Propeller Blade Element Momentum Theory with Vortex Wake Deflection

27TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

PROPELLER BLADE ELEMENT MOMENTUM THEORY WITH VORTEX WAKE DEFLECTION

M. K. Rwigema School of Mechanical, Industrial and Aeronautical Engineering University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa

Keywords: Blade Element Momentum, Propeller, Vortex Wake, Wake Deflection

Abstract propeller disk/cross-sectional slip-stream S area (m2) Various propeller theories are treated in developing a model that analyses the t time (s) aerodynamic performance of an aircraft U resultant velocity at blade element (m/s) propeller along with the construct and V flow velocity (m/s) behaviour of the resultant slip-stream. Blade thrust axis location downstream from x element momentum theory is used as a low- propeller (m) order aerodynamic model of the propeller and α local angle of attack (rad) is coupled with a vortex wake representation of β local element pitch angle (rad) the slip-stream to relate the vorticity distributed γ strength of distributed annular vorticity throughout the slip-stream to the propeller (m/s) forces. strength of distributed swirl vorticity γ s (m/s) strength of bound vorticity on propeller Nomenclature Γ (m2/s) blade φ angular coordinate around propeller disk a axial induction factor (rad) ϕ local inflow angle (rad) a′ tangential induction factor σ ′ local solidity c chord length (m) 3 ρ air density (kg/m ) C drag coefficient Ω propeller rotational speed (rad/s) D

C lift coefficient Subscripts: L C thrust coefficient T ∞ free-stream x at station x dr blade element and annulus width (m) p propeller/aircraft body dQ torque of element or annulus (Nm) s fully-developed slip-stream dT thrust of element or annulus (Nm) 0 at φ = π/2 F combined tip- and hub-loss coefficient between thrust line and direction of F I momentum of inertia resultant propeller force N in-plane normal force (N)

p pressure (P) 1 Introduction r radial position (m) An understanding of the propeller slipstream is R propeller radius (m) important in the design phase of propeller- R hub radius (m) hub driven aircraft. Aerodynamic performance, aircraft stability, and noise and vibration can be significantly affected by it’s interactions with

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structural and aerodynamic control surfaces. During the early stages of design, one of the inputs is the thrust curve of the power plant - available thrust versus flight velocity. The iterative optimisation procedure requires rapid analysis of prospective engines and propellers to compare aircraft performance and discover configuration characteristics.

A propeller producing thrust by forcing air Fig. 1: Propeller stream-tube behind the aircraft produces a slip-stream. This may rudimentarily be considered as a For the axial direction, the change in flow cylindrical tube of spiraling air propagating momentum along a stream-tube starting rearward over the fuselage and wings – having upstream, passing through the propeller, and detrimental and beneficial effects. Flow within then moving off into the slip-stream must equal the slip-stream is faster than free-stream flow the thrust produced by this propeller, [3] resulting in increased drag over the parts T = ρπ R2V (V − V ) (1.1) exposed to it’s trajectory. The rotary motion of s s s ∞ the slip-stream also causes the air to strike the Applying the conservation of mass to the tailplane at indirect angles, having an effect of stream-tube control volume yields, the stability and control of the aircraft. A V = AV = AV (1.2) ∞ ∞ s s This analysis shows compatibility between Far upstream the static pressure is p and the different approaches to propeller modeling ∞ velocity V . The static pressure falls to a value using blade element theory to predict the ∞ p- immediately upstream of the actuator disk propeller forces, momentum theory to relate the and rises discontinuously through the disk, to a flow momentum at the propeller to that of the value p+ immediately downstream, although the far wake, and a vortical wake model to describe velocity remains constant at V between these the slipstream deflection. two planes, which are an infinitesimal distance apart. The thrust is therefore given by, + − 2 Mathematical Models T = ( p − p )A (1.3) Since the flow is inviscid, the total pressure - in 2.1 Momentum Theory accordance with Bernoulli’s equation - is constant along any streamline, except for those Classical momentum theory, introduced by that pass through the disk. The axial and Froude [1] as a continuation of the work of tangential velocities are continuous as no fluid Rankine [2], imposes five simplifying is created within the disk; the increase in total assumptions. The flow is assumed to be: 1) pressure is therefore experienced as an increase inviscid, 2) incompressible, and 3) irrotational; in static pressure. Applying Bernoulli’s equation and both 4) the velocity and 5) the static upstream and downstream of the actuator disk, pressure are uniform over each cross section of 1 the disk and stream-tube. + − 2 2 p − p = ρ(Vs − V∞ ) (1.4) 2 From the equations above it can be shown that the velocity of the flow through the actuator disk is the average of the upstream and downstream velocities. V + V V = ∞ s (1.5) 2

