We revisit the modeling, using actuator line methods (ALM), of flow curvature effects on airfoils and their application to vertical-axis turbines (VAT), which is important when the ratio of airfoil chord c to arm length R is not small. The models can only use the aerodynamic coefficients of the airfoil in uniform flow (here obtained using wall-resolved CFD of a NACA0015 airfoil at Rec=6.0×106); they then consist of analytical modifications of these coefficients. The models for the normal force and moment coefficients use the classic analogy of potential flow with curved streamlines past an airfoil; their expression depends on the airfoil pitch angle and its attachment point to the arm. They are known, yet many authors have neglected the contribution of the aerodynamic moment. We here extend the models so as to cover all possibilities of attachment point and pitch angle, and up to stall. Furthermore, we develop a new model for the tangential force (the analogy with that of potential flow being flawed): its inviscid part, which corresponds to negative drag, is required to compensate for the aerodynamic moment.
The improved models are first validated in steady flow corresponding to the NACA0015 airfoil rotating with c/R=2/7. We cover the whole range of pitch angles, up to stall. The results are shown to compare well with those of the reference CFD data. This validation is carried out for two attachment points: at airfoil mid-chord and at quarter-chord.
An ALM incorporating the improved models is then also implemented in the CFD framework and is used to simulate the unsteady flow corresponding to a VAT configuration: the rotating NACA0015 airfoil without pitch placed in a free stream and operating at optimal tip speed ratio of TSR=3.25. This is also simulated for both attachment points. Here, a novel method is also developed to explicitly enforce the moment in an ALM. The various components of the ALM results are compared with those of the reference CFD data and are found to be in good agreement throughout the rotation cycle.
Introduction
The pursuit of sustainable development encourages the development of environmentally responsible technologies. One such innovation is the vertical-axis turbine (VAT), which allows for the extraction of energy from wind, river and tidal currents. This study specifically focuses on this technology. As highlighted in several studies , VATs offer several advantages, including the ability to generate energy regardless of flow direction, ease of installation, and adaptability to various environments. This makes them a reliable and consistent energy source. Furthermore, demonstrated that VATs can increase the energy density of turbine farms, further enhancing their potential to improve overall efficiency.
A precise approach for predicting and analyzing VAT performances is through high-fidelity numerical simulations, such as time-accurate wall-resolved (WR) simulations. These simulations can enable turbine optimization with parametric studies and design guideline recommendations . For instance, demonstrated that incorporating detached end-plates significantly enhances the efficiency of VATs, while a follow-up study further improved VATs efficiency by modifying blade support structures. However, WR simulations are computationally expensive, as they require resolving the flow within the boundary layers. This region requires fine computational cells to accurately capture the strong gradients. One approach to reducing computational costs is to use a moderate-fidelity approach. The actuator line method (ALM), which has been widely applied in rotorcraft studies and horizontal-axis turbines , is a moderate fidelity approach that enables realistic predictions of aerodynamic performance and accurately replicates wake behavior . Originally introduced by , the ALM replaces the body-meshed lifting surfaces with equivalent volumetric forces in the Navier–Stokes equations. These forces are determined for each blade section by estimating the local upstream velocity and angle of attack, then applying precomputed aerodynamic airfoil coefficients to compute the corresponding loads and moment. Thus, this method allows for bypassing the resolution of the boundary layers on the lifting surface and significantly reduces the required mesh resolution and overall computational cost while still maintaining realistic results. Consequently, once the ALM has been suitably adapted to VATs, it becomes particularly valuable for performing multi-turbine cluster simulations, which would be unrealistic using WR simulations .
Recently, several ALM models have been specifically developed for VATs . All these studies emphasize the importance of considering flow curvature effects in the calculation of volumetric forces.
Corrections for flow curvature
As mentioned in the work of , the primary challenge of VAT modeling using the ALM lies in the quality of the airfoil's aerodynamic coefficient data. Since the blades of a VAT operate in a curved flow, the curvature of the streamlines modifies the aerodynamic coefficients of the airfoil compared to those in uniform flow. As a result, predicting these aerodynamic coefficients in curved flow becomes the main challenge. The effects of flow curvature were first studied by . In their work, they explore these effects using the concepts of “virtual camber” and “virtual incidence”. They demonstrated that, concerning the lift and the moment, the aerodynamics of a symmetric airfoil in curved flow can be treated similarly to the aerodynamics of a cambered airfoil with a non-zero incidence in uniform flow. This conclusion is reached through a purely geometric transformation, which modifies the airfoil profile by converting the curved flow into a uniform flow, thereby unfolding the curved streamlines while preserving the local angles between the streamlines and the airfoil surface.
A bidirectional conformal mapping between the two flow fields was then also proposed by so that the flow around a symmetric airfoil in the curved flow is mapped to that around a cambered airfoil in the straight flow. They claim that the aerodynamic coefficients around the mapped airfoil show a good correlation with those of the original airfoil in the rotating condition and hence that they can approximately reproduce the local flow field around the rotating blade from the flow data around the mapped static wing in a straight flow. This is however incorrect as far as the tangential force is concerned: indeed, a straight potential flow past a cambered airfoil only has a normal force and an aerodynamic moment, whereas an inviscid flow past a rotating airfoil also has a tangential force. We will show this and also propose a model for it.
and demonstrated the effects of virtual camber and virtual incidence by comparing the experimental results of an airfoil in uniform flow, and with a camber similar to the virtual camber of a symmetric airfoil in curved flow, with the CFD results of a symmetric airfoil in rotation.
Since flow curvature effects are important when the ratio between blade chord c and arm length R is not small, and since the ALM relies on accurate aerodynamic coefficients to replicate the forces and moments exerted by the blade on the flow, it is essential that the modeled aerodynamic coefficients used in the ALM accurately account for the effects of flow curvature.
As stated in Sect. , the curvature effects modify the aerodynamic coefficients; also the aerodynamic moment coefficient. For symmetric airfoil in uniform flow, the ALM does not need to account for the moment around the aerodynamic center, xac, as it is essentially zero (since xac coincides closely with the center of pressure). However, in curved flow, the moment mac is no longer negligible . The ALM must therefore also be able to impose the aerodynamic moment to account for its effects on the flow. We will hence also develop a novel method for imposing a moment in the ALM, in addition to imposing forces.
Objectives and outline
This paper focuses on developing and validating improved models to accurately account for flow curvature effects on airfoils. The modeling approach only uses the aerodynamic coefficients as a function of the angle of attack (for lift, drag and moment), as obtained when considering the airfoil in a uniform flow. We here consider a NACA0015 airfoil operating at a high Reynolds number of Rec=6.0×106, and the aerodynamic coefficients are obtained using wall-resolved CFD, up to the stall angle. The curves obtained are then also fitted using polynomials; see Appendix .
The correction models are used to adapt the aerodynamic coefficients obtained in uniform flow to the curved flow conditions. Thus, the ALM user only needs to apply the corrections to usual aerodynamic coefficient data by adjusting them to the flow curvature based on the dimensions of the modeled VAT (the critical ratio being c/R) and on the attachment point of the airfoil to the arm.
The models for the normal force and moment are based on the analogy with the potential flow past an airfoil with curved streamlines; they have long been known, although many authors have neglected the contribution of the aerodynamic moment. Moreover, the model for the normal force is further improved here to remain valid for a wide range of angles, up to stall. The model for the tangential force is new; here, the analogy with a curved potential flow is shown to be flawed (i.e., the correct tangential force does not correspond to adding the parasitic drag to the potential flow prediction).
The models for the forces and the moment are developed in Sect. , also for various attachment points of the blade to the arm: at mid-chord in Sect. and at quarter chord in Sect. . For each case, the validity of the models is then also assessed by comparing the model predictions to the reference results of CFD for a rotating NACA0015 airfoil with c/R=2/7, put at various pitch angles; and here up to the two stall angles (i.e., those for negative pitch and positive pitch, which are indeed different due to flow curvature). The models are then further extended to an arbitrary location of the attachment point in Sect. .
The actuator line method (ALM) used to impose forces in the unsteady flow solver is then presented in Sect. . A new method for also imposing the aerodynamic moment into the ALM is then developed in Sect. .
The improved flow curvature correction models are then used in actual ALM simulations of a simplified vertical-axis turbine (VAT) configuration in Sect. for the purpose of further validating the adapted ALM in a more complex and unsteady flow. The case considered is that previously studied by : a single-blade VAT configuration with c/R=2/7 operating at TSR=3.25 (corresponding to its maximum power production and without boundary layer separation). Two cases are considered: Sect. for the blade attached at quarter-chord and with the ALM centered there; Sect. for the blade attached at mid-chord and for two sub-cases: ALM centered at quarter-chord and ALM centered at mid-chord. For each case, the predictions obtained using the adapted ALM are validated against those of the corresponding reference CFD simulation.
Corrections of flow curvature effects for various blade attachment points
We consider a blade section of a VAT that corresponds to an aerodynamic profile without camber. Its chord is denoted c and the arm that links the blade to the mast has a length denoted R. The airfoil is attached to the arm at some distance xp from its leading edge. The case where the attachment point is at mid-chord corresponds to xp=c/2; that where the attachment point is at quarter-chord corresponds to xp=c/4.
We consider cases where flow curvature effects are important. In particular, we compare the prediction of the simplified models developed below against the reference results from wall-resolved CFD of the rotating airfoil at a high Reynolds number of Rec=6.0×106 and for a case with strong flow curvature effects corresponding to c/R=2/7, the CFD simulations being performed in the rotating frame.
Curved potential flow past a flat plate
We consider an (x,y) coordinate system with a flat plate of chord c and centered at the origin. The plate is tilted by an angle αp relative to the horizontal. To produce curved streamlines, we put a point vortex of circulation Γ0<0 (i.e., inducing a clockwise velocity field) below the plate, at the position (0,-R) (i.e., at z=-iR). The exact solution for the curved potential flow past the flat plate can be obtained analytically and so can the force exerted by the flow on the plate: this is done in Appendix . The streamlines of the potential flow are shown in Fig. for the case c/R=2/7 and for different values of αp. The Kutta–Joukowski condition being satisfied, the streamline emanating from the trailing edge is tangent to it and the velocity is finite there. The case with αp=0 corresponds to a flow that is left/right symmetric; the normal force exerted by the flow on the plate acts downward. The case with αp=4° has a normal force close to zero. The case with αp=8° has a normal force acting upward.
Streamlines (blue) of curved potential flows past a flat plate (black) with c/R=2/7 and for various tilt angles: αp=-4, 0, 4 and 8°.
One can make an analogy between the potential flow and the flow past a flat plate that is attached at its center to an arm of length R that rotates at the angular velocity Ω. The upstream velocity seen by the plate is U=(ΩR) and its analog is the velocity U=Γ02πR induced by the vortex in the potential flow model; this sets Γ0=-2πΩR2.
