A schematic view of the experimental apparatus used in this study is shown in Figure 1. The air compressed by the compressor passes through the air dryer. It is cooled and dehumidified, and then supplied to the surge tank. The pulsating component is removed there. The compressed air steadily flows into the tanks, finally. The pressure in the tank is measured by a digital manometer, and can be adjusted by a valve installed in the pipe. The unnecessary air is discharged to the atmosphere.

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 Figure 2. The circular pipe have an inner diameter d₀ = 14 mm and an outer diameter d₁ = 16 mm. Also, the distance between the ends of the pipes is kept b₀ = 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. Figure 3 shows a schematic view of the visualization system. A light with a flashing time of 180 ns was used as the light source. The light emitted from the light source is condensed on the pinhole by the condenser lenses. A point light source from the pinhole becomes a parallel ray via a collimator lens, and it is reflected by the mirror “A” set at 45˚ to visualize the jet from the front. The light passing through the jet is reflected by a mirror “B” set at 45˚ and collected by a collimator lens to be photographed with the camera.

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. Figure 4 shows an acoustic measurement system. The microphone is attached to the microphone stand at a position separated by r = 750 mm from the slit nozzle exit. The angle was set at 60˚ with respect to the jet axis. In order to prevent reflection of jet noise from the experimental apparatus, a sound absorbing material is glued to the circular tubes comprising the nozzle and the wall surface of the experimental equipment.

Figure 5 shows jets visualized by shadowgraph method. These pictures are average images of 30 photos randomly taken, respectively. The dark region in the center of the picture is a shadow of the circular pipe and it is larger than the actual diameter due to the optical system. Shock waves cannot be seen when the pressure ratio PR is 2.2, while shock waves surrounding the nozzle can be observed at PR = 2.4. These annular shocks are called “shock ring”. These are referred to as the first shock ring and the second shock ring from the side of the nozzle exit, and it is possible to detect up to the fifth shock ring. At PR = 2.6 or more, the shock ring becomes unclear on the downstream side and one can find up to the second shock ring. As the pressure ratio PR increases, the distance between the adjacent shock rings increases and the first and the second shock rings become stronger because they can be seen clearly.

Figure 6 shows a photograph taken randomly at PR = 4.8 and it is not an average image. Since the shock wave ring is closely circular, some points are selected on the shock ring and a circle is specified using a least squares method. The radius of the nozzle is subtracted from the evaluated radius of the first shock ring to obtain the first cell length. In same way, the second cell length was estimated by subtracting the radius of the first shock ring from the radius of the second shock ring. Figure 7 shows the cell lengths. The abscissa is the pressure ratio, and the ordinate the length of the cell. The circle denotes the first cell length, and the square the second cell length. It can be seen that the cell length increases with increase in the pressure ratio as for both the first cell and the second cell. Also, the length of the second cell is always shorter than that of the first cell. This is the features of the radial jet.

FFT analysis of the sound pressure histories is carried out to find frequency characteristic of the noise radiating from the jet. Figure 8 shows the result obtained by FFT analysis. The ordinate represents the sound pressure, and the abscissa the frequency. At every pressure ratio, there is a mountain-type broadband frequency component in the low frequency side and a narrowband frequency component in the high frequency side. A turbulent mixed noise is considered to include in this broadband noise. The narrowband component is screech. The frequency where the sound pressure takes a maximum value at the apex of the mountain shape is measured as f_b and the screech as f_s as shown in Figure 8(a).

Figure 9 shows the measured frequencies f_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 Figure 8, along with increase in the pressure ratio rises, both broadband noise and screech develop. Thus, the characteristics of the broadband noise is very similar to those of the screech, this means that the broadband component is caused by the mechanism of feedback loop like the screech phenomena. Kawasaki et al. [10] reported that the single screech frequency was not based on the first nor the second cell but the third cell from the experiment of a radial jet with d₁ = 16 mm. The second cell length is shorter than the first cell length as shown in Figure 7, and the third one is much shorter. It is expected that the first cell or the second cell contributes to the radiation of the broadband noise if the broadband noise is due to the feedback loop via cell structure.

Krothapalli et al. [11] conducted the experiment of underexpanded jet issuing from two types of rectangular nozzle and estimated Strouhal number of screech frequency by the following equation.

S t = f ⋅ D U 0 (1)

where f is the screech frequency, D the nozzle width, and U₀ 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, b₀ the nozzle exit width. x 1 and x 2 are locations of a start and an end of any cell, respectively. In Equation (4), L₁ 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.

Figure 10 shows Strouhal number of frequency f_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 [11] .

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 [12] . The screech frequency f is given by

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 [13] . The advection velocity U_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 [2] [13] [14] [15] .

Assuming that the sound speed at stagnation point in the tank c₀ 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 L₁. The sound speed c_a at the temperature inside the laboratory (20˚C) was used.

Figure 11 shows coefficients related to the advection velocity of the calculated disturbance. The ordinate is the coefficients related to the advection velocity of disturbance, and the abscissa the pressure ratio. The coefficient is 0.37 at PR = 2.8 and it increases gradually and reaches 0.53 at PR = 4.8. The average value of

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 [14] and α = 0.55 by Raman [13] . From this also, it can be considered that screech phenomena with respect to the first cell length are closely related to the frequency f_b.