2 PROPELLER BLADE ELEMENT MOMENTUM THEORY WITH VORTEX WAKE DELFECTION

An axial induction factor, a, is customarily Additional equations [1], [5] governing the state defined as, of the flow are dependent on the characteristics V = V (1+ a) (1.6) of the propeller blades, such as airfoil shape and ∞ and from Equation (1.5), twist distribution. Blade Element (BE) theory uses these geometrical properties to determine V = V (1+ 2a) (1.7) s ∞ the forces exerted by a propeller on the flow- A real propeller, however, is never uniformly field. loaded as assumed by the Rankine-Froude actuator disk model. In order to analyse the radial load variation along the blades, the angular momentum being imparted to the wake by the propeller must be considered.

2.1.1 Effects of Wake Rotation The conservation of angular momentum Fig. 2: Blade element aerodynamic forces necessitates rotation of the slip-stream if the As the word element in the title suggests, BE propeller is to impart useful torque. Jonkman [4] theory, again, uses several annular stream-tube describes how the Rankine-Froude actuator disk control volumes. At the propeller plane, the model may be modified to account for this boundaries of these control volumes effectively rotation. As such, assumptions 4) and 5) from split the blade into a number of distinct the actuator disk model may be relaxed and elements, each of width dr. At each element, three supplementary assumptions added: blade geometry and flow-field properties can be 1. The flow entering the control volume far related to a differential propeller thrust, dT, and upstream remains purely axial and torque, dQ, if the following assumptions are uniform made:

2. The slip-stream can be split into a series 1. Just as the annular stream-tube control of non-interacting, annular stream-tube volumes used in the slip-stream rotation control volumes analysis were assumed to be non-

3. The angular velocity of the slip-stream interacting [assumption (1)], it is flow far downstream of the propeller is assumed that there is no interaction low so the static pressure can be between the analyses of each blade assumed to be the unobstructed ambient element static pressure. 2. The forces exerted on the blade elements Although wake rotation is now included in the by the flow stream are determined solely analysis, the assumption that the flow is by the two-dimensional lift and drag irrotational has not been lifted. Conserving characteristics of the blade element angular momentum about an axis consistent airfoil shape and orientation relative to with the slip-stream’s axis of symmetry can be the incoming flow. applied to determine the torque. As we are required to obtain the local angle-of- dL dIω dm Q = = = ωr 2 (1.8) attack to determine the aerodynamic forces on a dt dt dt blade element, we must first determine the inflow angle based on the two components of 2.2 Blade Element Theory the local velocity vector. V (1+ a) Unless some state of flow is assumed, tanϕ = ∞ (2.1) momentum theory does not provide enough Ωr(1− a′) equations to solve for the differential propeller thrust and torque at a given span location. From Fig. 2 the following relation is apparent,

