R&M : Aerodynamics

Prof M S Prasad

Aircraft Config

Elevons on delta wings, for pitch and roll control, if there is no horizontal tail; Flaperons, or trailing edge flaps-ailerons extended along the entire span: Tailerons, or stabillizers-ailerons (independently controlled); swing wings, with an articulation that allows sweep angle variation; canards, with additional pitch control and stabilization.

AIAE/AISST@ msprasad2011

Ailerons : control Roll rate ( differential movement. Elevators : Pitch rate & rudders : Yaw rate

AIAE/AISST@ msprasad2011

Missile co ordinates

AIAE/AISST@ msprasad2011

Lambda = incidence plane angle Alpha = incidence angle in Pitch plane beta = incidence angle in Yaw plane sigma = Total angle such that tan β=tanς sinλ tan α=tanς cosλ

Control surfaces Missiles

Elevator deflection η : ( l/2 (δ1- δ3) .. Positive control deflection (η) causes negative pitch. Rudder deflection : l/2 (δ2- δ4 ) Aileron def : ½(δ1+ δ3 + δ2+ δ4 ) or any set of two. Positive control def lection causes negative roll.

•Minimum weight -> sphere •Minimum drag -> slender body •Minimum axial load -> low thrust •Minimum lateral load -> sphere •Minimum gravity loss -> high thrust

Configuration Design Goals

•Maximum payload -> lightweight structure, high mass ratio, multiple stages, high specific impulse •Perceived simplicity, improved range safety -> single stage •Minimum cost -> low-cost materials, economies of scale •Minimum environmental impact -> non- toxic propellant

Configuration

C G

Simple model

M(..) moment due to control Surface def delta

Simple model

From aerodynamics point of view these values are to be found out with A given configuration.

Launch Vehicle Structural Loads • Static/quasi-static loads – Gravity and thrust – Propellant tank internal pressure – Thermal effects • Rocket • Cryogenic propellant • Aerodynamic heating • Dynamic loads – Bending and torsion – “Pogo” oscillations – Fuel sloshing – Aerodynamics and thrust vectoring • Acoustic and mechanical vibration loads – Rocket engine – Aerodynamic noise

Characteristics of Bodies of revolution a. Drag at supersonic speeds is primarily dependent on nose. And amount of Boat tailing. a. Base drag is greatly affected by the presence of jet. b. Lift major contributor is nose section. c. The resultant center of pressure for a conventional body varies between 10 -15% of the body length at low AOA. A t High AOA it moves aft of the nominal CP depending on the boat tailing For moderate boat tailing ( 7 deg ) the CP tends to shifts rewards with angle of attack The nose rounding of by r/R of 0.2 does not increase the drag considerably. for analysis purposes , we can assume complete body Cn(alpha) as 0.03 – 0.04 for a wide range of subsonic and supersonic flow.

Types of Drag

Type Cause Skin Friction Viscosity In Medium Pressure drag Shape of body Base Drag Exhaust & wake

The skin friction drag is the downstream resultant of all shear (viscous) forces experience by the fore body

1. Shear forces are tangential to the missile’s surface 2. It is dependent on the amount of wetted area

3. A quick estimate of the skin friction drag is to take the viscous drag of a flat plate with the same surface area, length and Reynolds number as the missile

Viscous drag coefficient for a flat plate

𝐶𝑑𝑓𝑝 = 0.043

𝑅𝑒 1/6

for Re no. 10^6 to 10 ^7

SKIN Friction Drag

Base Drag

Base drag is the drag resulting from the wake or “dead air” region behind the missile. Base drag is less of a problem during powered flight but during free flight it can account for as much as 50% of total Drag

Wave or Pressure drag

Mainly arises from Nose and after body and depends on Mach Number , shape and dimension of the nose & after body. 𝐶𝑑𝑤 = 𝐶𝑑𝑤 )Nose + cdw ( boat tail) Subsonic Flows it is generally small , However for transonic & supersonic it is significant % of the total drag. For wings the wave drag for an airfoil varies as thickness square. At supersonic speed the base experiences a great negative pressure Relative to free stream resulting in substantial drag. Chapman theory suggests :- C d b = Cpb Sb/ S where sb = base area C p b is base pressure co eff.

Transonic drag

As speed increase to M =1 the Drag increases considerably. This is termed as Transonic drag. Missiles have to pass through this transonic drag rise to get to supersonic

Drag & fineness

• Primarily Skin Friction Drag • Minimal Pressure Drag • No Wave Drag • No Interference Drag • No Induced Drag • Elliptical

• Increasing Fineness Ratio • Decreases Wave Drag • Increases Skin Friction Drag • Optimum Ratio is approximately 5

Low angles of attack a very small amount of Normal force is generated by the Mid section mainly carry over from nose., At large angles of attack some normal force is generated due to cross flow drag which acts normal to the body Cen Line. The total force is : C N = 2 α + Cd ( α = 90deg) Ap/ S α ^2 { Allen & Perkins ) , A p = planform area S = ref area.

MID SECTION

BOAT TAIL : Tapered section in the aft portion used for reducing the drag due to large base pressure and base area. Boat tailing reduces the base area hence the drag. Accurately to find the drag is difficult due to geometry of fore and aft portion and real fluid case. However linearized Prandtl equation is used for first hand approximation.

