Demystifying S1-S2 Optimization in Turbomachinery

  1. Historically turbomachinery development began with empirical rules postulated by early pioneers. With the need for jet engine for aircraft propulsion, dimensionless analysis became popular, followed by the 1 D mean line design and 2D meridional methods. Today 2D meridional methods with 3D blade to blade CFD/FEA methods are a necessity as efficiency and reliability requirements are further pushed.


  1. One key aspect of 2D meridional design is S1-S2 optimization, which is a time consuming, laborious task and hence subject to human errors. S1-S2 optimization is a task of reviewing, adjusting and optimizing the flow path in the Tangential (S1 or blade-to-blade or pitchwise) and the Meridional (S2 or span wise) planes. The main purpose is to:
  • Fit the flow path to specific meridional dimensional constraints
  • Adjust blade-to-blade parameters while taking into account structural constraints.

  1. The need for preserving the turbine performance (that has been selected) and then allowing alteration of flow path dimensions requires use of inverse solvers. The parameters that affect the blade-to-blade calculations are the number of blades and the chord. These can be optimized in the S1 optimization, while meeting structural requirements. For optimizing the spanwise distribution, the S2 optimization is chosen to modify/ fine tune the Mollier diagram (Heat drop or Enthalpy) to meet the desired performance. In the S1 optimization adjusting important pitch-wise parameters such as the chord, pitch and number of blades to preserve optimal “solidity”/”relative pitch” by  considering structural constraints. The S1 optimization can be performed either for the full turbine (cylinder) or for particular components (rotors/stators).


  1. An effective analysis requires six different criteria for optimization which is basically grouped into two categories. The groups are based on the chord that gives the maximum efficiency or minimum length that is required to meet the structural requirements. For both of these groups the optimization can be performed to:
  • Obtain the optimum chord, with a constant relative pitch
  • Get the optimum chord and optimum relative pitch
  • Obtain the optimum relative pitch while maintaining the chord constant


  1. If the max efficiency chord is selected then the chord optimization is done with first preference to the aerodynamic criteria and meeting the structural requirement for the given material. If the min length chord is selected, then the chord optimization is done with the objective to obtain the smallest value which meets the structural requirement and MSF (Margin of Safety) value nearer to “zero”.


  1. The AxSTREAM® software suite provides an  automated process for S1-S2 optimization enabling improved designs and reducing engineering hours.


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