must be equal to the clockwise circulation of the velocity va ds − vb ds, where va and vb are the velocity just above and beneath the sheet. Consequently, the density strength of the sheet results Note that, the velocity exhibits a jump across the sheet, so that the velocity at the sheet is defined as

4

Rodolfo Bontempo et al. / Energy Procedia 148 (2018) 360–367 R. Bontempo and M. Manna / Energy Procedia 00 (2018) 000–000

363

Furthermore, the total pressure also exhibits a jump across the sheet which, in fact, separates the flow swallowed by the disk (that is the wake) from that not passing through the disk. Since the total pressure above and beneath the sheet are respectively equal to p0∞ and p0w , the difference in the static pressure across the sheet (∆psheet ) can be expressed as ∆psheet = p0∞ − p0w + 12 ρ(v2b − v2a ). Hence, the force-free condition ∆psheet = 0 becomes p0∞ − p0w =

1 2 1 ρ(v − v2b ) = ρ(va − vb )(va + vb ) 2 a 2

(4)

which, using equations (1), (2) and (3) returns CT = −ˆγvˆ sheet , 2

(5)

where γˆ = γ/U∞ and vˆ sheet = vsheet /U∞ . Note that, in the following, quantities denoted with a hat (ˆ) are made dimensionless using the reference length R (radius of the disk) and the reference velocity U∞ (free stream velocity). Also note that the above equation is a field relation, that is, it is valid at any point on the wake edge. Equation (5) can be used to evaluate the density strength distribution along the sheet vortex once the thrust coefficient and the velocity distribution along the wake edge (vsheet ) are known. From equation (5), it is clear that γ must be negative, namely, using the previously described sign convention, the wake tangential vorticity is anticlockwise around the positive z axis. Equation (5) also states that the product γˆ vˆ sheet must be uniform all along the wake edge. A simplified version of equation (5) holds at downstream infinity. Specifically, as known through the classical Axial Momentum Theory [27], the static pressure has to reach the freamstream conditions at a large distance downstream from the disk. Therefore, equation (1) promptly returns the following expression for the axial velocity in the wake at downstream infinity:  vz,+∞ = 1 + CT . (6)

Since the axial velocity outside the wake tends to U∞ when z → +∞ (see Glauert [27]), the sheet velocity and the free-force condition (5) can be rearranged with the help of equations (3) and (6) to give at downstream infinity √  1 + 1 + CT vˆ sheet,+∞ = and γˆ +∞ = 1 − 1 + CT . (7) 2

3. The discrete vortex model z The flow induced by the tangential sheet vortex can be conveniently considered as the superposition of the flow induced by infinite ring vortices distributed along the wake surface. The discrete counterpart of this model is obtained dividing the wake edge into N panels with length ∆sn for n = 1, . . . , N. Following the classical panel method approach, the continuous sheet-vortex with elemental strength γds is replaced by a set of N ring vortices with strength γn ∆sn and located in the pivotal points of the panels. Figure 3 shows a meridional view of the discrete wake vorticity model. The crosses represent the pivotal points of the panels, the ends of which are denoted by the full diamonds. A cosine stretching function is used to cluster the panels in the disk proximity, since significant gradients in the flow field exist therein. The vertical arrows represent the traces in the meridional plane of the ring vortices which, as discussed in the previous section, have to be negative, that is anticlockwise around the positive z axis. Figure 4 shows the sign convention for a ring vortex with positive strength as well as some nomenclature used to define the characteristics of this kind of singularity. In particular, the figure shows the n-th ring vortex in the wake, where rn is the radius of the ring, zn is the axial coordinate of its centre, and P(zP , rP ) is the generic point where the velocity induced by the ring vortex is evaluated. Applying the Biot-Savart law, the dimensionless form of the axisymmetric axial and radial velocities induced by the n-th ring vortex at the generic point P(zP , rP ) can be written as (see for instance [26, 54])     γˆ n ∆ sˆn 2(˜r − 1) K(k) − 1 + 2 E(k) , (8) vˆ z,n→P = −  z˜ + (˜r − 1)2 2πˆrn z˜2 + (˜r + 1)2     γˆ n ∆ sˆn z˜/˜r 2˜r K(k) − 1 + 2 E(k) , (9) vˆ r,n→P =  z˜ + (˜r − 1)2 2πˆrn z˜2 + (˜r + 1)2

Rodolfo Bontempo et al. / Energy Procedia 148 (2018) 360–367 R. Bontempo and M. Manna / Energy Procedia 00 (2018) 000–000

