Baixe Projeto Aerodinâmico Aplicado à Aeromodelos e outras Manuais, Projetos, Pesquisas em PDF para Engenharia Aeronáutica, somente na Docsity! ESTIMATING R/C MODEL AERODYNAMICS AND PERFORMANCE Dr. Leland M. Nicolai, Technical Fellow Lockheed Martin Aeronautical company June 2002 I OVERVIEW The purpose of this white paper is to enlighten students participating in the SAE Aero Design competition on how to estimate the aerodynamics and performance of their R/C models. II DEFINITIONS LIFT: The aerodynamic force resolved in the direction normal to the free stream due to the integrated effect of the static pressures acting normal to the surfaces. DRAG: The aerodynamic force resolved in the direction parallel to the free stream due to (1) viscous shearing stresses, (2) integrated effect of the static pressures acting normal to the surfaces and (3) the influence of the trailing vortices on the aerodynamic center of the body. INVISCID DRAG-DUE-TO-LIFT: Usually called induced drag. The drag that results from the influence of trailing vortices (shed downstream of a lifting surface of finite aspect ratio) on the wing aerodynamic center. The influence is an impressed downwash at the wing aerodynamic center which induces a downward incline to the local flow. (Note: it is present in the absence of viscosity) VISCOUS DRAG-DUE-TO-LIFT: The drag that results due to the integrated effect of the static pressure acting normal to a surface resolved in the drag direction when an airfoil angle-of-attack is increased to generate lift. (Note: it is present without vortices) SKIN FRICTION DRAG: The drag on a body resulting from viscous shearing stress over its wetted surface. PRESSURE DRAG: Sometimes called form drag. The drag on a body resulting from the integrated effect of the static pressure acting normal to its surface resolved in the drag direction. INTERFERENCE DRAG: The increment in drag from bringing two bodies in proximity to each other. For example, the total drag of a wing-fuselage combination will usually be greater than the sum of the wing drag and fuselage drag independent of one another. PAGE 11 PROFILE DRAG: Usually taken to mean the sum of the skin friction drag and the pressure drag for a two-dimensional airfoil. TRIM DRAG: The increment in drag resulting from the aerodynamic forces required to trim the aircraft about its center of gravity. Usually this takes the form of added drag-due- to-lift on the horizontal tail. BASE DRAG: The specific contribution to the pressure drag attributed to a separated boundary layer acting on an aft facing surface. WAVE DRAG: Limited to supersonic flow. This drag is a pressure drag resulting from noncancelling static pressure components on either side of a shock wave acting on the surface of the body from which the wave is emanating. COOLING DRAG: The drag resulting from the momentum lost by the air that passes through the power plant installation (ie; heat exchanger) for purposes of cooling the engine, oil and etc. RAM DRAG: The drag resulting from the momentum lost by the air as it slows down to enter an inlet. AIRFOIL: The two-dimensional wing shape in the X and Z axes. The airfoil gives the wing its basic angle-of-attack at zero lift (F 06 1OL), maximum lift coefficient (Clmax), moment about the aerodynamic center (that point where CmF 0 6 1 = 0), Cl for minimum drag and viscous drag-due-to-lift. Two-dimensional airfoil test data is obtained in a wind tunnel by extending the wing span across the tunnel and preventing the formation of trailing vortices at the tip (essentially an infinite aspect ratio wing with zero induced drag). The 2D aerodynamic coefficients of lift, drag and moment are denoted by lower case letters (ie; Cl, Cd and Cm) III APPROACH We approximate the aircraft drag polar by the expression CD = CDmin + (K’ + K’’)( CL - CLmin)2 The CDmin is made up of the pressure and skin friction drag from the fuselage, wing, tails, landing gear, engine, etc. With the exception of the landing gear and engine, the CDmin contributions are primarily skin friction since we take deliberate design actions to minimize separation pressure drag (ie; fairings, tapered aft bodies, high fineness ratio bodies, etc). The second term in the CD equation is the drag-due-to-lift and has it two parts: K’ = inviscid or induced factor = 1/(F 07 0 AR e) K’’ = viscous factor = fn(LE radius, t/c, camber) The e in the K’ factor can be determined using inviscid vortex lattice codes. The e for low speed, low sweep wings is typically 0.9 – 0.95 (a function of the lift distribution). PAGE 