Arbitrary Mean Flow Research Articles

Unifying the stationary and convective Ffowcs Williams and Hawkings solutions for aeroacoustic predictions via a Lorentz transformation

The convective Ffowcs Williams and Hawkings (FW-H) equation and its integral solutions are often used for noise predictions of aeroacoustic phenomena in moving mediums. In this study, a set of unified integral solutions for stationary and convective FW-H equations are proposed by making use of the extended Lorentz transformation for arbitrary mean flow orientations. The new solutions in Euclidean coordinates have compact forms that resemble the well-known F1 and F1A solutions of Farassat, with the main differences from existing ones in the definitions of the equivalent monopole and dipole sources, and also their associated equivalent thickness noise and loading noise. Validations have been performed for two benchmark problems in moving mediums, one is a stationery monopole source and the other is a rotating dipole source. Apart from the well-recognized distorted acoustic fields due to the convective amplification, the results also show that the convective effects induce dipole-like acoustic components for a pure monopole source and monopole-like acoustic components for a pure dipole source. These are explained well by the newly-derived unified integral solutions and provide more insights on the noise source mechanisms.

Acoustic nonlinearities in a quasi 1-D duct with arbitrary mean properties and mean flow

Nonlinear temporal dynamics of acoustic oscillations in a quasi one-dimensional (1-D) duct are investigated using both numerical and analytical methods. The spatiotemporal nonlinear wave equation is derived for pressure oscillations in a quasi 1-D duct with axially varying cross-section and spatially inhomogeneous mean properties such as the velocity, temperature, density and pressure. Using the finite element method with quadratic interpolation functions, the linear Helmholtz equation is solved for the modal shapes and frequencies. With the modal shape as the weighting function, the standard Galerkin method is applied to transform the spatiotemporal wave equation into a second-order nonlinear ordinary differential equation (ODE) governing the time evolution of modal amplitudes. The limit-cycle amplitude and frequency of pressure oscillations are quantified analytically using the Lindstedt–Poincaré perturbation method. Furthermore, to capture the transient evolution to the limit cycle, the method of averaging is applied to the nonlinear temporal ODE for the modal amplitude. From the Lindstedt–Poincaré method, it is seen that a limit cycle exists when the linear damping coefficient μ in the nonlinear ODE is of the opposite sign as the quantity S that is a function of the coefficients of the quadratic and cubic terms in the ODE. For a given S, limit cycle is created or destroyed as μ changes sign, a scenario referred to as the “Hopf bifurcation.” The phase portraits and limit-cycle amplitudes obtained from numerical solution are in excellent agreement with those derived from the analytical techniques.

On the convective wave equation for the investigation of combustor stability using FEM-methods

Solving the Helmholtz equation with spatially resolved finite element method (FEM) approaches is a well-established and cost-efficient methodology to numerically predict thermoacoustic instabilities. With the implied zero Mach number assumption all interaction mechanisms between acoustics and the mean flow velocity including the advection of acoustic waves are neglected. Incorporating these mechanisms requires higher-order approaches that come at massively increased computational cost. A tradeoff might be the convective wave equation in frequency domain, which covers the advection of waves and comes at equivalent cost as the Helmholtz equation. However, with this equation only being valid for uniform mean flow velocities it is normally not applicable to combustion processes. The present paper strives for analyzing the introduced errors when applying the convective wave equation to thermoacoustic stability analyses. Therefore, an acoustically consistent, inhomogeneous convective wave equation is derived first. Similar to Lighthill’s analogy, terms describing the interaction between acoustics and non-uniform mean flows are considered as sources. For the use with FEM approaches, a complete framework of the equation in weak formulation is provided. This includes suitable impedance boundary conditions and a transfer matrix coupling procedure. In a modal stability analysis of an industrial gas turbine combustion chamber, the homogeneous wave equation in frequency domain is subsequently compared to the Helmholtz equation and the consistent Acoustic Perturbation Equations (APE). The impact of selected source terms on the solution is investigated. Finally, a methodology using the convective wave equation in frequency domain with vanishing source terms in arbitrary mean flow fields is presented.

