Viscous Analysis of Three-Dimensional Turbomachinery Flows on Body Conforming Grids Using an Implicit Solver

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)

A 3-D Euler analysis for turbomachinery flows on a C-type grid is presented. The analysis is based on the Beam and Warming implicit algorithm for solution of the unsteady Euler equations and is derived from the ARC3D code developed by Pulliam at NASA Ames Research Center. Modifications made to convert this code from external flow applications to internal turbomachinery flows are given in detail. These changes include the addition of inflow, outflow, and periodic boundary point calculation procedures. Also presented are the C-grid construction procedures. Finally, results of code experimental verification studies for 3-D compressor cascade and rotor flows are presented.

A new code aimed at the solution of three-dimensional internal laminar and turbulent compressible flows is presented. The solution procedure is based on the scalar form of the approximate factorization method. The Navier-Stokes equations in mass-averaged form are iterated in time until a steady solution is reached. Evidence is given to the implicit treatment of diffusive terms that proved able to speed up convergence and enhance the algorithm robustness. Turbulence effects are accounted for by the Baldwin-Lomax turbulence model and the q - ω two-equation model. For the first, an investigation on the model behavior in case of multiple boundaries is performed. The numerical code is validated by computing the flow in the Stanitz 90-deg turning elbow. Finally the code is applied to the computation of the flow in the VKI LS-82 three-dimensional linear cascade.

A two-equation k-omega turbulence model has been developed and applied to a quasi-three-dimensional viscous analysis code for blade-to-blade flows in turbomachinery. the code includes the effects of rotation, radius change, and variable stream sheet thickness. The flow equations are given and the explicit runge-Kutta solution scheme is described. the k-omega model equations are also given and the upwind implicit approximate-factorization solution scheme is described. Three cases were calculated: transitional flow over a flat plate, a transonic compressor rotor, and transonic turbine vane with heat transfer. Results were compared to theory, experimental data, and to results using the Baldwin-Lomax turbulence model. The two models compared reasonably well with the data and surprisingly well with each other. Although the k-omega model behaves well numerically and simulates effects of transition, freestream turbulence, and wall roughness, it was not decisively better than the Baldwin-Lomax model for the cases considered here.

Results from numerical simulations of air flow and heat transfer in a 'branched duct' geometry are presented. The geometry contains features, including pins and a partition, as are found in coolant passages of turbine blades. The simulations were performed using a multi-block structured grid system and a finite volume discretization of the governing equations (the compressible Navier-Stokes equations). The effects of turbulence on the mean flow and heat transfer were modeled using the Baldwin-Lomax turbulence model. The computed results are compared to experimental data. It was found that the extent of some regions of high heat transfer was somewhat under predicted. It is conjectured that the underlying reason is the local nature of the turbulence model which cannot account for upstream influence on the turbulence field. In general, however, the comparison with the experimental data is favorable.

The locally implicit scheme and a rigid-deformable dynamic mesh algorithm are introduced to solve the unsteady Navier-Stokes equations with Baldwin-Lomax turbulence model in the Cartesian coordinate system. In this scheme, a new treatment of viscous flux function is developed on the quadrilateral-triangular meshes. To validate the present scheme, the viscous flow over an oscillating flat plate is studied. By using the above solution approach, turbulent flows around transonic oscillating cascade of four biconvex blades with different oscillation amplitudes and interblade phase angles are investigated. From the distributions of magnitude and phase angle of the first harmonic dynamic pressure difference coefficient, which include the experimental data and numerical results obtained by linearized theory, Euler and Navier-Stokes solvers, the present solution approach is reliable and acceptable. The instantaneous pressure and/or Mach number contours indicate the effects of oscillation amplitude and interblade phase angle on the unsteady flow phenomena, such as vortex-shedding, pressure waves, shocks and boundary layer interactions.

This paper reports the progress being made towards complete turbo-pump simulation capability for liquid rocket engines. The Space Shuttle Main Engine (SSME) turbo-pump impeller is used as a test case for the performance evaluation of the MPI, hybrid MPI/Open-MP, and MLP versions of the INS3D code. Then, a computational model of a turbo-pump has been developed for the shuttle upgrade program. Relative motion of the grid system for rotor-stator interaction was obtained by employing overset grid techniques. Unsteady computations for SSME turbo-pump, which contains 101 zones with 31 million grid points, are carried on Origin 2000 systems at NASA Ames Research Center. The approach taken for these simulations, and the performance of the parallel versions of the code are presented.

Symbolic vector analysis in plasma physics

This work addresses the modeling of high pressure electric discharge in an arc-heated wind tunnel. The combined numerical solution of Poisson’s equation, radiative transfer equations, and the set of Favre-averaged thermochemical nonequilibrium Navier–Stokes equations allows for the determination of the electric, radiation, and flow fields, accounting for their mutual interaction. Semi-classical statistical thermodynamics is used to determine the plasma thermodynamic properties, while transport properties are obtained from kinetic principles with the Chapman–Enskog method. A multi-temperature formulation is used to account for thermal non-equilibrium. Finally, the turbulence closure of the flow equations is obtained by means of the Spalart–Allmaras model, which requires the solution of an additional scalar transport equation. A Streamline upwind Petrov–Galerkin stabilized finite element formulation is employed to solve the Navier–Stokes equation. The electric field equation is solved using the standard Galerkin formulation. A stable formulation for the radiative transfer equations is obtained using the least-squares finite element method. The developed simulation framework has been applied to investigate turbulent plasma flows in the 20 MW Aerodynamic Heating Facility at NASA Ames Research Center. The current model is able to predict the process of energy addition and re-distribution due to Joule heating and thermal radiation, resulting in a hot central core surrounded by colder flow. The use of an unsteady three-dimensional treatment also allows the asymmetry due to a dynamic electric arc attachment point in the cathode chamber to be captured accurately. The current work paves the way for detailed estimation of operating characteristics for arc-heated wind tunnels which are critical in testing thermal protection systems.

