Numerical simulation of the Francis-99 hydroturbine with correlation to experimental measurements are presented. Steady operation of the hydroturbine is analyzed at three operating conditions: the best efficiency point (BEP), high load (HL), and part load (PL). It is shown that global quantities such as net head, discharge and efficiency are well predicted. Additionally, time-averaged velocity predictions compare well with PIV measurements obtained in the draft tube immediately downstream of the runner. Differences in vortex rope structure between operating points are discussed. Unsteady operation of the hydroturbine from BEP to HL and from BEP to PL are modeled. It is shown that simulation methods used to model the steady operation produce predictions that correlate well with experiment for transient operation. Time-domain unsteady simulation is used for both steady and unsteady operation. The full-fidelity geometry including all components is meshed using an unstructured polyhedral mesh with body-fitted prism layers. Guide vane rotation for transient operation is imposed using fully-conservative, computationally efficient mesh morphing. The commercial solver STAR-CCM+ is used for all portions of the analysis including meshing, solving and post-processing.

An unsteady simulation of a two-stage, cooled, high pressure turbine cascade is achieved by applying the harmonic balance method, a mixed time domain and frequency domain computational fluid dynamic technique for efficiently solving periodic unsteady flows. A comparison of computed temperature and pressure profile predictions generated using the harmonic balance method and a conventional steady mixing plane analysis is presented. The predicted temperature and pressure profiles are also compared to experimental data at the stage exit plane. The harmonic balance solver is able to efficiently model unsteady flows caused by wake interaction and secondary flow effects due to cooling flows. It is demonstrated that modeling the unsteady effects is critical to the accurate prediction of time-averaged flow field quantities, particularly for cooled machines.

This study seeks to design the aerodynamic features a first stage vane for a 100 MW class. supercritical CO2 Brayton cycle turbomachine. For a turbine inlet temperature of 1350 K, the recuperated configuration is found to provide the highest cycle efficiency, and the corresponding cycle parameters are then used to design the turbine stages. A 6-stage turbine is selected and the "first stage is designed following a one-dimensional mean line approach. Initial mean line turbomachine parameters (work coefficient and flow coefficient) are selected to provide high thermodynamic efficiency and simple radial equilibrium equation principles. Turning loss correlations are utilized to define and optimize hub and casing velocity triangle parameters. Typical turbomachinery characteristic parameters are used to compare the carbon dioxide turbine with typical air combustion turbines. Detailed aerodynamic analysis is perfoimed on a complete three-dimensional model of the vane flow field using a commercial computational fluid dynamics code, STAR-CCM+. Actual properties of the working fluid are input to the model from the REFPROP database provided by the US National Institute of Standards and Technology (NIST). The detailed flow field is computed, from which aerodynamic loss coefficients are calculated. The computer model confirms that the design is successful in turning supercritical carbon dioxide at the prescribed angle and pressure. However, results of the real fluid simulation show that aerodynamic losses caused the stage efficiency to be about 4% below the design target.

A two-step approximate factorization technique for implementing a computationally stable nonlinear unsteady frequency-domain harmonic balance solution method within existing implicit computational fluid dynamic flow solver codes is presented. The approach uses an explicit discretization of the harmonic balance source term, and no new implicit code development is required. Both of these features enable the harmonic balance method to be implemented within existing implicit flow solver codes with minimal modification necessary to the underlying flow solver code. The resulting harmonic balance solver can then be used for modeling nonlinear periodic unsteady flows. The methodology is applied to the NASA OVERFLOW flow solver code, and results are presented for transonic viscous flow past an unsteady pitching airfoil section.

The harmonic balance method is a mixed time domain and frequency domain approach for efficiently solving periodic unsteady flows. The implementation described in this paper is designed to efficiently handle the multiple frequencies that arise within a multistage turbomachine due to differing blade counts in each blade row. We present two alternative algorithms that can be used to determine which unique set of frequencies to consider in each blade row. The first, an all blade row algorithm, retains the complete set offrequencies produced by a given blade row's interaction with all other blade rows. The second, a nearest neighbor algorithm, retains only the dominant frequencies in a given blade row that arise from direct interaction with the adjacent rows. A comparison of results from a multiple blade row simulation based on these two approaches is presented. We will demonstrate that unsteady blade row interactions are accurately captured with the reduced frequency set of the nearest neighbor algorithm, and at a lower computational cost compared to the all blade row algorithm.An unsteady simulation of a two-stage, cooled, high pressure turbine cascade is achieved using the present harmonic balance method and the nearest neighbor algorithm. The unsteady results obtained are compared to steady simulation results to demonstrate the value of performing an unsteady analysis. Considering an unsteady flow through a single blade row turbine blade passage, it is further shown that unsteady effects are important even if the objective is to obtain accurate time-averaged integrated values, such as efficiency.