However, it must be stressed that the analogy is limited. The potential flow being irrotational, the velocity induced by the vortex decreases when R increases; for the plate in solid body rotation, the velocity increases with increasing R. In particular, the analogy can only be used to obtain a model for the normal force fn exerted by the flow on the plate and for the moment mac around its aerodynamic center, only for moderate tilt angles. Properly modeling the tangential force ft will require another argument that is not compatible with potential flow theory, even for inviscid flows, and this will be developed in the paper.
When considering airfoils (thus with some thickness) instead of plates, the models for fn and mac based on the analogy with potential flow will also be acceptable if all points of the airfoil remain such that their distance to the airfoil center is small compared to the arm length R.
Transformation used for developing the simplified model
The complex potential due to the vortex is
f(z)=ϕ(z)+iψ(z)=-iΓ02πlogz+iRR=iURlogzR+i,
with z=x+iy the complex variable, ϕ(z) the potential and ψ(z) the streamfunction. As it is defined up to an arbitrary constant, we can subtract log(i)=π/2 and also write
f(z)=iURlog1-izR.
This then leads to the conformal transformation proposed in :
Z(z)=iRlog1-izR⇔z(Z)=iRe-iZR-1.
Now, for z points with |z|R small, which is the case when c/2R≪1, each conformal transformation can be approximated using the first term in its Taylor series:
Z(z)≃z+i2z2R⇔z(Z)≃Z-i2Z2R.
These simplified transformations are those retained in what follows for developing the correction models accounting for flow curvature effects.
The flat plate in the z plane is transformed into a curved plate in the Z plane, and the potential f(z) for the curved flow past the flat plate can be approximated from the potential F(Z) of a uniform flow past the cambered plate: this is done by writing f(z)=F(Z) for z=z(Z). We stress that this approximated potential flow is not the exact one; the exact one is that developed in Appendix . Nevertheless, we will see that the analogy between the flow of a rotating airfoil attached to an arm and this simplified potential flow already allows one to model quite well the effects of flow curvature on the force normal to the arm and on the aerodynamic moment. However, the analogy completely fails to model the force tangential to the arm (and so does the exact solution of Appendix ). The proper model for the tangential force will be obtained by considering the rotating airfoil attached to an arm and the net moment at the center of rotation (which must be zero in inviscid flow).
Flat plate or thin airfoil attached at its mid-chord (<inline-formula><mml:math id="M69" display="inline"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:math></inline-formula>)Flat plate without pitch angle in an inviscid flow
We first consider a flat plate of chord c in the z plane, which is centered at zc=0 and is also attached to the arm there: zp=0 (which corresponds to the case xp=c/2). The plate is horizontal: no pitch angle (αp=0): see Fig. a.
Using the transformation Z(z)=z+i2z2R, the plate is transformed into a horizontal and symmetric cambered plate in the Z plane: the attachment point zp=zc=0 is transformed into Zp=Zc=0, the trailing edge zT=c/2 is transformed into ZT=c/2+ic2/(8R) and the leading edge zL=-c/2 is transformed into ZL=-c/2+ic2/(8R); see Fig. b.
(a, d) Schematic of a flat plate attached at xp=c/2 without pitch angle in the z plane (solid black) and a background flow with curved streamlines (light gray). (b, c) Transformed cambered plate in the Z plane in a uniform background flow. Note that the flow curvature was here exaggerated for clarity, using a too-large value of c/R=1/2.
Using the approximate potential flow discussed above, the normal force, fn, exerted by the flow U on the cambered plate is obtained as
Cn(0)=-2πc4R.
The force fn(0) is negative, thus oriented inward.
On the other hand, we also have the exact value of Cn(0) obtained from the exact potential flow presented in Appendix , with the final result of Eq. (). The amplitude of Cn(0) is thus a little bit less than 2πc4R. Nevertheless: when c2R2≪1, which is most often the case and is the one considered in this paper, we can neglect this effect. We then simply associate the flow curvature effect with an effective shift in the angle of attack, by the angle αc defined as
sin(αc)=c4R⇒Cn(0)=-2πsin(αc).
The aerodynamic moment mc/2(0) acting at the center of the plate (i.e., at the mid-chord point) is here zero, by left–right symmetry of the flow around the plate, in the same way as for the potential flow developed in Appendix and shown in Fig. .
For an inviscid flow, it would take no energy to rotate the airfoil attached to the arm: the moment m0 around the center of rotation of the arm would therefore be zero. We stress that this is also confirmed in Appendix by performing an inviscid simulation of the flow past a rotating airfoil. As this moment is equal to m0=mc/2(0)+Rft(0), and since mc/2(0)=0, the tangential force exerted on the plate by an inviscid flow will also be zero: ft(0)=0 and hence Ct(0)=0. For a viscous flow, there will, of course, be some parasitic drag, and thus a value of Ct(0)>0; we will add this effect later, in Sect. .
We already stress that, in an inviscid flow, the tangential force will no longer be zero when the plate is pitched by some angle αp, because mc/2 will then be non-zero; see Sect. .
We also obtain the moment at the location of the aerodynamic center (which is located at a distance c/4 from the leading edge): since mc/2(0)=0, we obtain mac(0)=mc/4(0)=-(c/4)fn(0)>0; see Fig. c and d. The associated moment coefficient is thus
Cmac(0)=Cmc/4(0)=mc/4(0)12ρU2c2=-Cn(0)4≃π2sin(αc)≃π2c4R.
We note that the angle αp=0 considered here is the geometrical pitch angle of the profile attached at its mid-point relatively to the horizontal, not the angle that the profile makes with the curved flow streamline at the location of the aerodynamic center and which is used as the “effective angle of attack” α in an actuator line method (ALM) when the line is located at the aerodynamic center. At this location, the curved streamline makes an angle of α=c4R relative to the horizontal plate (see Fig. d); thus an angle also equal to αc.
The usual aerodynamic coefficients relative to the curved flow velocity vector considered at the location of the aerodynamic center are easily obtained using the angle αc of the local velocity vector relative to the plate:
Cl=Cn(0)cos(αc),Cd=Cn(0)sin(αc).
As Cn(0) is negative, both Cl and Cd are negative; as shown in Fig. d.
Flat plate with pitch angle in an inviscid flow
We now further pitch the flat plate at a moderate angle αp relative to the horizontal, as shown in Fig. a. The normal force coefficient will be modeled using
Cn(αp)≃2πsin(αp)-2πsin(αc).
We note that this formula is slightly different from the formula inspired from the potential flow theory past a cambered plate, which would be
Cn(αp)≃2πsin(αp-αc).
Obviously, both formulas provide results that are very close for small αc and moderate αp-αc; they both also provide Cn=0 when αp=αc. The reason why we will use Eq. () instead of Eq. () will become clear when we further extend the model to thin airfoils in real viscous flows and up to large pitch angles, even up to the stall of the airfoil, and compare the results to those of the reference CFD results.
(a, d) Schematic of a flat plate attached at xp=c/2 with a moderate pitch angle 0<αp<αc in the z plane (solid black) and a background flow with curved streamlines (light gray). (b, c) Transformed tilted cambered plate in the Z plane in a uniform background flow. Note that the flow curvature was here exaggerated for clarity, using a too-large value of c/R=1/2.
For αp<αc, Cn<0 and the normal force is oriented inward, as in Fig. a; for αp>αc, Cn>0 and the normal force is oriented outward.
When αp≠0, the flow around the plate is no longer left-right symmetric (as for the potential flow shown in Fig. ). The moment mc/2 around the center of the plate is, therefore, non-zero.
For the model, we use the prediction for a potential flow past the tilted cambered plate put in an horizontal stream at velocity U:
Cmc/2(αp)=mc/212ρU2c2=π4sin(2αp)=π2sin(αp)cos(αp).
The moment around the aerodynamic center is hence obtained as mc/4(αp)=mc/2(αp)-(cos(αp)c/4)fn(αp), and thus the associated moment coefficient is
Cmc/4(αp)=Cmc/2(αp)-Cn(αp)4cos(αp)≃π2sin(αc)cos(αp).
As the moment m0 around the center of rotation of the arm must still be zero in an inviscid flow (see also Appendix for a numerical confirmation of this), there must now also be a net tangential force component fti due solely to the pressure distribution, and which is such that
m0=mc/2(αp)+Rfti(αp)=0.
The model for the tangential force coefficient in inviscid flow is hence obtained as
Cti(αp)=-cRCmc/2(αp)≃-4sin(αc)Cmc/2(αp).
Of course, there will also be a parasitic drag coefficient to add to the model for Ct when modeling viscous flow; see Sect. .
Thin airfoil with pitch angle in a real viscous flow
The correction models above, obtained assuming an inviscid flow and a flat plate, are also used for thin aerodynamic profiles without camber and in real flows and thus with the added parasitic effects associated with viscosity.
For the normal force coefficient, the model becomes
Cn(αp)≃Clu(αp)-Clu(αc),
where Clu(β) is the usual curve for the lift coefficient of the airfoil in a uniform flow as a function of its angle of attack β relative to that uniform flow. This curve is obtained from experiments or using CFD. In the present case, it was obtained using CFD; see Appendix .
When the airfoil is not pitched, the moment at mid-chord, Cmc/20, is small yet non-zero. It can be estimated by performing CFD of the airfoil deformed using the transformation Z(z) and put in a uniform flow (for the deformed NACA0015 airfoil, we measure Cmc/20≃-0.00746: see Appendix ).
When the airfoil is pitched by an angle αp, we then use, by similarity with the model obtained for the plate,
Cmc/2(αp)≃Cmc/20+14Clu(αp)cos(αp).
We then obtain, for the moment around the quarter-chord point,
Cmc/4(αp)=Cmc/2(αp)-14Cn(αp)cos(αp)≃Cmc/20+14Clu(αc)cos(αp).
In the same way, the moment around the true aerodynamic center, which is located at xacc very close to 14 but not exactly there (for the NACA0015 airfoil, we measure xacc≃0.2476: see Appendix ), is obtained as
Cmac(αp)≃Cmc/20+12-xaccClu(αc)cos(αp).
To model the tangential force coefficient Ct, we add to the inviscid flow contribution, Cti=-cRCmc/2, the parasitic contribution Ctν associated with the viscous effects, i.e., due to the presence of boundary layers and their separation, resulting in pressure drag and friction drag:
Ct(αp)=Ctν(αp)+Cti(αp)=Ctν,p(αp)+Cti(αp)+Ctν,f(αp)=Ctp(αp)+Ctν,f(αp)≃Cdu,p(αp)-cRCmc/2(αp)+Cdu,f(αp+αc).
The term Ctν,p+Cti in parentheses represents the complete pressure component: we denote it Ctp.