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V (1+ a) the induced velocity distribution along the U = ∞ (2.2) propeller. To compensate for this deficiency in sinϕ BEM theory, a tip-loss (or correction) factor, F, The induced velocity components in Equations originally developed by Prandtl [7] is used. (2.1) and (2.2) are a function of the forces on the blades and blade element momentum theory As a blade has a suction-surface and a pressure- is used to calculate them. From Fig. 2, the thrust surface, air tends to flow over the blade tip from distributed around an annulus of width dr is the lower (pressure) surface to the upper equivalent to, (suction) surface, effectively reducing the resulting forces in the vicinity of the tip. This 1 2 dT = BρU (CL cosϕ − CD sinϕ)cdr (2.3) theory is summarized by a correction to the 2 induced velocity field and can be expressed and the torque introduced by each annular simply by the following: section is given by, 2 −1 − f 1 2 F = cos e (3.3) dQ = BpU (C sinϕ + C cosϕ)crdr (2.4) π 2 L D where, B R − r 2.3 Blade Element Momentum Theory ftip = (3.4) 2 r sinϕ Combining the results of the previous two analyses enables a model of the performance of Much like the tip-loss model, the hub-loss a propeller whose airfoil properties, size, and model serves to correct the induced velocity twist distribution are known. This analysis is resulting from a vortex being shed near the hub based on the differential propeller thrust, dT, of the propeller. The model uses a similar and torque, dQ, derived from momentum theory implementation to that of the tip-loss to describe and blade element theory, being equivalent. the effect of this vortex, replacing Equation (3.4) with, Equations (2.3) and (2.4) are made more useful B r − Rhub by noting that ϕ and U can be expressed in fhub = (3.5) terms of induction factors etc [3], [6]. 2 r sinϕ Substituting Equation (2.2) and carrying out For a given element, the local aerodynamics some algebra, may be affected by both the tip- and hub-loss, in which case the two correction factors are V 2 (1+ a)2 multiplied to create the total loss factor. dT ∞ C cos C sin rdr = σ′πρ 2 ( L ϕ − D ϕ) sin ϕ Now, to relate the induced velocities in the (3.1) propeller plane to the elemental forces of Equations (3.1) and (3.2), the conservation of V 2 (1+ a)2 momentum in the axial direction (between the dQ ∞ C sin C cos r 2dr far upstream and the far downstream section) = σ′πρ 2 ( L ϕ + D ϕ) sin ϕ must be considered. This states that the thrust (3.2) introduced by each propeller annulus is 2.3.1. Prandtl Loss Correction equivalent to the change in axial momentum The effect on induced velocity in the propeller flow rate. The correction factor is used to plane is most pronounced near the tips of the modify the momentum segment of the BEM blades. The original blade element momentum equations. From Equations (1.3), (1.4), and (1.7) theory does not consider the influence of , considering a radial differential, dT = 4πrρV 2 (1+ a)aFdr (3.6) vortices shed from the blade tips into the slip- ∞ stream on the induced velocity field. These tip- Considering a radial differential of Equation vortices create multiple helical structures in the (1.8) and substituting Equation (1.6), wake, (as seen in Fig. 3), and play a major role in

4 PROPELLER BLADE ELEMENT MOMENTUM THEORY WITH VORTEX WAKE DELFECTION

dQ = 4πr 3ρV Ω(1+ a)a′Fdr (3.7) ∞ Equalising Equations (3.6) and (3.7) with Equations (2.3) and (2.4), respectively, it is possible to obtain, −1 ⎡ 4F sin2 ϕ ⎤ a = ⎢ − 1⎥ (3.8) σ(C cosϕ − C sinϕ) ⎣ L D ⎦ −1 ⎡ 4F sinϕ cosϕ ⎤ a′ = ⎢ + 1⎥ (3.9) σ(C sinϕ − C cosϕ) ⎣ L D ⎦

