4. Lifting flow over a circle. Find the complex potential that gives a uniform f
ID: 2086447 • Letter: 4
Question
4. Lifting flow over a circle. Find the complex potential that gives a uniform flow (of strength A), a source and a sink (both of strength B). Show that a source and sink give rise to a doublet (dipole flow). Combine a doublet and uniform flow for various values of A and B. Now add a circulation of strength C to this flow. Draw their streamlines individually and then combined. Show that only in combination they give an asymmetric flow around a circle centered at origin. In the context of the above discussion, discuss how complex analysis can be used to study airflow around an airfoil.Explanation / Answer
In the field of fluid dynamics, an area of significant practical importance is the study of airfoils. An airfoil refers to the cross sectional shape of an object designed to generate lift when moving through a fluid. Fundamentally, an airfoil generates lift by diverting the motion of fluid flowing over its surface in a downward direction, resulting in an upward reaction forceUnderstanding lift at a higher level thus involves the physical modeling of the fluid flowing over an airfoil.
In particular, one common method of modeling the fluid flow around an airfoil requires an understanding of complex number mathematics.Before we develop a model for the fluid flow around airfoils, it is important to define airfoils geometrically and to acquaint ourselves with the nomenclature with which they are characterized.
This characterizing system defines airfoil shapes with a series of digits corresponding to non-dimensionalized airfoil properties. The number of digits used to describe an airfoil corresponds to the complexity of the airfoil.
. The principle of the conservation of mass is expressed for an incompressible flow by the equation
? · V = 0 .....................................................a)
An irrotational flow is defined as a flow where the vorticity ~? is zero at every point. Since vorticity is defined as the curl of the velocity field, this imposes the condition
? = ? × V = 0
? · (??) = ?2? = 0
The relation between the velocity potential ? and the stream function ? is given by the equations
?? ?x = ?? ?y
?? ?y = ? ?? ?x
The first step to modeling the fluid flow around an airfoil is solving for the lifting flow around a cylinder in the z plane. This lifting flow can be modeled with the superposition of three basic, elementary flows. Because the solution for the flow around a cylinder has been studied extensively
This uniform flow can be defined with the potential function
? = V?r cos (?)
Consider a flow where all the streamlines are straight lines converging or diverging from a central point O. If the flow is converging to the central point, the flow is referred to as a sink flow. Conversely, if the flow is diverging from the central point, the flow is referred to as a source flow. The resulting velocity field for these flows only has a radial component Vr, which is inversely proportional to the distance from O. With these boundary conditions in place, the potential and stream functions for a source and sink are
? = ? /2 ? ln (r)
? = ?/ /2? ?
The streamlines for this final superposition of three flows . Because there is a vortex flow, the cylinder is now rotating with a finite angular velocity. This rotation eliminates the symmetry along the horizontal axis, creating an uneven pressure distribution, which generates lift
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