2019 International Conference on Applied Mathematics, Modeling, Simulation and Optimization (AMMSO 2019) ISBN: 978-1-60595-631-2

Modeling and Analysis of Helmholtz Cavity Basing on One-dimensional Plane Wave Propagation

Zhen LI and Yong CHEN*

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, China

*Corresponding author

Keywords: Helmholtz resonance, Volumetric measurement, One-dimensional plane wave model.

Abstract. Propellant volumetric measurement based on the tank cavity resonant dynamics may be a

promising alternate for satellite to improve its performance. The tank could be modeled as

‘Helmholtz resonator’, and volume of the cavity can be calculated from the resonant frequency

according to the theoretical formula. Unlike the traditional Helmholtz resonation modelling, where

cavity is taken as an ideal acoustic component, this paper gives mathematical modelling based on

one-dimensional plane wave approximation, with compressibility of liquid being concerned.

Comparisons with simulated and experimental results validate the present model. Good consistency

could be found between theoretical and numerical results. Meanwhile, they are slightly larger than the

experimental results. Results prove that present model can accurately predict the resonance resonant

frequency of the tank, which may be used in the propellant volumetric measurement.

Introduction

High-accuracy volumetric measurement is important for satellite to improve its performance,

which means making full use of resources and increasing effective load. There have been several

methods to satisfy the measurement requirement according to literature, such as PVT (Pressure,

Volume, Temperature), PGS (Thermal Propellant Gauging) and BK (Book-Keeping) method.

Problems of above three methods are considered by Yendler[1], which shows that errors generate due

to nonideal gas in PVT method and accumulate with time in BK method. Meanwhile, accuracy of

PGS method is inversely proportional to the propellant fill level. Compared to these traditional

methods, cavity-resonance measurement, which is based on Helmholtz resonance, may be a possible

alternate for volumetric measurement. Corresponding ground experiment by Webster[2] shows that

the measurement accuracy is better than ±0.1% of the resonator’s volume (3L) and the repeatability is

generally ±1ml.

The tank could be modeled as ‘Helmholtz resonator’, which consists of an empty cavity and a

small opening neck, with external space is connected through neck. In traditional model, resonator is

commonly taken as mass-spring model, with air in cavity is supposed as spring and all mass

significant is concerned in neck[3]. Corresponding formula of resonant frequency is given by

neglecting parameters such as the specific cavity geometry, the existing propellant, to mention a

few[4]. Cavity volume and geometries of open neck determine the resonant frequency. Conversely, if

the geometries of the neck are determined, the volume of the cavity can be calculated from the

resonant frequency. There have been many researches on the resonant frequency of Helmholtz cavity

in the literature. Length factor is introduced that represent additional air motion on one side of plate,

which is based on infinite plate with cylindrical hole[5]. Ingard[6] given the end correction on the

side of aperture that opens into finite cavity, with several geometries of aperture and cavity is

selected. Alster[5] considered the motion effects of the mass particles inside the resonator, and

analyzed the influence of the shape of vessel, such as sphere, prism, frustum of a cone, and cylinder.

Attentions are concentrated on the effects caused by shape of neck and tank, while other parameters,

like the compressibility of the propellant, are neglected. Commonly, propellant is assumed to be

incompressible compared with the compressible gas and the propellant-gas boundary is model as

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rigid wall. However, compressibility of the propellant may also affect the cavity resonance and will

result in performance reduction of volumetric measurement.

This paper therefore theoretically models the resonator based on a one-dimensional wave

approximation, with compressible liquid being concerned. Theoretical formula is given by solving

the corresponding determinant problem of the coefficient matrix. Results of present model are

compared with the numerical simulations and experimental data. The structure of this paper is as

follows. Section 2 gives a basic model under the assumption that the propellant is compressible.

Section 3 compares theoretical predictions with numerical simulation, experimental results. Section 4

concludes the work presented in this paper.

Theoretical Modeling

Geometry of tank is shown in Figure 1, with the neck and cavity is modeled as cylinder. The tank

consists of three main parts, namely the opening neck, the gas segment and the propellant segment,

meanwhile the inlet of neck opens to external atmosphere. The cross-sectional area of neck and cavity

are 0S , 1S respectively, with corresponding length being 0L , 2L . Length of gas segment is 1L .