Note that we modeled Ctν,f(αp) using Cdu,f(αp+αc) with Cdu,f(β) the curve for the fraction due to wall friction of the drag coefficient for the airfoil in uniform flow as a function of its angle of attack β (see Appendix ). This choice is motivated by the fact that, in the reference CFD of the rotating airfoil, Ctν,f is seen to be minimum when αp≃-αc and not when αp≃0. As to Ctp=Ctν,p+Cti, it is found, in the CFD, to be symmetric relative to αp≃0, and hence we model Ctν,p(αp) using Cdu,p(αp). The model for the parasitic drag is thus taken as
Ctν(αp)=Ctν,p(αp)+Ctν,f(αp)≃Cdu,p(αp)+Cdu,f(αp+αc).
We also stress an important point: if the moment mc/2 is not enforced in the ALM (using the procedure developed in Sect. ), then only the parasitic tangential force ftν must be enforced in the ALM, while fti must be ignored.
Finally, the force coefficients can also be decomposed into the usual aerodynamic coefficients considered at c/4 by using the angle of attack αp+αc that the curved flow velocity vector makes there relative to the airfoil chord:
Cl=Cn(αp)cos(αc)-Ct(αp)sin(αc),Cd=Cn(αp)sin(αc)+Ct(αp)cos(αc).
Comparison of the model predictions against CFD data of a rotating NACA0015 airfoil
We test the curvature correction models proposed above for a case with significant curvature effects (with c/R=2/7) and for a NACA0015 airfoil (hence an airfoil that is not very thin) attached at xp=c/2. We then obtain sin(αc)=1/14≃0.07143 and αc=arcsin(1/14)≃0.07149rad (thus ≃4.096°).
We first consider the case αp=0. Using the lift curve of the NACA0015, the prediction is Cn≃-Clu(αc)≃-0.454 (Cn≃-2π(1/14)≃-0.449 when using the flat plate model) to compare with Cn≃-0.431 measured in CFD of the rotating airfoil.
For the flat plate, we have Cmc/2=0 and thus Cmc/4=Cmac=-Cn/4≃(π/2)(1/14)≃0.1122. For the NACA0015 airfoil, and if Cmc/20 is neglected, the predictions are Cmc/4≃0.1135 and Cmac≃0.1146. The values measured in CFD of the rotating airfoil are Cmc/2≃-0.0056, Cmc/4≃0.10225 and Cmac≃0.10327. When using the modeled Cmc/20≃-0.00746 (obtained using CFD of the airfoil deformed using Z(z) and put in a uniform flow; see Appendix ), the predictions are Cmc/4≃0.1079 and Cmac≃0.1090: indeed better.
The values measured in CFD of the rotating airfoil are also Ctp≃0.00387, Ctν,f≃0.00682 and thus Ct≃0.01069. When using the modeled Cmc/20 value, the predictions become Cti≃0.00211 and thus Ctp≃0.00401, which compares well with the reference CFD value. The model is again improved when we incorporate the small, yet non-zero, value of Cmc/20.
We present in Fig. the predicted values for the aerodynamic coefficients. The range of moderate pitch angles corresponds to -12≤αp≤12°. We see that the predictions are quite accurate within this range. The comparison is also made for the complete range, up to the two stall angles (they are slightly different due to the flow curvature effects). We see that the model developed above also performs well up to both stall angles and that the predicted values (of roughly -18° and 18°) are in fair agreement with those of the reference CFD (that are roughly -19° and 17 deg). We stress that this would not be the case if we had used
Cn(αp)≃Clu(αp-αc),
as model for Cn instead of Eq. (). Although both choices produce similar predictions for Cn, and hence also for all other coefficients, in the range of moderate tilt angles, using Eq. () would predict incorrect stall angles (at roughly -18+4=-14° and 18+4=22°: not in line with those of the reference CFD).
NACA0015 airfoil in rotation with c/R=2/7 and attached at xp=c/2: comparison between the model-predicted values (magenta) of Cn, Cmc/2, Cmc/4, Ctp, Ctν,f, Ct and the reference values obtained in CFD of the rotating airfoil (blue).
Finally, we stress that the model for Cmc/4 is based on the analogy of the inviscid potential flow as a cambered airfoil. As such, it cannot take into account deviations from when the pitch angle exceeds ≃10°, where viscous effects on the moment also become significant, resulting in deviations up to ≃0.025-0.03 near stall.
Flat plate or thin airfoil attached at its quarter-chord (<inline-formula><mml:math id="M232" display="inline"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:math></inline-formula>)
We consider next a flat plate of chord c in the z plane, which is attached to the arm at zp=0 but has its center at zc=c/4 (it corresponds to the case xp=c/4). The plate is attached at its aerodynamic center, hence its pitch angle αp is also the angle that the plate makes relative to the curved flow velocity vector at its aerodynamic center.
We first consider the case without pitch angle: αp=0. The horizontal plate is transformed into a tilted and cambered plate in the Z plane: its attachment point zp=0 is transformed into Zp=0, its center zc=c/4 is transformed into c/4+ic2/(32R), its trailing edge zT=3c/4 is transformed into ZT=3c/4+i9c2/(32R) and its leading edge zL=-c/4 is transformed into ZL=-c/4+ic2/(32R); see Fig. .
(a) Schematic of a flat plate attached at xp=c/4 and without pitch angle in the z plane (solid black) and a background flow with curved streamlines (light gray). (b) Transformed tilted cambered plate in the Z plane in a uniform background flow. Note that the flow curvature was here exaggerated for clarity, using a too-large value of c/R=1/2.
The slope of the tilted cambered plate at its center is then obtained as (dY/dX)c=c/(4R). This is the angle that the cambered plate makes relative to the uniform horizontal velocity U. We note that this angle also corresponds to αc.
When αp=0, the normal force coefficient is obtained as
Cn(0)≃2πsin(0-αc)-2πsin(αc)=-4πsin(αc).
When we pitch the flat plate by an angle of attack αp, the model becomes
Cn(αp)≃2πsin(αp-αc)-2πsin(αc).
We stress again that this formula is not exactly that obtained from the potential flow theory for the curved plate in uniform flow and pitched by the angle -αc and which would be
Cn(αp)≃2πsin(αp-2αc).
Both formulas provide results that are very close for moderate angles. The reason why we use Eq. () instead of Eq. () is again that it provides better results when we extend the model to thin airfoils in viscous flows and up to large angles, including stall, as will be shown in the next section.
The moment coefficient at the plate center is again obtained using the angle of attack αp-αc of the cambered plate in the uniform flow:
Cmc/2(αp)=π4sin(2(αp-αc))=π2sin(αp-αc)cos(αp-αc).
This moment is indeed zero, as it should be, when αp=αc, as the cambered plate of the transformed flow is then horizontal. The moment coefficient around the aerodynamic center is hence also obtained as
Cmc/4(αp)=Cmc/2(αp)-Cn(αp)4cos(αp-αc)≃π2sin(αc)cos(αp-αc).
In an inviscid flow, the net pressure force is not purely normal: there must also be a net tangential component in order to compensate for the moment at the attachment point so that the moment around the center of the arm is zero; see Appendix :
m0=mc/4+Rfti=0⟹Cmc/4(αp)+RcCti(αp)=0.
The inviscid tangential force coefficient is thus obtained as
Cti(αp)=-cRCmc/4(αp)≃-4sin(αc)π2sin(αc)cos(αp-αc)=-2πsin2(αc)cos(αp-αc).
The extension of the prediction models to thin airfoils in viscous flow is then also obtained as follows. For the lift coefficient, the model becomes
Cn(αp)≃Clu(αp-αc)-Clu(αc).
For the moment coefficient at mid chord, the model is obtained as
Cmc/2(αp)≃Cmc/20+14Clu(αp-αc)cos(αp-αc).
The moment coefficients Cmc/4 and Cmac are then obtained as
Cmc/4(αp)=Cmc/2(αp)-14Cn(αp)cos(αp-αc)≃Cmc/20+14Clu(αc)cos(αp-αc)
and
Cmac(αp)=Cmc/2(αp)-12-xaccCn(αp)cos(αp-αc).
We note that the constant term Cmc/20 is the same as that used previously when modeling an airfoil attached at xp=c/2. Indeed, the airfoil attached at xp=c/4 and with a pitch of αp=αc must have the same moment as the airfoil attached at xp=c/2 without pitch (αp=0).
We now add the parasitic drag coefficient Ctν to the inviscid tangential pressure force coefficient Cti=-cRCmc/4. The model is finally obtained as
Ct(αp)=Ctν(αp)+Cti(αp)=Ctν,p(αp)+Cti(αp)+Ctν,f(αp)=Ctp(αp)+Ctν,f(αp)≃Cdu,p(αp-αc)-cRCmc/4(αp)+Cdu,f(αp).
We modeled Ctν,f(αp) using Cdu,f(αp) instead of using Cdu,f(αp-αc) because the CFD data show that Ctν,f is minimum when αp≃0. As to Ctp=Ctν,p+Cti, it is found, in the CFD, to be symmetric relative to αp≃αc, and hence we model Ctν,p(αp) using Cdu,p(αp-αc). The model for the parasitic drag is thus taken as
Ctν(αp)=Ctν,p(αp)+Ctν,f(αp)≃Cdu,p(αp-αc)+Cdu,f(αp).
We again stress an important point: if the moment mc/4 is not enforced in the ALM (using the procedure developed in Sect. ), then only the parasitic tangential force ftν must be enforced in the ALM, while fti must be ignored.
Comparison of the model predictions against CFD data of a rotating NACA0015 airfoil
We test the curvature correction models proposed above for the same case with c/R=2/7 and a NACA0015 airfoil, but this time attached to the arm at xp=c/4.
For the case αp=0, the prediction is Cn≃-0.9079 (Cn≃-0.8976 when using the flat plate model) to be compared with Cn≃-0.893 measured in CFD of the rotating airfoil.
The predictions for the moment coefficients are Cmc/2≃-0.1132 and Cmc/4≃0.1132, the values measured in CFD being Cmc/2≃-0.1222 and Cmc/4≃0.1010.
We present in Fig. the predicted values for the aerodynamic coefficients compared to the reference CFD values. Here, the range of moderate pitch angles corresponds to -12+4=-8≤α≤12+4=16° (taking into account the shift by αc≃4° due to the attachment point being at xp=c/4 instead of xp=c/2). We see that the predictions are quite accurate within this range. The comparison is also made for the complete range, up to the two stall angles (that are here much different due to the flow curvature effects combined with the shift in attachment). We see that the models developed above also perform fairly well up to both stall angles and that the predicted values (of roughly -18+4=-14° and 18+4=22° are in fair agreement with those of the reference (that are -16° and 20-21°). We stress again that this would not be the case if we had used
Cn(αp)≃Clu(αp-2αc)
instead of Eq. (). Using Eq. () would provide similar results for moderate angles but would predict incorrect stall angles (at roughly -18+8=-10° and at 18+8=26°).
