Aerodynamics Fundamentals - II

MISSILE AERODYNAMICS INTRODUCTION 1. Study of the movement of a body in the presence of air is called aerodynamics and this study is vitally important for the design of aircraft, missiles and rockets. 2. Earth’s Atmosphere: The atmosphere is densest close to earth’s surface at sea level. As we go higher it becomes thinner i.e., the pressure and density are lower. The sensible atmosphere is up to a height of 90 km (Homosphere) beyond which is heterosphere. The temperature also varies with height. The layer of atmosphere nearest to earth is called troposphere. Above that are stratosphere, ionosphere and the last exosphere. The very high-speed fighter aircraft fly up to altitudes of about 30 km, while transport jets fly up to about 10-11 km. Aerodynamics: Classification 3.

The distinction between solids, liquids and gases is made as follows: (a) Put them in a large closed container, solid will not change i.e., its shape and boundaries will remain the same; whereas the liquid will change its shape to conform to that of the container and will take on the same boundaries as the container up to the maximum depth of the liquid and the gas will completely fill the container, taking on the same boundaries as the container. (b) Fluid denotes either a liquid or a gas. When a force is applied tangentially to the surface of a solid, the solid will experience a finite deformation, and the tangential force per unit area-the shear stress-will usually be proportional to the amount of deformation. In contrast, when a tangential shear stress is applied to the surface of a fluid, the fluid will experience a continuously increasing deformation, and the shear stress will be proportional to the rate change of momentum. (c) The most fundamental distinction is at atom and molecular level i.e., spacing between molecules. In solids, molecules are closely packed while in liquids and gases spacing is large. Hence intermolecular forces are much weaker and motion of molecules occurs freely particular throughout gases.

4. Fluid Dynamics: - The study of dynamics of fluid can be subdivided into three areas as follows: (a) (b) (c)

Hydrodynamics-flow of liquids Gas Dynamics-flow of gases Aerodynamics-flow of air.

5. Aerodynamics: - The applications in prediction of forces and moments on, and heat transfer to (aerodynamics heating), bodies moving through a fluid is called external aerodynamics since they deal with external flows over a body. In contrast, the applications in determination of flows moving internally through ducts, calculation and measurement of flow properties inside rocket and air-breathing engines, engine thrust or flow conditions in test section of wind tunnel is called internal aerodynamics. Definition Of Basic Aerodynamic Quantities 6. The four basic aerodynamic quantities are pressure, density, temperature and flow velocity. A fifth quantity is streamlines. (a) Pressure is the normal force per unit area exerted on a surface due to time rate of change of momentum of the gas molecules impacting or crossing that surface (point property). P = lim (dF/dA), dA tending to zero Where dA = elemental area dF = force on one side of dA due to pressure. (b)

Density is defined as the mass per unit volume (point property). ρ = lim(dm/dv), dv tending to zero where dv = elemental volume around a point dm = mass of fluid inside dv

(c) The temperature T of a gas is directly proportional to the average kinetic energy of the molecules of the fluid. In fact, if KE is the molecular kinetic energy, then temperature is given by KE = (3/2)kT, where k is Boltzmann constant. (d) The principal focus of aerodynamics is fluids in motion. Hence, flow velocity is important. The velocity of a flowing gas at any fixed point B in space can be defined as the velocity of an infinitesimally small fluid element as it sweeps through B. The flow velocity V has both magnitude and direction and hence is a vector quantity (p, ρ and T are scalar quantities). (e) A moving fluid element traces out a fixed path in space. As long as the flow is steady (no fluctuations with time), this path is called a streamline of the flow. Drawing the streamlines of the flow field is an important way of visualizing the motion of the gas. (f) If two streamlines are rubbing at each other, friction plays a role and exerts a force of magnitude dFf on one of the streamlines acting tangentially in the direction of the force. The shear stress τ is the limiting form of the magnitude of the frictional force dFf per unit area dA where dA is perpendicular to the y axis and has shrunk to nearly zero i.e., τ = lim(dFf /dA) as dA tends to zero.

In aerodynamic applications, the value of shear stress at a point on a streamline is proportional to the spatial rate of change of velocity of normal to the streamline at that point i.e., τ α dV/dy or τ = µ (dV/dy) where the constant of proportionality, µ , is defined as the viscosity coefficient and dV/dy is the velocity gradient. Aerodynamic Forces and Moments 7. No matter how complex the body shape may be, the aerodynamic forces and moments on the body are due entirely to only to two basic sources: (a) Pressure distribution over the body surface (force per unit area normal to the surface) (b) Shear stress distribution over the body surface (force per unit area tangential to the surface) 8. The net effect of p and τ distributions integrated over the complete body surface is a resultant aerodynamic force R and moment M on the body.