The adaptive nonlinear filtering and limiting in spatially high order schemes (Yee et al. J. Comput. Phys. 150, 199–238, (1999), Sjögreen and Yee, J. Scient. Comput. 20, 211–255, (2004)) for the compressible Euler and Navier–Stokes equations have been recently extended to the ideal and non-ideal magnetohydrodynamics (MHD) equations, (Sjögreen and Yee, (2003), Proceedings of the 16th AIAA/CFD conference, June 23–26, Orlando F1; Yee and Sjögreen (2003), Proceedings of the International Conference on High Performance Scientific Computing, March, 10–14, Honai, Vietnam; Yee and Sjögreen (2003), RIACS Technical Report TR03. 10, July, NASA Ames Research Center; Yee and Sjögreen (2004), Proceedings of the ICCF03, July 12–16, Toronto, Canada). The numerical dissipation control in these adaptive filter schemes consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is very general. The objective of this paper is to investigate the performance of three commonly used types of discontinuity capturing nonlinear numerical dissipation for both the ideal and non-ideal MHD.

In the numerical simulation of high Reynolds-number flow, one can frequently supply only enough grid points to resolve the viscous terms in a thin layer. As a consequence, a body-or stream-aligned coordinate system is frequently used and viscous terms in this direction are discarded. It is argued that these terms cannot be resolved and computational efficiency is gained by their neglect. Dropping the streamwise viscous terms in this manner has been termed the thin-layer approximation. The thin-layer concept is an old one, and similar viscous terms are dropped, for example, in parabolized Navier-Stokes schemes. However, such schemes also make additional assumptions so that the equations can be marched in space, and such a restriction is not usually imposed on a thin-layer model. The thin-layer approximation can be justified in much the same way as the boundary-layer approximation; it requires, therefore, a body-or stream-aligned coordinate and a high Reynolds number. Unlike the boundary-layer approximation, the same equations are used throughout, so there is no matching problem. Furthermore, the normal momentum equation is not simplified and the convection terms are not one-sided differenced for marching. Consequently, the thin-layer equations are numerically well behaved at separation and require no special treatment there. Nevertheless, the thin-layer approximation receives criticism. It has been suggested that the approximation is invalid at separation and, more recently, that it is inadequate for unsteady transonic flow. Although previous comparisons between the thin-layer and Navier-Stokes equations have been made, these comparisons have not been adequately documented.

A zonal method based on the iterative coupling between an in viscid panel method and a Navier-Stokes (NS) method has been developed to analyze the complex flowfield associated with the V/STOL flow environment in order to improve the prediction of aerodynamic characteristics. The low-order panel method, VSAERO, developed by Analytical Methods, Inc. is used to solve for the entire flow region, which includes the V/STOL configuration and jet boundary model. In two separate investigations presented here, ARC3D, a well-known compressible thin layer NS code developed at NASA Ames Research Center, and APPL, a parabolized NS code developed at Amtec Engineering, are used to solve for the details of the jet region subject to the boundary condition provided by VSAERO. Overlapping boundaries are used for the purpose of coupling. Comparisons with the available experimental data for a jet-in-crossflow and a V/STOL configuration with lift-jet in a subsonic freestream show qualitative agreement with experiment and improved correlations over those by inviscid computations alone.

Chapter 13 - Continuity and Navier-Stokes equation in different coordinate systems

This paper presents a numerical simulation method developed for separated flow in cascades using the Euler equations and demonstrates the feasibility of this method. MacCormack’s two-steps explicit finite difference scheme is used to discretize the equations in conservation form, and the artificial viscosity is added to the discretized inviscid equations by means of the self-adapted filter technique. The initial separation boundary is given according to simple experimental results. The numerical simulation results including subsonic and transonic turbine cascades flow with or without separation show that the fundamental idea of this numerical method is reasonable and simple. The present study indicates that for solving certain engineering problems it is a simple and effective tool for adding some viscosity corrections to inviscid flow model, especially the current when the Navier-Stokes equations have not been solved very effectively for various complicated flows in turbomachinery.

An unsteady, compressible, two-dimensional, parallelized Navier Stokes solver is developed to predict the flow around cascaded vibrating blades. The presented method applies the Advection Upstream Splitting Method to discretize the convective terms and central differences for the diffusive terms of the Navier-Stokes equations on a structured H-type mesh in boundary-fitted coordinates. Turbulence is modeled by using the Baldwin-Lomax turbulence model. The time accurate integration of the governing equations is performed by applying an explicit four-stage Runge-Kutta scheme. Some of the calculations are conducted on a IBM/SP2 parallel computer using the message passing system MPI. At the farfield boundary, viscous effects are assumed to be negligible and for steady state and the unsteady flow capacitive boundary conditions based on the theory of characteristics for the locally one-dimensional problem are used. Unsteady inviscid calculations are performed with the same code using non-reflecting boundary condtitions.