0L

2L

1L

Propellant

Gas

neck

inlet

atmoshpere

0S

1S

Figure 1. Geometry of tank with opening neck connected with atmosphere.

There is assumption that the wavelength of acoustic is far larger than diameter of cavity so that

wave propagates only along axial direction. Another important hypothesis is linearization. Any

quantity F in this paper is decomposed as F F F , where F is the base part and F is the

fluctuating part. Effects of viscous dissipation and thermal conduction are neglected, meanwhile gas

is assumed as ideal. Then we have the model based on one-dimensional plane wave approximation as

shown in Figure 2. Subscripts like ‘0’, ‘1’, ‘2’, ‘n’ are used to represent variables in the neck, gas,

propellant and arbitrary segment.

x1x L

u pR

d o w nR

2x L0x

0n 1n 2n

+

-

, , ,n n n np v T

bottom0x L

gas-propellant interface

0S 1S

inlet

Figure 2. One-dimensional resonator based on linearization hypothesis.

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The base part in the tank is stationary, then we reach the following constraints for the base flow

0 1 2 0 1 2 0 1 2 0 1, , 0, .p p p T T T v v v (1)

With , p ,T , v represent density, pressure, temperature and velocity respectively.

The acoustic disturbance could be considered as superposition of waves propagating upstream

(denoted by ‘-’) and downstream (denoted by ‘+’). As the plane wave is considered, the perturbations

in time domain(denoted by superscript ‘ ’ ) could be expressed by perturbations in frequency

domains(denoted by superscript ‘ ^ ’ )

ˆ ˆ, exp i , , exp in n n n n np x t p t x c p x t p t x c (2)

Disturbances of the acoustic velocity could get according to the pressure

1 pv dt

x

(3)

Combing Eq. (2) with (3), the disturbances of the acoustic pressure and velocity could be expressed

by pressure perturbations downstream and upstream

ˆ ˆ ˆ ˆT T

n n n n nv x p x x p p B (4)

With

1 i 1 iexp exp

i iexp exp

n n n n n n

n

n n

x x

c c c cx

x x

c c

B (5)

At the zero location 0x , there is a sudden change of cross section with area ratio ( 1 0/C C ). Noting

that the Mach number is zero, and there is no heat input to system, the velocity jump relation in terms

of volumetric flow rate could be expressed as [7, 8]

0 0 1 1 0 1, .C v C v p p (6)

Then we have the following expression

0

0 1

10 1

0 1

ˆ ˆ(0) (0) (0) , (0) .

ˆ ˆ1

Cp p

Cp p

A B B 0 A (7)

There is same relation like Eq. (6) at liquid-gas boundary 1x L , with cross section is unity

0 1 1 1 0 1 1 1, .v L v L p L p L (8)

Similar to Eq. (7), at the location 1x L , we have

1 1 1 1

12 1

2 1

2 1

ˆ ˆ 1( ) ( ) ( ) , ( ) .

ˆ ˆ 1L L L L

p p

p p

B A B A (9)

The acoustic reflections at the inlet ( 0x L ) and bottom ( 2x L ) should be considered. By

introducing reflection coefficients ( upR and downR ), we have the boundary conditions

0up up up

0 0 0 0 0 0

0

ˆ0, exp i / exp i / .

ˆ

pL L L c R L c

p

F F (10)

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2down down down

2 2 2 2 2 2

2

ˆ0, exp i / exp i / .

ˆ

pL L R L c L c

p

F F (11)

Inserting Eq. (11) and (9) yields

1 1 1

1 1down

2 2 1

1

ˆ( ) ( ) ( ) 0.

ˆL L L

pL

p

F B A B (12)

Combining Eqs. (7), (10)and (12) gives the model which considers the compressibility of the

propellant, written in matrix form

4 4 0 0 1 1ˆ ˆ ˆ ˆ0, , , , .

Tp p p p G X X (13)

With

1 1 1

up

0 1 2

4 4 0 1

1down

1 2 2 2 1

(0) (0) (0)

( ) ( ) ( )L L L

L

L

F 0

G A B B

0 F B A B

(14)

To get the resonance frequency, it means the matrix has a non-zero solution, leading to the vanishing

of the determinant 4 4det 0 G

Simulation and Validation

The validity of proposed model is demenstrated by comparing resonant frequency of theoretical

calculation with numerical simulations and experimental results by Webster[2]. Corresponding

experimental configurations are shown in Figure 1 with the neck’s length and diameter being 170mm

and 44mm respectively. The length and diameter of the cylindrical cavity are 190mm and 141.8mm .