364

5

r

γn ∆sn Semi-Infinite Vortex Cylinder

RSIVC

Disk

R

∆sn

z zSIVC Figure 3. Sketch of the discrete wake vortex model in a meridional view.

y

rP

rn

P(zP , rP )

x

zn

zP

z

Figure 4. 3D sketch of the ring vortex set-up.

where the following dimensionless quantities have been introduced:  γˆ n = γn /U∞ , ∆ sˆn = ∆sn /R, rˆn = rn /R, z˜ = (zP − zn )/rn and r˜ = rP /rn . In the above equations, the modulus k is 4˜r/[˜z2 + (˜r + 1)2 ], while K(k) and E(k) are the complete elliptic integral of the first and second kind, respectively. For 0 ≤ k ≤ 1, the value of these two special functions is numerically computed using the Arithmetic-Geometric Mean method proposed by King [36]. According to the nomenclature used in equations (8) and (9), the velocity components induced by the n-th ring vortex on the pivotal point of the m-th panel are named vˆ z,n→m and vˆ r,n→m . Then, the axial and radial velocities induced on N N ˆ z,n→m and n=1; ˆ r,n→m , respectively. the m-th pivotal point by all the other N − 1 ring vortices are simply n=1; nm v nm v In the case m = n, equations (8) and (9) have a singular behaviour, and they cannot be directly applied to evaluate the self-induced velocity of the m-th panel. For this reason, the approach proposed by Lewis [40] is used. More in detail, the continuous wake vorticity sheet (see Figure 2) exhibits a curvature in both the (r, θ) and (z, r) planes. This double curvature gives rise to two different contributions to the self-induced velocity on a ring vortex. Assuming that the ratio ∆sm /rm is small, Lewis [40] considered the (z, r) contribution equal to the self-induced velocity of a rectilinear vorticity element, that is vˆ (z,r) z,m→m = −

βm+1 − βm−1 γˆ m cos βm , 8π

vˆ (z,r) r,m→m = −

βm+1 − βm−1 γˆ m sin βm , 8π

(10)

where βm is the slope of the m-th panel. The contribution due to the (r, θ) curvature is evaluated adapting the Lamb [38] formula for the self-induced velocity of a smoke ring vortex. In particular, it can be shown that [40, 41, 49]     γˆ m ∆ sˆm 1 8πˆrm vˆ (r,θ) and vˆ (r,θ) = − (11) ln − r,m→m = 0. z,m→m 4πˆrm ∆ sˆm 4

6

Rodolfo Bontempo et al. / Energy Procedia 148 (2018) 360–367 R. Bontempo and M. Manna / Energy Procedia 00 (2018) 000–000

Hence the overall self-induced velocity components are      βm+1 − βm−1 ∆ sˆm 8πˆrm 1 vˆ z,m→m = − cos βm − γˆ m , ln − 8π 4πˆrm ∆ sˆm 4 vˆ r,m→m = −

βm+1 − βm−1 γˆ m sin βm . 8π

365

(12) (13)

Since in an inviscid formulation the wake can be considered fully developed moving towards downstream infinity, the far wake can be conveniently modelled by a straight semi-infinite vortex cylinder (SIVC) with constant radius (see Figure 3). By doing so, the total number of panel, as well as the computational cost, can be significantly reduced because only the near wake has to be discretised. Once again the strength of the cylinder is defined as clockwisepositive around the z axis. The velocity field induced by the SIVC can be directly evaluated integrating the Biot-Savart law along the cylinder surface [18], thus obtaining       γˆ SIVC  r∗ − 1 z∗   vˆ z,SIVC→P = − K(k) − ∗ δ+  Π(n, k)  , (14)     ∗2 ∗ 2 2π  r + 1 z + (r + 1)     2ˆγSIVC k2 vˆ r,SIVC→P = − K(k) , (15) E(k) − 1 −  2 πk2 z∗2 + (r∗ + 1)2

where z∗ = (zP − zSIVC )/RSIVC , r∗ = rP /RSIVC , RSIVC is the SIVC radius, zSIVC is the  axial coordinate representing the beginning of the cylinder (see Figure 3) and the modulus k is now defined as 4r∗ /[z∗2 + (r∗ + 1)2 ]. Moreover, in equation (15), Π(n, k) is the complete elliptic integral of the third kind whose characteristic is n = 4r∗ /(r∗ + 1), γˆ SIVC = γSIVC /U∞ is the SIVC density strength and, finally,    π r∗ < 1,     δ= (16) π/2 r∗ = 1,     ∗ 0 r > 1.

