A slug length calculation for a contraction with mean flow between two half cylinders

Wei Na 1

Department of Mechanics and Maritime Sciences, Chalmers University of Technology, SE-41296 Gothenburg, Sweden

Dong Yang Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, PR-518055 ShenZhen, China

Aswathy Surendran Department of Engineering Physics and Computation, Technical University of Munich, GE-85747 Munich, Germany

Susann Boij KTH Royal Institute of Technology, Marcus Wallenberg laboratory for Sound and Vibration Research, Deptartment of Engineering Mechanics, SE-10044 Stockholm, Sweden

Aimee Morgans Department of Mechanical Engineering, Imperial College London, SW72AZ London, United Kingdom

Huadong Yao Department of Mechanics and Maritime Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

ABSTRACT A slug length is widely used to describe additional mass inertia at low frequencies in duct acoustics. It is often used in acoustic energy analysis, e.g. as one of the inputs for semi-analytical or empirical models to obtain the acoustic reflection and transmission coe ffi cients. However, the calculation of slug length is usually empirical and limited to certain conditions, such as simple geometric configurations, low frequencies, no mean flow, etc. In this paper, the slug length at a contraction with mean flow between two half cylinders is calculated by di ff erent methods: solving the Laplace’s equation numerically, solving the Helmholtz equation, and using the Cummings – Fant equation based on the numerical results of frequency-domain linearized Navier-Stokes equations. Both the frequency-dependance and the mean flow e ff ect are discussed. The calculated slug length can also be used as a crucial input, for example, in the Dowling and Hughes slit model and modified Cummings slit model to predict the acoustic scattering at tube rows in the presence of a cross mean flow[ 7 ].

1 wein@chalmers.se

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

1. INTRODUCTION AND NUMERICAL CONFIGURATION

The use of heat exchangers in aeroengines can have a large impact on thermoacoustic instabilities, since the heat exchangers influence the heat transfer as well as the acoustic perturbations in the thermoacoustic feedback loop. Here, we only consider heat exchangers with the structure of tube rows subject to cross-flow as Fig. (1) shows, and focus on the acoustic response of heat exchanger tube rows without any heat transfer. Multiple physical phenomena are involved when there is an incident acoustic wave and flow going through the contraction of the tube row. Acoustic waves will be scattered at the cylinders, leading to both sound transmission and reflection; In addition, acoustic energy is absorbed within the boundary layer due to the wall friction on the cylinder surfaces; When flow passes through the contraction, a jet flow and recirculation zones are formed in the formation of the shear layers. Such shear layers are unstable to low-frequency acoustic perturbations, and energy may be absorbed or amplified due to coupling between acoustic waves and vorticity waves. Thus, it is of interest to make an analysis of flow-acoustic interactions for tube row heat exchangers.

Figure 2: The simplified two-dimensional numerical geometry of the tube rows investigated in the paper, where R is the diameter of the cylinder, R = 5.9mm, H p is the duct height H p = 17.2mm, yields the open area ratio η = 0 . 31. The duct length up- and downstream of the half cylinders are L u = 40 R , L d = 40 R , respectively.

Figure 1: Schematic of the three- dimensional tube rows in the presence of cross-flow.

2. SLUG LENGTH CALCULATION

The slug length is an important criteria for acoustic energy analysis [ 2 ]. Further, it is an input for the analytical model proposed by [ 1 ] to obtain the acoustic reflection and transmission coe ffi cients. However, the evaluation of the slug length is often empirical and limited by many factors, e.g., di ff erent geometries and flow conditions. For example, T. Luong [ 5 ] proposed a model for the slug length for a bias-flow aperture, that is ℓ = 2 l 0 + l ω , where l 0 is the end correction by the irrotational flow and l ω is the thickness of the orifice. In this case, the slug length is a constant value independent of frequency, and is thus limited to describing the behaviour of low-frequency incident acoustic waves when a jet flow is present. However, the value of the slug length changes at higher frequencies due to that the pulsations in the momentum flux cannot continue to be absorbed by acceleration of the incompressible jet [ 5 ]. Here, we propose a methodology to get the frequency dependent slug length, where the slug length is calculated from the numerical LNSE results through using the Cummings – Fant equation [ 6 ][ 3 ][ 4 ]. By the work conducted here, the slug length for the geometry depicted in Fig. (2) is obtained numerically considering a cross-flow with di ff erent inlet flow speeds and at di ff erent frequencies. This analysis establishes a link between the numerical model and the analytical model, providing a physical insight for the vortex-sound interaction at tube rows in the presence of flow. First, the slug

length in the absence of flow is calculated by solving the Laplace equation numerically, as presented in Section 2.1. The slug length as a function of frequency, and with cross-flow is then calculated from the LNSE numerical results in the presence of flow in combination with the use of the Cummings - Fant equation as described in Section 2.2.

