ARCHIVES OF ACOUSTICS Vol. 45, No. 4, pp. 633–645 (2020) DOI: 10.24425/aoa.2020.135251 Research Paper Finite Element Modelling of a Flow-Acoustic Coupling in Unbounded Domains Paweł ŁOJEK∗, Ireneusz CZAJKA, Andrzej GOŁAŚ AGH – University of Science and Technology Department of Power Systems and Environmental Protection Facilities Kraków, Poland ∗Corresponding Author e-mail: [email protected] (received June 29, 2020; accepted August 17, 2020 ) One of the main issues of design process of HVAC systems and ventilation ducts in particular is correct modelling of coupling of the flow field and acoustic field of the air flowing in such systems. Such a coupling can be modelled in many ways, one of them is using linearised Euler equations (LEE). In this paper, the method of solving these equations using finite element method and open source tools is decribed. Equations were transformed into functional and solved using Python language and FEniCS software. The non-reflective boundary condition called buffer layer was also implemented into equations, which allowed modelling of unbounded domains. The issue, influence of flow on wave propagation, could be adressed using LEE equations, as they take non-uniform mean flow into account. The developed tool was verified and results of simulations were compared with analytical solutions, both in one- and two-dimensional cases. The obtained numerical results are very consistent with analytical ones. Furthermore, this paper describes the use of the developed tool for analysing a more complex model. Acoustic wave propagation for the backward-facing step in the presence of flow calculated using Navier-Stokes equations was studied. Keywords: linearised Euler equations (LEE); FEniCS; finite element method; non-reflective boundary conditions; open source. 1. Introduction phenomena, associated with both the flow of fluid and the propagation of the wave. However, according When using HVAC systems in buildings, acoustic to (Colonius, 1997), the computational complexity comfort is one of the crucial features from the user’s of the DNS method is too large to be used in engineer- point of view. One of the main factors affecting acous- ing applications. Computational aeroacoustic (CAA) tic comfort is ambient noise caused in modern build- methods are often used for this purpose. The classi- ings by components of HVAC systems, such as fans, cal approach is to use large eddy simulation (LES) ventilation units, and air conditioning systems. The method, which is less accurate (and computationally noise that is produced by these devices could be further complex) than DNS, to compute the fluid flow. Then, enhanced by air ducts. Understanding the behaviour acoustic analogies are used to calculate the sound pres- of acoustic wave in the presence of the flow in ducts sure level in the far field (Wagner et al., 2006). Most is important in the design and operation of such sys- of the acoustic analogies are based on the acoustic tems. It requires experimental or numerical analysis, analogy derived by Lighthill (1952). The disadvan- of which the latter is more often used, because of eco- tage of this method is the lack of information about nomical reasons. phenomena occurring in the near field. This classi- The most widely used model for the behaviour cal, hybrid approach is implemented in state-of-the- of the fluid and the acoustic wave are Navier-Stokes art numerical software, such as Actran, ANSYS Flu- equations. However, using them involves many prob- ent, Comsol, or OpenFOAM (Epikhin et al., 2015). lems resulting from the lack of their exact solution for An alternative method used for computational aeroa- the general case (Wagner et al., 2006). The most ac- coustics is called Kirchoff’s integral method. It con- curate method is direct numerical simulation (DNS) sists of calculating the nonlinear near (flow) field using of the equations, which provides a full picture of the Navier-Stokes equations and then using Kirchhoff sur- 634 Archives of Acoustics – Volume 45, Number 4, 2020 face integral to find the acoustic pressures at far field 2. Mathematical model (Lyrintzis, George, 1989). Another hybrid method is to use Navier-Stokes Continuity (1) and Navier-Stokes (2) equations are equations to simulate the fluid flow, and then cou- mainly used to describe the behaviour of the fluid ple the fluid velocity field described by them with (Rienstra, Hirschberg, 2004): acoustic field described by one of the propagation @ρ @ρvi models. The most used acoustic propagation mod- 0; (1) @t @xi els are well known linearised Euler equations (LEE) + = described for example in (Bailly, Juvé, 2000) and @vi @Pji @vi Ewert, ρ ρvj fi; (2) acoustic perturbation equations proposed by ( @t @xj @xj Schröder, 2003). In this paper, this modelling ap- + + = proach was used. Acoustic propagation model given where p is the pressure, vi is the velocity, t is the time, by linearised Euler equations was used in combina- ρ is the density, fi is the