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This study focuses on a noise emitted from a radial underexpanded jet. The underexpanded jet is well known as one of supersonic jets and it is formed when the pressure ratio across a convergent nozzle is more than the critical value. The underexpanded jet has been used in many fields, such a propulsion of rocket, laser cutting and a glass tempering process. Such jet is exhausted from a circular nozzle. The expansion wave, the compression wave and the shock wave are periodically formed in the jet, which leads to quasi-periodical shock-cell structure. On the other hand, an underexpanded jet radially issues from intake and exhaust valves of an internal combustion engine and a pressure control valve. The radial underexpanded jet does not have the quasi- periodical shock-cell structure such as the jet from the circular nozzle, its cell length decreasing along jet axis. In this study, an underexpanded jet radially discharged from a circular slit nozzle, which consists of two circular tubes, is experimentally examined for different nozzle pressure ratios. The jet structure is visualized using Schlieren method and a noise emitted from the jet is measured. Typically, the shock associated broadband noise is analyzed and a relation between the jet structure and the radiation noise frequency is discussed. As a result, it is found that the broadband noise lies on lower frequency than a screech tone and that the 1st cell length plays an important role in broadband noise radiation.

The underexpanded jet, which issues form a rectangular or a circular nozzle, is well known as one of the supersonic jets and it is formed when the pressure ratio across a convergent nozzle is more than the critical value. The underexpanded jet has barrel-shaped cell structure which is quasi-periodical along the jet axis. And it is characterized by the radiation of sounds such as a narrowband screech with a high sound pressure. The quasi-periodical cell structure is responsible for the screech phenomena. Disturbance travels at the jet boundary and interacts with the shock present at each cell node, which leads to radiation of sound wave. The sound wave propagates upstream and interacts with the jet at the nozzle lip, and then new disturbance is originated and goes downstream again. This loop repeats itself and a noise with a narrowband frequency is strongly emitted. This phenomenon has been studied [

In the present study, the underexpanded jet radially spreads out like a disc. Thus, the underexpanded radial jet does not have quasi-periodic cell structure. The underexpanded radial jet is generated inside the pressure reducing valve of the piping and near the poppet valve of the internal combustion engine [

In this study, therefore, a visualization of the jet structure and an acoustic measurement were performed for the underexpanded radial jet. From the results, an influence of the jet structure on acoustic properties of broadband noise was discussed.

A schematic view of the experimental apparatus used in this study is shown in

In this study, the ratio P R ( = p 0 / p a ) was varied from 2.0 to 4.8 in increments of 0.2, where p 0 is the stagnation pressure in the tank and p a the atmospheric pressure.

The air supplied to two tanks goes through a circular pipe fixed on each tank. The circular pipes are arranged facing each other as shown in _{0} = 14 mm and an outer diameter d_{1} = 16 mm. Also, the distance between the ends of the pipes is kept b_{0} = 2 mm. In this study, the nozzle composed of these two circular pipes is called “slit nozzle”. The air flow through the pipe accelerates up to the nozzle exit and reaches a sonic speed. The air is underexpanded and radially discharged from the nozzle exit to the atmosphere.

Shadowgraph method was employed for visualization of jet.

The noise emitted from the jet is measured by the microphone. The sound pressure history of the noise is obtained through an A/D board with 1 MHz sampling.

FFT analysis of the sound pressure histories is carried out to find frequency characteristic of the noise radiating from the jet. _{b} and the screech as f_{s} as shown in

_{s} and f_{b}. The ordinate represents frequency and the abscissa pressure ratio. It can be seen that both f_{b} and f_{s} decrease as the pressure ratio rises. f_{b} is found at the pressure ratio of 2.8 or more. As shown in _{1} = 16 mm. The second cell length is shorter than the first cell length as shown in

Krothapalli et al. [

S t = f ⋅ D U 0 (1)

where f is the screech frequency, D the nozzle width, and U_{0} the calculated mean velocity at the nozzle exit for ideally expanded isentropic flow. Krothapalli shows that Strouhal number is approximately obtained by the following empirical equation with high accuracy.