11 Wing AR = 10, Wing taper = 0.5 Wing area = SRef = 1440 in2 = 10 ft2 Wing span = 120 in Landing gear: tricycle Item Planform Wetted Reference Area Area Length (in2) (in2) (in) Fuselage 151 605 25 Engine /mount 15 100 na Horiz Tail 126 252 7 (MAC) Tail Boom 14 28 48 + fuselage Landing gear 12 24 na Wing (exposed) 1360 2720 12.4 (MAC) Vert Tail 0 189 9.8 (MAC) Fuselage Re = 625,000, assume BL is turbulent Fuselage CDmin = FF Cf SWet/SRef Where FF is a form factor (Reference 1, pg 281 or Reference 2, page 11-21) representing a pressure drag contribution. Form factors are empirically based and can be replaced with CFD or wind tunnel data. FF = 1 + 60/(FR)3 + 0.0025 FR = 1.49 FR = fuselage fineness ratio = fuselage length/diameter = 25/5 = 5 Fuselage CDmin = 0.0032 Wing Re = 310,000 Wing CDmin = FF Cf SWet/SRef Where FF = [1 + L(t/c) + 100(t/c)4] R and L is the airfoil thickness location parameter (L = 1.2 for the max t/c located at F 0B 3 0.3c and L = 2.0 for the max t/c < 0.3c)and R is the lifting surface correlation parameter. Thus L = 1.2 and R is determined from Reference 1, page 281 or Reference 2, page 11-13 for a low speed, unswept wing to be 1.05. Since a wing Re = 310,000 could be either laminar or turbulent, we will calculate the minimum drag coefficient both ways and compare with the section Cdmin = 0.0145 (from Figure 1). If the BL is laminar, the wing Cf = 0.00239 and wing CDmin = 0.0057. PAGE 11 If the BL is turbulent, the wing Cf = 0.0059 and wing CDmin = 0.014. Thus the wing boundary layer must be turbulent and we will use wing CDmin = 0.0145. Horizontal Tail The Re = 175,000, therefore we’ll assume the BL is laminar. The tail (both horizontal and vertical) CDmin equation is the same as for the wing. For a t/c = 0.09 airfoil with L = 1.2 and R = 1.05, the horiz tail CDmin = 0.00046. Vertical Tail The Re = 245,000, therefore assume the BL is laminar. For a t/c = 0.09 airfoil with L = 1.2 and R = 1.05, the vert tail CDmin = 0.00039. Tail Boom The reference length for the tail boom is the fuselage length plus the boom length since the BL will start on the fuselage and continue onto the boom. Thus the tail boom Re = 1.825x106 and the BL is turbulent. Thus Tail Boom CDmin = 1.05 Cf SWet/SRef = 0.00009 Where the factor 1.05 accounts for tail/boom interference drag. Landing Gear From Reference 3, page 13.14 a single strut and wheel (4 inch diameter, 0.5 inch wide) has a CDmin = 1.01 based upon frontal area. Thus the tricycle gear CDmin = (3)(1.01)(2)/1440 = 0.0042. Engine From Reference 3, page 13.4, Figure 13 the engine CDmin = 0.34 based upon frontal area. For a 6 in2 frontal area the engine CDmin = 0.002. Total CDmin The total CDmin is the sum of all the components, thus total model CDmin = 0.02484 Total Drag Expression Assuming a wing efficiency e = 0.95 gives an induced drag factor K’ = 1/(F 07 0 AR e) = 0.0335. Notice that the often omitted viscous drag factor K’’ = 0.0137 is 40% of the induced drag factor. The total drag expression is PAGE 11 CD = 0.02484 + 0.0472(CL – 0.7)2 The untrimmed (neglecting the horizontal tail drag-due-to lift) model drag polar and L/D are shown on Figure 4. Figure 4 Notional model aircraft total drag polar and L/D VI ESTIMATING PERFORMANCE Takeoff The takeoff ground roll distance SG is the distance required to accelerate from V = 0 to a speed VTO, rotate to 0.8 CLmax and have L = W. The 0.8 CLmax is an accepted value to allow some margin for gusts, over rotation, maneuver, etc. Assuming a W = 45 lb (12 lb model and 33 lb payload), altitude = 3000 feet (standard day) and a CLmax = 1.67 (from Figure 1) gives the following VTO VTO = [2 W/(S F 07 2 0.8 CLmax)] ½ = 55.65 ft/sec = 38 mph The takeoff acceleration will vary during the ground roll and is given by the following expression (see discussion in Reference 2, Chapter 10) a = (g/W)[T – D - FC (W – L)] where g = gravitational constant = 32.2 ft/sec2 FC = coefficient of rolling friction = 0.03 A useful expression for the ground roll distance SG is given by the equation (from reference 2, page 10-7) SG = VTO2/(2 amean) where amean = acceleration at 0.7 VTO Using the notional model aircraft with the wing at 0º angle of incidence (F 06 1 for minimum drag during the ground run) and data from Figures 2 and 4 gives Ground roll CL = 0.7 Ground roll CD = 0.02484 CLTO = 1.34 @ F 06 1 = 7 F 0 B 0 The static thrust available is assumed to be 20 lb. This thrust will degrade with forward speed as shown on Figure 5. The data scatter represents measurements by different SAE Aero Design teams. PAGE 11