Thus, when we include two-dimensional airfoil Fig. 3: Vortex tube tables of lift and drag coefficient as a function Only the annular vorticity is considered initially. of the angle-of-attack, α, we have a set of This has strength, γ, and has to meet certain equations that can be iteratively solved for the compatibility conditions: induced velocities and the forces on each blade element. 1. The total pressure inside the wake is taken to be higher than the free-stream by the loading on the propeller. 2.4 Vortex Structure of the Wake However, the inner and outer flows must Propeller theory generally presumes that, after have the same static pressure, which an initial distortion, the vortex sheets shed from yields the condition, the trailing edges of the propeller blades form a 1 1 T V 2 V 2 (4.1) set of interleaved helicoidal sheets which ρ s = ρ ∞ + 2 2 2 π R propagate uniformly downstream parallel to the 2. From a consideration of the continuity of slip-stream axis without further deformation as mass, the radius of the fully-developed if they were rigid surfaces. In reality, these slip-stream, R is related to that of the sheets will soon roll up into a set of helical s, propeller, R, by, vortex filaments and a central vortex filament of VR2 = V R2 (4.2) opposite sense on the axis. s s 3. By considering vorticity, the velocity The helicoidal vortex sheets are floating freely jump across the wake boundary, in an irrotational field with equal velocity on γ = V − V (4.3) either side of the sheet, hence equal pressure. s ∞ There is also no discontinuity of normal velocity, only a discontinuity of tangential 2.4.1. The Propeller related to the Vortex Wake velocity with magnitude equivalent to the vortex From the previous description of air flowing strength of the sheet. from the aerodynamic pressure side to Since there is no pressure discontinuity across aerodynamic suction, this is now described with the sheets, we deduce that sheets move axially the trailing vorticity which includes a reduction backward without deformation. To approximate of the inflow angle towards the blade tip and, this we assume that the discrete bound vorticity resultantly, a reduction in blade lift [10]. is transformed in the slip-stream onto the The induced velocity may be considered to be surface of a vortex tube. There will be two the resultant velocity at a point due to the entire components of this vorticity: a tube of elemental system of bound and free vorticity. For annular vortices that account for the thrust, and simplicity we assume that the propeller blades lines of elemental axial velocity along the tube are narrow and are distributed along equally that account for swirl [8]. spaced radial lines. Considerations of symmetry further reveal that equally spaced radial vortex

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lines of equal strength induce no overall But from a simplification of Equation (4.5), velocity on any one of the lines. These integrated across the blade, and for a number of conditions therefore satisfy that the effect on blades, B, each blade due to bound vorticity on the other 2T B (4.10) blades can be ignored and only trailing vorticity ΓΩ = 2 ρR contributes to the resultant velocity at a blade. such that, The fundamental expressions for the forces T developed by the propeller may be described ργVw = 2 (4.11) most conveniently by applying the Kutta- π R Joukowsky theorem, This combined vortex model, ensures that the dL = ρU × Γdr (4.4) swirl is contained within the stream-tube and induces no flow outside the wake [9]. The cross-product relationship can be used to separate the two components of the resultant force, the thrust and the torque. 2.4.3. Propeller at Incidence dT = ρΓΩr(1− a′)dr (4.5) We now consider a propeller at incidence to the dQ = ρΓV (1+ a)rdr (4.6) slip-stream axis. In general, this will generate ∞ thrust and a tangential in-plane force N. Initially, thrust alone is considered; the 2.4.2. The Propeller Slip-stream inclusion of the in-plane force will follow as an The deflection of the wake is determined by incremental effect. Relative to the slip-stream, relating the propeller forces to the the down-going blade will have a forward characteristics of the fully-developed slipstream. velocity component, whereas the up-going side The aerodynamic forces generated by a will be retreating. propeller under any deviation from uniform flight parallel to the thrust axis will lead to deflection of the slip-stream. We begin by considering the swirl component in the wake, represented by the vorticity γs. The axial component of the bound vorticity shed from the tip of the propeller blades per unit time Fig. 4: Propeller slip-stream is related to elemental axial vorticity lines along Considering a simple propeller blade at a the stream-tube by, rotation angle φ (where φ = 0 is at the top) with BΓV Δt = γ 2π R V Δt (4.7) s s constant bound vorticity shed at the tip, the in- so that, plane force acting normal to the blade will be ρΓVR (for αp = 0). Taking an angular position φ 2π Rsγ s = BΓ (4.8) around the propeller disk, the vertical This satisfies the condition that the total component of the tip velocity is ΩRssin(φ), and vorticity shed at the tips, BΓ, must also be shed the forward component of this is ΩRsαssin(φ) in the opposite sense at the roots as a discrete for small αs. Equation (4.7) now becomes, vortex down the axis. BΓ[V + ΩR α sin(φ)]Δt = γ 2π R V Δt (4.12) s s s s Behind the propeller the vortex tube translates from which, downstream as it is embedded between the inner ⎡1+ ΩR α sin(φ) ⎤ and outer flows with a velocity, Vw. The annular s s 2π Rsγ s = BΓ ⎢ ⎥ (4.13) component of the bound vorticity shed per unit V ⎣ ⎦ time is related to the elemental annular vorticity distributed over the surface of the tube by, Compared with Equation (4.8), this shows an incremental sinusoidal variation of swirl γ 2πV = BΓΩ (4.9) w velocity that will induce a downwash. Now a