NACA0015 airfoil in rotation with c/R=2/7 and attached at xp=c/4: comparison between the predicted values (magenta) of Cn, Cmc/2, Cmc/4, Ctp, Ctν,f, Ct and the reference values obtained in CFD of the rotating airfoil (red).
Flat plate or thin airfoil attached at an arbitrary position
The case in which the attachment point is arbitrarily chosen can also be modeled. We then write
xp=(1+δ)c4,
where δ is a chosen parameter. The canonic cases xp=c/2 and xp=c/4 investigated and modeled so far correspond, respectively, to δ=1 and δ=0.
The opening angle, αv, between the attachment point xp and the airfoil center c/2 is obtained as
αv=12-xpccR=(1-δ)c4R=(1-δ)αc.
We first consider the case of a flat plate without pitch angle (αp=0). The horizontal plate is transformed into a tilted and cambered plate in the Z plane: its attachment point zp=0 is transformed into Zp=0, its center zc=(1-δ)c/4 is transformed into (1-δ)c/4+i(1-δ)2c2/(32R), its trailing edge zT=(3-δ)c/4 is transformed into ZT=(3-δ)c/4+i(3-δ)2c2/(32R) and its leading edge zL=-(1+δ)c/4 is transformed into ZL=-(1+δ)c/4+i(1+δ)2c2/(32R). The slope of the tilted cambered plate at its center is then obtained as
dYdXzc=(1-δ)c4R=(1-δ)αc=αv.
The angle of the cambered plate relative to the uniform horizontal velocity U is therefore also equal to αv.
When further pitching the flat plate by an angle αp, the angle α that the curved flow velocity vector at c/4 makes relative to the plate is now
α=αp+(αc-αv)=αp+δαc.
We also obtain
αp-αv=α-αc.
Using the same procedure as used for the cases with xp=c/2 and xp=c/4, we obtain the general models: for the flat plate and for thin airfoils without camber. We write them here for the airfoils. For the normal force coefficient Cn at the attachment point, we obtain
Cn(αp)≃Clu(αp-αv)-Clu(αc).
The moment coefficient at the plate center is again obtained from the potential flow theory as
Cmc/2(αp)≃Cmc/20+14Clu(αp-αv)cos(αp-αv).
This moment is indeed zero, as it should, for a plate (that has no Cmc/20) when αp=αv as the cambered plate of the transformed flow is then horizontal.
The aerodynamic moment coefficients, Cmc/4 and Cmac, are then obtained as
Cmc/4(αp)=Cmc/2(αp)-14Cn(αp)cos(αp-αv)≃Cmc/20+14Clu(αc)cos(αp-αv)
and
Cmac(αp)=Cmc/2(αp)-12-xaccCn(αp)cos(αp-αv).
In the same way, the model for the moment coefficient around the attachment point is obtained as
Cmxp(αp)≃Cmc/2(αp)-12-xpcCn(αp)cos(αp-αv).
As the moment m0=mxp+Rfti around the center of rotation of the arm must still be zero in inviscid flow, the model for the tangential force coefficient in inviscid flow becomes
Cti(αp)≃-cRCmxp(αp),
and hence the model for viscous flow is now taken as
Ct(αp)=Ctν(αp)+Cti(αp)=Ctν,p(αp)+Cti(αp)+Ctν,f(αp)=Ctp(αp)+Ctν,f(αp)≃Cdu,p(αp-αv)-cRCmxp(αp)+Cdu,f((αp-αv)+αc).
Again, if the moment mxp is not enforced in the ALM, then only ftν must be enforced, while fti must be ignored.
Finally, the force coefficients can also be decomposed into the usual aerodynamic coefficients considered at c/4 by using the angle of attack αp+δαc that the curved flow velocity vector makes there relative to the airfoil chord:
Cl=Cn(αp)cos(δαc)-Ct(αp)sin(δαc),Cd=Cn(αp)sin(δαc)+Ct(αp)cos(δαc).
Actuator line method for imposing the aerodynamic forces
We briefly recall the actuator line method (ALM) which is used here to model the effect of the airfoil forces on the flow. It was detailed in as implemented in a fourth-order incompressible flow solver using finite differences. It was also used in for aero-elastic studies of a 15 MW horizontal-axis turbine.
At each time step, the vector force per unit span -f that the airfoil exerts on the flow (i.e., the opposite of the vector force f that the flow exerts on the airfoil) is distributed over the neighboring grid points of the flow solver. In all that follows, the term “force” is understood as “force per unit span”. This is achieved using a 2-D Gaussian kernel centered at the airfoil AC location and whose (x,y) frame is fixed relative to the airfoil.
The force field applied to the flow solver is as follows:
fs(x,y)=-η2D(x,y)fwheref=fxe^x+fye^y
and
η2D(x,y)=ησc(x)ησt(y)=1πσcexp-x2σc21πσtexp-y2σt2,
with σc the core size in the airfoil chord direction and σt that in the thickness direction. Here, we use an isotropic kernel (σc=σt=σ) and with σ=c/4 as recommended by . The kernel is discretized using a 2-D template that consists of (2n+1)(2n+1) cells of size h×h. A node is located at the center of each cell (xi,yj)=(ih,jh), and the kernel is integrated analytically over that cell to obtain the weight. For the integration in x, we obtain, for 0≤i<n,
wi=∫xi-h/2xi+h/2ησ(x)dx=12erfxi+h/2σ-erfxi-h/2σ.
For the cell i=n, we include the part outside (to ensure that the sum of the weights is unity):
wn=∫xn-h/2∞ησ(x)dx=121-erfxn-h/2σ.
By symmetry, we have w-i=wi. The weights in y are wj=wi for j=i. The weight associated with the (i,j) cell is then wi,j=wiwj and the associated force is
fi,js=-wiwjf.
We use h=σ/2 and n=4. The weights wi are provided in Table . As the kernel was integrated over each cell of the template, we ensure that the forces are completely imposed to the flow solver.
Weights wi associated with the 1-D Gaussian kernel for the case with h=σ/2 and using n=4. The numbers provided have been rounded so that ∑i=-44wi=1 exactly.
i0±1±2±3±4wi0.276340.217410.105870.031890.00666
In the present implementation of the ALM used for modeling a VAT configuration, an overset mesh technique is employed, in which the flow solver mesh covering the turbine region, including the AL and its template, rotates together with the blade; see Fig. . Note that, when the AL and its template move through the flow solver grid, as in , the fraction fi,js of the force on each cell of the template must be further distributed to the nearest flow solver grid points. This is done using a bi-linear distribution kernel, ensuring that each fraction is distributed to the nearest grid points.
The effective velocity vector to be used in the ALM is obtained by using a weighted average of the velocity vectors provided by the flow solver around the control point (equal to the center of the Gaussian kernel, typically taken as the location of the aerodynamic center), their relative weighting being determined using the same template as that used for the force distribution. This procedure corresponds to “integral sampling” (as opposed to “pointwise sampling”) and has been shown to provide slightly better results . The effective velocity vector at the control point, v, is thus measured using
v=∑i=-nxnx∑j=-nynywiwjv(xi,yj).
When the airfoil is moving relative to the fixed reference frame, as is the case in the present work, we must add this relative velocity to the flow solver velocity field since the ALM requires the apparent velocity seen by the airfoil, V.
Vector field gs(x,y)/mac=g(r)e^θ used for the moment distribution (top) and its components (bottom): gxs(x,y)/mac along e^x (left) and gys(x,y)/mac along e^y (right).
Moreover, when the parasitic drag is significant, the apparent velocity vector, V, is not yet the suitable vector to use in the ALM, as it also contains the spurious velocity component induced by the wake vorticity associated with the parasitic drag. This induced velocity can be estimated using the procedure described in . Assuming that the wake is shed primarily in the direction of V, the corrected apparent velocity is
Vcorr=V1-14πcσCdν.
As Cdν is used in the above correction, a few iterations are required to determine all values; alternatively, the value of the previous time step can be used.
Once we have Vcorr, we measure the angle that this vector makes relative to the airfoil chord. Using this angle and the norm Vcorr of the velocity, we can compute, using the airfoil polar model, the lift and drag forces exerted by the flow on the airfoil.
Template used for the moment distribution for the case with h=σ/2 and using n=4. The color intensity of each (i,j) cell is proportional to its weights: wiqj (left) and qiwj (right).
Actuator line method for imposing an aerodynamic moment
The previous section was devoted to presenting our particular implementation of the ALM aimed at efficiently distributing the aerodynamic force acting on the airfoil onto the nearby flow solver grid points, using a 2D template corresponding to the discretization of a Gaussian kernel. The same template is also used to evaluate the effective velocity required to compute the aerodynamic forces.
Weights qj for the case with h=σ/2 and using n=4. By anti-symmetry, q-j=-qj.
j01234qj00.104270.101600.045940.01320
As such, however, the ALM cannot handle any aerodynamic moment. We extend the actuator line method to enable the distribution of an aerodynamic moment. Of course, a pure aerodynamic moment cannot be imposed, as the ALM uses a distribution kernel of some size σ. We hence replace the aerodynamic moment by a force field that has the same net effect, as presented in Fig. ; we then discretize this force field using the 2D template.
Computational domain and mesh used for the ALM simulations. The turbine region, including the template of the ALM, is implemented using an overset mesh approach to enable the 2-D template as described in Sects. and . The weights for the application of the forces are also illustrated in blue.
VAT with NACA0015 airfoil attached at xp=c/4 and operating at TSR=3.25: comparison between the ALM with mc/4 and fti explicitly enforced (red), the ALM without mc/4 and fti explicitly enforced (blue) and the wall-resolved reference simulation (black) of Cmc/2, Cmc/4, Cn and Ct. The results of the ALM without curvature corrections are also shown for comparison (green).
VAT with NACA0015 airfoil attached at xp=c/4 and operating at TSR=3.25: comparison between the results of the ALM with mc/4 and fti explicitly enforced (red), the ALM without mc/4 and fti explicitly enforced (blue) and the wall-resolved reference simulation (black) of Cp. The results of the ALM without curvature corrections are also shown for comparison (green).
Assume that the flow exerts a pitching moment per unit span mac around the airfoil aerodynamic center. Recall that the pitching moment is, by convention, positive when it acts to pitch the airfoil in the nose-up direction. The airfoil hence exerts a moment -mac on the flow. This effect can be achieved by imposing the following annular force field:
gs(x,y)=g(r)e^θmac
with
g(r)=-12πσ2ddrexp-r2σ2=1πσ3rσexp-r2σ2,
which indeed satisfies ∬rg(r)dS=1 where dS=rdθdr:
∬rg(r)dS=2π∫0∞g(r)r2dr=2∫0∞exp-r2σ2r3σ3drσ=2∫0∞exp-s2s3ds=∫0∞exp(-p)pdp=1,
where we have used the changes in variable s=r/σ and p=s2 and where the last integral has been evaluated by parts.
