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Journal of Theoretical and Computational Acoustics

Vol. 29, No. 1 (2021) 2050032 (27 pages)

c

The Author(s)

DOI: 10.1142/S2591728520500322

Application Limits of Conservative Source Interpolation

Methods Using a Low Mach Number Hybrid

Aeroacoustic Workflow

Stefan Schoder

Institute of Fundamentals and Theory in Electrical Engineering

Graz University of Technology, 8010 Graz, Austria

[email protected]

Andreas Wurzinger, Clemens Junger and Michael Weitz

Institute of Mechanics and Mechatronics, TU Wien

1060 Vienna, Austria

Clemens Freidhager, Klaus Roppert and Manfred Kaltenbacher

Institute of Fundamentals and Theory in Electrical Engineering

Graz University of Technology, 8010 Graz, Austria

Received 5 November 2020

Accepted 16 December 2020

Published 24 February 2021

In low Mach number aeroacoustics, the well-known disparity of scales allows applying hybrid simulation

models using different meshes for flow and acoustics, which leads to a fast computational

procedure. The hybrid workflow of the perturbed convective wave equation involves three steps:

(1) perform unsteady incompressible flow computations on a subdomain; (2) compute the acoustic

sources and (3) simulate the acoustic field, using a mesh specifically suited. These aeroacoustic

methods seek for a robust, conservative and computational efficient mesh-to-mesh transformation

of the aeroacoustic sources. In this paper, the accuracy and the application limitations of a cellcentroid-

based conservative interpolation scheme is compared to the computationally advanced

cut-volume cell approach in 2D and 3D. Based on a previously validated axial fan model where

spurious artifacts have been visualized, the results are evaluated systematically using a grid convergence

study. To conclude, the monotonic convergence of both conservative interpolation schemes is

demonstrated. Regarding arbitrary mesh deformation (for example, the motion of the vocal folds

in human phonation), the study reveals that the computationally simpler cell-centroid-based conservative

interpolation can be the method of choice.

Keywords: Aeroacoustics; Acoustic analogy; Hybrid CAA; Fan noise; Battery cooling fan.

This is an Open Access article published by World Scientific Publishing Company. It is distributed under

the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

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S. Schoder et al.

1. Introduction

A key challenge in computational aeroacoustics (CAA) is the large disparity of scales

between flow structures and audible acoustic wavelengths in low Mach number applications.

1 A direct numerical simulation based on the compressible flow equations, resolving

both scales from the sound generating vortices to the desired evaluation position, not only

leads to a high number of cells but also to a small time-step size to minimize dissipation of

the acoustic waves. Hybrid schemes in CAA separate the flow from the acoustic computation

using aeroacoustic analogies (see, e.g. Refs. 2 and 3). Each physical field is discretized by

an optimal grid. As a result, the two grids may be quite different according to the following

criteria: (1) near walls, the flow grid needs refinement to resolve boundary layers; (2) the

flow grid is mostly coarsened toward the outflow boundaries to dissipate vortices and (3) the

acoustic grid has to transport waves and therefore it is beneficial to use a uniform grid size

all over the computational domain. The two simulations are connected by an accurate data

transfer from the flow to the acoustic grid to minimize interpolation errors. Different strategies

can be applied to cope with this task, starting from low complexity nearest-neighbor

interpolation4,5 to complex volume intersections between flow and acoustic grid.6,7

Hybrid CAA schemes, which consider just a forward coupling of the flow to the acoustic

field, are valid at low Mach number if there is no feedback. For low Mach number Ma

flows, the scaling between the acoustic wavelength λ and the characteristic length of a vortex

l is given by λ ∼ l/Ma. Therefore, the sizes of the two grids are quite different. A

simple nearest-neighbor interpolation for computing the acoustic sources has shortcomings

(see, e.g. Refs. 4 and 5). In Refs. 4, 8 and 9, the acoustic right-hand side is computed by

summing the contributions of all flow cells belonging to finite elements (FEs) surrounding

an acoustic FE node. Thereby, the method assumes that the flow quantities used for

the acoustic source term computation are constant over each flow cell. This approach has

also been used in Ref. 10 for three-dimensional (3D) problems, where a grid dependency

has been observed, resulting in an underestimated sound pressure level over the whole frequency

spectrum. With respect to the constant values over each flow cell, any function-based

interpolation technique to construct an enriched function space (e.g. radial basis function

interpolation,11–15 FE interpolation16) will improve the interpolation result and increase the

computational effort.

Particularly important for the FE method (FEM) interpolation is that interpolation

of the FE sources can be done in two ways. First, the field variable is interpolated to the

integration points of the FE and then processed by the FE. This procedure has the drawback

that it does not conserve energy.4 Second, the whole source integral (right-hand side, RHS)

RHS(x, t) =



Ω

ψfadξ (1)

is interpolated. Therefore, the integral properties and energy are conserved. This procedure

leads to a natural conservative interpolation having the same computational burden as an

ordinary field interpolation. Therefore, the conservative approaches presented follow the

second idea that the integrated properties are conserved; however, we assume that the

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Application Limits of Conservative Source Interpolation Methods

lowest-order function space (space of constant functionsa) for the flow field is sufficient,

which leads to a fast and efficient algorithm.