6 PROPELLER BLADE ELEMENT MOMENTUM THEORY WITH VORTEX WAKE DELFECTION sinusoidal vorticity distribution of the form Following the analysis above, this is dispersed γ0sin(φ) induces a uniform downwash of over the trailing wake stream0tube with a form γ strength γ0/2 inside the wake, which governs the = γ0sin(φ), where, from Equation (4.8), deflection angle of the wake as it is 2π Rsγ 0 = BΓ0 (4.20) counterbalanced by the up wash from the free- This will add to the downwash in the wake, stream, V θp. ∝ balanced by the up wash from the free-stream, From Equation (4.13), with φ = π/2, γ0 = BΓΩαs V∝θp as above. The incremental effect is / (2πV), so that therefore, γ BΓΩα N V θ = 0 = s (4.14) ΔV θ = (4.21) ∞ p 2 4πV ∞ p 2πρRR V s From Equation (4.10) we deduce that, Expressing the above equation in terms of the Tα s (4.15) V∞θ p = 2 force coefficient and the propeller incidence, αp, 2ρπ R V and adding this to the right hand side of From Fig. 4, αs = αp - θp. Considering this along Equation (4.15), the analysis proceeds until with Equation (1.1), Equation (4.17) becomes, Tα = ρπ R2V (V + V )θ (4.16) θ ⎛ V ⎞ 1 dC V 2 p s ∞ p p = 1− ∞ + N s (4.22) Where T is the tangential component of thrust α ⎜ V ⎟ 4 dα V 2 αp p ⎝ ⎠ p and must be equivalent to the tangential momentum flux in the slip-stream, which is But, from actuator theory, 2 2 2 represented by ρπR VVsθp and resolved through Vs 4Vs 4 2 2 = 2 = 2 (4.23) θp. The other term, ρπR VV∝θp, can be re- V V + V ⎛ V ⎞ 2 ( s ∞ ) 1 ∞ written, from Equation (4.2), as ρπRs VsV θp, ⎜ + ⎟ ∝ ⎝ V ⎠ and represents the momentum flux of the part of s the free-stream displaced by the cylindrical The incremental term due to the in-plane force wake. now has a similar form to the contribution from From the above it can be shown that thrust alone, and Equation (4.17) becomes [9], dCN θ p V C C + = 1− ∞ = T (4.17) θ T dα α V 2 p p (4.24) p 1 1 C = 2 + − T α ( ) p 1+ 1− C ( T ) When αp ≠ 0, to account for the net normal force it is assumed that, in addition to any constant bound vorticity that may be present to 2.4.4. Slip-stream Trajectory generate thrust, there is a component of bound The trajectory of the slip-stream, and hence it’s vorticity that varies sinusoidally around the lateral displacement at any streamwise station, is governed by it’s angle to the thrust line (or propeller disk, Γ = Γ0sin(φ). The effect on the slip-stream by the tangential in-plane force may free-stream). The deflection from the datum can then be obtained by integration. The incidence be determined by resolving the normal-to-blade force vertically, case is considered and all angles are assumed small. N(φ) = ρVRΓ sin2 (φ) (4.18) 0 The approach adopted is to consider momentum Integration around the propeller disk, allowing in a direction perpendicular to the resultant for the number of blades, B, gives an average of force F, see Fig. 4 [10]. In this direction there is 1 no force and the momentum must remain N VBR (4.19) = ρ Γ0 constant. 2

