Efficient Frequency-Domain Aeroacoustic Simulation of Gust Response Using Harmonic Balance Method

In CFD, unsteady computations are cost-intensive. The HB method represents a cost-efficient alternative. Here, for time-periodic flows, the governing equations are recasted in the Fourier domain.
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\nIn the low Mach regime, the compressible governing equations are stiff. Therefore, density-based solvers converge slowly. Low Mach preconditioning equalizes the eigenvalues of the system of equations, to improve the condition number and remove the stiffness of the system.
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\nIn this paper, low Mach preconditioning is applied to the HB method, with emphasis on the NRBCs. These boundary conditions have a crucial impact on the flow inside the truncated computational domains used in CFD. Improper boundary conditions reflect waves exiting the computational domain and deteriorate the quality of the solution. However,NRBCs avoid spurious reflections.
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\nWe explain that to precondition the NRBC its formulation in terms of characteristics has to be adapted. An academic wave propagation test case is computed for different wave configurations to validate the preconditioned boundary conditions.
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\nThe use of non-preconditioned NRBCs in a preconditioned computation leads to instabilities and reflections at the boundaries of the domain. A consistent setup with preconditioned NRBCs improves the stability and leads to good non-reflecting properties for all presented wave configurations.

Smoothed particle hydrodynamics (SPH) is a meshfree method and has been widely used in the field of fluid dynamics. However, there is still a lack of robust approach to implement the far-field boundary condition in the simulation of free stream flow past the rigid body. To solve this problem, on the basis of Lastiwka et al.’s work, an improved non-reflecting boundary condition, called mixed characteristic boundary condition is proposed in this paper. The mixed characteristic boundary condition is tested by two-dimensional incompressible flows with a perturbation at the initial moment. The numerical results show that the mixed characteristic boundary condition can effectively suppress the reflection of waves on far-field boundaries. Finally, flows past a hydrofoil are simulated within the framework of the kernel gradient free SPH in two kinds of non-reflecting boundary condition. The results indicate that the present boundary condition is more robust and suitable for SPH than the one adopted by Lastiwka et al.

Abstract. The present work is focused on the sensitivity analysis of analytically--corrected acoustic metamaterials with respect to the variation of relevant parameters for aeronautical applications. The performance decay of acoustic metamaterials designed in static conditions when operating in a moving flow can be mitigated using a design process involving coordinate transformations capable of recasting the convective wave equation in the form of the static D'Alembertian. To this aim, a spacetime reformulation of the problem is needed in order to ensure the formal invariance of equations under the action of spacetime coordinate transformations. The considered test case sees a circumferential domain occupied by a metafluid embedded by a conventional medium characterized by a uniform background flow and within which a monopole source is located. All the numerical simulations are done in the frequency domain using the commercial finite element method solver. A parametric study is conducted to analyze the influence of the acoustic source position with respect to the background flow and the variation of the intensity of the latter.

Harmonic balance (HB) method is a pure frequency domain simulation method. Harmonic balance works well for mildly nonlinear RF circuits; however, its performance can be a critical issue as the circuit size increases and the circuit nonlinearity becomes dominant. With the advent of multi-core computers, multi-threading (MT) technique is a good choice to improve efficiency of the traditional harmonic balance method. Since the harmonic balance simulation is very memory-bandwidth-bounded, it is hard to achieve a good MT performance when the CPU number is large. In this paper, we present a harmonic balance technique, with great MT scalability, for many-core computers. We implemented the proposed method, tested it on real radio frequency (RF) circuits, and show that the method gives great MT performance on 4-core and 8-core computers.