Generalized acoustic Helmholtz equation and its boundary conditions in a quasi 1-D duct with arbitrary mean properties and mean flow

We derive the generalized Helmholtz equation governing the acoustic pressure field in a quasi one-dimensional duct with axially varying cross-section and arbitrary (axially inhomogeneous) mean properties such as the velocity, temperature, density and pressure. To express the Helmholtz equation exclusively in terms of the fluctuating pressure field pˆ, we developed an expression relating density and pressure fluctuations, which is a differential equation for the generalized case of a duct with arbitrary mean properties. The “classical” algebraic expression for the density fluctuation field, ρˆ=pˆ/c̄2, is strictly valid for a constant cross-section duct with uniform mean properties and uniform or zero mean flow (c̄ is the mean sound speed). We show that using the classical ρˆ–pˆ relation in deriving the Helmholtz equation may lead to significant errors in both the phase and amplitude of the acoustic field pˆ. These errors arise because the Helmholtz equation thus obtained fails to account for the boundary condition on density fluctuations at the duct inlet. Furthermore, a linearly-exact derivative boundary condition to the Helmholtz equation of the form dpˆdx=f(pˆ,uˆ,ρˆ;ω) is developed, where uˆ is the velocity fluctuation field and ω is the angular frequency. The pˆ field obtained by solving the generalized Helmholtz equation in conjunction with the derivative boundary condition shows excellent agreement with that obtained through the solution of the linearized Euler equations.

A modal boundary element method for axisymmetric acoustic problems in an arbitrary uniform mean flow

This paper presents a new numerical analysis approach based on an improved Modal Boundary Element Method (MBEM) formulation for axisymmetric acoustic radiation and propagation problems in a uniform mean flow of arbitrary direction. It is based on the homogeneous Modal Convected Helmholtz Equation (MCHE) and its convected Green’s kernel using a Fourier transform method. In order to simplify the flow terms, a general modal boundary integral solution is formulated explicitly according to two new operators such as the particular and convected kernels. Through the use of modified operators, the improved MBEM approach with flow takes a convective form of the general MBEM approach and has a similar form of the nonflow MBEM formulation. The reference and reduced Helmholtz Integral Equations (HIEs) are implicitly taken into account a new nonreflecting Sommerfeld condition to solve far field axisymmetric regions in a uniform mean flow. For isolating the singular integrations, the modal convected Green’s kernel and its modified normal derivative are performed partly analytically in terms of Laplace coefficients and partly numerically in terms of Fourier coefficients. These coefficients are computed by recursion schemes and Gauss-Legendre quadrature standard formulae. Specifically, standard forms of the free term and its convected angle resulting from the singular integrals can be expressed only in terms of real angles in meridian plane. To demonstrate the application of the improved MBEM formulation, three exterior acoustic case studies are considered. These verification cases are based on new analytic formulations for axisymmetric acoustic sources, such as axisymmetric monopole, axial and radial dipole sources in the presence of an arbitrary uniform mean flow. Directivity plots obtained using the proposed technique are compared with the analytical results.

Perfectly matched layer absorbing boundary conditions for Euler equations with oblique mean flows modeled with smoothed particle hydrodynamics.

Absorbing boundary conditions (ABCs) play a critical role in the simulation of sound or wave propagation problems. This paper proposes a technique of space-time transformed perfectly matched layer (PML) boundary condition implemented in a widely used mesh-free method called smoothed particle hydrodynamic (SPH) method, to absorb the outgoing sound waves with oblique shear mean flow. Special consideration is given to the particle features of the SPH, and the PMLs are formulated to correct the truncation error of SPH and absorb the outgoing wave at the same time, aiming to reduce the storage and computational cost in the infinite computational domain. Because the group velocity and phase velocity of the outgoing sound waves in the PMLs may be in different directions, exponentially growing pseudo reflections can result. The authors thus employ space-time transformation to eliminate the reflections effectively in PML boundaries for stable solutions. Moreover, a uniform framework of PML absorbing boundary conditions for Euler equations in the cases of arbitrary oblique mean flow and static media is derived. Finally, the present PML-SPH method with this stable absorbing boundary is applied to simulate sound waves propagating with mean flow. The obtained numerical results agree very well with the reference results.

