AERODYNAMICS STUDY NOTES UNIT I REVIEW OF BASIC FLUID MECHANICS. Continuity, Momentum and Energy Equations. Applications of Bernouli s theorem

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1 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 in Ideal 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 shapes for 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 - elliptic wing 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 REFERENCES: 1. Houghton, E.L., and Carruthers, N.B., Aerodynamics for Engineering students, Edward Arnold Publishers Ltd., London, Milne Thomson, L.H., Theoretical Aerodynamics, Dover Publication, Clancy, L.J., Aerodynamics, Pitman, 1986.

3 Continuity Equation: Integral Form Let us consider a control volume the control surface S is given by bounded by the control surface S. The efflu x of mass across where is the velocity vector at an elemental area( which is treated as a vect or by considering its 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 the control surface S. Hence, the continuity equation becomes (according to the st atement given by Eq. (9.1)) The second term of the EQN c an be converted into a volume integral by the use of the Gauss divergence theorem as

4 Since the volume does not change with time, the sequence of differentiation and integration in the 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 we can write Conservation of Momentum: Momentum Theorem In Newtonian mechanics, the conservation of momentum is defined by Newton s second law of motion. Newton s Second Law of Motion The rate of change of momentum of a body is proportional to the impressed action and takes 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 the control volume concept with that of a control mass system in terms of a general property of the system. Statement of Reynolds Transport Theorem The theorem states that "the time rate of increase of property N within a control mass system is equal to the time rate of increase of property N within the control volume plus the net rate of efflux 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

5 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 mass system 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) is zero. 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 accordingly η as the velocity. Then it becomes and

6 The velocity defining the linear momentum in Eq. (10.14) is described in an inertial frame of reference. 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 direct application 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 to describe 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 fixed coordinate system) which may or may not be a function of time. From Newton s law of motion

7 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

8 The angular momentum or moment of momentum theorem is also derived from Eqin consideration of the property N as the angular momentum and accordingly η as the angular momentum per unit mass. Thus, where A Control mass system is the angular momentum of the control mass system.. It has to be noted 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 a control mass system, while the first and second terms on the right hand side of the equation are the time rate of increase of angular momentum within a control volume and rate of net efflux of angular 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 applied to 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

9 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 imply work interaction. Therefore an equation for conservation of energy along a streamline can be 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 interaction between 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 per unit mass and the third term represents the potential energy per unit mass. Therefore the

10 sum of three terms in the left hand side of Eq. (13.8) can be considered as the total mechanical energy per unit mass which remains constant along a streamline for a steady inviscid and incompressible flow of fluid. Hence the Eq. (13.8) is also known as Mechanical energy equation. This equation was developed first by Daniel Bernoulli in 1738 and is therefore referred to as Bernoulli s equation. Each term in the Eq. (13.8) has the dimension of energy per unit mass. 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 13.9 can 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 excluded from the scope of the book. However, in many practical situations, problems related to real fluids can be analysed with the help of a modified form of Bernoulli s equation as where, h f represents the frictional work done (the work done against the fluid friction) per unit weight of a fluid element while moving from a station 1 to 2 along a streamline in the direction of flow. The term h f 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 1 and 2 on a streamline due to the effect of fluid friction or viscosity. It physically signifies that the difference in the total mechanical energy between stations 1 and 2 is dissipated into intermolecular or thermal energy and is expressed as loss of head h f in Eq. (13.10). The term head loss, is conventionally symbolized as h L instead of h f in dealing with practical problems. For an inviscid flow h L = 0, and the total mechanical energy is constant along a streamline

11 UNIT II TWO DIMENSIONAL FLOWS Uniform Flow Velocity does not change with y-coordinate There exists only one component of velocity which is in the x direction. Magnitude of the velocity is U 0. Since or, Thus, Using stream function ψ for uniform flow so The constants of integration C 1 and K 1 are arbitrary. The values of ψ and Φ for different streamlines and velocity potential lines may change but flow pattern 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

12 These are plotted in Fig..2(a) and consist of a rectangular mesh of straight streamlines and orthogonal straight potential-liines (remember streamlines and potential lines are always orthogonal ). It is conventional t o put arrows on the streamlines showing the dire ction of flow. This is a SAMPLE (Few pages have been extracted from the complete notes:-it s meant to show you the topics covered in the full notes and as per the course outline Download more at our websites: To get the complete notes either in softcopy form or in Hardcopy (printed & Binded) form, contact us on: Call/text/whatsApp / naarocom@gmail.com info@naarocom.com sales@naarocom.com Get news and updates by liking our page on facebook and follow us on Twitter Sample/preview is NOT FOR SALE

FUNDAMENTALS OF AERODYNAMICS

*A \ FUNDAMENTALS OF AERODYNAMICS Second Edition John D. Anderson, Jr. Professor of Aerospace Engineering University of Maryland H ' McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas