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Abstract

A hybrid method was explored to investigate the generation and near-field radiation of aerodynamic sound from an unsteady turbulent flow over a two-dimensional open cavity and three-dimensional jet flow. A two-dimensional cavity model was established to study the unsteady flow and radiated jet sound. It was revealed that the radiated sound that generated by the boundary layer separation and vortex impact cavity wall intervened in the front of the cavity, and an obvious interference phenomenon appeared. The far-field radiated sound generated by the cavity presented obvious directivity, and the sound pressure in the area located at 45°–135° interval was much higher. Then, the unsteady turbulence jet noises of the elliptical and rectangular nozzles were analyzed. It was revealed that the scale and intensity of the vortexes generated by the elliptical nozzle were larger than those by the rectangular nozzle. The jet noise of the elliptical nozzle is lower than that of the rectangular nozzle. Besides, the sound pressure distributions of the two nozzles presented obvious directivity. The sound pressure in the short-axis direction of the nozzle section was higher than that in the long-axis direction.

Keywords

Aerodynamic noise hybrid method two-dimensional open cavity three-dimensional jet flow numerical simulation

Introduction

The vortex structure is the main aerodynamic sound source in the flow field.¹ The accuracy of the sound source prediction has great influence on the calculation of the aerodynamic sound. Lighthill² was the first to describe the aerodynamic sound source using mathematical method. For some complex flow structures, the calculation of the aerodynamic sound source using the Lighthill stress tensor would be extremely complex and the prediction cost of some engineering noise would become unacceptable. With the rapid development of high-speed computer, some scholars employ the direct numerical simulation (DNS) method to study the jet noise. Freund and colleagues^3,4 used the DNS method to simulate the supersonic and subsonic jet noise with the Reynolds number of 2000 and 3600 and the Mach number of 1.92 and 0.9, correspondingly. The simulated flow field data and the total radiated sound pressure level were in good agreement with the experimental results carried out with the same Mach numbers. The superiority of the DNS method in jet noise calculation was fully proved.

The DNS method propounds high-performance computer equipment when used to simulate the high Reynolds number turbulence. Therefore, the “hybrid method” is often used to study the radiated turbulence jet noise.^5–8 Its core idea is to calculate the unsteady viscous compressible or incompressible flow first and then to calculate the radiated sound field by the inviscid linear Euler equation (LEE) or the wave equation. It not only greatly reduces the cost of the aerodynamic noise simulation but also makes the far-field acoustic calculation possible. As the calculation cost of the large eddy simulation (LES) method is much smaller than that of the DNS method, it is often adopted in the hybrid method to calculate the unsteady turbulence jet flow and the results are then used to forecast the radiated noise. The studies by Wang and Moin⁹ and Zhao et al.¹⁰ showed that the LES method can be used to simulate the low frequency and part of the high-frequency radiated jet noise at high Reynolds numbers with higher accuracy. The process of generation of the jet noise is an unsteady process, while the LES method performs well in the calculation of the unsteady data varying with time.¹¹ Therefore, the LES method would be the most effective computing tool to study the turbulence jet noise instead of the DNS method in a period of time in the future.⁵

Bechara et al.⁸ added the extracted sound source term of the turbulence field to the right of the Euler equation to calculate the sound reflection and transmission in the shear flow. Bailly et al.⁶ employed a simplified LEE to study the radiated sound flow of two symmetrical rotating vortexes. The method was found to be effective to avoid the growth of the instability wave, and the calculated results were in good agreement with the classical solutions. McMullan et al.¹² applied the LES method to simulate the flow field and obtained the sound source in the near field, then solved the far-field LEE in frequency domain, and next the radiated sound under different jet initial conditions was simulated. The results verified that the method was reliable.

A primary objective of this work was to develop a hybrid method to provide a potential prediction tool for numerical simulation of aerodynamic noise. The LES simulation was used to study the unsteady turbulence field and then the calculation of the sound field was carried out accordingly. The radiated sound in the near field was obtained using the LEE with acoustic source terms. And the Ffowcs Williams and Hawkings (FW-H) equation was used to solve the far-field acoustic. A calculation model was established based on the two-dimensional cavity and the simulation of the unsteady flow, and jet noise was conducted based on the established method. Then, the unsteady turbulence jet noises of the elliptical and rectangular nozzles were analyzed using the developed hybrid method.

Mathematical model

The flow field model

The Reynolds-averaged Navier–Stokes (RANS) turbulence model is operated based on time homogenization model. It is suitable only when the turbulent flow pulsation frequency is lower than the time scale. For the calculation of unsteady turbulent flow with transition process, the RANS turbulence model cannot catch the special flow structure accurately. Usually, the quasi steady-state flow field is calculated using the RANS model, and then the transient flow characteristics are studied by the LES model. Finally, the simulation of aerodynamic sound is carried out according to the simulation results of the LES model. Both the computational cost and the counting period are reduced.