external force density, Pij tion with Navier-Stokes equations. From the mathe- pδij τij is the fluid stress tensor, τij is the viscous matical point of view, linearised Euler equations are stress tensor, i; j 1; 2; 3 are dimensions. = a system of partial differential equations. The ex- Euler− equations can be derived from Navier-Stokes act solution of the equations are known only for ba- equations by the= assumption of inviscid and homoen- sic, one or two-dimensional cases. For more complex tropic flow (Åbom, 2006; Mechel, 2008): cases and geometries, approximate methods, such as @ρ @ρvi 0; (3) finite element (FEM), difference (FDM), and volume @t @x (FVM) methods are commonly used. The paper pro- + = poses the implementation of solver for linearised Euler @ρvi @ ρvivj pδij 0: (4) equations in Python using the finite element software @t @xj FEniCS (Alnaes et al., 2015). Linearised Euler equa- + ( + ) = tions in the form which takes the non-uniform veloc- Equations can be linearised around the mean value ity field into account were implemented, which ulti- of the parameters describing the flow, such as pressure, mately allowed describing the coupling of flow and density, or velocity. This allows to obtain linearised Eu- acoustic fields. This represents the suitability of the ler equations which allow determining acoustic wave tool for modelling the behaviour of HVAC systems propagation (Rienstra, Hirschberg, 2004). and air in ducts. In order for created tool to be used Assuming that each variable describing the flow more widely, for modelling unbounded domains, it was can be divided into a mean and acoustic component, necessary to propose and implement a non-reflecting it can be written (Dykas et al., 2010): boundary condition. A buffer zone condition has been ′ ′ ′ p p p ; vi vi v ; ρ ρ ρ ; (5) selected for this purpose. i ′ ′ ′ The created tool has been tested and verified in two where p; =v; ρ+are mean variables,= + p ; vi;= ρ are+ acoustic test cases for which analytical solution is known. It variables (pressure, velocity in i-th direction, density). was then used to calculate the behaviour of a sinu- Inserting the above into the equations (3) and (4), soidal acoustic wave in the presence of a non-uniform assuming the mean values p; vi 0; ρ const and using flow, calculated using the Navier-Stokes equations. The ideal gas equation of state given by: chosen geometric model for these simulations is a well = = p′ c2ρ′ known and researched backward-facing step. This ge- (6) ometry can be used to model, for example, channel allows to derive linearised= Euler equations which de- discontinuity. scribes acoustic wave propagation without a mean FEniCS software was chosen due to the fact that flow. The equations are written below: it is open source software, easy to use from a pro- ′ ′ gramming point of view, and allowing connection with @p 2 @vj c ρ 0; (7) pre- and post-processing tools. Due to its structure @t @xj and workflow, FEniCS allows to focus on the parts + = of mathematical and physical modelling, it is not ne- @v′ @p′ ρ i 0; (8) cessary to go into the details of the implementation @t @xi of FEM. For pre-processing, i.e. geometry and mesh Linearised Euler equations+ can= be reduced to the generation, the Salome software was used (Ribes, classic wave equation by taking the time derivative of Caremoli, 2007). The post-processing of the results Eq. (7), divergence of Eq. (8), and combining them was performed using self-developed Python scripts and (Rienstra, Hirschberg, 2004): Paraview tool (Ahrens et al., 2005). Paraview was also used for visualisation of results. @2p′ 1 @2p′ 0: 2 2 2 (9) @t c @xi − = P. Łojek et al. – Finite Element Modelling of a Flow-Acoustic Coupling in Unbounded Domains 635 ′ 2.1. Non-uniform mean flow ′ @p n p 0; (15) @n Equations (7) and (8) allow for description of wave • non-reflective boundary⋅ ∇ = – the= wave is damped/ propagation without a mean flow. If the derivation as- absorbed, the definition of non-reflective bound- sumes the presence of mean velocity distribution vi, ary condition is more complicated and cannot be equations take on a more complex form. reduced to simple Dirichlet or Neumann condi- In one-dimensional case equations take the form ′ ′ tions. (v1 u, v1 U, p p, p 0, x1 x): The main purpose of non-reflective boundary con- @p′ @u @p @U = = ρc=2 =U =p ; (10) dition (NRBC) is to mimic unlimited physical domain @t @x @x @x in bounded numerical domain. It can be done in many ways, including characteristic NRBC (Giles, 1990; @u = −1 @p − @u − @U pU @U U u ; Atkins, Casper, 1994; Koloszár et al., 2019), ra- 2 (11) @t ρ @x @x @x c ρ @x diation boundary condition (Hagstrom, Goodrich, ′ ′ while in two-dimensional= − − case− (v1 −u, v2 v, v1 U, 2003), absorbing boundary conditions (Givoli, 2008; ′ v2 V , p p, p 0, x1 x, x2 y)(Povitsky, 2000): Kosloff, Kosloff, 1986), not to mention Perfectly = = = Matched Layer (Berenger, 1994; Bermudez et al., @p @u @v @p @p = = ρc2= = U= V 2008).