The harmonic balance method implemented within STAR-CCM+ is a mixed frequency/time domain computational fluid dynamic technique, which enables the efficient calculation of time-periodic flows. The unsteady solution is stored at a small number of fixed time levels over one temporal period of the unsteady flow in a single blade passage in each blade row; thus the solution is periodic by construction. The individual time levels are coupled to one another through a spectral operator representing the time derivative term in the Navier-Stokes equation, and at the boundaries of the computational domain through the application of periodic and nonreflecting boundary conditions. The blade rows are connected to one another via a small number of fluid dynamic spinning modes characterized by nodal diameter and frequency. This periodic solution is driven to the correct solution using conventional (steady) CFD acceleration techniques, and thus is computationally efficient. Upon convergence, the time level solutions are Fourier transformed to obtain spatially varying Fourier coefficients of the flow variables. We find that a small number of time levels (or; equivalently, Fourier coefficients) are adequate to model even strongly nonlinear flows. Consequently, the method provides an unsteady solution at a computational cost significantly lower than traditional unsteady time marching methods.The implementation of this nonlinear harmonic balance method within STAR-CCM+ allows for the simulation of multiple blade rows. This capability is demonstrated and validated using a 1.5 stage cold flow axial turbine developed by the University of Aachen. Results produced using the harmonic balance method are compared to conventional time domain simulations using STAR-CCM+, and are also compared to published experimental data. It is shown that the harmonic balance method is. able to accurately model the unsteady flow structures at a computational cost significantly lower than unsteady time domain simulation.

Presented are flutter-onset trends of the F-16 fighter computed using a harmonic balance version of the NASA OVERFLOW 2 flow solver code. The harmonic technique enables one to model unsteady aerodynamics and aeroelastic response in the frequency domain with reduced computational cost compared to time marching solutions. Computed results compare well to flutter results computed using a separate harmonic balance computational flow solver code developed at Duke University. Select aeroelastic limit cycle oscillation results are also presented.

A novel approach for implementing a nonlinear unsteady frequency domain harmonic balance solution technique about existing implicit computational fluid dynamic flow solvers is presented. This approach uses an explicit discretization of the harmonic balance source term, which enables the harmonic balance method to be applied to existing implicit flow solvers with minimal need for modification to the underlying implicit flow solver code. The resulting harmonic balance solver can then be used for modeling nonlinear periodic unsteady flows. The methodology is applied to the OVERFLOW 2 flow solver code, and results are presented for transonic viscous flow past an unsteady pitching airfoil section. Unsteady aerodynamic and aeroelastic results for the F-16 fighter wing are also presented.

Abstract. A National Aeronautics and Space Administration computational fluid dynamics code, OVERFLOW 2, was modified to utilize a harmonic balance solution method. This modification allows for the direct calculation of the nonlinear frequency-domain solution of a periodic unsteady flow while avoiding the time consuming calculation of long physical transients that arise in aeroelastic applications. With the usual implementation of this harmonic balance method, converting an implicit flow solver from a time marching solution method to a harmonic balance solution method results in an unstable numerical scheme. However, a relatively simple and computationally inexpensive stabilization technique has been developed and is utilized in this paper. With this stabilization technique, it is possible to convert an existing implicit time-domain solver to a nonlinear frequencydomain method with minimal modifications to the existing code. This new frequencydomain version of OVERFLOW 2 utilizes the many features of the original code, such as various discretization methods and several turbulence models. The use of Chimera overset grids in OVERFLOW 2 requires care when implemented in the frequency-domain. This paper presents a harmonic balance version of OVERFLOW 2 capable of solving on overset grids for sufficiently small unsteady amplitudes.

The Harmonic Balance (HB) method developed by Hall et al. is a frequency domain method used for calculating the flow field around periodically oscillating bodies. When compared to the time domain equivalent, the HB method often offers a significant reduction in the computational cost associated with solving for the unsteady flow field. Unique to this HB method is how the governing equations are recast in the frequency domain. The system of equations solved in the frequency domain mimic closely the original time domain system of equations. This allows for methods originally developed for time domain Computational Fluid Dynamics (CFD) to be used when solving the system in the frequency domain. In-house codes at Duke University have been written to implement this HB method. These codes have been used to study complex flows ranging from turbomachinery to helicopter rotors. The research group has also implemented the HB method for the study of Limit Cycle Oscillation behavior of aircraft such as the F-16 fighter. 5 This research shows that it is possible to not only implement the same techniques as used with time domain methods, but it is also possible convert a complex time domain code to the HB method. This saves vast amounts of software production time and allows state of the art software to be the engine of the frequency domain solver. With the HB method, the spatial residual operator is unchanged from the time domain representation. This means that any discretization method implemented in a time domain CFD code can be used in the HB version. Also, within the HB method separate sub-time levels are being solved to a steady state. Thus, the time discretizations used in the original code can be used to drive the HB sub-time levels to steady state. With such similarity to the time domain code, the HB version can be created with relatively few changes to the original time domain code. The CFD code OVERFLOW was successfully converted to the HB method. Since OVERFLOW employs implicit discretization techniques, special treatment is required when converting to the HB method. In the past the research group at Duke University had used explicit discretization techniques in conjunction with the HB method. A von Neumann stability analysis shows that the HB method used in conjunction with an explicit method is unstable. However, that instability is in the long wavelength modes. Thus, the inclusion of the outer boundary stabilizes the scheme. When used in conjunction with an implicit method, the analysis shows that the the HB method is still unstable, but the instability is no longer confined to the long wavelength modes. As such, the inclusion of the outer boundary will not result in a stable scheme. To overcome this challenge a simple and computationally inexpensive stabilization technique was developed. For validation, trials were run using the original time domain OVERFLOW and the HB version of OVERFLOW for a pitching airfoil. Figure 1 shows the real and imaginary parts of the unsteady coefficient of pressure for a NACA 0012 airfoil oscillating in pitch about the quarter chord defined by α(t) = 1.0 cos(0.5t)[degrees] for a Mach number of 0.8. The agreement between the methods is excellent.