The LES model is a turbulent description method between the DNS method and the Renault time-averaged method. The filter function is adopted to classify the flow structures, and the turbulence is calculated directly by the unsteady Navier–Stokes equation. The sub-grid stress (SGS) model is set up to link large-scale vortexes and small-scale vortexes.

In the LES model, the unsteady Navier–Stokes equations are filtered by the spatial grids, as expressed by the following equations

\frac{\partial}{\partial t} (ρ {\bar{u}}_{i}) + \frac{\partial}{\partial x_{i}} (ρ {\bar{u}}_{i} {\bar{u}}_{j}) = - \frac{\partial \bar{p}}{\partial x_{i}} + \frac{\partial}{\partial x_{i}} (μ \frac{\partial {\bar{u}}_{i}}{\partial x_{j}}) - \frac{\partial τ_{ij}}{\partial x_{j}}

(1)

\frac{\partial ρ}{\partial t} + \frac{\partial}{\partial x_{i}} (ρ {\bar{u}}_{i}) = 0

(2)

where the variables with over line are the filtered variables, $τ_{ij} = ρ {\bar{u}}_{i} {\bar{u}}_{j} - ρ {\bar{u}}_{i} {\bar{u}}_{j}$ is the lattice stress, ρ is the density, u_i is the velocity, and p is the pressure.

Equations (1) and (2) are not closed as the SGS stress is unknown. Therefore, the lattice model is established using the related physical quantities to solve the equations. In this article, the Smagorinsky–Lilly lattice model is adopted which supposing the SGS stress to be

τ_{ij} - \frac{1}{3} τ_{kk} δ_{ij} = - 2 μ_{t} {\bar{S}}_{η}

(3)

where $μ_{t}$ is the turbulent viscosity of the lattice scale, $μ_{t} = {(C_{s} Δ)}^{2} | \bar{S} |$ , $| \bar{S} | = \sqrt{2 {\bar{S}}_{ij} {\bar{S}}_{ij}}$ , ${\bar{S}}_{ij} = (1 / 2) (\partial {\bar{u}}_{i} / \partial x_{j}) + (\partial {\bar{u}}_{j} / \partial x_{i})$ , $Δ = (Δ_{x} Δ_{y} Δ_{z})^{1 / 3}$ is the grid scale of i axis, and $C_{s} = 0.1$ is the Smagorinsky constant.

LEE with source term

The variables in the flow field can be decomposed into time-averaged terms and fluctuation terms. Then, the variables are inserted in the unsteady Navier–Stokes equation, and the viscosity fluctuation sources are omitted. The following equations can be obtained after rearranging

\frac{\partial ρ'}{\partial t} + {\bar{u}}_{j} \frac{\partial ρ'}{\partial x_{j}} + \bar{ρ} \frac{\partial u'_{j}}{\partial x_{j}} + u'_{j} \frac{\partial (\bar{ρ} + ρ')}{\partial x_{j}} + ρ' \frac{\partial ({\bar{u}}_{j} + u'_{j})}{\partial x_{j}} = {\bar{R}}_{c}

(4)

\frac{\partial {u^{'}}_{j}}{\partial t} + {\bar{u}}_{j} \frac{\partial {u^{'}}_{i}}{\partial x_{j}} + \frac{1}{\bar{ρ}} \frac{\partial p^{'}}{\partial x_{i}} + \frac{ρ^{'}}{\bar{ρ}} \frac{\partial ({\bar{u}}_{i} + {u^{'}}_{i})}{\partial t} + ({u^{'}}_{j} + \frac{ρ^{'}}{ρ} ({\bar{u}}_{j} + {u^{'}}_{j})) \frac{\partial ({\bar{u}}_{i} + {u^{'}}_{i})}{\partial x_{j}} = {\bar{R}}_{m}

(5)

where ${\bar{u}}_{i}$ , $\bar{ρ}$ , and $\bar{p}$ are time-averaged terms, ${\bar{R}}_{c}$ and ${\bar{R}}_{m}$ are the corresponding source terms. Because the magnitude of flow field variables is much larger than that of the fluctuation variables, the two items, $u'_{j} (\partial (\bar{ρ} + ρ')) / \partial x_{j} + ρ' (\partial ({\bar{u}}_{j} + u'_{j})) / \partial x_{j}$ and $(ρ' / \bar{ρ}) (\partial ({\bar{u}}_{i} + u'_{i}) / \partial t) + (u'_{j} + (ρ' / ρ) ({\bar{u}}_{j} + u'_{j})) (\partial ({\bar{u}}_{i} + u'_{i}) / \partial x_{j})$ , in equations (4) and (5) can be negligible. Then, equations (4) and (5) can be simplified as