A coaxial contra-rotating open rotor belongs to the next generation of aero-engines. It has an efficiency that is about 30% higher than that of a conventional turbojet engine. However, because of the high noise level, the open rotor has not been introduced into the commercial aviation market. Although there have been numerous efforts to reduce its noise level, the need to accurately predict the unsteady and complex flow field around the open rotor makes it difficult to apply the conventional design methodologies. In this paper, we introduce a state-of-the-art design methodology for solving the unsteady flow field problem of the lownoise open rotor design. A harmonic balance method that is an order of magnitude more efficient than the conventional time accurate CFD method is used to predict the aerodynamic performance of the open rotor. With the accurate formulation of the governing equations through the harmonic balance method, a design method that uses a surrogate model is employed to find optimum configuration that minimizes the noise level and total power at a constant thrust level. A noise prediction is made using the Farassat formula, derived from the Ffowcs-Williams-Hawking’s equation. To efficiently search for the optimum configuration, the design optimization is divided into the rotor topology design level and blade planform design level. In a previous study, we investigated the optimum rotor topology parameters such as the blade radii, rotor spacing, and pitch angle of the aft rotor. In this paper, an investigation is conducted to determine the optimum planform variables such as the twist angle, chord length for several design sections, and tip shape control parameters for the aft rotor. A genetic algorithm is used as a multi-objective optimization algorithm in combination with the Kriging surrogate model. Through the planform design for the aft rotor, the noise level and power consumption of the optimum rotor are reduced by 0.6 dB and 6.8% respectively.

This is the first part of a series of three papers on the simulation of turbulence and transition effects in a multistage low pressure turbine. In this first part, the extension, verification and validation of a Harmonic Balance (HB) method recently proposed by the authors to fully include established turbulence and transition models in the method is presented. As an alternating frequency/time-domain type method the implemented HB solver has the advantage of being able to utilize models (e.g. boundary conditions or residual functions) formulated in either the frequency- or time-domain. On the one hand this allows highly accurate nonreflecting boundary conditions formulated in the frequency-domain to be used along entry, exit or interface boundaries, and on the other hand complex nonlinear terms formulated in the time-domain to be used to describe nonlinear effects. Nevertheless, the wish to minimize the number of harmonics used to describe a given time-periodic unsteady flow, coupled with the often highly nonlinear nature of turbulence and transition models makes the full inclusion of such models in the HB method challenging. In this work the integration of Menter’s SST two-equation k–ω turbulence model along with Menter and Langtry’s two-equation γ–Reθ transition model in the context of a general framework for transport equations in the CFD solver TRACE is described in detail. Following the basic verification of the underlying transport equation framework, the implemented models are used to compute the well known high lift, low pressure turbine airfoil T106C and results are compared with the available experimental data as well as results from more conventional time-domain simulations. Alongside the basic validation of the models this testcase is furthermore used to investigate the importance of including higher harmonics, as opposed to only the zeroth harmonic, of the turbulence and transition models for the accurate prediction of the time-mean flow.

PurposeThe high computational effort of steady-state simulations limits the optimization of electrical machines. Stationary solvers calculate a fast but less accurate approximation without eddy-currents and hysteresis losses. The harmonic balance approach is known for efficient and accurate simulations of magnetic devices in the frequency domain. But it lacks an efficient method for the motion of the geometry.Design/methodology/approachThe high computational effort of steady-state simulations limits the optimization of electrical machines. Stationary solvers calculate a fast but less accurate approximation without eddy-currents and hysteresis losses. The harmonic balance approach is known for efficient and accurate simulations of magnetic devices in the frequency domain. But it lacks an efficient method for the motion of the geometry.FindingsThe three-phase symmetry reduces the simulated geometry to the sixth part of one pole. The motion transforms to a frequency offset in the angular Fourier series decomposition. The calculation overhead of the Fourier integrals is negligible. The air impedance approximation increases the accuracy and yields a convergence speed of three iterations per decade.Research limitations/implicationsOnly linear materials and two-dimensional geometries are shown for clearness. Researchers are encouraged to adopt recent harmonic balance findings and to evaluate the performance and accuracy of both formulations for larger applications.Practical implicationsThis method offers fast-frequency domain simulations in the optimization process of rotating machines and so an efficient way to treat time-dependent effects such as eddy-currents or voltage-driven coils.Originality/valueThis paper proposes a new, efficient and accurate method to simulate a rotating machine in the frequency domain.

