Kivanc Ekici

Nonlinear Analysis of Unsteady Flows in Multistage Turbomachines Using Harmonic Balance

A harmonic balance technique for the analysis of two-dimensional linear (small-disturbance) and nonlinear unsteady flows in multistage turbomachines is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of multistage machines to both blade vibration (the flutter problem) and wake interaction (the forced response problem). In general, the flowfield may have multiple excitation frequencies that are not integer multiples of each other, so that the unsteady flow is (sometimes) aperiodic in time. Using our approach, we model each blade row using a computational grid spanning a single blade passage. In each blade row, we store several subtime level solutions. For flows that are periodic in time, these subtime levels span a single time period. For aperiodic flows, the temporal period spanned by these subtime level solutions is sufficiently long to sample the relevant discrete frequencies contained in the aperiodic flow. In both cases, these subtime level solutions are related to each other through the time-derivative terms in the Euler or Navier-Stokes equations and boundary conditions; complex periodicity conditions connect the subtime levels within a blade passage, and interrow boundary conditions connect the solutions among blade rows. The resulting discretized equations, which are mathematically steady because time derivatives have been replaced by a pseudospectral operator in which the excitation frequencies appear as parameters, can be solved very efficiently using multigrid acceleration techniques. In this paper, we apply the technique to both flutter and wake-interaction problems and illustrate the influence of neighboring blade rows on the unsteady aerodynamic response of a blade row.

Computationally fast harmonic balance methods for unsteady aerodynamic predictions of helicopter rotors

A harmonic balance technique for the analysis of unsteady flows about helicopter rotors in forward flight and hover is presented in this paper. The aerodynamics of forward flight are highly nonlinear, with transonic flow on the advancing blade, subsonic flow on the retreating blade, and stalled flow over the inner portion of the rotor. Nevertheless, the unsteady flow is essentially periodic in time making it well suited for frequency domain analysis. The present method uses periodic boundary conditions that allows one to model the flow field on a computational grid around a single helicopter blade, no matter the actual blade count. Using this approach, we compute several solutions, each one corresponding to one of several instants in time over one period. These time levels are coupled to each other through a spectral time derivative operator in the interior of the computational domain and through the far-field and periodic boundary conditions around the boundary of the domain. In this paper, we apply the method to the three-dimensional Euler equations (although the method can also be applied to three-dimensional viscous flows), and examine the steady and unsteady aerodynamics about wings and rotors.

Three-Dimensional Unsteady Multi-stage Turbomachinery Simulations Using the Harmonic Balance Technique

In this paper, we propose an extension of the Harmonic Balance method for threedimensional, unsteady, multi-stage turbomachinery problems modeled by the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations. This time-domain algorithm simulates the true geometry of the turbomachine (with the exact blade counts) using only one blade passage per blade row, thus leading to drastic savings in both CPU and memory requirements. Modified periodic boundary conditions are applied on the upper and lower boundaries of the single passage in order to account for the lack of a common periodic interval for each blade row. The solution algorithm allows each blade row to resolve a specified set of frequencies in order to obtain the desired computation accuracy; typically, a blade row resolves only the blade passing frequencies of its neighbors. Since every blade row is setup to resolve different frequencies the actual Harmonic Balance solution in each of these blade rows is obtained at different instances in time or time levels. The interaction between blade rows occurs through sliding mesh interfaces in physical time. Space and time interpolation are carried out at these interfaces and can, if not properly treated, introduce aliasing errors that can lead to instabilities. With appropriate resolution of the time interpolation, all instabilities are eliminated. This new procedure is demonstrated using both two-and three dimensional test cases and can be shown to significantly reduce the cost of multi-stage simulations while capturing the dominant unsteadiness in the problem.