The Carlson duplication algorithm [19] is used to compute Π(n, k). The latter exhibits a singular behaviour for r∗ → 1. However, the product (r∗ − 1)Π(n, k), appearing in equation (14), vanishes as r∗ tends to the unity, so that the following asymptotic relation holds for the axial velocity:     1 z∗  K(k) for r∗ → 1. vˆ z,SIVC→P = −ˆγSIVC  + (17)  4 2π z∗2 + 4 Summarising, the overall velocity induced on the generic m-th panel by the whole wake structure can be expressed as the superposition of the velocity induced by itself, by all the other N − 1 ring vortices and, finally, by the SIVC, that is vˆ z,m =

N  n=1

vˆ z,n→m + vˆ z,SIVC→m ,

vˆ r,m =

N  n=1

vˆ r,n→m + vˆ r,SIVC→m ,

(18)

where equations (8) and (9) hold for n  m, while equations (12) and (13) are used when n = m. Up to now the discrete wake vortex model as well as the equations for the overall induced axial and radial velocities have been presented. However, to finalise the description of the model, the discrete counterparts of the two wake boundary conditions have to be discussed too. As shown in the previous section, the force-free condition ∆p sheet = 0 leads to equation (5) all along the wake, and to equations (7) at downstream infinity. Therefore, from equation (7) it can be easily inferred that the SIVC density strength γˆ SIVC can be written as  γˆ SIVC = 1 − 1 + CT . (19)

Yet, the density strength γˆ m of the m-th ring vortex reads −

CT = γˆ m vˆ m , 2

(20)

366

Rodolfo Bontempo et al. / Energy Procedia 148 (2018) 360–367 R. Bontempo and M. Manna / Energy Procedia 00 (2018) 000–000

7

 where vˆ m = vˆ 2z,m + vˆ 2r,m . The second wake boundary condition compels the wake edge to be aligned with the overall flow field. Thus, in order to evaluate the wake shape, the induced flow field needs to be computed through equations (18) which, however, require the knowledge of the wake shape, that is of the coordinates (ˆzn , rˆn ) for n = 1, . . . , N. This problem is solved using an iterative solution procedure which, for the sake of brevity, is described in the second part of the paper. 4. Conclusions This paper has presented the theoretical foundations of a free-wake uniformely-loaded actuator disk approach without wake rotation. The method can be considered as the limiting case of the Joukowsky rotor vortex model when the number of blades tends to infinity. Then, the wake has been modelled by a contoured cylindrical sheet-vortex originating from the disk edge. Moreover, in order to have a zero tangential velocity in the wake, no axial and radial components of the vorticity have been introduced. Applying the Bernoulli principle, it has been proven that, in order to ensure a zero pressure jump across the wake edge, the product between the velocity along the sheet and its density strength has to be uniform and equal to half of the thrust coefficient. Successively, the discrete counterpart of the model has been presented. The vortex sheet has been divided into ring vortices located at the pivotal points of N panels. The far wake has been modelled through a constant-radius semi-infinite vortex cylinder. The self-induced velocity of each panel has been computed taking into account the wake curvature both in the meridional and cross-plane. The