  Parameter extension simulation (PES) as a mathematical method for simulating
turbulent flows has been proposed in the study. It is defined as a calculation
of the turbulent flow for the desired parameter values with the help of a
reference solution. A typical PES calculation is composed of three consecutive
steps: Set up the asymptotic relationship between the desired solution and the
reference solution; Calculate the reference solution and the necessary
asymptotic coefficients; Extend the reference solution to the desired parameter
values. A controlled eddy simulation (CES) method has been developed to
calculate the reference solution and the asymptotic coefficients. The CES
method is a special type of large eddy simulation (LES) method in which a
weight coefficient and an artificial force distribution are used to model part
of the turbulent motions. The artificial force distribution is modeled based on
the eddy viscosity assumption. The reference weight coefficient and the
asymptotic coefficients can be determined through a weight coefficient
convergence study. The proposed PES/CES method has been used to simulate four
types of turbulent flows. They are decaying homogeneous and isotropic
turbulence, smooth wall channel flows, rough wall channel flows, and compressor
blade cascade flows. The numerical results show that the 0-order PES solution
(or the reference CES solution) has a similar accuracy as a traditional LES
solution, while its computational cost is much lower. A higher order PES method
has an even higher model accuracy.