Harmonic Balance Analysis of Limit Cycle Oscillations in Turbomachinery

A harmonic balance technique for the analysis of limit cycle oscillations of turbomachinery blades is presented. This method couples a computational fluid dynamics model to a single-degree-of-freedom structural dynamic model of the turbomachinery blades. The computational fluid dynamics solver uses a nonlinear frequency-domain (harmonic balance) approach that allows one to model the blade row of a turbomachine on a computational grid spanning a single blade passage. Using the harmonic balance approach, several solutions, each one corresponding to a different subtime level of the periodic unsteady flow, are computed simultaneously. These subtime-level solutions are coupled to each other in the computational field by a spectral approximation of the time-derivative term in the Navier―Stokes equation and also by application of far-field and periodic boundary conditions. The structural dynamic model is based on a similar approach in which a single vibratory mode of interest is modeled using the harmonic balance technique. The two solvers are coupled together through the upwash condition on the surface of the blade and the resulting generalized aerodynamic forces. In the proposed approach, the limit cycle oscillation frequency is treated as another unknown, which is solved iteratively, together with the governing equations of fluid flow and structural dynamics, thereby driving the residual of the aeroelastic problem to convergence in a single computational fluid dynamics run. The accuracy of the new method is compared with two other techniques and it is shown to offer significant computational savings.

Harmonic balance methods applied to computational fluid dynamics problems

In this paper, we briefly review the classical harmonic balance method, and describe the adaptation of the method required for its application to computational fluid dynamics models of unsteady time periodic flows. We describe several variations of the method including a classical balancing method with pseudo time relaxation, the nonlinear frequency domain form and the time spectral form. We show that the maximum stable Courant–Friedrichs–Lewy (CFL) number for explicit schemes is dependent on the grid reduced frequency, a non-dimensional parameter that depends on the cell size, characteristic wave speed, and the highest frequency retained in the harmonic balance analysis. We apply the harmonic balance methods to several nonlinear unsteady flow problems and show that even strongly nonlinear flows can be modelled accurately with a small number of harmonics retained in the model.

Harmonic Balance Analysis of Blade Row Interactions in a Transonic Compressor

In this paper we apply the harmonic balance technique to analyze an inlet guide vane and rotor interaction problem, and compare the computed flow solutions to existing experimental data. The computed results, which compare well with the experimental data, demonstrate that the technique can accurately and efficiently model strongly nonlinear periodic flows, including shock/vane interaction and unsteady shock motion. Using the harmonic balance approach, each blade row is modeled using a computational grid spanning just a single blade passage regardless of the actual blade counts. For each blade row, several subtime level solutions that span a single time period are stored. These subtime level solutions are related to each other through the time derivative term in the Euler (or Navier―Stokes) equations, which is approximated by a pseudo-spectral operator, by complex periodicity conditions along the periodic boundary of each blade rows computational domain, and by the interface boundary conditions between the vane and rotor. Casting the governing equations in harmonic balance form removes the explicit dependence on time. Mathematically, the equations to be solved are similar in form to the steady Euler (or Navier―Stokes) equations with an additional source term proportional to the fundamental frequency of the unsteadiness. Thus, conventional steady-state computational fluid dynamics techniques, including local time stepping and multigrid acceleration, are used to accelerate convergence, resulting in a very efficient unsteady flow solver.

Aerodynamic Asymmetry Analysis of Unsteady Flows in Turbomachinery

A nonlinear harmonic balance technique for the analysis of aerodynamic asymmetry of unsteady flows in turbomachinery is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of turbomachinery blades to self-excited vibrations. Traditionally, researchers have investigated the unsteady response of a blade row with the assumption that all the blades in the row are identical. With this assumption the entire wheel can be modeled using complex periodic boundary conditions and a computational grid spanning a single blade passage. In this study, the steady/unsteady aerodynamic asymmetry is modeled using multiple passages. Specifically, the method has been applied to aerodynamically asymmetric flutter problems for a rotor with a symmetry group of two. The effect of geometric asymmetries on the unsteady aerodynamic response of a blade row is illustrated. For the cases investigated in this paper, the change in the diagonal terms (blade on itself) dominated the change in stability. Very little mode coupling effect caused by the off-diagonal terms was found.Copyright