2.1. Numerical solution of the Laplace equation The slug length is an e ff ective length of the ‘slug’ of fluid flowing through the cylinder contraction. For the current heat exchanger configuration, the diameter of the heat tubes is much smaller than the acoustic wavelength. At low frequencies, the contraction region can be considered acoustically compact and the flow can be regarded as incompressible, and can be described by the Laplace equation in the limit of zero frequency. The slug length is an e ff ective length of the annular ‘slug’ of fluid flowing through the cylinder contraction. The inertia of this ‘slug’ equals the localised high speed irrotational component of the unsteady flow through the contraction [ 6 ]. In Cartesian coordinates, considering a two-dimensional, irrotational, incompressible flow, the stream function then satisfies the Laplace equation (1):

∂ψ 2

∂ x + ∂ψ 2

∂ y = 0 (1)

where the stream function ψ ( x , y ) is assumed to satisfy the boundary condition ∂ψ/∂ y = 1 both at the inlet and outlet of the duct, indicating the dimensionless inlet flow speed U in = 1. Besides, ψ = 0 and ψ = Hp / 2 are given as boundary conditions for the centerline and the upper duct wall, respectively. The slug length ℓ can be determined from the numerical solution of the Laplace’s equation by the formula: A K 0 = ℓ = Z L

∂ψ

∂ y − 1 ! dx (2)

− L

where A is the cross-sectional area of the duct, L is the half length of the tube and K 0 is the Rayleigh conductivity.

Figure 3: Streamlines for the flow past a half cylinder.

2.2. Numerical analysis from LNSE simulations In order to include the e ff ect of mean flow and of frequency, the slug length is now determined based on the LNSE results for the heat exchanger. The slug length is calculated from the LNSE numerical results using Cummings-Fant equation:

ℓ = A ( ˆ p + − ˆ p − )

i ωρ 0 ˆ Q (3)

where ˆ Q is the volume flux, ˆ p + and ˆ p − are the pressure perturbation amplitudes before and after the tube contraction.

0.026

0.02

0

0.024

-0.02

0.022

-0.04

0.02

Imag (l)

Real (l)

-0.06

0.018

-0.08

0.016

-0.1

LNSE U0=6.1 m/s LNSE U0=8.38 m/s LNSE U0=9.82 m/s LNSE U0=12.18 m/s LNSE U0=14.52 m/s LNSE U0=16.3 m/s

LNSE U0=6.1 m/s LNSE U0=8.38 m/s LNSE U0=9.82 m/s LNSE U0=12.18 m/s LNSE U0=14.52 m/s LNSE U0=16.3 m/s

0.014

-0.12

0.012

-0.14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strouhal number (St)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strouhal number (St)

(a) real( ℓ )

(b) imag( ℓ )

Figure 5: Slug length ℓ calculated from the LNSE numerical results in the presence of flow.

0.025

0.02

0

0.02

-0.02

-0.04

0.015

Imag (l)

Real (l)

-0.06

0.01

-0.08

Hemholtz - TMM Numerical Laplace LNSE U0=6.1 m/s LNSE U0=8.38 m/s LNSE U0=9.82 m/s LNSE U0=12.18 m/s LNSE U0=14.52 m/s LNSE U0=16.3 m/s

Hemholtz - TMM Numerical Laplace LNSE U0=6.1 m/s LNSE U0=8.38 m/s LNSE U0=9.82 m/s LNSE U0=12.18 m/s LNSE U0=14.52 m/s LNSE U0=16.3 m/s

-0.1

0.005

-0.12

0

-0.14

0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency

0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency

(a) real( ℓ )

(b) imag( ℓ )

Figure 4: Slug length ℓ calculated from the LNSE numerical results in the presence of flow.