This force field corresponds to applying a net moment of -mac to the flow, without applying any net force. The applied force field per unit mac is illustrated at the top of Fig. ; it is indeed counter-clockwise when mac is positive. It can be decomposed as
gs(x,y)=g(r)e^θmac=g(r)-sinθe^x+cosθe^ymac=1πσ3rσexp-r2σ2-sinθe^x+cosθe^ymac.
Since x=rcosθ and y=rsinθ, we finally obtain gs(x,y)=gxs(x,y)e^x+gys(x,y)e^y with
gxs(x,y)=-1πσ3yσexp-x2+y2σ2mac=-ησ(x)yσησ(y)macσ,gys(x,y)=1πσ3xσexp-x2+y2σ2mac=xσησ(x)ησ(y)macσ.
These components are illustrated at the bottom of Fig. .
Mean Cmc/2, Cmc/4, Cn, Ct and Cp over a blade cycle for a VAT with airfoil attached at xp=c/4 and operating at TSR=3.25. The results of the ALM without curvature corrections are also reported for comparison.
CoefficientsCmc/2‾Cmc/4‾Cn‾Ct‾Cp‾Wall-resolved simulation-0.1770.136-1.255-0.1870.482ALM without mc/4 and fti explicitly enforced-0.1610.166-1.325-0.1960.485ALM with mc/4 and fti explicitly enforced-0.1620.164-1.320-0.1990.494ALM without curvature corrections-0.009-0.002-0.027-0.1500.487
The above equations are then integrated analytically over each cell of the 2D template to obtain the values of the weights. We obtain
∫xi-h/2xi+h/2∫yj-h/2yj+h/2gxs(x,y)dxdy=-∫xi-h/2xi+h/2ησ(x)dx∫yj-h/2yj+h/2yσησ(y)dymacσ.
The first integral has already been evaluated: it is wi. Let us evaluate the second integral, which we denote qj. For 0≤j<n, we obtain
qj=∫yj-h/2yj+h/2yσησ(y)dy=1π∫yj-h/2yj+h/2yσexp-y2σ2dyσ=1π∫(yj-h/2)/σ(yj+h/2)/σsexp-s2ds=12π∫(yj-h/2)2/σ2(yj+h/2)2/σ2exp(-p)dp=12πexp-(yj-h/2)2σ2-exp-(yj+h/2)2σ2.
By anti-symmetry, we have q-j=-qj and q0=0. For the weight associated with the boundary cell j=n, the integration again includes the part of the kernel outside:
qj=∫yn-h/2∞yσησ(y)dy=12πexp-(yj-h/2)2σ2.
We hence finally obtain, for the (i,j) cell,
∫xi-h/2xi+h/2∫yj-h/2yj+h/2gxs(x,y)dxdy=-wiqjmacσ.
Following the same procedure, we also obtain
∫xi-h/2xi+h/2∫yj-h/2yj+h/2gys(x,y)dxdy=qiwjmacσ.
The weights wiqj and qiwj are illustrated in Fig. . We stress that our method imposes the net moment whatever the choice of σ. We hence can use the same value of σ as that used for imposing the forces, also for simplicity.
In summary, the ALM that imposes both the aerodynamic force f and moment mac amounts to distributing the following force on each (i,j) cell:
-wiwjfx+wiqjmacσe^x-wiwjfy-qiwjmacσe^y.
As the two components of the kernel were integrated over each cell of the template, we ensure that the moment is completely imposed to the flow solver.
VAT with NACA0015 airfoil attached at xp=c/4 and operating at TSR=3.25: comparison between the ALM with mc/4 and fti explicitly enforced (red) and the reference wall-resolved simulation (black) of the pressure contribution Ctp (with its components) and of the friction contribution Ctf to the tangential force coefficient.
Vorticity field for the wall-resolved simulation (left), the ALM simulation without mc/4 enforced (center) and the ALM simulation with mc/4 enforced (right).
Results: corrected ALM applied to a VAT configuration and comparison with CFD data
We consider the single-blade VAT configuration that was studied by . The blade profile is the NACA0015 airfoil considered previously. It is attached to the arm without pitch angle. The blade is rotating counterclockwise at the angular velocity Ω and the upstream flow, at velocity Uw, is coming from the left. The tip speed ratio is TSR=ΩRUw. The simulated case corresponds to TSR=3.25, which is the value at which the produced power is maximized. This value is also high enough that the flow past the blade does not separate dynamically at any position over the cycle; as is confirmed when examining the vorticity field of the wall-resolved simulation in Fig. . Hence, using the static airfoil polar data, solely corrected for flow curvature effects (i.e., no added dynamic polar model), is expected to be sufficient.
All simulations are conducted under the incompressible flow assumption. The ALM and wall-resolved simulations are performed using the finite-volume solver Siemens® STAR-CCM+®. The URANS equations are solved in conjunction with the Spalart–Allmaras turbulence closure model. A uniform velocity is imposed at the inlet, and a turbulent to molecular viscosity ratio of νt/ν=0.2 is used, representative of a clean flow. A uniform static pressure is imposed at the outlet. The far lateral boundaries are symmetry planes.
Second-order schemes are employed for the spatial discretization of both the convective and diffusive fluxes. The temporal integration is carried out using the second-order implicit unsteady scheme, and one turbine revolution is discretized into 1000 time steps. The pressure–velocity coupling is handled using the coupled flow algorithm for the ALM simulations to enhance numerical robustness, while the segregated approach SIMPLE is adopted for the wall-resolved simulations. The results reported correspond to simulations for which the relative variation in the cycle-averaged power coefficient is below 1 %, thereby ensuring statistical convergence.
For the wall-resolved configurations, an overset mesh technique is used to accommodate the blade motion. To ensure adequate boundary layer resolutions, prismatic layers are used near the blade surface, with a target value of y+≃1 for the height of the first cell, and a maximum wall-normal growth rate of 1.2. Two levels of mesh refinement are employed around the turbine: one for the turbine region, where the cell size is constrained by the overset mesh, and another one for the wake.
VAT with NACA0015 airfoil attached at xp=c/2 and operating at TSR=3.25: comparison between the ALM with mc/4, fti and fni explicitly enforced (red), the ALM without mc/4, fti and fni explicitly enforced (blue) and the reference wall-resolved simulation (black) of Cmc/2, Cmc/4, Cn and Ct. Both ALM cases are centered at c/4.
Case of airfoil attached at <inline-formula><mml:math id="M548" display="inline"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:math></inline-formula>
We first consider the case in which the airfoil is attached to the arm at its quarter-chord.
The velocity vector v of the flow is measured at xp=c/4 using the integral sampling; its components are (vt,vn). The components of the apparent velocity vector V as seen by the rotating blade are thus (Vt=ΩR+vt,Vn=vn). The angle of the velocity vector V relative to the airfoil chord is denoted α.
The mesh used for the ALM simulations is shown in Fig. . A rectangular domain is used, identical to that of the wall-resolved reference case. An overset technique is implemented to maintain a hexahedral mesh with the same template for the ALM (h=σ/2) as used in Sects. and . The overset approach also enables the rotation of the actuation zone. Two refinement regions are used: one for the turbine region, including the template, and another one for the wake. The cell size within the turbine region is chosen so as to ensure accurate interpolation between the region and the background mesh. Notably, the wall-resolved reference case mesh is more refined (≃328000 cells) compared to the ALM mesh (≃37000 cells), in line with the intended efficiency of the ALM approach. The methodology is applied to all ALM cases in this work.
Case without explicitly enforcing <inline-formula><mml:math id="M559" display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M560" display="inline"><mml:mrow><mml:msubsup><mml:mi>f</mml:mi><mml:mi mathvariant="normal">t</mml:mi><mml:mi mathvariant="normal">i</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula> in the ALM
We first consider the case where we do not explicitly enforce the moment mc/4 in the ALM; hence we also cannot enforce the inviscid tangential force fti in the ALM. Only the lift and parasitic drag contributions are enforced.
The velocity vector is first corrected to compensate for the velocity induced by the shed vortex wake produced by the parasitic drag. As this wake is assumed to be shed in the direction of V, we correct its amplitude:
Vcorr=V1-14πcσCdν.
The angle that the corrected velocity vector makes relative to the chord is therefore still α.
In terms of modeling, the lift force (i.e., force in the direction orthogonal to the apparent velocity) exerted by the velocity Vcorr on the airfoil attached at xp=c/4 and without pitch is equivalent to the normal force (= lift force) exerted by a horizontal flow of velocity Vcorr on the airfoil attached at xp=c/4 and pitched at an angle αp=α. The lift coefficient is thus modeled using
Cl(α)≃Clu(α-αc)-Clu(αc).
Likewise, the parasitic drag (i.e., force in the direction of the apparent velocity) is modeled using
Cdν(α)=Cdν,p(α)+Cdν,f(α)≃Cdu,p(α-αc)+Cdu,f(α).
for the parasitic drag coefficient.
The moment coefficients can also be estimated. For the moment coefficient at mid-chord, the model gives
Cmc/2(α)=Cmc/20+14Clu(α-αc)cos(α-αc).
The moment coefficient Cmc/4 is then also obtained as
Cmc/4(α)=Cmc/2(α)-14Cl(α)cos(α-αc)≃Cmc/20+14Clu(αc)cos(α-αc).
Further adding the inviscid tangential force coefficient Cti=-cRCmc/4, we finally obtain
Ct(α)=Cdν(α)cos(α)-Cl(α)sin(α)-cRCmc/4(α)=Cdν,p(α)cos(α)-Cl(α)sin(α)-cRCmc/4(α)+Cdν,f(α)cos(α),Cn(α)=Cdν(α)sin(α)+Cl(α)cos(α)=Cdν,p(α)sin(α)+Cl(α)cos(α)+Cdν,f(α)sin(α).
We have intentionally split each term into pressure and friction contributions. The net forces ft and fn (each split in pressure and friction contributions) and the moments, mc/2 and mc/4, are then finally obtained using Vcorr, c and ρ. They can be compared to the values measured in CFD.
As we here do not explicitly impose mc/4 in the ALM, we also do not impose fti. The tangential force that we impose in the ALM is hence simply the tangential projection of the lift and parasitic drag forces.
VAT with NACA0015 airfoil attached at xp=c/2 and operating at TSR=3.25: comparison between the results of the ALM with mc/4, fti and fniexplicitly enforced (red), the ALM without mc/4, fti and fni explicitly enforced (blue) and the reference wall-resolved simulation (black) of Cp. Both ALM cases are centered at c/4.