The numerical simulation of hydrodynamic stability and aeroacoustic problems requires the use of high-order, low-dispersion and low-dissipation numerical methods. It also requires appropriate boundary conditions to avoid reflections of outgoing waves at the boundaries of the computational domain. There are many different methods to avoid wave reflection at the boundaries such as the buffer zone and boundary conditions based on characteristic equations. This paper considers the use of a methodology called perfectly matched layer (PML). The PML is evaluated for the simulation of an acoustic pulse in a uniform flow and the Kelvin–Helmholtz instability in a mixing layer using the linear and nonlinear form of the Euler equation. PML results are compared with other non-reflecting boundary condition methods in terms of effectiveness and computational cost. The other non-reflecting boundary conditions implemented were the buffer zone (BZ), widely used in aeroacoustic and hydrodynamic problems, and the energy transfer and annihilation (ETA), a very simple boundary condition to be implemented. The results show that the PML is an effective boundary condition method, but can be computationally expensive. The PML is also more complex to implement and requires careful stability analysis. The other boundary conditions, the BZ and the ETA, are also effective and may perform better than the PML depending on the flow conditions. These two methods have an advantage in terms of robustness and are much simpler to implement than the PML.

The problem of the diffraction of sound by two-dimensional rigid cylindrical objects has been formulated in generalized curvilinear coordinates. The governing equations for the aeroacoustic field are derived by perturbing the unsteady Euler equations and then extracting the steady mean flow equations from them. The governing equations are then solved by an explicit, four-stage Runge-Kutta, time-marching, finite-volume numerical scheme. The scheme has been implemented with far-field radiation and time-accurate wall boundary conditions. The cases of diffraction by a knife edge and a cylinder have been computed and compared against their corresponding analytical solutions in order to demonstrate the prediction capability of the numerical scheme. The acoustic field resulting from the interaction of the incident, reflected, and diffracted fields has been predicted very well in both the shadow and the illuminated zones. The scheme has also been applied to compute the diffraction of plane waves as well as that of a monopole acoustic field by a GA(W)-1 airfoil. I. Introduction W HEN sound waves impinge upon large rigid obstacles in their path, the resultant aeroacoustic field comprises of the incident waves, the reflected waves, and the diffracted waves. The diffraction of sound waves in the shadow zone plays a key role in assessing the acoustic shielding capability of aerodynamic bodies and forms a problem of fundamental importance in the field of aeroacoustics . The various fluctuating quantities that describe the aeroacoustic field are governed by the Euler equations which are, of course, nonlinear. Even if the governing equations are linearized, it is extremely difficult to obtain analytical solutions in the cases of the realistic twodimensional obstacles such as airfoils without some unacceptable drastic simplifications. The difficulty can be essentially attributed to the complexities associated with the treatment of the boundary conditions. These difficulties pertaining to the analytical solutions have been appreciated by many investigators14 and have led to the emergence of a new field of computational aeroacoustics (CAA). Progress in the methods of CAA has now advanced because of the considerable advances in the fields of computational fluid dynamics (CFD) and computer technology. The aim of both CFD and CAA is to find solutions to the practical problems numerically, but the main differences between the two lie in the nature of the problems one deals with and in the type of the information one wishes to obtain. CFD treats steady as well as unsteady problems, whereas all the CAA problems are, by definition, unsteady. In current CFD problems, interests are focused primarily on the flowfield in the immediate vicinity of an aerodynamic body, and the objectives are to determine aerodynamic quantities such as pressure distributions, skin friction, lift and drag, etc. On the other hand, in CAA problems, attention is directed to both the near-field acoustic vibrations and the far-field noise radiation away from the body, and the objectives are to find the directivity and the spectral characteristics of the acoustic field. Therefore, while the well-established techniques of CFD can be utilized in order to develop the