In Figure 4 on page 4, the real part and imaginary part of the slug length calculated from the LNSE numeral results with various incoming flow speeds are presented as function of frequency. The real part of the slug length represents the inertial e ff ect and the imaginary part represents the added drag e ff ect caused by the small perturbations. The numerical solution of Laplace equation described in Section 2.1, is denoted by blue stars. A slug length for no-flow condition is calculated as well by solving the Helmholtz equation, plotted as a solid black line in Figure 4 on page 4. It is observed that, the slug length calculated from Laplace equation (1) does not show any frequency dependency. It agrees with the solution from Helmholtz equation at lower frequencies, as expected. We expect the slug length calculated from the LNSE numerical results in the presence of flow to agree with that calculated from the Laplace equation and the Helmholtz equation when there is no flow, at low frequencies. However, in Figure 4 on page 4, the slug length calculated in the presence of flow is smaller than the one obtained without flow. With the increasing flow speeds, the slug length increases, and tend to the no-flow results. In Figure 5 on page 4, the real and imaginary parts of the slug length calculated from the LNSE numeral results are also plotted as a function of the Strouhal number. Usually in the small Strouhal number region, the hydrodynamic e ff ect dominates and these are not modelled by the Helmholtz equation or the Laplace equation. For higher Strouhal numbers in combination with higher frequencies, the acoustic e ff ects together with convection e ff ects will dominate. In Figure 5 on page 4, we can observe these di ff erent regions and it is also clear that at Strouhal number around 1, the slug length values for di ff erent flow speeds scales with the Strouhal number. It should be noted

that the flow separation and the sound propagation are in the di ff erent time scale. LNSE numerical methodology is a liner approach based on a steady mean flow field in frequency domain, thus the information about the development of the fluid, such as, unsteady flow separation and the generation or decay of the vorticity, is missing in the numerical simulations. For small Strouhal numbers, however, the time scale of the flow field is shorter than the acoustic time scale, implying that the time dependence in the flow might play a role. Further studies are needed to investigate this parameter region.

3. SUMMARY

In this paper, di ff erent numerical methods to calculate slug length are introduced for both with and without flow. The frequency dependence of slug length in the presence of mean flow is discussed. Further usage of slug length to calculate the acoustic scattering coe ffi cients will be presented in a extented journal paper.

ACKNOWLEDGEMENTS

This research is financially supported by the EU project ’Installed adVAnced Nacelle uHbr Optimisation and Evaluation (IVANHOE)’, with the grant agreement number 863415.

REFERENCES

[1] Charles Boakes, Aswathy Surendran, Dong Yang, and Aimee Morgans. Acoustic scattering in arrays of orifices, slits and tube rows with mean flow: A comparison. in: Proceedings of the 23rd International Congress on Acoustics: Integrating 4th EAA Euroregio 2019 , 2019. [2] Susann Boij. Flow e ff ects on the acoustic end correction of a sudden in-duct area expansion. The Journal of the Acoustical Society of America , 126(3):995–1004, 2009. https://doi.org/10. 1121/1.3177263 . [3] MS Howe and RS McGowan. On the generalised fant equation. Journal of sound and vibration , 330(13):3123–3140, 2011. https://doi.org/10.1016/j.jsv.2011.01.017 . [4] MS Howe and RS McGowan. Production of sound by unsteady throttling of flow into a resonant cavity, with application to voiced speech. Journal of fluid mechanics , 672:428, 2011. https: //doi.org/10.1017/S0022112010006117 . [5] T Luong, Michael S Howe, and Richard S McGowan. On the Rayleigh conductivity of a bias- flow aperture. Journal of Fluids and Structures , 21(8):769 – 778, 2005. https://doi.org/10. 1016/j.jfluidstructs.2005.09.010 . [6] RM Schleicher and MS Howe. On the interaction of sound with an annular aperture in a mean flow duct. Journal of Sound and Vibration , 332(21):5594–5605, 2013. https://doi.org/10. 1016/j.jsv.2013.05.006 . [7] Aswathy Surendran, Wei Na, Charles Boakes, Dong Yang, Aimee Morgans, and Susann Boij. A low frequency model for the aeroacoustic scattering of cylindrical tube rows in cross-flow. Journal of Sound and Vibration , 527:116806, 2022.