Case with explicitly enforcing <inline-formula><mml:math id="M597" display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M598" display="inline"><mml:mrow><mml:msubsup><mml:mi>f</mml:mi><mml:mi mathvariant="normal">t</mml:mi><mml:mi mathvariant="normal">i</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula> in the ALM
We next consider the case where we explicitly impose mc/4 in the ALM (using the procedure developed in Sect. ); hence we also include the contribution of fti in the imposition of the tangential force.
The values of Cl, Cd, Cmc/4 and Cti are the same as those obtained in the above section.
Here, however, the velocity vector V must be corrected to compensate for the velocity induced by the shed vortex wake produced by the total net force imposed in the direction of V, thus including the projection of fti in this direction (as it will also shed some vorticity into the wake, even though this vorticity is not present in the real flow). The correction is thus
Vcorr=V1-14πcσCdν+Cticos(α).
Results and comparison
We present the results of both types of simulations using ALM in Figs. 10–11 and in Table 3. The normalization used here is different from that used for the airfoil in rotation: here, the forces fn and ft are normalized using the wind velocity Uw and the turbine diameter D=2R, thus using 12ρUw2D. To normalize the moments mc/4 and mc/4, we use 12ρUw2Dc. As for the power coefficient Cp of the VAT, it is obtained using
Cp=-(ftR+mc/4)Ω12ρUw3D.
It is also interesting to compare the ALM-predicted contributions of pressure and friction to the tangential force ft with those from the wall-resolved simulation. Given that the two ALM cases should be similar, we chose to perform the comparison only for the ALM with mc/4 and fti explicitly enforced; see Fig. 12.
Finally, the vorticity fields from the ALM methods are also compared to that of the wall-resolved simulation in Fig. .
Mean Cmc/2, Cmc/4, Cn, Ct and Cp over a blade cycle for a VAT with airfoil attached at xp=c/2 and operating at TSR=3.25. Both ALM cases are centered at c/4.
CoefficientsCmc/2‾Cmc/4‾Cn‾Ct‾Cp‾Wall-resolved simulation-0.0140.149-0.650-0.1470.490ALM without mc/4, fti and fni explicitly enforced-0.0110.169-0.729-0.1490.493ALM with mc/4, fti and fni explicitly enforced-0.0140.167-0.733-0.1500.500
VAT with NACA0015 airfoil attached at xp=c/2 and operating at TSR=3.25: comparison between the ALM with mc/4, fti and fni explicitly enforced (red) and the reference wall-resolved simulation (black) of the pressure contribution Ctp (with its components) and of the friction contribution Ctf to the tangential force coefficient. The ALM is centered at c/4.
Case of airfoil attached at <inline-formula><mml:math id="M658" display="inline"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:math></inline-formula>
We consider next the case where the blade is attached to the arm at xp=c/2, again without pitch.
Case with ALM centered at <inline-formula><mml:math id="M660" display="inline"><mml:mrow><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:math></inline-formula>
In the usual case where the ALM is strictly applied at the aerodynamic center, the airfoil attached at xp=c/2 and without pitch must be considered “equivalent to the same airfoil attached at xp=c/4 and with a pitch angle of αc”.
The components of the flow velocity v measured at c/4 are (vt,vn). The components of the apparent velocity vector V are thus
Vt=ΩRcos(αc)cos(αc)+vt=ΩR+vt,Vn=ΩRcos(αc)sin(αc)+vn=ΩRtan(αc)+vn,
where R/cos(αc) corresponds to the slightly extended effective arm length when considering the circle passing through this point.
The angle of the velocity vector V relative to the airfoil chord is denoted α. The airfoil chord makes an angle αc relative to the curved flow streamline considered at c/4. The angle of the velocity vector relative to the curved flow streamline at c/4 is thus αe=α-αc. Again, there are then two ways of applying the ALM, and these are investigated below.
We first consider the case where we do not explicitly enforce the moment mc/4 in the ALM; hence we also cannot enforce the associated inviscid force in the ALM. Only the lift and the parasitic drag contributions will be enforced in the ALM.
VAT with NACA0015 airfoil attached at xp=c/2 and operating at TSR=3.25: comparison between the ALM with mc/2 and fti explicitly enforced (red), the ALM without mc/2 and fti explicitly enforced (blue) and the reference wall-resolved simulation (black) of Cmc/2, Cmc/4, Cn and Ct. Both ALM cases are centered at xp=c/2.
VAT with NACA0015 airfoil attached at xp=c/2 and operating at TSR=3.25: comparison between the results of the ALM with mc/2 and fti explicitly enforced (red), the ALM without mc/2 and fti explicitly enforced (blue) and the reference wall-resolved simulation (black) of Cp. Both ALM cases are centered at xp=c/2.
The amplitude of the velocity must be corrected to compensate for the velocity induced by the shed vortex wake produced by the parasitic drag:
Vcorr=V1-14πcσCdν.
The angle of this corrected velocity vector relative to the curved flow streamline at c/4 is therefore still α-αc.
In terms of modeling, the lift force (i.e., force in the direction orthogonal to the apparent velocity) exerted by the velocity Vcorr on the airfoil attached at c/4 and with pitch αc is equivalent to the normal force (= also lift force) exerted by a horizontal flow of velocity Vcorr on the airfoil attached at c/4 and pitched by an angle αp=αe+αc. The lift coefficient is thus modeled using
Cl(α)≃Clu((αe+αc)-αc)-Clu(αc)=Clu(α-αc)-Clu(αc).
Likewise, the parasitic drag (i.e., force in the direction of the apparent velocity) is modeled using
Cdν(α)=Cdν,p(α)+Cdν,f(α)≃Cdu,p(α-αc)+Cdu,f(α).
These expressions are the same as those obtained for the case where the airfoil was attached at xp=c/4. This makes sense since the apparent velocity vector is that considered at c/4 in both cases and since the angle α of this vector is defined relative to the airfoil chord in both cases. The only difference lies in the apparent velocity vector; see Eq. ().
The models for the moment coefficients are also identical:
Cmc/2(α)=Cmc/20+14Clu(α-αc)cos(α-αc)
and
Cmc/4(α)=Cmc/2(α)-14Cl(α)cos(α-αc)≃Cmc/20+14Clu(αc)cos(α-αc).
The coefficient for the inviscid force on the airfoil “as if it were attached at c/4” still counterbalances the moment mc/4. This inviscid force now acts in the direction of αc, and thus we obtain Cti=-cRCmc/4cos(αc) and Cni=-cRCmc/4sin(αc). We finally obtain
Ct(α)=Cdν(α)cos(α)-Cl(α)sin(α)-cRCmc/4(α)cos(αc)=Cdν,p(α)cos(α)-Cl(α)sin(α)-cRCmc/4cos(αc)+Cdν,f(α)cos(α),Cn(α)=Cdν(α)sin(α)+Cl(α)cos(α)-cRCmc/4(α)sin(αc)=Cdν,p(α)sin(α)+Cl(α)cos(α)-cRCmc/4(α)sin(αc)+Cdν,f(α)sin(α).
When we do not explicitly impose mc/4 in the ALM, we also do not impose fti and fni.
When we explicitly enforce mc/4 in the ALM, we must also explicitly enforce fti and fni. The correction of the measured velocity V must now also account for the component of the inviscid force in this direction:
Vcorr=V1-14πcσCdν+Cticos(α-αc).
We present the results of both types of simulations using the ALM centered at c/4 in Figs. 14–16 and in Table 4.
Mean Cmc/2, Cmc/4, Cn, Ct and Cp over a blade cycle for the three cases studied of a VAT at TSR=3.25 with the blade attached at xp=c/2. Both ALM cases are centered at xp=c/2.
CoefficientsCmc/2‾Cmc/4‾Cn‾Ct‾Cp‾Wall-resolved simulation-0.0140.149-0.650-0.1470.490ALM without mc/2 and fti explicitly enforced-0.0110.167-0.712-0.1510.502ALM with mc/2 and fti explicitly enforced-0.0190.167-0.744-0.1460.492
VAT with NACA0015 airfoil attached at xp=c/2 and operating at TSR=3.25: comparison between the ALM with mc/2 and fti explicitly enforced (red) and the reference wall-resolved simulation (black) of the pressure contribution Ctp (with its components) and of the friction contribution Ctf to the tangential force coefficient. The ALM is centered at xp=c/2.
Case with ALM centered at <inline-formula><mml:math id="M760" display="inline"><mml:mrow><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:math></inline-formula>
One could however also consider that an ALM can be used as a model to represent the airfoil at its attachment point, thus centering the ALM at xp=c/2. In this case, we also measure the flow velocity vector v at xp=c/2: its components are (vt,vn). The components of the apparent velocity vector V are thus (Vt=ΩR+vt,Vn=vn). The angle it makes relative to the airfoil chord is α.
In terms of modeling, the force exerted in the direction orthogonal to the apparent velocity Vcorr on the airfoil attached at xp=c/2 and without pitch is equivalent to the normal force exerted by a horizontal flow of velocity Vcorr on the airfoil attached at xp=c/2 and pitched by an angle αp=α; the parasitic force exerted in the direction parallel to the apparent velocity is equivalent to the parasitic tangential force exerted by a horizontal flow of velocity Vcorr on the airfoil pitched by an angle αp=α. The related force coefficients are thus modeled using
C⟂(α)=Clu(α)-Clu(αc),C∥(α)=Cdu,p(α)+Cdu,f(α+αc).
The normal and tangential components are hence obtained as
Cn(α)=C⟂(α)cos(α)+C∥(α)sin(α),Ct(α)=C∥(α)cos(α)-C⟂(α)sin(α).
The corrected velocity vector is
Vcorr=V1-14πcσC∥.
The angle that Vcorr makes relative to the curved flow streamline at the location of the aerodynamic center is αe=α-αc. The moment mc/4 is hence obtained as
Cmc/4(α)≃Cmc/20+14Clu(αc)cos(α-αc).
The moment coefficient Cmc/2 is thus obtained as
Cmc/2(α)≃Cmc/4+14Cn(α).
The associated inviscid tangential force is then fti=-mc/2/R, thus with coefficient
Cti(α)=-cRCmc/2(α).
If we do not explicitly enforce the moment mc/2 in the ALM, we also do not enforce fti. We then only enforce the normal and tangential forces associated with Cn and Ct of Eq. ().
If we explicitly enforce the moment mc/2 in the ALM, we must also enforce fti; hence the component Cti(α) must be added to the formula for Ct in Eq. (). The corrected velocity vector must also be modified:
Vcorr=V1-14πcσC∥+Cticos(α).
We present the results of both type of simulations using the ALM centered at xp=c/2 in Figs. 17–19 and in Table 5.
In this configuration (ALM centered at xp=c/2), it can be observed that the upstream prediction (0°–180°) is excellent. In fact, when the ALM is centered at c/4, a small shift in the peak power coefficient is noticeable in the upstream part of the cycle. This shift is eliminated when using the ALM centered at c/2. For the downstream region (180°–360°), small discrepancies persist, as the flow is disturbed by the wake of the upstream blade, making the effective angle of attack prediction more challenging.
