Assessing Convergence in Predictions of Periodic-Unsteady Flowfields

Abstract

Predictions of time-resolved flowfields are now commonplace within the gas-turbine industry, and the results of such simulations are often used to make design decisions during the development of new products. Hence it is necessary for design engineers to have a robust method to determine the level of convergence in design predictions. Here we report on a method developed to determine the level of convergence in a predicted flowfield that is characterized by periodic unsteadiness. The method relies on fundamental concepts from digital signal processing including the discrete Fourier transform, cross correlation, and Parseval’s theorem. Often in predictions of vane–blade interaction in turbomachines, the period of the unsteady fluctuations is expected. In this method, the development of time-mean quantities, Fourier components (both magnitude and phase), cross correlations, and integrated signal power are tracked at locations of interest from one period to the next as the solution progresses. Each of these separate quantities yields some relative measure of convergence that is subsequently processed to form a fuzzy set. Thus the overall level of convergence in the solution is given by the intersection of these sets. Examples of the application of this technique to several predictions of unsteady flows from two separate solvers are given. These include a prediction of hot-streak migration as well as more typical cases. It is shown that the method yields a robust determination of convergence. Also, the results of the technique can guide further analysis and∕or post-processing of the flowfield. Finally, the method is useful for the detection of inherent unsteadiness in the flowfield, and as such it can be used to prevent design escapes.

Unsteady Aerodynamic Interaction in a Closely Coupled Turbine Consistent With Contrarotation

“…The Power Spectral Density (PSD) was used to determine the percent magnitude of the unsteadiness at a given frequency relative to the sum of the magnitudes over the entire frequency range by invoking Parseval's Theorem. This usage of the PSD was in keeping with that of Clark and Grover [7]. Mathematically, the PSD at a given frequency is the product of the Fourier component at that frequency and its complex conjugate divided by the number of samples.…”

“…Clark and Grover [7] detail the design methodology used to reduce the unsteady magnitude on the blade, as well as stresses the importance of ensuring periodic convergence in order to assess accurately the unsteady flowfield. A fuzzy set convergence criteria was implemented in order to quantify periodic convergence of the unsteady flowfield prior to post-processing the numerical analysis to obtain resonant stresses.…”

Effect of Inlet Temperature Non-Uniformity on High-Pressure Turbine Performance

“…Figure 6.7 shows plots of convergence levels for the mean pressure, crosscorrelation factor (CCF) at zero lag, and fractional power of the power spectrum, as well as the overall convergence level (OCE, i.e., the minimum value of the previous convergence levels). All examined convergence levels reached values much closer to 1 (perfect periodicity) than the convergence threshold (CT) of 0.95, as suggested by Clark and Grover (2007) to be sufficient for overall convergence. Once the reference case had converged, the simulation files were copied and the inlet boundary conditions were set to reference the UDFs which describe the inlet total temperature fields of the VI and MP cases.…”

Effects of Non-Uniform Inlet Temperature Distribution on High-Pressure Turbine Blade Loading

“…In addition, a number of quantitative tests, based on the discussion by Clark and Grover [30], were performed to determine the convergence level of periodicity. Figure 2 shows plots of convergence levels at the stator outlet and the rotor outlet for the mean pressure, cross-correlation factor (CCF) at zero lag, and fractional power of the power spectrum, as well as the overall convergence level (OCE, i.e., the minimum value of the previous convergence levels).…”