Average-passage Equation System Research Articles

Investigation of Unsteady Rotor–Stator Interaction and Deterministic Correlation Analysis in a Transonic Compressor Stage

Abstract Unsteady blade row interaction (UBRI) has a large impact on performance in high-speed axial compressors. Thus, the influence of UBRI should be accurately predicted in compressor routine design. In this study, the unsteady rotor–stator interaction in a transonic single-stage compressor, NASA Stage 35, is studied by steady and unsteady Reynolds-Averaged Navier–Stokes methods. First, the UBRI is analyzed. Periodic separation bubbles on the stator suction surface induced by the rotor wake are observed. Comparisons between steady and unsteady results show that the radial distribution of flow angle near the hub and tip region of the stator is strongly influenced by UBRI. Second, deterministic correlations are analyzed based on the average-passage equation system. The distribution of deterministic stress and the relationship between spatial correlations and deterministic correlations are analyzed. Results show that a strong nonlinear interaction caused by the rotor wake and the stator potential field is found in the rotor–stator gap, which is responsible for the generation of spatial–time correlation. Then, the anisotropy of deterministic correlations is analyzed using the Lumley triangle. Results show that the deterministic stress is highly anisotropic in the stator passage, which can be considered in the modeling using a similar strategy of turbulence modeling. At last, the relationships between the harmonics and decay rate of deterministic correlations are analyzed. The decay rate of deterministic correlation is highly related to the corresponding harmonics, and the higher-order harmonic exhibits a higher decay rate. A modified exponential decay model is proposed for deterministic correlation based on harmonics, which provides improved performance.

An Exponential Decay Model for the Deterministic Correlations in Axial Compressors

The unsteady blade row interaction (UBRI) is inherent and usually has a large effect on performance in multistage axial compressors. The effect could be considered by using the average-passage equation system (APES) in steady-state environment by introducing the deterministic correlations (DC). How to model the DC is the key in APES method. The primary purpose of this study is to develop a DC model for compressor routine design. The APES technique is investigated by using a 3D viscous unsteady and time-averaging Computational fluid dynamics (CFD) flow solver developed in our previous studies. Based on DC characteristics and its effects on time-averaged flow, an exponential decay DC model is proposed and implemented into the developed time-averaging solver. Steady, unsteady, and time-averaging simulations are conducted on the investigation of the UBRI and the DC model in the first transonic stage of NASA 67 and the first two stages of a multistage compressor. The DC distributions and mean flow fields from the DC model are compared with the unsteady simulations. The comparison indicates that the proposed model can take into account the major part of UBRI and provide significant improvements for predicting compressor characteristics and spanwise distributions of flow properties in axial compressors, compared with the steady mixing plane method.

Investigation of model development for deterministic correlations associated with impeller-diffuser interactions in centrifugal compressors

The average-passage equation system (APES) provides a rigorous framework to account for the deterministic unsteady effects by the so-called deterministic correlations (DC), which include both deterministic stress correlations (DCS) and deterministic total enthalpy correlations (DCH). These correlations should be modeled to close the system of equations. In this paper, the distribution of DC in a transonic centrifugal compressor is presented, and its relative importance is revealed. The assumption made by Adamczyk that the pure unsteady fluctuation is significantly smaller than the spatial fluctuation is verified at the impeller-diffuser interface. The decomposition of DCH is also discussed to determine its two different physical mechanisms. Finally, the transport equations in terms of DCS in cylindrical coordinates are derived, and the terms are evaluated to determine the ones that are necessary to model. All these analyses significantly contribute to our model development for DC in centrifugal compressors.

Numerical investigations of impeller–diffuser interactions in a transonic centrifugal compressor stage using nonlinear harmonic method

Numerical investigations have been conducted on a transonic centrifugal compressor stage with a vaned diffuser. Due to the high loading and small radial gap between the impeller and diffuser, the unsteady impeller–diffuser interactions can be very large compared with axial compressors, and this has a significant impact on the time-averaged performance of a centrifugal compressor stage. Steady, unsteady and NonLinear Harmonic (NLH) simulations have been carried out to study the interactions in a transonic centrifugal compressor. Comparisons have been made against the overall performance and flow details. The distribution of deterministic correlations are extracted and compared to show the effect of unsteadiness due to impeller–diffuser interactions. All of these analyses show that the NLH method gains more accuracy compared with the steady-state mixing plane simulation and saves CPU time compared with an unsteady time-accurate computation. NLH and unsteady methods are in good agreement in overall performance predictions compared with the steady-state mixing plane method while small differences still exist in flow details. In general, the NLH method obtains good accuracy in terms of both deterministic stress and enthalpy correlations which are sufficiently accurate to be directly used in model development in the average-passage equation system instead of the more time-consuming unsteady simulation.

Study of modeling unsteady blade row interaction in a transonic compressor stage part 1: code development and deterministic correlation analysis

The average-passage equation system (APES) provides a rigorous mathematical framework for accounting for the unsteady blade row interaction through multistage compressors in steady state environment by introducing deterministic correlations (DC) that need to be modeled to close the equation system. The primary purpose of this study is to provide insight into the DC characteristics and the influence of DC on the time-averaged flow field of the APES. In Part 1 of this two-part paper, firstly a 3D viscous unsteady and time-averaging flow CFD solver is developed to investigate the APES technique. Then steady and unsteady simulations are conducted in a transonic compressor stage. The results from both simulations are compared to highlight the significance of the unsteady interactions. Furthermore, the distribution characteristics of DC are studied and the DC at the rotor/stator interface are compared with their spatial correlations (SC). Lastly, steady and time-averaging (employing APES with DC) simulations for the downstream stator alone are conducted employing DC derived from the unsteady results. The results from steady and time-averaging simulations are compared with the time-averaged unsteady results. The comparisons demonstrate that the simulation employing APES with DC can reproduce the time-averaged field and the 3D viscous time-averaging flow solver is validated.

