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Invention of Lift Generating Shapes. P M V Subbarao Professor Mechanical Engineering Department. Creation of A Solid Object to Positively Perturb the Flow ……. New Complex Potential.
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Invention of Lift Generating Shapes P M V Subbarao Professor Mechanical Engineering Department Creation of A Solid Object to Positively Perturb the Flow ……
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.
Pressure Distributation Pressure distribution on the surface of the cylinder can be found by using Benoulli’s equation. Thus, if the flow is steady, and the pressure at a great distance is pinf,
THE VORTEX In the case of a vortex, the flow field is purely tangential. The picture is similar to that of a source but streamlines and equipotential lines are reversed. The complex potential is There is again a singularity at the origin, this time associated to the fact that the circulation along any closed curve including the origin is nonzero and equal to g. If the closed curve does not include the origin, the circulation will be zero.
Uniform Flow Past A Doublet with Vortex The superposition of a doublet and a uniform flow gives the complex potential
Angle of Attack Unbelievable Flying Objects
Natural Genius Vs Artificial Machines Lift = ? L = V
Anatomy of an airfoil • The straight line that joins the leading and trailing ends of the mean camber line is called the chord line. • An airfoil is defined by first drawing a “mean” camber line. • The length of the chord line is called chord, and given the symbol ‘c’. • To the mean camber line, a thickness distribution is added in a direction normal to the camber line to produce the final airfoil shape. • Equal amounts of thickness are added above the camber line, and below the camber line. • An airfoil with no camber (i.e. a flat straight line for camber) is a symmetric airfoil. • The angle that a freestream makes with the chord line is called the angle of attack.
Transformation for Inventing a Machine A large amount of airfoil theory has been developed by distorting flow around a cylinder to flow around an airfoil. The essential feature of the distortion is that the potential flow being distorted ends up also as potential flow. The most common Conformal transformation is the Jowkowski transformation which is given by z plane plane y = K1 = K3 = C2 = C1 = C3 = C2 = K3 = C3 x In the diagram a uniform flow in the z plane is transformed into an equivalent form in the w plane using a transform of the form = f(z). To see how this transformation changes flow pattern in the z (or x - y) plane, substitute z = x + iy into above expression.
Theoretical Derivation of Ideal Flow Past a Flat Plate • Consider a circle drawn in the z plane, z = cei. • The Joukowski transform maps the circle into a flat plate,
y z plane plane b -2c 2c x Flow Past A Flat Plate A circle of radius b is mapped into a straight line in the w plane entirely on the real axis between -2cand 2c. If a uniform flow had been drawn over the circle, the transform would have mapped that flow into the flow over a flat plate in the plane.
Transformation of Flow Past A Larger Circle If the circle originally had a radius slightly larger than the transform constant c, z = Rei, with R > c, the circle would have formed an ellipse instead of the flat plate. This gives For a circle of radius r in z plane is transformed in to an ellipse in z - planes:
If the flow over a cylinder had been transformed it would have created the flow over an ellipse -R R
Translation Transformations • If the circle is centered in (0, 0) and the circle maps into the segment between -2c to 2c and lying on the x axis; • If the circle is centered in (xc ,0), the circle maps in an airfoil that is symmetric with respect to the x axis; • If the circle is centered in (0,yc ), the circle maps into a curved segment; • If the circle is centered in and (xc , yc ), the circle maps in an asymmetric airfoil.
y z plane plane R x -2c 2c ec c In turbines point of view, the most interesting application of the Joukowski transform is to an offset circle. If we consider a circle slightly offset from the origin along the negative real axis, one obtains a symmetric Joukowski airfoil Joukowski Airfoils The equation of the offset circle is z = Rei-ec, where the constant e is a small number.
y z plane plane R A x -2c 2c ec c A B B Lift Generating Blades If the cylinder is displaced slightly along the complex axis as well, one obtains a cambered airfoil shape.
Final Remarks • One of the troubles with conformal mapping methods is that parameters such as xc and yc are not so easily related to the airfoil shape. • Thus, if we want to analyze a particular airfoil, we must iteratively find values that produce the desired section. • A technique for doing this was developed by Theodorsen. • Another technique involves superposition of fundamental solutions of the governing differential equation. • This method is called thin airfoil theory.
The thickness problem Let the thickness disturbance field be represented by a distribution of mass sources along the x -axis in the range 0 <x < C .
Forces and moments on a thin cambered airfoil at zero angle of attack