Harmonic balance analysis of limit cycle oscillations in turbomachinery

DOI: 10.2514/1.J050858 A harmonic balance technique for the analysis of limit cycle oscillations of turbomachinery blades is presented. Thismethodcouplesacomputational fluiddynamicsmodelto asingle-degree-of-freedom structural dynamicmodel of the turbomachinery blades. The computational fluid dynamics solver uses a nonlinear frequency-domain (harmonic balance) approach that allows one to model the blade row of a turbomachine on a computational grid spanningasinglebladepassage.Usingtheharmonicbalanceapproach,severalsolutions,eachonecorrespondingto a different subtime level of the periodic unsteady flow, are computed simultaneously. These subtime-level solutions are coupled to each other in the computational field by a spectral approximation of the time-derivative term in the Navier–Stokes equation and also by application of far-field and periodic boundary conditions. The structural dynamic model is based on a similar approach in which a single vibratory mode of interest is modeled using the harmonic balance technique. The two solvers are coupled together through the upwash condition on the surface of the blade and the resulting generalized aerodynamic forces. In the proposed approach, the limit cycle oscillation frequency is treated as another unknown, which is solved iteratively, together with the governing equations of fluid flow and structural dynamics, thereby driving the residual of the aeroelastic problem to convergence in a single computational fluid dynamics run. The accuracy of the new method is compared with two other techniques and it is shown to offer significant computational savings.

Nonlinear Analysis of Unsteady Flows in Multistage Turbomachines Using the Harmonic Balance Technique

A harmonic balance technique for the analysis of two and three-dimensional linear (small-disturbance) and nonlinear unsteady flows in multistage turbomachines is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of multistage machines to both blade vibration (the flutter problem) and wake interaction (the forced response problem). In general, the flow field may have multiple excitation frequencies that are not integer multiples of each other, so that the unsteady flow is (sometimes) aperiodic in time. Using our approach, we model each blade row using a computational grid spanning a single blade passage. In each blade row, we store several sub-time level solutions. For flows that are periodic in time, these sub-time levels span a single time period. For aperiodic flows, the temporal “period” spanned by these sub-time level solutions is suciently long to sample the relevant discrete frequencies contained in the aperiodic flow. In both cases, these sub-time level solutions are related to each other through the time derivative terms in the Euler or Navier-Stokes equations, and boundary conditions ‐ complex periodicity conditions connect the sub-time levels within a blade passage, and inter-row boundary conditions connect the solutions among blade rows. The resulting discretized equations ‐ which are mathematically “steady” because time-derivatives have been replaced by a pseudo-spectral operator in which the excitation frequencies appear as parameters ‐ can be solved very eciently using multi-grid acceleration techniques. In this paper, we apply the technique to both flutter and wake interaction problems and illustrate the influence of neighboring blade rows on the unsteady aerodynamic response of a blade row.

Modeling and Analysis of the W7-X High Heat-Flux Divertor Scraper Element

The Wendelstein 7-X stellarator experiment is scheduled for the completion of device commissioning and the start of first plasma in 2015. At the completion of the first two operational phases, the inertially cooled test divertor unit will be replaced with an actively cooled high heat-flux divertor, which will enable the device to increase its pulse length to steady-state plasma performance. Plasma simulations show that the evolution of bootstrap current in certain plasma scenarios produce excessive heat fluxes on the edge of the divertor targets. It is proposed to place an additional scraper element in the 10 divertor locations to intercept some of the plasma flux and reduce the heat load on these divertor edge elements. Each scraper element may experience a 500-kW steady-state power load, with localized heat fluxes as high as 20 MW/m2. Computational analysis has been performed to examine the thermal integrity of the scraper element. The peak temperature in the carbon-carbon fiber composite, the total pressure drop in the cooling water, and the increase in water temperature must all be examined to stay within specific design limits. Computational fluid dynamics modeling is performed to examine the flow paths through the multiple monoblock fingers as well as the thermal transfer through the monoblock swirl tube channels.