Effective and efficient simulation and verification techniques are always highly demanded in the electronic design automation (EDA) community. However, existing modeling and simulation approaches could no longer fulfill this growing demand due to the presence of large-scale distributed components and strongly nonlinear devices. In this thesis, two novel techniques are presented for fast simulation and verification of analog and radio-frequency (RF) circuits. Emphasis is placed on developing an analytical approach for periodic steady-state (PSS) analysis and an effective model order reduction (MOR) technique for nonlinear, especially highly nonlinear, systems arising in analog/RF circuit applications.
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\nThe first approach, named autonomous Volterra (AV), achieves efficient PSS analysis of nonlinear circuits. With elegant analytic forms and availability of efficient solvers, AV constitutes a competitive steady-state algorithm besides the two mainstream PSS algorithms, namely, shooting Newton (SN) and harmonic balance (HB). Nonlinear systems are first captured in nonlinear differential algebraic equations (DAEs), followed by expansion into linear Volterra subsystems. A key step of steady-state analysis lies in modeling each Volterra subsystem with autonomous nonlinear inputs. The PSS solution of these subsystems then proceeds with a series of Sylvester equation solves, completely avoiding the guesses of initial condition and time stepping as in SN, as well as the uncertain length of Fourier series as in HB. Error control in AV is also straightforward by monitoring the norms of the Sylvester equation solutions. It is further demonstrated that AV is readily parallelizable with superior scalability towards large-scale problems.
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\nBesides PSS analysis, another important step in analog/RF design is transient as well as general time simulation. To this end, the second part of this thesis features a tensorbased nonlinear model order reduction (TNMOR) algorithm that allows efficient simulation of nonlinear circuits via the emerging techniques utilizing tensors, namely, a multidimensional generalization of matrices. Unlike existing nonlinear model order reduction (NMOR) methods, high-order nonlinearities are captured using tensors in TNMOR, followed by decomposition and reduction to a compact tensor-based reduced-order model (ROM). Consequently, TNMOR completely avoids the dense reduced-order system matrices, which in turn permits faster simulation and less memory requirement. Numerical experiments on transient and PSS analyses confirm the superior accuracy and efficiency of TNMOR, particularly in highly nonlinear scenarios.

Having the limitations like Grid requirements and complexity and enabling the advantages like high accuracy, computational efficiency as well as numerical stability Compact Finite Differential Methods has applications like Navier-Stokes equations, Burger?s equation, Boundary layer flows and Advection-dispersion equations in the field of fluid dynamics, a field where Compact Finite Methods (CFMs) seems to better fit for finding solutions. In this paper we are going to study about usefulness of compact finite derivatives in certain conditions like Navier-Stokes equations. Implementation of Navier-Stokes equation in the field of fluid dynamics results to the accuracy attribute of CFMs.

Convective correction of metafluid devices based on Taylor transformation

Assume that a fluid is inviscid, incompressible, and irrotational. A nonlinear Schrödinger equation (NLSE) describing the evolution of gravity waves in finite water depth is derived using the multiple-scale analysis method. The gravity waves are influenced by a linear shear flow, which is composed of a uniform flow and a shear flow with constant vorticity. The modulational instability (MI) of the NLSE is analyzed, and the region of the MI for gravity waves (the necessary condition for existence of freak waves) is identified. In this work, the uniform background flows along or against wave propagation are referred to as down-flow and up-flow, respectively. Uniform up-flow enhances the MI, whereas uniform down-flow reduces it. Positive vorticity enhances the MI, while negative vorticity reduces it. Hence, the influence of positive (negative) vorticity on MI can be balanced out by that of uniform down (up) flow. Furthermore, the Peregrine breather solution of the NLSE is applied to freak waves. Uniform up-flow increases the steepness of the free surface elevation, while uniform down-flow decreases it. Positive vorticity increases the steepness of the free surface elevation, whereas negative vorticity decreases it.

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.

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 row's 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.