Planforms

Square/Rect Delta straight Taper

Bluntness

Optimal ratio is .15 • Provided length remains constant Applicability dependent upon fineness ratio and velocity • Fineness ratio ≤ 5 • Below Hypersonic

Supersonic case • Pressure Drag Decreases • Wave Drag Decreases • Fineness Ratio of 5 is Critical

Fin design Mission Dependence • Stability (CP, CG, Roll, …) Independent Variables • Atmospheric Density • Temperature • Wind Conditions • Surface Finish (Assumed Constant) • Angle of Attack (Assumed Zero)

Minimize Drag Maintain Structural Integrity • Minimize Divergence • Minimize Bending-Torsion Flutte • Minimize Mass Maximize Fin Joint Strength Maintain Passive Stability

Thinner Symmetrical Fins Result in Lower CD in Sub, Trans, and Supersonic Regions

FIN SHAPE Supersonic Data • Trapezoidal (Clipped Delta) Lower CD than Delta • Delta and Diamond have Similar CD

Pogo osciallation

Pogo oscillation – Longitudinal resonance of the launch vehicle structure • flexing of the propellant-feed pipes induces thrust variation – Gas-filled cavities added to the pipes, damping oscillation –

Fuel slosh – Lateral motion of liquid propellant in partially empty tank induces inertial forces – Resonance with flight motions can occur – Problem reduced by baffling

Transient Loads at cut off

Drag Reduction

Near Mach 1, the drag of a slender wing-body combination is equal to that of a body of revolution having the same cross-sectional area distribution.

Nose Drag : Transonic case

1. Good 2. Fair 3. Inferior

Nose cone geometry Conical Elliptical Ogive (Tangent) Parabolic Power Series Sears-Haack (Von Karman)

L is the overall length of the nosecone R is the radius of the base of the nosecone y is the radius at any point x, as x varies from 0 at the tip of the nosecone to L The full body of revolution of the nosecone is formed by rotating the profile around the centerline (CL)

Nose cone Parameters

Parabolic shape nose

The Parabolic Series nose shape is generated by rotating a segment of a parabola around a line parallel to its axis of symmetry. y=R{(2[x/L]-K[x/L]^2)/(2-K)} for 0≤K≤1 • K= 0 for a CONE • K= .5 for a 1/2 PARABOLA • K= .75 for a 3/4 PARABOLA • K= 1 for a PARABOLA (base tangent to airframe) Cp=L/2 V= πR^2 L/2 S=R^2/4L

Power series Nose The Power Series shape is characterized by its (usually) blunt tip, and by the fact that its base is not tangent to the body tube. The Power series nose shape is generated by rotating a parabola about its major axis. The base of the nosecone is parallel to the latus rectum of the parabola, and the factor n controls the ‘bluntness’ of the shape. As n decreases towards zero, the Power Series nose shape becomes increasingly blunt; at values of n above about .7, the tip becomes sharp. y=R(x/L)^n for 0≤n≤1 • n = 1 for a CONE • n = .75 for a ¾ POWER • n = .5 for a ½ POWER (PARABOLA) • n = 0 for a CYLINDER

Conical

The sides of a cone are straight lines, so the diameter equation is simply, y = Rx/L Cones are sometimes defined by their ‘half angle’, φ = tan-1(R/L) and y = x tan φ Cp= L/3 V= πR^2 L/3 S= πR(R^2 + L^2)^ 0.5

Elliptical Nose cones

The profile of this shape is one-half of an ellipse, with the major axis being the centerline and the minor axis being the base of the nosecone • This shape is advantageous for subsonic flight due to its blunt nose and tangent base • It is defined by: y = R(1-x2 /L2)½ • Cp=3L/2 • V=2πR2L/3 • S=πL2+*πR2/ς*ln,(1+ς)/(1-ς)-++/2 where ς=(L2+R2)/L

OGIVE

This shape is formed by a circle segment where the base is on the circle radius and the airframe is tangent to the curve of the nosecone at its base. The radius of the circle that forms the ogive is: ρ = (R^2 + L^2)/2R The radius y at any point x, as x varies from 0 to L is: y = (ρ^2-(x- L)^2)½+R- ρ where L≤ρ C p = V/4 pi r ^2

v

The Parabolic Series nose shape is generated by rotating a segment of a parabola around a line parallel to its axis of symmetry. y=R{(2[x/L]-K[x/L]2)/(2-K)- for 0≤K≤1 • K= 0 for a CONE ,• K= .5 for a 1/2 PARABOLA ,• K= .75 for a 3/4 PARABOLA • K= 1 for a PARABOLA (base tangent to airframe) Cp=L/2 V= πR2L/2 S=R2/4L

Aspect ratio

Aspect ration of a Missile config is dependent on :- a. Load factor b. Wing area c. Trim angles of attack d. L/D ratio e. Max permissible span. f. The effect of increasing the aspect ration is ;- g. Increase in Cn , Cd0 h. Increase L/D)max i. Reduce wing area j. Increase structural weight k. Neg effect of CP travel