7 M.K. Rwigema

For a segment of slip-stream at station x ⎛ ⎞ downstream of the propeller, the longitudinal ⎜ x ⎟ moment is ρSxVxdx, and the tangential 1 ⎜ ⎟ f (x) = 1+ D (4.32) ⎜ 2 ⎟ momentum due to the outer flow is (ρSxV∝dx)θx. 2 ⎜ 1 ⎛ x ⎞ ⎟ Momentum is constant along the normal to F, ⎜ + ⎟ 4 ⎝⎜ D⎠⎟ such that ⎝ ⎠ ρS dx[V (α + θ ) + V θ ] = x x x F ∞ x (4.25) Equation (4.28) can be re-written as ρS dx[V (α + θ ) + V θ ] s s s F ∞ p α (V − V )V = α (V − V )V + (V − V )V α where the conditions on the right are those in x x ∞ s s s ∞ x x s ∞ p the fully-developed slip-stream. (4.33) 2 2 2 and from Equation (4.31) Writing Sx = πRx , Ss = πRs and S = πR , V − V = (V − V ) f (x) (4.34) S V = S V = SV (4.26) x ∞ s ∞ x x s s and So finally, ⎡ V ⎤ α α 1 V V ⎛ 1 ⎞ S Vdx ∞ x = s x − ∞ − 1 = ρ p ⎢(α x + θ F ) + θx ⎥ = ⎜ ⎟ V α p α p f (x) Vs Vs ⎝ f (x) ⎠ ⎣ x ⎦ (4.27) (4.35) ⎡ V ⎤ α ⎡ V ⎛ 1 ⎞ ⎤ V ⎛ 1 ⎞ ∞ s 1+ ∞ − 1 − ∞ − 1 ρS pVdx ⎢ α s + θ F + θ p ⎥ ⎢ ⎥ ( ) V α V ⎝⎜ f (x) ⎠⎟ V ⎝⎜ f (x) ⎠⎟ ⎣ ⎦ p ⎣ s ⎦ s

Now, θx = αp - αs and θp = αp - αs, so that The diameter of the slip-stream at any station x ⎛ V ⎞ V ⎛ V ⎞ V is given by 1 ∞ ∞ 1 ∞ ∞ α x ⎜ − ⎟ + α p = α s ⎜ − ⎟ + α p ⎝ Vx ⎠ Vx ⎝ Vs ⎠ Vs D 1+ a x = (4.36) (4.28) D 1+ s From the equations of motion the streamwise distribution of velocity in the slip-stream is In order to calculate the displacement, z, of the derived in [11] as, slip-stream it is only necessary to integrate the angular deflection. V s = 1+ s (4.29) x V z = D α dx (4.37) ∞ ∫ x in which 0 ⎡ ⎤ ⎢ x ⎥ s = a 1+ (4.30) 3. References ⎢ 2 ⎥ ⎢ x2 + d ⎥ 4 [1] Froude RE. Trans Inst Naval Architects, ⎣ ⎦ 1889;30:390. Equations (4.29) and (4.30) can be re-arranged [2] Rankine WJM. On the mechanical principles of the to give action of propellers. Trans Inst Naval Architects (British), 1865;6(13).

Vx V∞ ⎛ V∞ ⎞ [3] Glauert H. Airplane Propellers, vol iv, div. L. In: = + ⎜1− ⎟ f (x) (4.31) V V ⎝ V ⎠ Durand WF, editor. Aerodynamic Theory. Berlin: s s s Julius Springer, 1935 [Reprinted 1963 by Dover where Publications, Inc., New York]. [4] Jonkman JM. Modeling of the UAE Wind Turbine for Refinement of FAST_AD. NREL Technical Report [NREL/TP-500-34755], 2003. [5] Drzewiecki S. Bulletin de L’Association Technique Maritime, Paris. A Second Paper in 1901, 1892.

8 PROPELLER BLADE ELEMENT MOMENTUM THEORY WITH VORTEX WAKE DELFECTION

[6] Betz A. with Appendix by L. Prandtl, 1919, Schraubenpropeller mit Geringstem Energieverlust, Göttinger Nachrichten, Göttingen, p. 193-217. Reprinted in Vier Abhandlungen uber Hydro- und Aerodynamik, L. Prandtl & A. betz, 1927 [7] Wald QR. The distribution of circulation on propellers with finite hubs. ASME Paper 64- WA/UNT-4, Winter Annual Meeting, New York, 1964. [8] Wald QR. The aerodynamics of propellers. Progress in Aerospace Sciences 42, pg 85-128, 2006. [9] ESDU. Propeller slipstream modeling for incidence and sideslip. Item No. 06013, Engineering Sciences Data Unit, London, 2006. [10] ESDU. Introduction to installation effects on thrust and drag for propeller-driven aircraft. Item No. 85015, Engineering Sciences Data Unit, London, 1985.

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