Analytical solutions for the acoustic field in a thin annular duct with temperature gradient and mean flow

The analytical solution for the one-dimensional acoustic field in a straight duct with arbitrary mean temperature gradient and mean flow has been recently derived based on an adapted WKB approximation (J. Li, A.S. Morgans, J. Sound Vib. 400 (2017) 248–269 [12]). These solutions reproduce the acoustic field very accurately across a wide range of flow conditions. These approaches are applied herein to a more complex configuration – a thin annular duct with arbitrary mean temperature gradient and mean flow; this is a common element (e.g., the annular combustion chamber) used in gas turbines and aero-engines. The derived analytical solutions are also validated on similar configurations and by using the same methods as our previous work.

Comment on “the one dimensional acoustic field with arbitrary mean axial temperature gradient and mean flow” (J.Li and A.S.Morgans, Journal of Sound and Vibration 400 (2017) 248–269)

Comment on “the one dimensional acoustic field with arbitrary mean axial temperature gradient and mean flow” (J.Li and A.S.Morgans, Journal of Sound and Vibration 400 (2017) 248–269)

Commentary on manuscript “Comment on ‘The one-dimensional acoustic field with arbitrary mean axial temperature gradient and mean flow’ (J. Li and A. S. Morgans, Journal of Sound and Vibration 400 (2017) 248–269)”

Commentary on manuscript “Comment on ‘The one-dimensional acoustic field with arbitrary mean axial temperature gradient and mean flow’ (J. Li and A. S. Morgans, Journal of Sound and Vibration 400 (2017) 248–269)”

An improved time-dependent Boundary Element Method for two-dimensional acoustic problems in a subsonic uniform flow

In this paper, we have developed an improved formulation of two-dimensional Convected Boundary Element Method (CBEM) for radiation and propagation acoustic problems in a uniform mean flow with arbitrary orientation. The improved CBEM approach is derived from an advanced form of time-space two-dimensional Boundary Integral Equation (BIE) according to new Sommerfeld Radiation Conditions (SRC) with arbitrary mean flow. The acoustic variables of these formulations are expressed only in terms of the acoustic field as well as its normal and tangential derivatives. The multiplication operators are based explicitly on the two-dimensional Green's function and its convected normal derivative kernel. The proposed terms significantly reduce the presence of flow quantities incorporated in the classical integral formulations. Precisely, the convected kernel only requires the evaluation of two terms instead of several terms in conventional formulations due to the flow effects in the temporal and spatial derivatives. Also, for the singular integrations, the kernels containing logarithmic and weak singularities are converted to regular forms and evaluated partially analytically and numerically. The formulation is derived to be easy to implement as a numerical tool for computational codes of acoustic mediums with arbitrary mean flow. The accuracy and robustness of this technique is assessed through several examples such as the two-dimensional monopole, dipole and quadrupole sources in a uniform mean flow. An application of the improved formulation coupled with Particular Dirichlet-to-Neumann operators (PDtN) has been presented to describe the acoustic field inside two-dimensional infinite ducts in a uniform mean flow. The numerical results are compared to analytical, conventional BEM and Finite Element Method (FEM) formulations.

The one-dimensional acoustic field in a duct with arbitrary mean axial temperature gradient and mean flow

This paper presents an analytical solution for the one-dimensional acoustic field in a duct with arbitrary mean temperature gradient and mean flow. A wave equation for the pressure perturbation is derived which relies on very few assumptions. An analytical solution for this is derived using an adapted WKB approximation. The solution is a superposition of waves travelling in either direction and thus provides physical insight. It is also very easy to calculate. The proposed solution is applied to ducts with a mean temperature profile which varies axially with (i) a linear and (ii) a partial sine wave profile: predictions are compared to those obtained by numerically solving the linearised Euler equations (LEEs). The analytical solution reproduces the acoustic field very accurately across a wide range of flow conditions which span both low and moderate-to-high subsonic Mach numbers. It always performs well when the frequency exceeds a certain value; when the mean temperature profile is linear, it also performs well to very low frequencies. This increased frequency range for linear mean temperature profiles leads to its application to more complicated profiles in a piecewise linear manner, axially segmenting the temperature profile into regions that can be approximated as linear. The acoustic field is predicted very accurately as long as enough segmentation points are used and the condition for the linear mean temperature profile is satisfied: |k0|>|α|, where k0 is the local wave number when there is no mean flow and α is the normalised mean density gradient. The proposed solution is extensively compared to previous analytical solutions, and is found to be more accurate and reliable, especially at higher Mach numbers. The entropy wave generated by communication between the acoustic waves and the distributed mean temperature zone is calculated using the LEEs. It is found to remain very small across all operating conditions, such that both the entropy wave and its impact on the acoustic field can be neglected.