N L R

α α

M D

V∞

α c

(a)

A

The resultant R can be split into components L = lift = component perpendicular to the relative wind V∞ (also called free stream velocity) D = drag = component of R parallel to V∞

(b) The chord c is the linear distance from the leading edge to the trailing edge of the body. Sometimes, R is split into components perpendicular and parallel to the chord and by definition N = Normal force = component of R perpendicular to c A = Axial force = component of R parallel to c. The angle of attack α is defined as the angle between c and V∞. Hence, α I is also the angle between L and N and between D and A. The geometrical relation D between these two sets of components is L = N cos α - A sin α α D = N sin α + A cos α L (c)

N

D α

cos α = adjacent/hypotenuse = L/N = D/A sin α = opposite/hypotenuse = -L/A(since α =-ve) = D/N

L

A

(d) If we consider a two-dimensional model (N will be denoted as N’ and A as A’ in this case i.e., force per unit span), we are interested in the contribution to the total normal force N’ and the total axial force A’ due to the pressure and shear stress on the elemental area dS. N’u

θ

pu A’u θ

LE

u

A’l

α V∞

τ

TE

θ

pl θ N’ l

τ l

(i) The elemental normal and axial forces for upper and lower body surface will be given as dNu’ = -pudsucos θ - τ udsusin θ dAu’ = -pudsusin θ + τ udsucos θ dNl’ = pldslcos θ - τ ldslsin θ

dAl’ = pldslsin θ + τ ldslcos θ cos θ = adj/hyp = -dNu’/pu = dAu’/τ u sin θ = opp/hyp= dNu’ / τ u = -dAu’/ pu (ii) The total normal and axial forces per unit span are obtained by integrating above equations from leading edge (LE) to the trailing edge (TE): N’ = ∫ dNu’ + ∫ dNl’ A’ = ∫ dAu’ + ∫ dAl’ (iii) The aerodynamic moment exerted on the body depends on the point about which moments are taken and leading edge is taken. By convention, moments that tend to increase α (pitch up) are positive and the vice versa and y is a positive number above the chord and negative below the chord. Thus the moment per unit span about the leading edge due to p and τ on the elemental area dS on the upper surface is dMu’ = (pucos θ - τ usin θ )x dsu + (-pusin θ + τ ucos θ )y dsu dMl’ = (-plcos θ + τ lsin θ )x dsl + (plsin θ + τ lcos θ )y dsl (iv) The moment about the leading edge per unit span is obtained by integrating the above equations from the leading to trailing edges. MLE’ = ∫ dMu’ + ∫ dMl’ (v) The above equations demonstrate that the sources of the aerodynamic lift, drag and moments on a body are the pressure and shear stress distributions integrated over the body. A major goal of theoretical aerodynamics is to calculate p(s) and τ (s) for a given body shape and free stream conditions thus yielding the aerodynamic forces and moments. (e) Dimensionless force and moment coefficients are quantities even more fundamental in nature than the aerodynamic forces and moments themselves. (i) Let ρ ∞ and V∞ be the density and velocity, respectively, in the free stream far ahead of the body. We define a dimensional quantity called the free stream dynamic pressure as q∞ = ½(ρ ∞ V∞2) (ii) Let S be a reference area and l be the reference length. (iii) The dimensionless force and moment coefficients are defined as follows: (aa) Lift coefficient: CL = L/( q∞ S) (bb) Drag coefficient: CD = D/( q∞ S) (cc) Normal force coefficient: CN = N/( q∞ S) (dd) Axial force coefficient: CA = A/( q∞ S) (ee) Moment coefficient: CM = M/( q∞ Sl)

(f) The symbols in capital letters above denote force and moment coefficients for a complete three-dimensional body. In contrast, for a two-dimensional body the forces and moments are per unit span and the coefficients are denoted in lowercase letters (c is the chord length): (i) Lift coefficient: cl = L’/( q∞ c) (ii) Drag coefficient: cd = D’/( q∞ c) (iii) Moment coefficient: cm = M’/( q∞ c2) (iv) Pressure coefficient: Cp = p-p∞/ q∞ where p∞ is the free stream pressure (v) Skin friction coefficient: cf = τ /q∞ (g) Substituting dx = ds cos θ , dy = -(ds sin θ ) and dividing by S = c(1) and q∞ , the eqns for N’ and A’ can be re-written in terms of dimensionless coefficients as below: -

9.

Centre of Pressure The normal and axial forces on the body are due to the distributed loads imposed by pressure and shear distributions. Moreover, these distributed loads generate a moment about the leading edge as given in eqn 8 (d) (iv). Therefore, N’ and A’ must be placed on the airfoil at such a location to generate the same moment about the leading edge. If A’ is placed on the chord line as shown in fig below, then N’ must be located a distance xcp downstream of the leading edge such that MLE’ = - (xcp)N’ or xcp = - (MLE’/N’) (since MLE’ is shown as pitch-up, N’ is – ve) N’

MLE ’

A’ xcp

(a) Thus center of pressure (xcp ) is defined as the location where the resultant of a distributed load effectively acts on the body. If moments were taken about the center of pressure, the integrated effect of the distributed loads would be zero. (b) An alternate definition of the center of pressure is that point on the body about which the aerodynamic moment is zero.