iterative solution procedure used to evaluate the wake shape and the analysis of the overall flow field characteristics are reported in the companion paper. References [1] Bontempo, R., Cardone, M., Manna, M., 2016. Performance analysis of ducted marine propellers. Part I - decelerating duct. Applied Ocean Research 58, 322–330. doi:10.1016/j.apor.2015.10.005. [2] Bontempo, R., Cardone, M., Manna, M., Vorraro, G., 2014. Ducted propeller flow analysis by means of a generalized actuator disk model. Energy Procedia 45C, 1107–1115. doi:10.1016/j.egypro.2014.01.116. [3] Bontempo, R., Cardone, M., Manna, M., Vorraro, G., 2015a. A comparison of nonlinear actuator disk methods for the performance analysis of ducted marine propellers, in: Proceedings of the 11th European Turbomachinery Conference. [4] Bontempo, R., Cardone, M., Manna, M., Vorraro, G., 2015b. A comparison of nonlinear actuator disk methods for the performance analysis of ducted marine propellers. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 229, 539–548. doi:10.1177/0957650915589673. [5] Bontempo, R., Manna, M., 2013. Solution of the flow over a non-uniform heavily loaded ducted actuator disk. Journal of Fluid Mechanics 728, 163–195. doi:10.1017/jfm.2013.257. [6] Bontempo, R., Manna, M., 2014. Performance analysis of open and ducted wind turbines. Applied Energy 136, 405–416. doi:10.1016/j. apenergy.2014.09.036. [7] Bontempo, R., Manna, M., 2016a. Analysis and evaluation of the momentum theory errors as applied to propellers. AIAA Journal 54, 3840–3848. doi:10.2514/1.J055131. [8] Bontempo, R., Manna, M., 2016b. Effects of duct cross section camber and thickness on the performance of ducted propulsion systems for aeronautical applications. International Journal of Aerospace Engineering 2016. doi:10.1155/2016/8913901. [9] Bontempo, R., Manna, M., 2016c. Effects of the duct thrust on the performance of ducted wind turbines. Energy 99, 274–287. doi:10.1016/ j.energy.2016.01.025. [10] Bontempo, R., Manna, M., 2016d. A nonlinear and semi-analytical actuator disk method accounting for general hub shapes: part i - open rotor. Journal of Fluid Mechanics 792, 910–935. doi:10.1017/jfm.2016.98. [11] Bontempo, R., Manna, M., 2017a. Actuator disc methods for open propellers: assessments of numerical methods. Engineering Applications of Computational Fluid Mechanics 11, 42–53. doi:10.1080/19942060.2016.1234978. [12] Bontempo, R., Manna, M., 2017b. The axial momentum theory as applied to wind turbines: some exact solutions of the flow through a rotor with radially variable load. Energy Conversion and Management 143, 33–48. doi:10.1016/j.enconman.2017.02.031. [13] Bontempo, R., Manna, M., 2017c. Effects of the approximations embodied in the momentum theory as applied to the nrel phase vi wind turbine. International Journal of Turbomachinery Propulsion and Power 2, 9. doi:10.3390/ijtpp2020009. [14] Bontempo, R., Manna, M., 2017d. Effects of the approximations embodied in the momentum theory as applied to the nrel phase vi wind turbine, in: Proceedings of the 12th European Turbomachinery Conference. [15] Bontempo, R., Manna, M., 2017e. Highly accurate error estimate of the momentum theory as applied to wind turbines. Wind Energy 20, 1405–1419. doi:10.1002/we.2100.



