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Propeller Blade Element Momentum Theory with Vortex Wake Deflection

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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 1 M.K. Rwigema 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, 3 M.K. Rwigema 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.
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