In the previous article on screw propellers, I acquainted the readers with the basic terminologies related to screw propeller geometry and its slip phenomenon. In this article, I will impart an insight into the core helicoidal geometry of screw propeller emphasising on the different views of the propeller.
SCREW PROPELLER- THE REFERENCE SYSTEM
To describe the propeller geometry we generally take reference of the cylindrical coordinate system. This is because in propeller we talk about radial sections and helix formed on a cylinder. Thus a cylindrical coordinate system is appropriate. The coordinate system is as shown.
The major planes in the coordinate system are z=0, Θ=0 and r=0. Any offset of the propeller blade is taken in 3-dimensional space with reference to these planes.
SCREW PROPELLER- THE GEOMETRY
To be specific the geometry of the propeller is a bit complex. I have tried to simplify the geometry as much as possible.
If we take a propeller and cut a radial section, then it will look as shown.
The face line of the radial section is a part of a helicoidal surface with some offset at the leading and trailing edge. What does this mean?
This means that suppose I have 2 sticks orthogonally arranged such that one stick is rotated about the second stick as an axis and the first stick advances along with rotation. Thus the loci of the tip of the stick as shown will trace out a helix on the imaginary cylinder as shown. The face of the propeller at a radial section is a part of the helix where the radius of the imaginary cylinder is the radial section. Also, the face offsets from the helix at the leading and trailing edge. The back surface of the propeller blade depends on the aerofoil section profile that each radial section of the propeller blade is made up of.
Image courtesy: Google Images.
Now if we cut the imaginary cylinder longitudinally and open the cylinder to form a rectangle, the helix forms the diagonal of the rectangle such that the face of the radial section lies on the diagonal as shown.
Image courtesy: Google Images.
This denoted the actual section of the propeller. It can be thought that this actual section of the propeller is bent and giver a definite curvature and this is done for all radial sections. Then these sections are joined to form the complete propeller blade. This brings us to different views how we look at the propeller which will give us a firm idea about the various curvatures that are imparted to a plane aerofoil section to form a propeller radial section.
Before we jump onto propeller views it must be noted that the propeller blade radial section has curvature in 2 planes. If we see the propeller blade face and just concentrate on the radial section then we can see one curvature (the radial curvature) as the section is a part of the circular arc cut off from the total blade. Also as the face is a part of the helix, so looking at the propeller blade from the top view and concentrating on the radial section cut off from the blade, the section has a second curvature due to the helicoidal surface as shown.
Thus if we take a radial section of the propeller and straighten the 2 curvature one by one then the different views of the propeller are formed.
Suppose we have a propeller blade. This is made by various radial sections as told earlier. If we take the orthogonal projection of the blade radial sections on a plane perpendicular to the z-axis (z=0 plane), the view formed is called the projected view. These projections taken for all radial sections form the projected outline of the propeller blade. This can be understood by taking a propeller blade and projecting a light on the blade along the z-axis. The shadow cast on the wall which is perpendicular to the z-axis is called the projected view of the propeller. The area within the projected outline of the blade is called the projected area AP.
Image Courtesy: Team Learn Ship Design.
This view is a bit difficult to understand and has to be concentrated. Now again we take a particular radial section of the propeller blade which has curvature in 2 planes. We have learned the definition of pitch and considering a propeller having uniform pitch distribution along the radius of the blades, we can infer that the pitch angles of all radial sections will be different that comes from simple calculations. We know that the face of each radial section is a part of the helix chord and thus it will have a midpoint (C say). By definition developed view of a propeller is the projection of the radial section on a plane which is through the point c and makes an angle equal to the pitch angle (∅ for that radial section) with z=0 plane.
This is similar to that of projected view but difference being, the light is projected at an angle equal to pitch angle for that radial section with the z-axis.
The physical significance of this view is that the helicoidal curvature of the radial section is straightened in this view. Thus unlike projected view, this view is not like a circular arc but has offsets laterally forming an ellipse due to the straightening of the helicoidal curvature.
This process done for all radial section gives the developed outline and the area within the developed outline is called the developed area AD.
Image Courtesy: Google Images.
This view is easy to understand. If we open up the imaginary cylinder for a particular radial section then actually both the helicoidal and radial curvature of the section is straightened and it gives a proper aerofoil section with no curvature along the diagonal of the rectangle as shown.
Image Courtesy: Google Images.
This aerofoil section makes an angle of pitch angle (∅ for that radial section) with Θ=0 plane. If we rotate it by the same pitch angle, the expanded section is obtained. This view done for all radial sections give us the expanded outline and the area within the expanded outline is called expanded area AE.
Image Courtesy: Team Learn Ship Design
It must be mentioned that these different views and areas are characteristics of a propeller and thus are very important to understand.
This article wraps up the screw propeller basics, giving useful insights into the reference coordinate system used to define a propeller, the helix concept and the various propeller views and how they characterise the geometry of a screw propeller.
Article by: Rijay Majee.