8

Rodolfo Bontempo et al. / Energy Procedia 148 (2018) 360–367 R. Bontempo and M. Manna / Energy Procedia 00 (2018) 000–000

367

[16] Bontempo, R., Manna, M., 2018a. Performance analysis of ducted marine propellers. part ii–accelerating duct. Applied Ocean Research 75, 153–164. doi:10.1016/j.apor.2018.03.005. [17] Bontempo, R., Manna, M., 2018b. A ring-vortex free-wake model for uniformly loaded propellers. part ii - solution procedure and analysis of the results. Energy Procedia (in press) . [18] Branlard, E., Gaunaa, M., 2015. Cylindrical vortex wake model: right cylinder. Wind Energy 18, 1973–1987. doi:10.1002/we.1800. [19] Carlson, B.C., 1979. Computing elliptic integrals by duplication. Numerische Mathematik 33, 1–16. doi:10.1007/BF01396491. [20] Coleman, R.P., Feingold, A.M., Stempin, C.W., 1945. Evaluation of the induced-velocity field of an idealized helicoptor rotor. Technical Report ARR L5E10. NACA. [21] Conway, J.T., 1998. Exact actuator disk solutions for non-uniform heavy loading and slipstream contraction. J. Fluid Mech. 365, 235–267. [22] Drzewiecki, S., 1893. M´ethode pour la d´etermination des e´ l´ements m´ecaniques des propulseurs h´elico¨ıdaux. Bulletin del’Association Technique Maritime . [23] Froude, R.E., 1889. On the part played in propulsion by differences of fluid pressure. Transactions of the Institute of Naval Architects 30, 390–405. [24] Froude, W., 1878. On the elementary relation between pitch, slip and propulsive efficiency. Transactions of the Institute of Naval Architects 19, 47–65. [25] Gaggero, S., Villa, D., Tani, G., Viviani, M., Bertetta, D., 2017. Design of ducted propeller nozzles through a ranse-based optimization approach. Ocean Engineering 145. [26] Gibson, I.S., 1972. Application of vortex singularities to ducted propellers. Ph.D. thesis. University of Newcastle upon Tyne. [27] Glauert, H., 1935. Airplane propellers, in: Durand, W.F. (Ed.), Aerodynamic theory. Springer. volume IV, Division L, pp. 169–360. [28] Greenberg, M.D., 1972. Nonlinear actuator disk theory. Zeitschrift fuer Flugwissenschaften 20, 90–98. [29] Greenberg, M.D., Powers, S.R., 1970. Nonlinear actuator disk theory and flow field calculations, including nonuniform loading. Technical Report CR-1672. NASA. [30] Gur, O., Rosen, A., 2008. Comparison between blade-element models of propellers. Aeronautical Journal 112, 689. [31] Hanson, D.B., 1985. Compressible lifting surface theory for propeller performance calculation. Journal of aircraft 22, 19–27. [32] Hess, J.L., Valarezo, W.O., 1985. Calculation of steady flow about propellers using a surface panel method. Journal of Propulsion and Power 1, 470–476. [33] Jha, P.K., Schmitz, S., 2018. Actuator curve embedding–an advanced actuator line model. Journal of Fluid Mechanics 834. [34] Joukowsky, N.E., 1912. Vortex theory of screw propeller, i. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 16, 1–31. [35] Kim, Y.H., Park, S.O., 2009. Navier-stokes simulation of unsteady rotor-airframe interaction with momentum source method. International Journal of Aeronautical and Space Sciences 10, 125–133. [36] King, L.V., 1927. On the direct numerical calculation of elliptic functions and integrals. Cambridge University Press. [37] Kominer, S., Rosen, A., 2013. New actuator-disk model for a skewed flow. AIAA journal 51, 1382–1393. [38] Lamb, H., 1932. Hydrodynamics. Cambridge University Press. [39] Le Chuiton, F., 2004. Actuator disc modelling for helicopter rotors. Aerospace Science and Technology 8, 285–297. [40] Lewis, R.I., 1991. Vortex element methods for fluid dynamic analysis of engineering systems. Cambridge University Press. [41] Lewis, R.I., Ryan, P.G., 1972. Surface vorticity theory for axisymmetric potential flow past annular aerofoils and bodies of revolution with application to ducted propellers and cowls. Journal of Mechanical Engineering Science 14, 280–296. [42] Morgado, J., Silvestre, M.A.R., P´ascoa, J.C., 2014. Validation of new formulations for propeller analysis. Journal of Propulsion and Power 31, 467–477. [43] Øye, S., 1990. A simple vortex model, in: Proc. of the third IEA Symposium on the Aerodynamics of Wind Turbines, ETSU, Harwell, pp. 1–15. [44] Palmiter, S.M., Katz, J., 2010. Evaluation of a potential flow model for propeller and wind turbine design. Journal of Aircraft 47, 1739–1746. [45] Rand, O., Rosen, A., 1984. Efficient method for calculating the axial velocities induced along rotating blades by trailing helical vortices. Journal of aircraft 21, 433–435. [46] Rankine, W.J.M., 1865. On the mechanical principles of the action of propellers. Transactions Institute of Naval Architects 6, 13–39. [47] Rosen, A., Gur, O., 2005. Propeller performance at low advance ratio. Journal of Aircraft 42, 435–441. [48] Rosen, A., Gur, O., 2008. Novel approach to axisymmetric actuator disk modeling. AIAA journal 46, 2914–2925. [49] Ryan, P.G., 1970. Surface vorticity distribution techniques applied to ducted propeller flows. Ph.D. thesis. University of Newcastle upon Tyne. [50] Schulten, J.B.H.M., 1996. Advanced propeller performance calculation by a lifting surface method. Journal of propulsion and power 12, 477–485. [51] van Kuik, G.A.M., 2014. The relationship between loads and power of a rotor and an actuator disc. Journal of Physics: Conference Series 555, 012101. doi:10.1088/1742-6596/555/1/012101. [52] van Kuik, G.A.M., Lignarolo, L.E.M., 2016. Potential flow solutions for energy extracting actuator disc flows. Wind Energy 19, 1391–1406. doi:10.1002/we.1902. [53] Wu, T.Y., 1962. Flow through a heavily loaded actuator disc. Schiffstechnik 9, 134 – 138. [54] Yoon, S.S., Heister, S.D., 2004. Analytical formulas for the velocity field induced by an infinitely thin vortex ring. International journal for numerical methods in fluids 44, 665–672. doi:10.1002/fld.666.