S t = K ⋅ P R − 3 2 (2)

where PR is the pressure ratio, p 0 / p a and K is the constant value, 0.89.

In the present study, the Strouhal number can also be obtained by the following Equation (3).

S t = f ⋅ b U j (3)

where f is the frequency, b the cell width, and U_{j} the jet velocity at fully expansion.

Unlike the rectangular jet, the cell width of the radial jet narrows toward the downstream. Let b ¯ be the average cell width and b ¯ can be expressed by the following Equation (4) using a continuous equation.

b ¯ = 1 L ∫ x 1 x 2 r 0 b 0 r 0 + x d x (4)

where L is the cell length, r 0 = d 1 / 2 the radius of the nozzle, b_{0} the nozzle exit width. x 1 and x 2 are locations of a start and an end of any cell, respectively. In Equation (4), L_{1} is substituted for L and the average first cell width b ¯ 1 is calculated, and the average second cell width b ¯ 2 is calculated in the same way.

_{b} calculated with cell width b ¯ 1 or b ¯ 2 . The ordinate is the Strouhal number, and the abscissa the pressure ratio. The red line is the Strouhal number of the screech obtained by the empirical formula, Equation (2), of Krothapalli [

The Strouhal number with the cell width b ¯ 1 shows good agreement with the empirical formula. From this, it can be considered that the frequency f_{b} is influenced by the screech phenomena related to the first cell rather than that related to the second cell.

The screech phenomena have been reported to be caused by feedback mechanism due to disturbance [

1 f = L U c + L c a (5)

where U_{c} is the advection velocity of the disturbance, L is the cell length, and c_{a} is the sound speed in the surrounding atmosphere [_{c} of the disturbance is given by the following equation, Equation (6).

U c = α U j = α M j c j (6)

where U_{j}, M_{j} and c_{j} are the jet velocity at fully expansion, the jet Mach number, and the sound speed of jet, respectively. And α is a coefficient for the advection velocity of disturbance and is taken to be α < 1 from various papers [

Assuming that the sound speed at stagnation point in the tank c_{0} is equal to c_{a}, Equations (5) and (6) yield the following Equation (7).

α = 1 M j 2 + κ − 1 2 ( c a f ⋅ L − 1 ) 2 (7)

where κ is the specific heat ratio, and κ = 1.40 for air. From Equation (6), α was calculated for each pressure ratio with frequency f_{b} and first cell length L_{1}. The sound speed c_{a} at the temperature inside the laboratory (20˚C) was used.

α was 0.47. The value is very suitable, compared with the values of α reported so far, i.e. α = 0.42 by Suda [_{b}.

The underexpanded radial jet is experimentally simulated. The visualization of the jet structure and an acoustic measurement were carried out. From the results, an influence of the jet structure on acoustic properties of broadband noise was discussed and the following conclusions are drawn.

1) The cell length increases as the pressure ratio rises. And it was found that the length of the second cell is shorter than that of the first cell.

2) As the pressure ratio increases, both f_{b} and f_{S} decrease and the broadband noise is developed.

3) The frequency f_{b} is affected by the first cell length and, it is conceivable that screech phenomena of the first cell are closely related to the broadband noise.

It is well known that the jet issuing from a rectangular or a circular nozzle has quasi-periodic cell structure. This is mainly responsible for the screech phenomena. On the other hand, although the radial jet does not have quasi-periodic structure and the cell length becomes shorter along the jet axis, the jet emits the broadband noise and the screech. Therefore, we will need to study what kind of mechanism exists in radiation of the broadband noise and the screech from radial jet without quasi-periodic structure.

The authors declare no conflicts of interest regarding the publication of this paper.

Hiramoto, Y., Suzuki, H., Endo, M. and Sakakibara, Y. (2019) A Study on Broad Band Noise Emitted from Underexpanded Radial Jet. Open Journal of Fluid Dynamics, 9, 72-81. https://doi.org/10.4236/ojfd.2019.91005