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Explore the history of airfoil development and the need for lift generating devices. Learn about the fascinating Kutta-Joukowski theorem and the analysis of aerofoils using conformal mapping and complex potentials. Discover the engineering qualities of functions satisfying Laplace's equation and the elementary irrotational plane flows.
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Mathematics to Innovate Blade Profile P M V Subbarao Professor Mechanical Engineering Department Also a Fluid Device, Which abridged the Globe into Global Village…. Creation of A Solid Object to Positively Perturb the Flow ……
History of Airfoil Development Creation of Lift Generating Devices
The Basic & Essential Cause for Generation of Lift • The experts advocate an approach to lift by Newton's laws. • Any solid body that can force the air downward clearly implies that there will be an upward force on the airfoil as a Newton's 3rd law reaction force. • From the conservation of momentum for control Volume • The exiting air is given a downward component of momentum by the solid body, • and to conserve momentum, • something must be given an equal upward momentum to solid body. • Only those bodies which can give downward momentum to exiting fluid can experience lift ! • Kutta-Joukowski theorem for lift.
Fascinating Vortex Phenomena : Kutta-Joukowski Theorem The Joukowsky transformation is a very useful way to generate interesting airfoil shapes.
Analysis of Aerofoils • Several methods are available for developing an aerofoil and predicting the pressure distribution around the same. • These are categorized as the following:. • 1. Conformal Mapping uses the fact that solutions to Laplace's equation can be easily constructed using analytic functions of a complex variable. • 2. Thin Airfoil Theory leads to some very important insights about airfoils. • 3. Surface Panel Methods provide a more accurate analysis method for airfoils and are l widely used.
It is possible to demonstrate that the condition of irrotational field implies the existence of a velocity potential such that The Velocity Potential • On substituting the definition of potential into the incompressible continuity equation we obtain • The velocity potential must then satisfy the Laplace equation and it consequently is a harmonic function of space. • Solution of the Laplace equation, with an appropriate set of boundary conditions, leads then to the determination of the flow field. • Laplace equation has been widely studied in many fields, and shows some interesting properties.
Engineering Qualities of Functions satisfying Laplace Equation • Among the latter, one of the most important is its linearity. • Given two solutions of the Laplace equation, any linear combination of them (and in particular their sum and difference) is again a valid solution. • The potential of a complex flow can then be obtained by superimposing potentials of simpler flows.
Conformal Mapping Through Complex Analysis Any analytic function of a complex variable () satisfies the equation for incompressible, irrotational flow: It is possible to relate one flow field to another by setting: Where is related to z by an analytic function of z, = f(z). The idea behind airfoil analysis by conformal mapping is to relate the flow field around one shape which is already known (by whatever means) to the flow field around an airfoil.
THE COMPLEX POTENTIAL A Flow field can be represented as a complex potential. In particular we define the complex potential In the complex (Argand-Gauss) plane every point is associated with a complex number In general we can then write
The Cauchy-Riemann conditions are: Differentiating the first equation with respect to x and the second with respect to y and adding gives: Any analytic function of a complex variable is a solution to Laplace's equation . May be used as part of a more general solution. W = φ+ i ψ is called the complex potential.
Fertile Complex Potentials Now, if the function f is analytic, this implies that it is also differentiable, meaning that the limit • The derivative of the complex potential W in the complex z plane gives the complex conjugate of the velocity. • Knowledge of the complex potential as a complex function of z leads to the velocity field through a simple derivative.
ELEMENTARY IRROTATIONAL PLANE FLOWS The uniform flow The source and the sink The vortex
THE UNIFORM FLOW The first and simplest example is that of a uniform flow with velocity U directed along the x axis. In this case the complex potential is The streamlines are all parallel to the velocity direction. Equi-potential lines are obviously normal to stream lines. If U is real, the flow is along a direction with a speed U. The flow direction can be adjusted by changing real and imaginary parts.
THE SOURCE OR SINK: The Perturbation Functions Source (or sink), the complex potential of which is • This is a pure radial flow, in which all the streamlines converge at the origin, where there is a singularity due to the fact that continuity can not be satisfied. • At the origin there is a source, m > 0 or sink, m < 0 of fluid. • Traversing any closed line that does not include the origin, the mass flux (and then the discharge) is always zero. • On the contrary, following any closed line that includes the origin the discharge is always nonzero and equal to m.
Iso f lines Iso y lines • The flow field is uniquely determined upon deriving the complex potential W with respect to z.
THE DOUBLET The complex potential of a doublet
Uniform Flow Past A Doublet : Perturbation of Uniform Flow The superposition of a doublet and a uniform flow gives the complex potential
Find out a stream line corresponding to a value of steam function is zero
There exist a circular stream line of radium R, on which value of stream function is zero. • Any stream function of zero value is an impermeable solid wall. • Plot shapes of iso-streamlines.
Note that one of the streamlines is closed and surrounds the origin at a constant distance equal to
New Complex Potential Recalling the fact that, by definition, a streamline cannot be crossed by the fluid, this complex potential represents the irrotational flow around a cylinder of radius R approached by a uniform flow with velocity U. Moving away from the body, the effect of the doublet decreases so that far from the cylinder we find, as expected, the undisturbed uniform flow. In the two intersections of the x-axis with the cylinder, the velocity will be found to be zero. These two points are thus called stagnation points.
Velocity Flow Field due to Flow past a Cylinder To obtain the velocity field, calculate dW/dz.
with Equation of zero stream line:
V2Distribution of flow over a circular cylinder The velocity of the fluid is zero at = 0o and = 180o. Maximum velocity occur on the sides of the cylinder at = 90o and = -90o.