\frac{\partial p}{\partial t} + {\bar{u}}_{j} \frac{\partial p}{\partial x_{j}} + \bar{ρ} c^{2} \frac{\partial u_{i}}{\partial x_{j}} = {\bar{R}}_{c}

(6)

\frac{\partial u_{i}}{\partial t} + {\bar{u}}_{j} \frac{\partial u_{i}}{\partial x_{j}} + \frac{1}{\bar{ρ}} \frac{\partial p}{\partial x_{i}} = {\bar{R}}_{m}

(7)

The FW-H equation

The volume integrals are dropped and replaced by a so-called permeable FW-H formulation in Fluent. The sound pressure at an observer location, x, can be written as¹³

p^{'} (\overset{⇀}{x}, t) = {p^{'}}_{T} (\overset{⇀}{x}, t) + {p^{'}}_{L} (\overset{⇀}{x}, t)

(8)

where

\begin{matrix} 4 π p'_{T} (\overset{⇀}{x}, t \end{matrix}) = \int_{f = 0} [\frac{ρ_{0} ({\overset{\cdot}{U}}_{n} + U_{\overset{\cdot}{n}})}{r {(1 - M_{r})}^{2}}] ds + \int_{f = 0} [\frac{ρ_{0} U_{n} {r {\overset{\cdot}{M}}_{r} + a_{0} (M_{r} - M^{2})}}{r^{2} {(1 - M_{r})}^{3}}] ds

(9)

4 π {p^{'}}_{L} (\overset{⇀}{x}, t) = \frac{1}{a_{0}} \int_{f = 0} [\frac{{\dot{L}}_{r}}{r {(1 - M_{r})}^{2}}] d s + \int_{f = 0} [\frac{L_{r} - L_{M}}{r^{2} {(1 - M_{r})}^{2}}] d s + \frac{1}{a_{0}} \int_{f = 0} [\frac{L_{r} {r {\dot{M}}_{r} + a_{0} (M_{r} - M^{2})}}{r^{2} {(1 - M_{r})}^{3}}] d s

(10)

Limited boundary truncation technique

In the aerodynamic noise simulation, the accuracy requirements both in space and time should be satisfied. As a fact, the sound wave propagates in free space, while calculation area is limited, and the boundary conditions of fluid field cannot be considered to be far-field free boundary. Hence, the vortexes generated by unsteady flow field shock the boundary and cause large reflection, which would produce “false wave” phenomenon in the flow field. If the “false wave” is greater than the real sound waves in the flow field, the error will be beyond the allowed range. Therefore, special treatment must be taken on the flow calculation boundary of the flow field.

In order to reduce the influence of the outlet boundary on the acoustic calculation, a damp truncation region is set in the front of the outlet boundary of the calculation region. In the damp truncation region, the grids are stretched gradually, as shown in Figure 1. An additional source term is introduced into the control equation of this region to force the transient calculation results to be infinitely close to the corresponding reference value of the quasi steady flow field.

Figure 1.

Schematic diagram of the damp truncation region.

The additional source term introduced into the control equation of the damp truncation region is as follows

S (\vec{x}) = - α (\vec{x}) (q (\vec{x}) - q_{ref} (\vec{x}))

(11)

where $α (\vec{x})$ is the truncation function which is related to the damping coefficient and the location of the truncation region, $q (\vec{x})$ is the transient flow variable, $q_{ref} (\vec{x})$ is the quasi steady flow variable, and $q_{ref} (\vec{x}) = \sum_{i = n}^{m} q_{i} (\vec{x}) Δ t / \sum_{i = n}^{m} Δ t$ , where $q_{i} (\vec{x})$ is the calculated flow variable of each time step and $Δ t$ is the time step.

