Model equation for simulating flows in multistage turbomachinery

A steady, three-dimensional average-passage equation system is derived for use in simulating multistage turbomachinery flows. These equations describe a steady, viscous flow that is periodic from blade passage to blade passage. From this system of equations, various reduced forms can be derived for use in simulating the three-dimensional flow field within multistage machinery. It is suggested that a properly scaled form of the average-passage equation system would provide an improved mathematical model for simulating the flow in multistage machines at design and, in particular, at off-design conditions

An electrostatic analog for generating cascade grids

Accurate and efficient numerical simulation of flows through turbomachinery blade rows depends on the topology of the computational grids. These grids must reflect the periodic nature of turbomachinery blade row geometries and conform to the blade shapes. Three types of grids can be generated that meet these minimal requirements: through-flow grids, O-type grids, and C-type grids. A procedure which can be used to generate all three types of grids is presented. The resulting grids are orthogonal and can be stretched to capture the essential physics of the flow. A discussion is also presented detailing the extension of the generation procedure to three dimensional geometries

Semidirect computations for transonic flow

A semidirect method, driven by a Poisson solver, was developed for inviscid transonic flow computations. It is an extension of a recently introduced algorithm for solving subsonic rotational flows. Shocks are captured by implementing a form of artificial compressibility. Nonisentropic cases are computed using a shock tracking procedure coupled with the Rankine-Hugoniot relationships. Results are presented for both subsonic and transonic flows. For the test geometry, an unstaggered cascade of 20 percent thick circular arc airfoils at zero angle of attack, shocks are crisply resolved in supercritical situations and the algorithm converges rapidly. In addition, the convergence rate appears to be nearly independent of the entropy and vorticity production at the shock

Simulation of turbomachinery flows

With the interest in jet propulsion at the end of World War II, aerodynamicists were challenged to develop mathematical models which could be used to design turbomachinery components for jets. NASA Lewis engineers and scientists played a major role in meeting this challenge. Some of their accomplishments are highlighted as well as those of others. Several problems are addressed which must be solved if jet propulsion technology is to advance

Unsteady transonic flow in cascades

There is a need for methods to predict the unsteady air loads associated with flutter of turbomachinery blading at transonic speeds. The results of such an analysis in which the steady relative flow approaching a cascade of thin airfoils is assumed to be transonic, irrotational, and isentropic is presented. The blades in the cascade are allowed to undergo a small amplitude harmonic oscillation which generates a small unsteady flow superimposed on the existing steady flow. The blades are assumed to oscillate with a prescribed motion of constant amplitude and interblade phase angle. The equations of motion are obtained by linearizing about a uniform flow the inviscid nonheat conducting continuity and momentum equations. The resulting equations are solved by employing the Weiner Hopf technique. The solution yields the unsteady aerodynamic forces acting on the cascade at Mach number equal to 1. Making use of an unsteady transonic similarity law, these results are compared with the results obtained from linear unsteady subsonic and supersonic cascade theories. A parametric study is conducted to find the effects of reduced frequency, solidity, stagger angle, and position of pitching axis on the flutter

A semi-direct solver for compressible 3-dimensional rotational flow

An iterative procedure is presented for solving steady inviscid 3-D subsonic rotational flow problems. The procedure combines concepts from classical secondary flow theory with an extension to 3-D of a novel semi-direct Cauchy-Riemann solver. It is developed for generalized coordinates and can be exercised using standard finite difference procedures. The stability criterion of the iterative procedure is discussed along with its ability to capture the evolution of inviscid secondary flow in a turning channel

Full potential solution of transonic quasi-3-D flow through a cascade using artificial compressability

The three-dimensional flow in a turbomachinery blade row was approximated by correcting for streamtube convergence and radius change in the throughflow direction. The method is a fully conservative solution of the full potential equation incorporating the finite volume technique on body fitted periodic mesh, with an artificial density imposed in the transonic region to insure stability and the capture of shock waves. Comparison of results for several supercritical blades shows good agreement with their hodograph solutions. Other calculations for these profiles as well as standard NACA blade sections indicate that this is a useful scheme analyzing both the design and off-design performance of turbomachinery blading

Supersonic Stall Flutter of High Speed Fans

An analytical model is developed for predicting the onset of supersonic stall bending flutter in axial flow compressors. The analysis is based on a modified two dimensional, compressible, unsteady actuator disk theory. It is applied to a rotor blade row by considering a cascade of airfoils whose geometry and dynamic response coincide with those of a rotor blade element at 85 percent of the span height (measured from the hub). The rotor blades are assumed to be unshrouded (i.e., free standing) and to vibrate in their first flexural mode. The effects of shock waves and flow separation are included in the model through quasi-steady, empirical, rotor total-pressure-loss and deviation-angle correlations. The actuator disk model predicts the unsteady aerodynamic force acting on the cascade blading as a function of the steady flow field entering the cascade and the geometry and dynamic response of the cascade. Calculations show that the present model predicts the existence of a bending flutter mode at supersonic inlet Mach numbers. This flutter mode is suppressed by increasing the reduced frequency of the system or by reducing the steady state aerodynamic loading on the cascade. The validity of the model for predicting flutter is demonstrated by correlating the measured flutter boundary of a high speed fan stage with its predicted boundary. This correlation uses a level of damping for the blade row (i.e., the log decrement of the rotor system) that is estimated from the experimental flutter data. The predicted flutter boundary is shown to be in good agreement with the measured boundary

Supersonic unstalled flutter

Flutter analyses were developed to predict the onset of supersonic unstalled flutter of a cascade of two-dimensional airfoils. The first of these analyzes the onset of supersonic flutter at low levels of aerodynamic loading (i.e., backpressure), while the second examines the occurrence of supersonic flutter at moderate levels of aerodynamic loading. Both of these analyses are based on the linearized unsteady inviscid equations of gas dynamics to model the flow field surrounding the cascade. These analyses are utilized in a parametric study to show the effects of cascade geometry, inlet Mach number, and backpressure on the onset of single and multi degree of freedom unstalled supersonic flutter. Several of the results are correlated against experimental qualitative observation to validate the models

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 average 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 counter rotating propeller and a high speed fan stage. Results from these simulations are presented