Calculation of theta slope for Angle Definition Type “Theta with LE/TE Beta”

    • Christian Woiczinski

      Hi everyone!

      I’m using a Bezier curve with four control points and the Angle Definition Type “Theta with LE/TE Beta” for defining the shape of a blade’s camberline (see section 10.2.4 in Ansys TurboSystem User’s Guide).

      It is intuitively clear that the blade angle at the leading edge (betaLE) determines the slope of the theta curve at m = 0% (analogously, the blade angle at the trailing edge (betaTE) determines the slope at m = 100%).

      But how exactly can I calculate this slope?

      What I understand is that the slope of the control polygon in the first control point equals the slope of the Bezier curve at m = 0% (analogously for the last point, i.e. m = 100%).

      Therefore, when fixing the x-coordinate of the second control point, then one should be able to calculate the y-coordinate of the second control point only with the help of betaLE (assuming that the first control point is completely given).

      The basic relation between beta and theta is described in section “9.1.11. Definitions” in Ansys TurboSystem User’s Guide. But how exactly can I deduce from this relation the slope in the coordinate system with “% m” as x-axis?

      To put it differently: When I consider theta as a function of m in this coordinate system, how can I calculate the first derivative of this function in dependence of the beta angle?
          theta'(m) = ??? (some function of beta)

      And especially
          theta'(0) = ???     (some function betaLE)
          theta'(100) = ???     (some function betaTE)

      I tried various calculations (considering the corresponding integrals in section 9.1.11 and their derivatives with respect to m), but I don’t get the same slope as the BladeGen does. (I fear I miss some factor or additional calculation step due to the change of the coordinate systems…) 

      Background of my question: I want to do some pre-analysis / optimization of the angle distribution *before* feeding it to BladeGen. Therefore, I want to reproduce the calculation of the theta Bezier points based on betaLE/TE.

      (Unfortunately, I must to use the Angle Definition Type “Theta with LE/TE Beta”, switching to another way of defining the angle is not possible for me.)

      Your help would be greatly appreciated!


    • rfblumen
      Ansys Employee

      Given a certain blade span fraction (e.g. 50% span), the blade section can be converted from 3-D Cartesian space in (x,y,z) to 2-D developed coordinates (m,s), where m is the meridional distance (integral(sqrt(dx^2+dz^2)))  and s represents the integrated arc length (integral(r*dtheta)).  Meridional distance can be normalized by radius to give m'.

      If we look at the developed view of the blade in (m,s) coordinates, the slope of the blade mean camberline is defined as the blade angle beta.  The relationshipo between beta and theta is:


              ~= r*del*theta)/del(m)

      Knowing radius, r, and beta you can calculate the slope of del(theta)/del(m) in the Angle plots in BladeEditor.

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