AERODYNAMICS

STUDY NOTES

UNIT I REVIEW OF BASIC FLUID MECHANICS .

Continuity, Momentum and Energy Equations. Applications of Bernouli’s theorem

UNIT II TWO DIMENSIONAL FLOWS

Complex Potential, Point Source and Sink, Potential Vortex, Uniform Parallel flow and their Combinations, Pressure and Velocity Distributions on Bodies with and Without Circulation inIdeal and Real fluid Flows. Kutta Joukowski’s Theorem.

UNIT III CONFORMAL TRANSFORMATION

Complex potential function, Blasius theorem, principles of conformal transformation, Kutta -Juokowaski transformation of a circle into flat plate, airfoils & ellipses.

UNIT IV AIRFOILS

Joukowski, Karman Trefftz Profiles, Von Mises airfoils -Glauert’s thin airfoil theory, symmetrical airfoil, cambered airfoil, flapped airfoil, determination of mean camber line shapesfor uniform & linear distribution of circulation. elements of panel method

UNIT V SUBSONIC WING THEORY .

Downwash & induced drag, Vortex line, Horse Shoe Vortex, Biot and Savart Law, Flow past finite wings - vortex model of the wing - induced drag – Prandtl’s lifting line theory - ellipticwing –influence of taper and twist applied to wings – effect of sweep back – delta wings.

TEXT BOOK: 1. Anderson, J.D., “Fundamentals Of Aerodynamics”, McGraw Hill Book Co.,

New York 2005.

REFERENCES: 1. Houghton, E.L., and Carruthers, N.B., “Aerodynamics for Engineering

students”, Edward Arnold Publishers Ltd., London, 1989. 2. Milne Thomson, L.H., “Theoretical

Aerodynamics”, Dover Publication, 2008 3. Clancy, L.J., Aerodynamics”, Pitman, 1986.

Continuity Equation: Integral Form

Let us consider a control volume bounded by the control surface S. The efflu x of mass acrossthe control surface S is given by

where is the velocity vector at an elemental area( which is treated as a vect or by consideringits positive direction along the n ormal drawn outward from the surface).

Fig 10.2 A Control Volum e for the Derivation of Continuity Equation (integral form)

The rate of mass accumulation w ithin the control volume becomes

where d is an elemental volu me, ρ is the density and is the total volume bounded by thecontrol surface S. Hence, the continuity equation becomes (according to the st atement given byEq. (9.1))

The second term of the EQN c an be converted into a volume integral by the use of the Gaussdivergence theorem as

Since the volume does not change with time, the sequence of differentiation and integration inthe first term of Eq.(10.6) can be interchanged.Therefore Eq. (10.6) can be written as

Equation (10.7) is valid for any arbitrary control volume irrespective of its shape and size. So wecan write

Conservation of Momentum: Momentum Theorem

In Newtonian mechanics, the conservation of momentum is defined by Newton’s second law ofmotion.

Newton’s Second Law of Motion

· The rate of change of momentum of a body is proportional to the impressed action andtakes place in the direction of the impressed action.

· If a force acts on the body ,linear momentum is implied.· If a torque (moment) acts on the body,angular momentum is implied.

Reynolds Transport Theorem

A study of fluid flow by the Eulerian approach requires a mathematical modeling for a control volume either in differential or in integral form. Therefore the physical statements of the principle of conservation of mass, momentum and energy with reference to a control volume become necessary.

This is done by invoking a theorem known as the Reynolds transport theorem which relates thecontrol volume concept with that of a control mass system in terms of a general property of thesystem.

Statement of Reynolds Transport Theorem

The theorem states that "the time rate of increase of property N within a control mass system isequal to the time rate of increase of property N within the control volume plus the net rate ofefflux of the property N across the control surface”.

Equation of Reynolds Transport Theorem

After deriving Reynolds Transport Theorem according to the above statement we get

In this equation

N - flow property which is transported

η - intensive value of the flow property

pplication of the Reynolds Transport Theorem to Conservation of Mass and Momentum

Conservation of mass The constancy of mass is inherent in the definition of a control masssystem and therefore we can write

To develop the analytical statement for the conservation of mass of a control volume, the Eq.(10.11) is used with N = m (mass) and η = 1 along with the Eq. (10.13a).

This gives

The Eq. (10.13b) is identical to Eq. (10.6) which is the integral form of the continuity equation derived in earlier section. At steady state, the first term on the left hand side of Eq. (10.13b) iszero. Hence, it becomes

Conservation of Momentum or Momentum Theorem The principle of conservation of momentum as applied to a control volume is usually referred to as the momentum theorem.

Linear momentum The first step in deriving the analytical statement of linear momentum

theorem is to write the Eq. (10.11) for the property N as the linear - momentum and

accordingly η as the velocity . Then it becomes

The velocity defining the linear momentum in Eq. (10.14) is described in an inertial frame ofreference. Therefore we can substitute the left hand side of Eq. (10.14) by the external

forces on the control mass system or on the coinciding control volume by the directapplication of Newton’s law of motion. This gives

This Equation is the analytical statement of linear momentum theorem.