Efficient Mode Matching Based on Closed-Form Integrals of Pridmore-Brown Modes

A new mode-matching method is developed for acoustic modes in axially sectioned lined ducts with parallel, but otherwise arbitrary mean flow. The method is evaluated in detail for a circularly symmetric configuration (where the mean flow depends only on the radial coordinate), with the so-called Pridmore-Brown modes, satisfying the radial Pridmore-Brown equation. Classically, the matching of the modal representations at the interfaces between the sections is based on continuity of pressure and velocity in combination with projection to a suitable but unrelated set of test functions by means of a standard integral inner product. The alternative proposed here uses essentially the same Pridmore-Brown modes, but instead of the standard inner product it applies a particular bilinear form that can be evaluated in closed form. Apart from numerical efficiency, this approach features also a higher accuracy because it avoids the inherently inaccurate numerical quadrature of oscillating functions. The results are compared with results obtained by an implementation of the classical approach, and the agreement is excellent, with higher accuracy and with greater computational efficiency.

Computational fluid dynamics simulation of sound propagation through a blade row

The propagation of sound waves through a blade row is investigated numerically. A wave splitting method in a two-dimensional duct with arbitrary mean flow is presented, based on which pressure amplitude of different wave mode can be extracted at an axial plane. The propagation of sound wave through a flat plate blade row has been simulated by solving the unsteady Reynolds average Navier-Stokes equations (URANS). The transmission and reflection coefficients obtained by Computational Fluid Dynamics (CFD) are compared with semi-analytical results. It indicates that the low order URANS scheme will cause large errors if the sound pressure level is lower than -100 dB (with as reference pressure the product of density, main flow velocity, and speed of sound). The CFD code has sufficient precision when solving the interaction of sound wave and blade row providing the boundary reflections have no substantial influence. Finally, the effects of flow Mach number, blade thickness, and blade turning angle on sound propagation are studied.

Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics

AbstractFor the numerical simulation of time harmonic acoustic scattering in a complex geometry, in presence of an arbitrary mean flow, the main difficulty is the coexistence and the coupling of two very different phenomena: acoustic propagation and convection of vortices. We consider a linearized formulation coupling an augmented Galbrun equation (for the perturbation of displacement) with a time harmonic convection equation (for the vortices). We first establish the well-posedness of this time harmonic convection equation in the appropriate mathematical framework. Then the complete problem, with Perfectly Matched Layers at the artificial boundaries, is proved to be coercive + compact, and a hybrid numerical method for the solution is proposed, coupling finite elements for the Galbrun equation and a Discontinuous Galerkin scheme for the convection equation. Finally a 2D numerical result shows the efficiency of the method.

A low-Mach number model for time-harmonic acoustics in arbitrary flows

This paper concerns the finite element simulation of the diffraction of a time-harmonic acoustic wave in the presence of an arbitrary mean flow. Considering the equation for the perturbation of displacement (due to Galbrun), we derive a low-Mach number formulation of the problem which is proved to be of Fredholm type and is therefore well suited for discretization by classical Lagrange finite elements. Numerical experiments are done in the case of a potential flow for which an exact approach is available, and a good agreement is observed.

Efficient Computation of Spinning Modal Radiation Through an Engine Bypass Duct

The aim of this work is to accurately and efficiently predict sound radiation out of a duct with flow. The sound propagation inside a generic engine bypass duct, refractions by the shear layer of the exhaust flow, and propagation in the near field are the main focus of the study. The prediction uses either a modified form of linearized Euler equations or an alternative model based on acoustic perturbation equations, which were extended to cylindrical coordinates. The two models were compared on a canonical case of sound propagation out of a semi-infinite duct with flow. Good agreements between the predictions were achieved. The more general case of a generic aircraft engine bypass duct with flow was then investigated with the technique of adaptive mesh refinement to increase the computational efficiency. The results show that both linearized Euler equations and acoustic perturbation equations models can predict the near-field sound propagation and far-field directivity. The acoustic perturbation equations model, however, is more adaptive for its suitability to an arbitrary background mean flow.

