A nonlinear harmonic methodology has been developed to calculate unsteady viscous flows through turbomachinery blades. Flow variables are decomposed into time-averaged variables and unsteady perturbations, resulting in the time-averaged equations with extra nonlinear stress terms depending on the unsteady perturbations. An efficient evaluation of an unsteady flowfield is obtained by solving the first-order harmonic equations. The nonlinear interactions between the time-averaged flowfield and the unsteady perturbations were included by a strong coupling approach. The basic computational methodology was applied to the two-dimensional Navier-Stokes equations, and the method was validated against several test cases. Computational results show that this method is much more efficient than the nonlinear time-marching methods while still modeling dominant nonlinear effects

Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines

Fundamentals of fluid motion in erosion by solid particle impact

This paper presents a numerical analysis of wake/rotor interaction using a time-marching solution of the unsteady, nonlinear Euler equations. The incoming wakes are specified through the unsteady inflow boundary conditions. The lagged periodic boundary condition that arises due to unequal rotor/stator pitches is handled by a new technique that inclines the computational plane in time. Comparison of results for a flat-plate cascade with results using a linear analytic theory demonstrates the method's capability to accurately predict unsteady forces, moments, and radiated sound. Results are also presented for a low-speed turbine.

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Calculation of Unsteady Wake/Rotor Interaction

The primary focus of this paper is convective heat transfer in axial flow turbines. Research activity involving heat transfer generally separates into two related areas: predictions and measurements. The problems associated with predicting heat transfer are coupled with turbine aerodynamics because proper prediction of vane and blade surface-pressure distribution is essential for predicting the corresponding heat transfer distribution. The experimental community has advanced to the point where time-averaged and time-resolved three-dimensional heat transfer data for the vanes and blades are obtained routinely by those operating full-stage rotating turbines. However, there are relatively few CFD codes capable of generating three-dimensional predictions of the heat transfer distribution, and where these codes have been applied the results suggest that additional work is required. This paper outlines the progression of work done by the heat transfer community over the last several decades as both the measurements and the predictions have improved to current levels. To frame the problem properly, the paper reviews the influence of turbine aerodynamics on heat transfer predictions. This includes a discussion of time-resolved surface-pressure measurements with predictions and the data involved in forcing function measurements. The ability of existing two-dimensional and three-dimensional Navier-Stokes codes to predict the proper trends of the time-averaged and unsteady pressure field for full-stage rotating turbines is demonstrated. Most of the codes do a reasonably good job of predicting the surface-pressure data at vane and blade midspan, but not as well near the hub or the tip region for the blade. In addition, the ability of the codes to predict surface-pressure distribution is significantly better than the corresponding heat transfer distributions. Heat transfer codes are validated against measurements of one type or another. Sometimes the measurements are performed using full rotating rigs, and other times a much simpler geometry is used. In either case, it is important to review the measurement techniques currently used. Heat transfer predictions for engine turbines are very difficult because the boundary conditions are not well known. The conditions at the exit of the combustor are generally not well known and a section of this paper discusses that problem. The majority of the discussion is devoted to external heat transfer with and without cooling, turbulence effects, and internal cooling. As the design community increases the thrust-to-weight ratio and the turbine inlet temperature, there remain many turbine-related heat transfer issues. Included are film cooling modeling, definition of combustor exit conditions, understanding of blade tip distress, definition of hot streak migration, component fatigue, loss mechanisms in the low turbine, and many others. Several suggestions are given herein for research and development areas for which there is potentially high payoff to the industry with relatively small risk.

Convective Heat Transfer and Aerodynamics in Axial Flow Turbines

This paper presents calculations of a stator/rotor interaction in a highly loaded transonic first turbine stage. Of particular interest is the propagation and reflection of shocks which originate at the trailing edge of the upstream stator. These produce a 40% variation in the lift on the rotor, which would cause structural vibrations and increased losses. Also, the unsteady shocks would cause temporary boundary-layer separation near the leading edge of the suction surface. The numerical procedure solves the inviscid unsteady Euler equations, including quasi-three-dimensional terms. The use of a conservative treatment guarantees the correct treatment of the moving shocks. A simple technique is used to couple the calculations on the stator and rotor grids. A key feature of the paper is the use of "time-incline d" computational planes to allow the analysis of cases in which the ratio of stator and rotor pitches is not equal to unity or the ratio of two small integers.