Study of modeling unsteady blade row interaction in a transonic compressor stage part 2: influence of deterministic correlations on time-averaged flow prediction

The average-passage equation system (APES) provides a rigorous mathematical framework for accounting for the unsteady blade row interaction through multistage compressors in steady state environment by introducing deterministic correlations (DC) that need to be modeled to close the equation system. The primary purpose of this study was to provide insight into the DC characteristics and the influence of DC on the time-averaged flow field of the APES. In Part 2 of this two-part paper, the influence of DC on the time-averaged flow field was systematically studied. Several time-averaging computations were conducted with various boundary conditions and DC for the downstream stator in a transonic compressor stage, by employing the CFD solver developed in Part 1 of this two-part paper. These results were compared with the time-averaged unsteady flow field and the steady one. The study indicated that the circumferential-averaged DC can take into account major part of the unsteady effects on spanwise redistribution of flow fields in compressors. Furthermore, it demonstrated that both deterministic stresses and deterministic enthalpy fluxes are necessary to reproduce the time-averaged flow field.

Three-dimensional flow field downstream of an embedded stator in a multistage axial flow compressor: Part 2: Deterministic stress and heat flux distribution and average-passage equation system

The results from the area traverse measurements of the unsteady velocity and total temperature downstream of the second stator of a three-stage axial flow compressor have been correlated to derive various deterministic stress and heat flux terms. These terms are consistent with those arising from the average-passage equation system of Adamczyk. The deterministic periodic stress and heat flux terms were found to be larger than the aperiodic terms for both the normal and the shear components. Consequently the terms involving the aperiodic components in the average-passage equations could be neglected for stator exit and rotor inlet flow modelling. The deterministic periodic normal and shear stresses were seen to be most significant in the stator wakes away from the end-wall regions. The most significant shear stress correlation was between the axial and the tangential velocity components. Since the correlations involving the radial component were small, it is postulated that the dominant mechanism for mixing (in the radial direction) is due to the steady deterministic radial velocity. All three components of deterministic heat flux were found to be significant in this flow field, especially in the wakes. The dominant terms in the average-passage equation system away from the end walls were due to the tangential gradient compared with the radial gradient terms and both the terms were found to be of equal importance in the hub and casing end-wall regions.

Impact of deterministic correlations on the steady flow field

A time-dependent numerical simulation of a single-stage transonic turbine is used as a direct simulation of stator-rotor interactions. The different parts of the deterministic correlations are computed and compared. The present work aims to provide insight into the influence of these correlations on the steady flow field of the average-passage equation system. This is related to the deterministic models of the literature. Several steady computations are performed with the various correlations. These results are compared with the time-averaged flow field. The impact of boundary conditions is also studied to highlight the importance of the non-uniform circumferential distribution at the so-called ‘mixing plane’. This work establishes that a non-uniform circumferential distribution is a key to ensure the success of steady computation. Furthermore, it demonstrates that full deterministic correlations are necessary to reproduce the time-averaged flow field. Spatial correlations are thus insufficient to model the stator wakes that interact with the rotor. Steady computations using only spatial correlations give less accurate results than those obtained without any correlation. This is a serious flaw of most of the literature's models.

The Numerical Simulation of a High-Speed Axial Flow Compressor

The advancement of high-speed axial flow multistage compressors is impeded by a lack of detailed flow field information. Recent developments in compressor flow modeling and numerical simulation have the potential to provide needed information in a timely manner. This paper, which consists of two parts, will explore this topic. The first part will address the development of a computer program to solve the viscous form of the average-passage equation system for multistage turbomachinery. Programming issues such as in-core versus out-of-core data storage and CPU utilization (parallelization, vectorization, and chaining) will be addressed. Code performance will be evaluated through the simulation of the first four stages of a five-stage, high-speed, axial flow compressor on a CRAY Y-MP8/8128 computer. The second part will address the flow physics, which can be obtained from the numerical simulation. In particular, an examination of the end-wall flow structure will be made, and its impact on blockage distribution assessed.

Simulation of Three-Dimensional Viscous Flow Within a Multistage Turbine

This work outlines a procedure for simulating the flow field within multistage turbomachinery, which includes the effects of unsteadiness, compressibility, and viscosity. The associated modeling equations are the average passage equation system, which governs the time-averaged flow field within a typical passage of a blade row embedded within a multistage configuration. The results from a simulation of a low aspect ratio stage and one-half turbine will be presented and compared with experimental measurements. It will be shown that the secondary flow field generated by the rotor causes the aerodynamic performance of the downstream vane to be significantly different from that of an isolated blade row.

Errata: Kinetic Energy Equations for the Average-Passage Equation System

Errata: Kinetic Energy Equations for the Average-Passage Equation System

Kinetic energy equations for the average-passage equation system

Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.

A Model for Closing the Inviscid Form of the Average-Passage Equation System

A mathematical model is proposed for closing or mathematically completing the system of equations which describes the time-averaged flow field through the blade passages of multistage turbomachinery. These equations, referred to as the average-passage equation system, govern a conceptual model which has proven useful in turbomachinery aerodynamic design and analysis. The closure model is developed so as to insure a consistency between these equations and the axisymmetric through-flow equations. The closure model was incorporated into a computer code for use in simulating the flow field about a high-speed counterrotating propeller and a high-speed fan stage. Results from these simulations are presented.