Computation of Indirect Combustion Noise by a CAA Method

Indirect combustion noise, which is generated by the acceleration of hot gases from a turbulent swirling flame, has become an important issue to understand turbo-machinery core noise. This indirect combustion noise it is dominated by a strong interaction of hydrodynamic and acoustic perturbations due to the transonic base flow in the turbine stages and combustion chamber exit nozzle. Current numerical simulation methods for the phenomenon based on compressible Large Eddy Simulation (LES) are very expansive. Thus LES applications are still limited to one design or few variations with immense computational costs. The most simple model fully covering indirect combustion noise, are the linearized non-isentropic Euler equations over an arbitrary mean flow. Therefore the gap between theory and LES is closed by a method adopted from Computational Aeroacoustics (CAA). It combines a high cost efficiency in terms of required points per wavelength with a high adaptability to realistic mean flow conditions. The CAA method is first validated with a simplified experiment and then applied to predict the acoustic properties of two model combustion chambers, one of them with transonic termination nozzle. By means of the acoustic intensity the sources of the noise are located.

A computational mode-matching approach for sound propagation in three-dimensional ducts with flow

A finite element (FE) mode-matching approach is presented for duct acoustics with flow and circumferentially varying liners. The primary application of the method is the prediction of sound attenuation in turbofan inlets and bypass ducts including the effects of liner splices. A fully numerical procedure has been developed to determine the acoustic modes in ducts of arbitrary cross-section and mean flow profile. A numerical mode-matching method is proposed using a modified matching technique in order to deal more accurately with liner discontinuity with flow. An analytic radiation model can also be integrated with the mode-matching procedure in order to obtain far-field directivity for tones and broadband multimode noise. The accuracy of the mode-matching method is demonstrated by comparison with exact solutions and high-resolution FE solutions. The influence of non-axisymmetric liners on noise radiation is also illustrated.

A coupled finite-element, boundary-integral method for simulating ultrasonic flowmeters

Today's most popular technology of ultrasonic flow measurement is based on the transit-time principle. In this paper, a numerical simulation technique applicable to the analysis of transit-time flowmeters is presented. A flowmeter represents a large simulation problem that also requires computation of acoustic fields in moving media. For this purpose, a novel boundary integral method, the Helmholtz integral-ray tracing method (HIRM), is derived and validated. HIRM is applicable to acoustic radiation problems in arbitrary mean flows at low Mach numbers and significantly reduces the memory demands in comparison with the finite-element method (FEM). It relies on an approximate free-space Green's function which makes use of the ray tracing technique. For simulation of practical acoustic devices, a hybrid simulation scheme consisting of FEM and HIRM is proposed. The coupling of FEM and HIRM is facilitated by means of absorbing boundaries in combination with a new, reflection-free, acoustic-source formulation. Using the coupled FEM-HIRM scheme, a full three-dimensional (3-D) simulation of a complete transit-time flowmeter is performed for the first time. The obtained simulation results are in good agreement with measurements both at zero flow and under flow conditions.

Comparison of a finite element model with a multiple-scales solution for sound propagation in varying ducts with swirling flows

A multiple-scales (MS) solution is proposed to study sound propagation in slowly varying ducts with mean swirling flows. Instead of the standard linearized Euler equations, the MS method is applied to the so-called Galbrun’s equation. This equation is based on an Eulerian–Lagrangian description and corresponds to a wave equation written only in terms of the Lagrangian perturbation of the displacement. This yields simpler differential equations to solve for the MS model as well as simpler boundary conditions. In this paper, Galbrun’s equation is also solved by a mixed pressure-displacement finite element method (FEM). The proposed FEM model has already been tested in authors’ previous papers. This model is quite general and is extended here to arbitrary mean flows, including compressibility and swirling flow effects. Some MS and FEM solutions are then compared in order to validate both models.