In the analysis of finite control volumes pertaining to practical problems, it is convenient todescribe all fluid velocities in a frame of coordinates attached to the control volume. Therefore,an equivalent form of Eq.(10.14) can be obtained, under the situation, by substituting N as and

accordingly η as , we get

where is the rectilinear acceleration of the control volume (observed in a fixedcoordinate system) which may or may not be a function of time. From Newton’s law of motion

Therefore,

The Eq. (10.16) can be written in consideration of Eq. (10.17) as

At steady state, it becomes

In case of an inertial control volume (which is either fixed or moving with a constant rectilinear

velocity), and hence Eqs (10.18a) and (10.18b) becomes respectively

and

In general, the external forces in Eqs (10.14, 10.18a to 10.18c) have two components - the body force and the surface force. Therefore we can write

where is the body force per unit volume and is the area weighted surface force.

Angular Momentum

The angular momentum or moment of momentum theorem is also derived from Eqinconsideration of the property N as the angular momentum and accordingly η as the angularmomentum per unit mass. Thus,

where AControl mass system is the angular momentum of the control mass system. . It has to benoted that the origin for the angular momentum is the origin of the position vector

The term on the left hand side of Eq.(10.19) is the time rate of change of angular momentum of acontrol mass system, while the first and second terms on the right hand side of the equation arethe time rate of increase of angular momentum within a control volume and rate of net efflux ofangular momentum across the control surface.

The velocity defining the angular momentum in Eq.(10.19) is described in an inertial frame

of reference.Therefore, the term can be substituted by the net moment ΣM appliedto the system or to the coinciding control volume. Hence one can write as

At steady state

Application of momentum theorem

· orces acting due to internal flows through expanding or reducing pipe bends.· Forces on stationary and moving vanes due to impingement of fluid jets.· Jet propulsion of ship and aircraft moving with uniform velocity.

Steady Flow Energy Equation

The energy equation for a control volume is given by Eq. At steady state, the first term on the right hand side of the equation becomes zero and it becomes

In consideration of all the energy components including the flow work (or pressure energy) associated with a moving fluid element, one can substitute ’e’ in Eq. (13.4) as

and finally we get

The Eq. (13.5) is known as steady flow energy equation.

· Bernoulli's Equation· Energy Equation of an ideal Flow along a Streamline· Euler’s equation (the equation of motion of an inviscid fluid) along a stream line for a

steady flow with gravity as the only body force can be written as

· Application of a force through a distance ds along the streamline would physically implywork interaction. Therefore an equation for conservation of energy along a streamline canbe obtained by integrating the Eq. (13.6) with respect to ds as

· Where C is a constant along a streamline. In case of an incompressible flow, Eq. (13.7)can be written as

· The Eqs (13.7) and (13.8) are based on the assumption that no work or heat interactionbetween a fluid element and the surrounding takes place. The first term of the Eq. (13.8)represents the flow work per unit mass, the second term represents the kinetic energy perunit mass and the third term represents the potential energy per unit mass. Therefore the

sum of three terms in the left hand side of Eq. (13.8) can be considered as the totalmechanical energy per unit mass which remains constant along a streamline for a steadyinviscid and incompressible flow of fluid. Hence the Eq. (13.8) is also known asMechanical energy equation.

· This equation was developed first by Daniel Bernoulli in 1738 and is therefore referred toas Bernoulli’s equation. Each term in the Eq. (13.8) has the dimension of energy per unitmass. The equation can also be expressed in terms of energy per unit weight as

· In a fluid flow, the energy per unit weight is termed as head. Accordingly, equation 13.9can be interpreted as

· Pressure head + Velocity head + Potential head =Total head (total energy per unit weight).· Bernoulli's Equation with Head Loss· The derivation of mechanical energy equation for a real fluid depends much on the

information about the frictional work done by a moving fluid element and is excludedfrom the scope of the book. However, in many practical situations, problems related toreal fluids can be analysed with the help of a modified form of Bernoulli’s equation as

· where, hf represents the frictional work done (the work done against the fluid friction) perunit weight of a fluid element while moving from a station 1 to 2 along a streamline in

the direction of flow. The term hf is usually referred to as head loss between 1 and 2,since it amounts to the loss in total mechanical energy per unit weight between points 1and 2 on a streamline due to the effect of fluid friction or viscosity. It physically signifiesthat the difference in the total mechanical energy between stations 1 and 2 is dissipatedinto intermolecular or thermal energy and is expressed as loss of head hf in Eq. (13.10).The term head loss, is conventionally symbolized as hL instead of hf in dealing withpractical problems. For an inviscid flow hL = 0, and the total mechanical energy isconstant along a streamline

UNIT II TWO DIMENSIONAL FLOWS

Uniform FlowVelocity does not change with y-coordinateThere exists only one component of velocity which is in the x direction.

Magnitude of the velocity is U0 .

Since

or,

Thus,

Using stream function ψ for uniform flow

so

The constants of integration C1 and K1 are arbitrary.The values of ψ and Φ for different streamlines and velocity potential lines may change but flowpattern is unaltered. The constants of integration may be omitted, without any loss of generality and it is possible to write

Fig 2 (a) Flownet for a Uniform Stream (b) Flownet for uniform Stream with an Angles with x-axis

These are plotted in Fig. .2(a) and consist of a rectangular mesh of straight streamlines andorthogonal straight potential-liines (remember streamlines and potential lines are alwaysorthogonal ). It is conventional t o put arrows on the streamlines showing the dire ction of flow.

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