Optimization and Computational Fluid Dynamics - Department of ...

6 Numerical Optimization for Advanced Turbomachinery Design 157
Pmass
˙mreq
Fig. 6.6 Variation of penalty function for incorrect mass flow
Pperf is the penalty for non-optimum performance and increases with decreasing
efficiency (η)
Pperf =max[|ηreq − η| , 0.0]
Minimizing this term corresponds to maximizing the efficiency. The required
efficiency ηreq is set to an unachievable value (for instance 1.0) so
that this penalty never goes to zero. The argument is that, after all other
requirements are met, the efficiency should still be maximized.
PAeroBC is the penalty for violating the aerodynamic boundary conditions.
The purpose of this penalty is to enforce the boundary conditions and requirements
at the inlet and outlet of the computational domain that cannot be
imposed such as: the outlet flow angle (β2), the mass flow or pressure ratio,
etc. The penalties for not respecting the boundary conditions start increasing
when the actual values differ from the target values by more than a predefined
tolerance. Following penalty for incorrect mass flow increases when the
mass flow differs more than 2% from the required value (Fig. 6.6):
Pmass =
� � ��2 | ˙mact − ˙mreq|
max
− 0.02 , 0.
˙mreq
Pmech is the penalty for not respecting the mechanical constraints. The
latter must be satisfied without compromise because exceeding the maximum
stress level cannot be tolerated as it may destroy the device. A rigorous
respect of the minimum stress limits requires a Finite Element stress Analysis
(FEA) and will be discussed in detail in Sect. 6.6. The computational effort
can be drastically reduced if one can replace the mechanical constraints by
simpler geometrical ones that are much easier to verify.
The large stresses in the blade root section of radial impellers are a complex
function of the blade curvature and lean. The subsequent deformations can
reduce the tip clearance to zero which may lead to the destruction of the
optimized geometry. Traditional design systems limit the lean (Fig. 6.7) to a
maximum value based on experience and simple stress models.
Prescribing the radial variation of the cross section area of a fan blade or
low-pressure (LP) turbine blade is a common way to control the centrifu-