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Stator/rotor interaction in a transonic turbine

A numerical method of solution of the inviscid, compressible, two-dimensional, unsteady flow on a blade-toblade stream surface through a stage (rotor and stator), or a single blade row, of an axial flow compressor or fan is described. A cyclic procedure has been developed for representation of adjacent blade-to-blade passages, which asymptotically achieves the correct phase between all passages of a stage. A shock-capturin g finitedifference method is employed in the interior of the passage, and a method-of-characteristiics technique is used at the boundaries. The blade slipstreams form two of the passage boundaries, and are treated as moving contact surfaces capable of supporting jumps in entropy and tangential velocity. The Kutta condition is imposed by requiring the slipstreams to originate at the trailing edges which are assumed to be sharp. Results are presented for a transonic fan stage. The rotor tip section solution is compared with an experimental pressure contour map. The subcritical stator solution is compared with results from a stream function method. Finally, the periodic solution for the stage, which has 44 rotor blades and 46 stator blades, is discussed.

Numerical Solution of Periodic Transonic Flow through a Fan Stage

Current computational methods for analyzing flows in turbomachinery and other related internal propulsion components are presented The methods are divided into two classes The inviscid methods deal specifically with turbomachinery applications Viscous methods, deal with generalized duct flows as well as flows in turbomachinery passages Inviscid methods are categorized into the potential, stream function, and Euler approaches Viscous methods are treated in terms of parabolic, partially parabolic, and elliptic procedures Various grids used in association with these procedures are also discussed

Review—Computational Methods for Internal Flows With Emphasis on Turbomachinery

This paper describes Fortran programs that give the solution to the two-dimensional, subsonic, nonviscous flow problem on a blade-to-blade surface of revolution of a turbomachine. Flow may be axial, radial, or mixed. There may be a change in stream channel thickness in the through-flow direction. Either single, tandem, or slotted blades may be handled as well as blade rows with splitter vanes. Also, small regions may be magnified to give more detail where desired, such as around a leading or trailing edge or through a slot. The method is based on a finite difference solution of the stream function equations. Numerical examples are shown to illustrate the type of blades which can be analyzed, and to show results which can be obtained. Results are compared with experimental data.Copyright © 1969 by ASME

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Programs for Computation of Velocities and Streamlines on a Blade-to-Blade Surface of a Turbomachine

A numerical method of solution of the inviscid, compressible, two-dimensional unsteady flow on a blade-to-blade stream surface through a stage (rotor and stator), or a single blade row, of an axial flow compressor or fan is described. A cyclic procedure has been developed for representation of adjacent blade-to-blade passages, which asymptotically achieves the correct phase between all passages of a stage. A shock-capturing finite-difference method is employed in the interior of the passage, and a method-of-characteristics technique is used at the boundaries. The blade slipstreams form two of the passage boundaries, and are treated as moving contact surfaces capable of supporting jumps in entropy and tangential velocity. The Kutta condition is imposed by requiring the slipstreams to originate at the trailing edges, which are assumed to be sharp. Results are presented for several transonic fan rotors, and compared with available experimental data consisting of holographic observations of shock structure and pressure contour maps. A subcritical stator solution is also compared with results from a relaxation method. Finally, a periodic solution for a stage consisting of 44 rotor blades and 46 stator blades is discussed.

Numerical solution of periodic transonic flow through a fan stage

Three-dimensional unsteady transonic flow through an axial turbomachine stage is described in terms of a pair of two-dimensional formulations pertaining to orthogonal surfaces, namely, a blade-to-blade surface and a hub-to-casing surface. The resulting systems of nonlinear, inviscid, compressible equations of motion are solved by an explicit finite-difference technique. The blade-to-blade program includes the periodic interaction between rotor and stator blade rows. Treatment of the boundary conditions and of the blade slipstream motion by a characteristic type procedure is discussed in detail. Harmonic analysis of the acoustic far field produced by the blade row interaction, including an arbitrary initial transient, is outlined. Results from the blade-to-blade program are compared with experimental measurements of the rotating pressure field at the tip of a high-speed fan. The hub-to-casing program determines circumferentially averaged flow properties on a meridional plane. Blade row interactions are neglected in this formulation, but the force distributions over the entire blade surface for both the rotor and stator are obtained. Results from the hub-to-casing program are compared with a relaxation method solution for a subsonic rotor. Results are also presented for a quiet fan stage which includes transonic flow in both the rotor and stator and a normal shock in the stator.

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William D. McNally

Papers

Numerical Solution of Periodic Transonic Flow through a Fan Stage

Review—Computational Methods for Internal Flows With Emphasis on Turbomachinery

Programs for Computation of Velocities and Streamlines on a Blade-to-Blade Surface of a Turbomachine

Numerical solution of periodic transonic flow through a fan stage

Time-Dependent Transonic Flow Solutions for Axial Turbomachinery