Water is used to represent propellant. The gas and liquid density are 31.29kg/m and 31000kg/m ,

corresponding sound speed are 0 1 343m/sc c and 2 1500m/sc .

At the bottom ( 2x L ), fully reflection leads to the rigid boundary condition, with down 1R . At the

inlet of the tank, the pressure disturbance will not vanish immediately, but propagate to the infinity

where the pressure disturbance reduces to zero. According to literature contributions, end correction

i is introduced to model such effect. The neck length therefore changes to

0 0 i

*L L (15)

Where pressure disturbance vanishes at the inlet up 1R . Ingard[6] gave a theoretical prediction of

the end correction with 1/2

i 0= 8 / 3 ( / )C . When 1L is determined, resonant frequency of present

model could be given by calculating the determinant of Eq. (14). It should be noticed that classical

model introduce another end correction o for outlet that opens into cavity, called o . In present

model, the correction length i should be introduced in the present model to accept the scattering at

the inlet of the opening neck, but o is not required as the perturbation dynamics in cavity has been

modeled.

Numerical simulations are implemented by COMSOL. In the numerical simulations, the

configurations keep consistent with experimental rig. To consider the end correction, we impose a

large sphere around the model, with the diameter being extremely larger than that of the neck, to

represent the external environment. Pulse vibration is used to excite the resonance frequency of the

system to mimic the flow-excited noise generations[9]. Two fixed point is chosen, with one as

acoustic source meanwhile another point is used for recording the pressure perturbation. The

spectrum peak is obviously the resonate frequency of resonator.

Figure 3a shows results of theoretical calculation, numerical simulation and experiment under

different volume of liquid. Physically speaking, the volume of the gas segment decreases with the

increase of the water filling level, which leads to enlarge of corresponding resonant frequency.

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Results in Figure 3a shows that the resonant frequency becomes larger along with the increase of the

water filling level. Good consistency could be found between the present model and numerical

simulation, while the experimental results seem a little lower than the theoretical and numerical

predictions.

Specifically, We define deviations as /p w wf f f and /s w wf f f , where pf , sf and wf are

the resonant frequencies, obtained from theoretical calculation. numerical simulations and

experimental results conducted by Webster respectively. Figure 3b plot the deviations of the

theoretical predictions and experiment, numerical simulations and experiment. Obviously, Figure 3b

shows that both of two deviations decrease when the water fills the cavity. A possible reason is that

the value of fraction is inversely proportional to that of the denominator, which is the growing

experimental results wf . Meanwhile, deviations of the theoretical predictions and experiment are

slightly larger than deviations of numerical simulations and experiment. Besides, the close lines of

deviation show good consistency between theoretical and numerical results. Results shown in Figure

3 prove the theoretical model based on one-dimensional plane wave propagation well predicts

resonant frequency of Helmholtz resonator with compressibility of liquid being considered.

Figure 3. Resonant frequencies obtained from theoretical prediction, numerical simulation and experimental test under

different water filling level. a. resonant frequencies for different water filling level. b. the changing deviation

for different water filling level.

Conclusions

This paper theoretically models Helmholtz resonator based on the one-dimensional plane wave

approximation, with compressibility of liquid being considered. Predicted resonant frequency by the

present model is compared with numerical simulation and experimental results. Good consistency

can be found among the present model and numerical simulation, while the experimental results seem

a little lower than the theoretical and numerical predictions. Results prove that one-dimensional

model can accurately predict the resonance resonant frequency of the tank, which may be used in the

propellant volumetric measurement.

References

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Meeting and Exhibit, 2006.

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and solids, Sensors, 10 (2010) 10663-10672.

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liner, Journal of Sound and Vibration, 333 (2014) 3536-3549.

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Journal of Sound and Vibration, 178 (1994) 337-348.

[5] M. Alster, Improved calculation of resonant frequencies of Helmholtz resonators, Journal of

Sound and Vibration, 24 (1972) 63-85.

[6] U. Ingard, On the theory and design of acoustic resonators, The Journal of the acoustical society of

America, 25 (1953) 1037-1061.

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flame at zero and non-zero Mach numbers, Combustion and Flame, 162 (2015) 60-67.

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