This paper presents an efficient method of approximating unsteady flows using a blockwise discrete spatial Fourier series for the modeling of three-dimensional non-axisymmetric flows without making any hypothesis about its temporal periodicity. The method aims at capturing the long wavelength flow patterns which are present in many unsteady problems of industrial interest, such as compressor stability, with a drastic reduction in computational resources. The method is intended to be used to compute flows exhibiting large-scale instabilities and where the fundamental frequency of the problem is not known beforehand. The approach discretizes the domain using a finite number of blocks or passages, where the flow variables at the supposedly periodic boundaries are continuously updated using the spatial Fourier coefficients of a uniformly spaced set of reduced-passage domains. The NASA rotor 67 under stall conditions has been used as verification validation case to demonstrate the effectiveness and viability of the proposed modeling strategy. The comparison between the solutions obtained with the discrete Fourier series and the full-annulus solution shows that accurate solutions can be obtained with a low number of harmonics. The new method has been applied to investigate the rotating stall inception of the NASA rotor 67 for clean and distorted inlet flow near stall operating conditions. The method is shown to accurately reproduce the full-annulus solution with a few spatial harmonics, capturing the characteristic features of the complex flow induced by the tip leakage vortex breakdown. The computational cost in this application has been reduced by a factor of between three and seven, although this number heavily depends on the ratio between the number of retained harmonics and the number of blades.

This paper presents an efficient method of approximating unsteady flows using a block-wise discrete spatial Fourier series for the modeling of three-dimensional non-axisymmetric flows without making any hypothesis about its temporal periodicity. The method aims at capturing the long wavelength flow patterns which are present in many unsteady problems of industrial interest. The method is intended to be used to compute flows exhibiting large-scale instabilities and where the fundamental frequency of the problem is not known beforehand. The approach discretizes the domain using a finite number of blocks or passages, where the flow variables at the supposedly periodic boundaries are continuously updated using the spatial Fourier coefficients of a uniformly spaced set of reduced-passage domains. The NASA rotor 67 under the effect of distorted inflow conditions has been used as verification case to demonstrate the effectiveness and viability of the method. The comparison between the Passage-Spectral Method and the full-annulus solution shows that accurate solutions can be obtained with a low number of harmonics. The new method has also been applied to investigate the rotating stall inception of the NASA rotor 67 for distorted inlet flows near stall operating conditions. The method is shown to accurately reproduce the full-annulus solution with a few spatial harmonics. The computational cost in this application has been reduced by a factor of between three and seven, although this number heavily depends on the ratio between the number of retained harmonics to the number of blades.

The unsteady laminar flow between two large rotating disks when one of them is impulsively started is described using the von Kármán similarity analysis to reduce the solution of the Navier–Stokes equations to a set of ordinary differential equations. It is found that the transient phenomenon consists of three distinct phases. Firstly, the development of the Ekman boundary layer, where a quasi-steady Stewartson-type of flow is created. Secondly, angular momentum is initially diffused in the axial direction until the inward radial convection of angular momentum becomes dominating. Thirdly, a Batchelor flow is created once the Bödewadt boundary layer is developed and the entrainment of disk and stator boundary layers are balanced. The dependences of the characteristic dimensionless times on the Reynolds number based on the cavity gap for the second and third stages have been identified numerically and analytically. It is found that the diffusive part of the transient is bypassed if the flow, initially at rest, is perturbed with a small circumferential velocity. The flow and heat transfer transient during a ramp of finite duration has been computed numerically. The integral angular momentum and energy balances of the cavity have been performed in order to understand the energy and momentum budget of the cavity. It is concluded that the spin-up and the spin-down process of a rotor–stator cavity using a quasi-stationary approximation of the fluid, where the time derivatives are neglected, is questionable for realistic gas turbine applications. Finally, the self-similar solution is compared against two-dimensional axisymmetric, steady and unsteady, laminar simulations to assess the limitations and validity of the self-similar analysis. It has been identified that in a closed squared cavity the outer shroud modifies the physics of the transient, but the general conclusions drawn from the one-dimensional model are still valid.

The dependence of the aerodynamic stability of fan blades with the nodal diameter and amplitude of the inlet perturbations is studied. The analysis is conducted using a block-wise spatial Fourier decomposition of reduced-passages to reconstruct the full-annulus solution. The explicit spatial Fourier approximation is exploited to characterize the relevance of each nodal diameter of the inlet perturbation in the fan stall process and study the nonlinear stability in a harmonic by harmonic basis. This approximation allows studying the contribution to stall of each circumferential mode separately. The methodology has been assessed for the NASA rotor 67. The maximum amplitude of total pressure distortion at which the compressor becomes unstable and triggers a stall process has been mapped. It has been proven that despite the complexity of a screen-induced total pressure distortion the only relevant parameter for the nonlinear stability of the fan is the most unstable nodal diameter. Full-annulus simulations have been conducted to assess the accuracy of the simplified nonlinear stability limit. It is concluded that performing a nonlinear simulation with the proper single harmonic perturbation is enough to assess fan stability. It is shown that for the NASA rotor 67 running at the nominal speed the most unstable nodal diameter is the first. This study not only shows a reduction in computational time to assess nonlinear fan stability by a factor of seven but also creates an efficient methodology for understanding the nonlinear instability of fans due to inlet distortion profiles.

The dependence of the aerodynamic stability of fan blades with the nodal diameter and amplitude of the inlet perturbations is studied. The analysis is conducted using a block-wise spatial Fourier decomposition of reduced-passages to reconstruct the full-annulus solution. The method represents very efficiently the unsteady flow generated by inlet non-uniformities. The explicit spatial Fourier approximation is exploited to characterize the relevance of each nodal diameter of the inlet perturbation in the fan stall process and study the nonlinear stability in a harmonic by harmonic basis. This approximation allows studying the contribution to stall of each circumferential mode separately. The methodology has been assessed for the NASA rotor 67. The maximum amplitude of total pressure distortion at which the compressor becomes unstable and triggers a stall process has been mapped. It has been proven that despite the complexity of a screen-induced total pressure distortion the only relevant parameter for the nonlinear stability of the fan is the most unstable nodal diameter. The equivalence in terms of stability between realistic distortion screens and single harmonic distortions has been assessed. Full-annulus simulations have been conducted to assess the accuracy of the simplified nonlinear stability limit. It is concluded that performing a nonlinear simulation with the proper single harmonic perturbation is enough to assess fan stability. The total pressure error with respect the full annulus simulation including a screen-induced pressure deficit at the intake is below 10%. It is shown that for the NASA rotor 67 running at the nominal speed the most unstable nodal diameter is the first. This study not only shows a reduction in computational time to assess nonlinear fan stability by a factor of seven but also creates an efficient methodology for understanding the nonlinear instability of fans due to inlet distortion profiles.