Regarding the conservation idea, a globally conservative approach (cell-centroid-based

approach) has been developed in Ref. 16, where the acoustic sources within the FE formulation

are first computed on the fine flow grid. These so-called loads are then interpolated

to the acoustic grid with a nearest-neighbor interpolation. This approach works with high

accuracy in cases where the flow grid has smaller cells than the acoustic grid. However, in

cases where the flow grid gets coarser than the acoustic grid, an approach like this induces

a phase error into the interpolated field. Then, the sources look like a chequerboard (Fig. 1

on the right-hand side). Regions, where the flow grid is coarser than the acoustic mesh,

occur mostly for the coarsened flow grid toward inflow and outflow boundaries since the

acoustic grid has to transport waves and therefore needs a uniform grid size all over the

computational domain.

To overcome such problems in the context of FEs, we have developed the cut-volume

cell weight interpolation and applied it to the aeroacoustic computation of an axial fan.17

Regarding finite volume (FV) schemes for acoustic propagation, similar investigations were

performed in Ref. 5. To accelerate the calculation for volume sources within a boundary element

(BE) method solving Lighthill’s inhomogeneous wave equation, the authors in Ref. 18

spatially condense the acoustic source data.

The first application of the cut-volume cell weight interpolation was performed in Ref. 17.

Building upon the idea, we study the acoustic effects of the current approach compared to

the simpler cell-centroid-based approach.16 Therefore, the algorithms are outlined stepby-

step using pseudocode. We aim to prove the following four hypotheses: First the cellcentroid-

based approach works with high accuracy in cases where the flow grid has smaller

cells than the acoustic grid. Second, we aim to detect a lower bound of the acoustic element

size concerning the flow cell for both methods. Third and for computational efficiency,

Fig. 1. Comparison of the nodal values using the cell-centroid (right) based and the cut-volume cell (left)

interpolation algorithm in areas where the flow grid is coarser than the acoustic mesh.

aIf a higher-order representation is desired, an additional regression step must be executed to construct the

higher-order representation,6 which reduces the computational efficiency.

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S. Schoder et al.

we would like to detect the maximum potential coarsening of the acoustic element size

concerning the flow field. Fourth, we study the influence of an awkward chequerboard

visualization on the sound. Having investigated these hypotheses, we can provide a general

guideline for choosing the most suitable method for your hybrid aeroacoustic workflow.

This study clarifies when an application of the cut-volume cell weight interpolation

is more beneficial, considering the computational performance and the accuracy of the

interpolation.

This paper is organized as follows: In Secs. 2 and 3, we provide the aeroacoustic model,

describe the two interpolation approaches and their limitations considering a variation of

the CAA grid. In Secs. 4 and 5, we study the effects of the two different conservative

interpolation schemes on the acoustic results concerning the acoustic discretization. Upon

previous findings, we exclusively concentrate on the interpolation between the flow and

the acoustic mesh through a detailed mesh study. We show the increase in computational

performance through coarsening the acoustic mesh without losing numerical accuracy and

discuss the suitability of the applied interpolation methods. Finally, in Sec. 6, we conclude

the findings and give additional remarks for future research.

2. Formulation

Having the definitions of acoustic perturbation equations (APEs)19–23 and introducing an

arbitrary Lagrangian–Eulerian (ALE) description for the operators D

Dt = ∂

∂t + (v − vr) · ∇,

where vr is the relative velocity of the grid, we arrive at the perturbed convective wave

equation (PCWE) for rotating systems (see Ref. 17)

1

c2

D2ψa

Dt2

− Δψa = − 1

¯ρc2

Dpic

Dt

. (2)

This scalar convective wave equation is computationally efficient and describes aeroacoustic

sources generated by incompressible flow structures and wave propagation through moving

media. This formulation reduces the number of unknowns (acoustic pressure pa = ¯ρDψa

Dt and

particle velocity va = −∇ψa) to just a scalar unknown, the acoustic velocity potential ψa.

It should be noted that (2) is an exact reformulation of the APEs for low Mach numbers.21

Based on the PCWE, we verify the source term interpolation of aeroacoustic sources for

nonmoving and moving meshes.

The derivation of the weak formulation (1) for the FEM involves a multiplication with

a test function ψ ∈ H1(Ω) and an integration over the computational domain Ω. Here, the

aeroacoustic source fa := − 1

¯ρc2

Dpic

Dt enters the formulation. To compute the aeroacoustic

sources on the acoustic grid in a hybrid workflow, (1) is approximated on the flow grid

(superscript f) and then interpolated to the acoustic grid (superscript a). Naturally, this

leads to an interpolation that conserves the integral property of the aeroacoustic sources

used by the FE simulation.

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Application Limits of Conservative Source Interpolation Methods

2.1. Cell-centroid-based source term interpolation

Starting from (1), the cell-centroid-based approach conserves the energy globally but

approximates the local energy conservation of the FE right-hand side by

Fa

i =



Ea

Na

i (ξ)fadξ ≈



k∈Mf

Na

Thank you, I will look that reference up!