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2.) The Joukowsky Transform is defined by f(z) = 2 + =. Z ainama.) Show that the region in the upper half-plane outside the unit circle maps to the full upper half-plane under the Joukowsky Transform. The points below are B = -1, C = i, and D = 1 (A and E are generic points on the real axis on either side of the unit circle). Be sure to find the image points B', C', and D', show that the upper unit circle gets mapped to the real axis, and any generic point in the gray region on the left (i.e. |z| = |x + iy| > 1 and y > 0) gets mapped into the gray region on the right (w = u+iv with v > 0). Y f AB DE a 1 ข U A' B' C' D'E' b.) Using Mathematica, find the image under the Joukowsky Trans- 1 1 5 form of the circle with center-+-i which passes through z = 1. The resulting image is called a Joukowsky airfoil. These airfoils were historically important in understanding airfoil design. (You should investigate the Mathematica commands ComplexExpand and ParametricPlot.)

2.) The Joukowsky Transform is defined by f(z) = 2 + =. Z ainama.) Show that the region in the upper half-plane outside the unit circle maps to the full upper half-plane under the Joukowsky Transform. The points below are B = -1, C = i, and D = 1 (A and E are generic points on the real axis on either side of the unit circle). Be sure to find the image points B', C', and D', show that the upper unit circle gets mapped to the real axis, and any generic point in the gray region on the left (i.e. |z| = |x + iy| > 1 and y > 0) gets mapped into the gray region on the right (w = u+iv with v > 0). Y f AB DE a 1 ข U A' B' C' D'E' b.) Using Mathematica, find the image under the Joukowsky Trans- 1 1 5 form of the circle with center-+-i which passes through z = 1. The resulting image is called a Joukowsky airfoil. These airfoils were historically important in understanding airfoil design. (You should investigate the Mathematica commands ComplexExpand and ParametricPlot.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2.) The Joukowsky Transform is defined by f(z) = 2 + =. Z ainama.) Show that the region in the upper half-plane outside the unit circle maps to the full upper half-plane under the Joukowsky Transform. The points below are B = -1, C = i, and D = 1 (A and E are generic points on the real axis on either side of the unit circle). Be sure to find the image points B', C', and D', show that the upper unit circle gets mapped to the real axis, and any generic point in the gray region on the left (i.e. |z| = |x + iy| > 1 and y > 0) gets mapped into the gray region on the right (w = u+iv with v > 0). Y f AB DE a 1 ข U A' B' C' D'E'
Transcribed Image Text:2.) The Joukowsky Transform is defined by f(z) = 2 + =. Z ainama.) Show that the region in the upper half-plane outside the unit circle maps to the full upper half-plane under the Joukowsky Transform. The points below are B = -1, C = i, and D = 1 (A and E are generic points on the real axis on either side of the unit circle). Be sure to find the image points B', C', and D', show that the upper unit circle gets mapped to the real axis, and any generic point in the gray region on the left (i.e. |z| = |x + iy| > 1 and y > 0) gets mapped into the gray region on the right (w = u+iv with v > 0). Y f AB DE a 1 ข U A' B' C' D'E'
b.) Using Mathematica, find the image under the Joukowsky Trans- 1 1 5 form of the circle with center-+-i which passes through z = 1. The resulting image is called a Joukowsky airfoil. These airfoils were historically important in understanding airfoil design. (You should investigate the Mathematica commands ComplexExpand and ParametricPlot.)
Transcribed Image Text:b.) Using Mathematica, find the image under the Joukowsky Trans- 1 1 5 form of the circle with center-+-i which passes through z = 1. The resulting image is called a Joukowsky airfoil. These airfoils were historically important in understanding airfoil design. (You should investigate the Mathematica commands ComplexExpand and ParametricPlot.)
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