Conclusions
The modeling of flow curvature effects on airfoils was further improved and implemented in an actuator line method (ALM), the rule being that such modeling can only use, as input, the aerodynamic coefficients of the airfoil in a uniform flow as a function of its angle of attack.
The basic models for the normal force and moment coefficients are not new and are obtained using the usual analogy with potential flow past an airfoil with curved streamlines. The model for the normal force was further improved here, to remain valid up to large pitch angles, even close to stall. The effect of the aerodynamic moment induced by the flow curvature is also found to be important and cannot be ignored. The curvature effects depend on both the attachment point of the airfoil to the arm and the pitch angle, and we have developed models that cover all possibilities.
The model for the tangential force is new (here, the analogy with a potential flow is flawed). The model is based on the fact that, for an airfoil attached to a rotating arm, the moment at the root of the arm must be zero if the flow is assumed inviscid. This was verified numerically by performing inviscid CFD simulations at various airfoil pitch angles. The inviscid tangential force that is compatible with the aerodynamic moment can, hence, be computed. For modeling viscous flows, we then add the parasitic drag to this inviscid tangential force.
The improved models were first validated against reference data corresponding to wall-resolved CFD of a rotating NACA0015 airfoil with cR=27 at various pitch angles, for angles up to stall and also for two attachment points of the airfoil to the arm (at mid-chord and quarter-chord).
The ALM with the improved models was then implemented in the CFD framework and used to simulate the unsteady flow corresponding to a VAT configuration: a rotating airfoil without pitch placed in a free stream at TSR=ΩRUw=3.25. For this, a novel method was also developed to enforce the aerodynamic moment in the ALM. The various components of the ALM results were compared with those of the reference CFD data. They were found to be in good agreement throughout the rotation cycle.
It was also shown that the ALM can here be used in various ways: if one ignores the contribution of the aerodynamic moment in the ALM, then one must also ignore the inviscid part of the tangential force in the ALM; if one enforces the contribution of the moment in the ALM, then one must also enforce the inviscid part of the tangential force.
The above validations give us confidence that the proposed models, together with their implementation in an ALM simulation framework (including the new method for enforcing a moment), constitute useful simplified tools for modeling complex unsteady flows with good accuracy, such as VAT configurations where the flow curvature modifies both the normal and tangential forces and also produces an aerodynamic moment.
We stress that the normal force model is, so far, obtained using the analogy with curved potential flow past a flat plate put at some pitch angle, as developed in Appendix . The model was however tested with good success on a NACA0015 airfoil: an airfoil that is not very thin and thus already challenges this model. Given that the ALM results obtained for the airfoil are quite good, it is likely that they would still be good when simulating a thicker airfoil, such as a NACA0018, and maybe still acceptable when simulating a NACA0021. One could also further improve the normal force model by taking into account the combined effects of flow curvature and airfoil thickness. Obtaining the exact solution of curved potential flow past a thick airfoil put at some pitch angle would constitute a good step in this direction.
We also note that the method proposed here for enforcing the aerodynamic moment in an ALM, in addition to enforcing the forces, can also be used to model unsteady flows involving cambered airfoils (thus with intrinsic aerodynamic moment), hence also VAT using cambered blades.
Finally, it is recognized that the proposed models are, so far, limited to flows without fast unsteady effects or dynamic stall. A next step in their development should focus on modeling such effects.
NACA0015 airfoil in uniform flow and associated aerodynamic coefficients
The aerodynamic coefficients of the NACA0015 airfoil in uniform flow were here obtained using wall-resolved CFD in RANS at Rec=6.0×106: a high Reynolds number case where the boundary layers are turbulent. We note that the simulation setup has also been validated against experimental data of the pressure coefficient distribution along the airfoil chord, at different spanwise location of a 3D wing, and up to near the wing tip; this is reported in .
The coefficients are denoted using a superscript “u” (which stands for “uniform” flow) so as to avoid any confusion with the coefficients measured in CFD of the rotating NACA0015 airfoil. The variation in the coefficients as a function of the angle of attack (denoted β to avoid any confusion with the notation α used for the angle of attack of a rotating airfoil) is shown in Fig. .
Coefficients Clu, Cdu (with its Cdu,f (green) and Cdu,p (blue) contributions) and Cmc/2u as a function of the angle of attack β for the NACA0015 airfoil in uniform flow: data for the complete range (top), data for the range |β|≤12° (bottom) and with the mathematical fit (magenta).
For the range with |β|≤12°, the lift coefficient can be fitted well using
Clu(β)≃6.390β-7.8β3=6.390β1-1.22β2,
with β in rad. The slope of 6.390 at β=0 was fitted using the CFD data at small angle (|β|≤4°): it is slightly larger than 2π due to the beneficial effect of thickness (also predicted by potential flow theory). The degradation when β increases (due to parasitic viscous effects on the pressure field) is however significantly faster than that of a potential flow (where it would go as sinβ≃β1-β2/6 for small β).
Using the same procedure, the moment coefficient around the mid-chord point, x=c/2, is fitted well using
Cmc/2u(β)≃1.614β-1.9β3.
The parasitic drag coefficient can also be fitted using
Cdu(β)≃0.00889+0.116β2+0.82β4.
Contrary to the lift and moment coefficients (where it is mostly the pressure field that contributes; the contribution due to wall friction being negligible), there are two important contributions to the parasitic drag: the friction drag and the pressure drag. Each can be fitted independently. We obtain
Cdu,f(β)≃0.00701-0.014β2-0.05β4,Cdu,p(β)≃0.00188+0.130β2+0.87β4.
The friction contribution is dominant at small angles (it represents 79 % of the drag when β=0), and it decreases slowly as the angle increases. The pressure contribution is moderate at small angles, and it increases rapidly as the angle increases; it even becomes dominant beyond about 10 °.
The precise position of the aerodynamic center is then further obtained using the CFD data for Clu(β) and Cmc/2u(β) with |β|≤4°, and the result is xac≃0.2476c. The distance between the aerodynamic center and the mid-chord point is thus =(0.5-0.2476)c=0.2524c. The aerodynamic moment coefficient is then obtained using
Cmacu(β)=Cmc/2u(β)-0.2524cos(β)Clu(β).
This is shown in Fig. . It is indeed constant for small angles (i.e., no linear term, as it should) and it is well fitted for moderate angles using
Cmacu(β)≃0.710β3+2.4β5.
The moment coefficient around x=c/4 (i.e., the approximate location of the aerodynamic center) is also shown: as expected, it is not totally flat for small angles. It can be fitted using
Cmc/4u(β)≃0.0144β+0.81β3.
The coefficient of the linear term is non-zero yet very small.
Coefficients Cmacu and Cmc/4u as a function of the angle of attack β: data for the complete range (top), data for the range |β|≤12° (bottom) and with the mathematical fit (magenta).
NACA0015 airfoil in rotation and associated aerodynamic coefficients
The NACA0015 airfoil in rotation was also simulated using wall-resolved CFD at Rec=6.0×106. Two cases were simulated: case with attachment point at mid-chord (xp=c/2) and case with attachment point at quarter-chord (xp=c/4).
Simulation setup
The simulation setup, with the key-hole mesh, is shown in Fig. . A close-up of the body-fitted mesh is also provided. The vorticity field associated with the boundary layers and the wake is also shown.
Simulation setup and key-hole mesh used for the simulation of the NACA0015 airfoil in rotation with c/R=2/7. Case with airfoil attached at mid-chord (xp=c/2) and tilting angle αp=6°. A close-up of the body-fitted mesh around the airfoil is also shown.
Airfoil attached at mid-chord (<inline-formula><mml:math id="M845" display="inline"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:math></inline-formula>)
The obtained aerodynamic coefficients are shown in Figs. B2–B4.
Cn, Cmc/2 and Ct as a function of the pitching angle αp for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/2.
Ctp and Ctν,f as a function of the pitching angle αp for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/2.
Cmc/4 and Cmac as a function of the pitching angle αp for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/2.
Airfoil attached at quarter-chord (<inline-formula><mml:math id="M862" display="inline"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:math></inline-formula>)
The obtained aerodynamic coefficients are shown in Figs. B5–B7.
Cn, Cmc/2 and Ct as a function of the pitching angle αp for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/4.
Ctp and Ctν,f as a function of the pitching angle αp for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/4.
Cmc/4 and Cmac as a function of the pitching angle αp for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/4.
Comparison
The coefficients Cl, Cmc/2, Cd, Cdp, Cdν,f, Cmc/4 and Cmac expressed as a function of the angle of attack α at c/4 are compared in Figs. –. For the case where xp=c/2, α=αp+αc, and for the case where xp=c/4, α=αp.
It is seen that the overlap is quite good. This supports the discussion made in the core of the paper that the “effective angle of attack” to consider in all cases is the angle that the airfoil chord makes relative to the streamline of the curved flow at the location of the aerodynamic center, in absence of the profile.
Comparison of the coefficients Cl, Cmc/2 and Cd as a function of the angle of attack α for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/2 (blue) or xp=c/4 (red).
Comparison of the coefficients Cdp and Cdν,f as a function of the angle of attack α for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/2 (blue) or xp=c/4 (red).
Comparison of the coefficients Cmc/4 and Cmac as a function of the angle of attack α for the NACA0015 airfoil in rotation with c/R=2/7 and xp=c/2 (blue) or xp=c/4 (red).
Deformed NACA0015 airfoil in uniform flow and associated aerodynamic coefficients
The flow past the NACA0015 airfoil deformed using the transformation Z(z)=z+i2z2R was also obtained in uniform flow, using wall-resolved CFD at Rec=6.0×106. This allows one to measure the moment coefficient at mid-chord Cmc/20 and then also account for it in the corrections for flow curvature effects. Table provides the results for the deformed NACA0015 airfoil in uniform flow and the NACA0015 airfoil in curved flow, as well as an ALM simulation of the deformed NACA0015 airfoil in uniform flow with moment distribution.
As predicted using the analogy presented in Sect. , the coefficients of the deformed airfoil in uniform flow match those of the airfoil in curved flow. We also see that the ALM with distribution of the aerodynamic moment is able to successfully reproduce the results of the deformed airfoil in uniform flow. Figure presents the vorticity field.
Vorticity field near the airfoil for the deformed airfoil in uniform flow (left) and the airfoil in curved flow (right).
Coefficients for the deformed NACA0015 airfoil in uniform flow, the NACA0015 airfoil in curved flow and the ALM results of the deformed NACA0015 airfoil in uniform flow.
CaseCnCtCtpCtfCmc/2Cmc/4Deformed NACA0015 in uniform flow-0.4230.009260.002480.00678-0.007460.0983NACA0015 in curved flow-0.4310.01070.003870.00682-0.005510.102ALM of the deformed NACA0015-0.4420.009610.002680.00693-0.004500.106Inviscid flow simulations
The purpose of the inviscid flow simulations was to verify that the moment m0 around the center of rotation of the arm is indeed zero when the flow is inviscid. We employ the same setup as that presented in Fig. . Additionally, a second-order implicit coupled flow solver is used to obtain the solution. The case studied is the NACA0015 airfoil attached at xp=c/4. Each term of Cm0(αp)=Cmc/4(αp)+RcCti(αp) is provided in Table . We see that Cm0 is indeed very close to zero; thereby confirming the argumentation that was used in developing the model for the inviscid contribution to the tangential force. Simulations were also attempted at higher pitch angles, but convergence became severely challenging due to the inviscid flow assumption.
Ct, Cmc/4 and Cm0 as functions of the pitch angle αp for the NACA0015 airfoil in inviscid curved flow with c/R=2/7. These values are obtained directly from the simulation, which explains why Cmc/4(αp)+RcCt(αp) does not exactly equal Cm0.
αp [°]CtiCmc/4Cm0-5-0.03880.13800.000640-0.03610.12810.000455-0.03350.11720.00003Solution of curved potential flow past a plate with pitch
We develop the exact solution for curved potential flow past a flat plate at various pitch angles. For this, we first consider the circle of radius a centered at the origin in a Z plane with (X,Y) coordinate system. We add a point vortex of circulation Γ0<0 below the circle, at Z0=-iR0eiα0=R0sinα0-iR0cosα0. For the circle to be a streamline, we must add an image vortex of circulation -Γ0>0 inside the circle, at Z1=-i(a2/R0)eiα0. The complex potential is thus
F(Z)=Φ(Z)+iΨ(Z)=-iΓ02πlogZ-Z0a-logZ-Z1a=-iΓ02πlogZ+iR0eiα0a-logZ+i(a2/R0)eiα0a,
with Φ the potential and Ψ the streamfunction. The velocity field is obtained as
dFdZ(Z)=U(Z)-iV(Z)=-iΓ02π1(Z-Z0)-1(Z-Z1)=-Γ02πR0-a2/R0eiα0Z2+iR0+a2/R0Zeiα0-a2ei2α0.
The circle in the Z plane can be transformed into a horizontal flat plate in the z plane, with (x,y) coordinate system, using the conformal transformation:
z(Z)=Z+a2Z.
The point ZL=-a is transformed into the plate leading edge zL=-2a, and the point ZT=a is transformed into the plate trailing edge zT=2a. The chord of the plate is thus c=4a. This transformation is also invertible, and we obtain
Z(z)=12z+z2-4a2.
The vortex located at Z0=-iR0eiα0 in the Z plane in now located at z0=Z0+a2/Z0=-iReiαp in the z plane, with
R0+a2R0sinα0=RsinαpandR0-a2R0sinα0=Rsinαp.
To each choice of distance R and tilt angle αp in the z plane, there is a corresponding distance R0 and angle α0 in the Z plane. For this, we write
z02-4a2=-R2cos(2αp)+4a2+iR2sin(2αp)=-R̃2ei2β.
The expressions for R̃ and β are obtained by identification:
R̃=R2cos(2αp)+4a22+R2sin(2αp)214,β=12arctanR2sin(2αp)R2cos(2αp)+4a2.
We hence finally obtain the position of Z0Z0=12z0+z02-4a2=-i2Reiαp+R̃eiβ.
Once Z0 is known, its modulus R0 and its angle α0 can be easily calculated.
The baseline case α0=0 is immediate as it corresponds to αp=0, and thus also R=R01-a2R02. Inverting this relation gives
R0=12R+R2-4a2=R21+1+2aR2=R21+1+c2R2.
We note that the obtained values of R0 and α0 are quite close to R and αp when the vortex is far enough from the plate (i.e., when the ratio c/R is “small enough”). For the case with significant curvature effects considered in this paper, this ratio is c/R=2/7≃0.286. Clearly, this not very small; nevertheless, we obtain, for the case without tilt angle,
R0c≃3.518,
which is still quite close to 3.50.
The complex potential of the flow past the plate in the z plane is
f(z)=ϕ(z)+iψ(z),
and it is obtained by writing f(z)=F(Z) for z=z(Z). The velocity field is then obtained using
dfdz(z)=u(z)-iv(z)=dFdZ(Z)dzdZ(Z)for z=z(Z).
To enforce the condition that the flow leaves the plate smoothly at its trailing edge, with a finite velocity there (i.e., the “Kutta–Joukowski condition”), we must add another vortex of circulation Γc at the center of the circle, the circulation of which being computed so that dfdz(zT) is finite. As dzdZ(ZT=a)=0 (since ZT=a is a singular point of the transformation z(Z)), we must impose that dFdZ(ZT=a)=0. The complex potential with the added vortex is, hence,
F(Z)=-iΓ02πlogZ-Z0a-logZ-Z1a+ΓcΓ0logZa,
and thus
dFdZ(Z)=-iΓ02π1(Z-Z0)-1(Z-Z1)+ΓcΓ01Z=-iΓ02πia2/R0-R0eiα0Z2+iR0+a2/R0eiα0Z-a2ei2α0+ΓcΓ01Z.
Imposing dFdZ(a)=0 determines the circulation of the added vortex:
Γc=R0-a2/R0R0+a2/R0-2asinα0Γ0.
The total circulation of the flow around the plate in the z plane (and also around the circle in the Z plane) is the sum of the circulations of the two vortices inside the circle:
Γ=Γc+(-Γ0)=-2a2/R0+asinα0R0+a2/R0-2asinα0Γ0.
For the case α0=0, we obtain
Γ=-2a2R021+a2R02Γ0>0.
We also note that the circulation Γ is zero when sin(α0)=-a/R0; this corresponds to the case without force exerted by the flow on the plate.
Finally, the forces exerted by the flow on the plate can be obtained using Blasius formula:
fx-ify=iρ2∮Czdfdz(z)2dz=iρ2∮CZdFdZ(Z)2dzdZ(Z)dZ.
We have used the change in variable, z(Z), as the integral must be performed in the Z plane using a contour CZ that contains the circle but excludes the outer vortex. Since dzdZ(Z)=(Z2-a2)Z2, we obtain
fx-ify=-iρ2Γ02π2∮CZ1(Z-Z0)-1(Z-Z1)+ΓcΓ01Z2Z2(Z2-a2)dZ=-iρ2Γ02π2∮CZG(Z)dZ.
The contour integral must be performed using the residue theorem:
∮CZG(Z)dZ=2πi∑residues of G(Z) inside CZ.
We thus finally obtain
fx-ify=ρΓ024π∑residues of G(Z) inside CZ.
It turns out that the only non-zero residues of G(Z) are those evaluated at Z1=-i(a2/R0)eiα0.
For the case α0=0 with the vortex just below the plate, and thus αp=0, the flow past the plate is left/right symmetric; hence fx(0)=0 and fy(0) is also the force normal to the plate, fn(0). The sum of the residues is then obtained as
i1R01-a2R024a2R021+a2R022,
and we finally obtain
fy(0)=fn(0)=-ρΓ02πR01-a2R022a2R021+a2R022Γ0=ρΓ02πRΓ1+a2R02=-ρUΓ1+a2R02<0,
where we recalled the horizontal velocity U=-Γ02πR>0 induced by the vortex and evaluated at the plate center. The lift of the curved flow past the plate with the vortex directly below it is thus almost equal to -ρUΓ: the usual formula for a plate or airfoil in a free stream at velocity U, with circulation Γ of the flow around it and without anything else nearby.
Using Eq. (), we obtain
aR0=c2R1+1+c2R2,
and the normal force coefficient is finally obtained as
Cn(0)=fn12ρU2c=-πc2R1+c2R2=-2π1+c2R2c4R.
Consider the straight line that goes from the vortex to the plate leading (or trailing) edge: it makes an angle ϕc with the line that goes from the vortex to the center of the plate, such that tan(ϕc)=c2R. We thus obtain
c2R1+c2R2=tan(ϕ)1+tan2(ϕc)=12sin(ϕc)cos(ϕc)=14sin(2ϕc),
and hence
Cn(0)=-2πsin(2ϕc)4.
We also define the half angle (used in the curvature correction models), αc=ϕc/2, and write this as
Cn(0)=-2πsin(4αc)4.
We have
c2R=tan(ϕc)=tan(2αc)=2tan(αc)1-tan2(αc),
which constitutes a quadratic equation for tan(αc). The solution is
tan(αc)=1+c2R2-1c2R.
The streamlines of the flows obtained are shown in Fig. for the case c/R=2/7 and for different values of the tilt angle αp of the plate relative to the vortex. To clearly show the flow relative to the plate, we present the streamlines in the coordinate system used in the core of the paper: the vortex Γ0 is located at (0,-R) below the plate and the plate is tilted by an angle αp relative to the horizontal. For the case with αp=0, the flow is left/right symmetric. For the case with αp=-4°, the force exerted by the flow is close to zero (it is exactly zero when αp≃4.185°).
All the results obtained above are exact within the framework of potential flow theory. Now, for cases with c2R2≪1, as with those considered in this paper, we can simply write
Cn(0)≃-2πsin(αc)withsin(αc)≃αc≃ϕc2≃12c2R.
Code availability
The reference and ALM results were obtained using the flow solver Siemens® STAR-CCM+®. This code is proprietary and cannot be distributed.
Data availability
All reference data are available at 10.14428/DVN/ZZAYDQ.
We stress that the aerodynamic coefficients of the NACA0015 airfoil in uniform flow are the only data required to perform the presented ALM simulations of flows with curvature effects using the correction models presented in the paper.
Author contributions
GW developed the curvature correction models for the normal and tangential forces acting on a rotating airfoil attached to an arm at an arbitrary position and for pitch angles up to stall, the ALM for distributing an aerodynamic moment, the potential flow solution of curved flow past a plate and the curvature correction models for the normal and tangential forces of VAT configurations. PR and TV carried out, using the software StarCCM+, the wall-resolved simulations to obtain all reference results and the ALM simulations used for comparison and validation of the curvature correction models; they also used an overset mesh for the VAT simulations. FT performed ALM simulations using an in-house code where the AL moves through the flow solver grid, also supporting the validation of the models. MD supported the development of the curvature correction models and a first version of the tangential force model. GD supervised the work of PR and participated in the validation of the ALM simulations as compared to the wall-resolved simulations. The figures were mostly produced by PR and FT.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. The authors bear the ultimate responsibility for providing appropriate place names. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.
Acknowledgements
We acknowledge the support of Otis Van Kerm for computations concerning Appendix and for producing the associated figure.
Review statement
This paper was edited by Emmanuel Branlard and reviewed by two anonymous referees.
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