For compressible viscous fluids, the control equations of the damp truncation region are

\frac{\partial ρ}{\partial t} + \frac{\partial ρ u_{j}}{\partial x_{j}} = - α (\vec{x}) (ρ - ρ_{ref})

(12)

\frac{\partial ρ u_{i}}{\partial t} + \frac{\partial}{\partial x_{j}} (ρ u_{i} u_{j} + p δ_{ij} - τ_{ij}) = - α (\vec{x}) (ρ u_{i} - {(ρ u_{i})}_{ref})

(13)

\frac{\partial ρ E}{\partial t} + \frac{\partial}{\partial x_{j}} ((ρ E + p) u_{j} + q_{j} - τ_{i j} u_{i}) = - α (\vec{x}) (ρ E - {(ρ E)}_{r e f})

(14)

where E is the fluid energy, $q_{j}$ is the heat flux, the truncation function is defined as $α (\vec{x}) = \max (α_{x}, α_{y}, α_{z})$ , and

α_{x} = \max (θ_{1} (x, x_{\max}), θ_{2} (x, x_{\min}))

(15)

α_{y} = \max (θ_{1} (y, y_{\max}), θ_{2} (y, y_{\min}))

(16)

α_{z} = \max (θ_{1} (z, z_{\max}), θ_{2} (z, z_{\min}))

(17)

where $x_{\max}$ , $x_{\min}$ , $y_{\max}$ , $y_{\min}$ , $z_{\max}$ , and $z_{\min}$ are the maximum and minimum values of the damp truncation region of the x, y, and z boundary and $θ_{1} (x_{i}, {x_{i}}_{\max})$ and $θ_{2} (x_{i}, {x_{i}}_{\min})$ are the function of the truncation boundary

θ_{1} (x_{i}, {x_{i}}_{\max}) = \frac{1}{2} β [1 - \tanh (\frac{{x_{i}}_{\max} - h_{i} - x_{i}}{C \cdot h_{i}})]

(18)

θ_{2} (x_{i}, {x_{i}}_{\min}) = \frac{1}{2} β [1 - \tanh (\frac{- {x_{i}}_{\min} - h_{i} + x_{i}}{C \cdot h_{i}})]

(19)

where $β$ is the damp intensity coefficient which is related to the convergence speed and the sound pressure amplitude, $h_{i}$ is the width of the damp truncation region, $x_{i}$ is the node coordinate function of the damp truncation region, and C is the smoothing constant, C = 0.28. On the premise of guarantee the convergence speed, the magnitude of the damp intensity coefficient of the damp truncation region can be 10⁴.

Solution strategy

In order to solve the flow field and aerodynamic noise field, the finite volume method is used to discrete the calculation region, then the unsteady Navier–Stokes equations are solved. The calculated flow results are substituted into the LEE with source item to compute the acoustic pressure field of near field after proper modification. Thereafter, a control surface is set in the flow field and the sound pressure of arbitrary points of far field employing the FW-H equation. The solution strategy is as follows:

The turbulence field is described by RANS model. When the flow field calculation converges stably, the calculated pressure p, velocity $u_{i}$ , and density $ρ$ are taken as the reference values of the damp truncation region.

The simulations on the unsteady flow field are carried out using the LES simulation on the basis of the steady flow calculation results. Then, the LEE with source terms is employed to modify the transient simulation results. A program is written using the user-defined function (UDF) to solve the liner Euler equation, and the noise sources of the unsteady flow are determined and the sound pressure propagation of near field can be calculated.

Corresponding control surface is set in the flow field and the calculation time step and the grid number of the control plane are adjusted to guarantee the veracity of the sound field calculation. Then, the FW-H equation is applied to solve the field variables of the control surface and to acquire the sound pressure of arbitrary points of far field. A damp truncation region is arranged in the simulation region, and appropriate source terms are introduced into the control equation of the damp truncation region to reduce the influence of the limited flow field boundary on the noise signal.

Numerical simulation of two-dimensional cavity flow noise

A two-dimensional cavity was taken as the calculation model to conduct the unsteady flow dynamic numerical calculation. The length-to-depth ratio of the cavity was $L / D = 2.0$ . The cavity front edge was 5D away from the import boundary of the calculation area and the trailing edge of the cavity was 10D away from the export boundary of the calculation area. The schematic of the cavity calculation area is shown in Figure 2. The import boundary of the calculation area was set to be pressure inlet and the export boundary was set to be pressure outlet, and the pressures were set as the same as the ambient pressure. The damp truncation region was also given in the schematic.

Figure 2.

Schematic of the cavity calculation area.

The whole calculation domain was meshed using quadrilateral grids. The grids had to be fine enough to resolve all the important physics, and the boundary conditions must not affect the solution significantly. In order to verify the grid independence and accuracy, three grid schemes with different grid densities (200 × 200, 300 × 300, and 400 × 400 in the cavity calculation area) were established. The grid node number of the other region was 720 × 460. The sound pressure in the cavity calculation area at different times is shown in Figure 3(a). It can be seen that calculated sound pressure using the three grids was almost the same except only slight changes in pressure amplitude at some times. Figure 3(b) shows the partial enlarged detail during the time of $t U_{0} / D = (2.5 - 5.0)$ . It can be seen that the grid (200 × 200) was sufficiently accurate for this study, and the main simulation work in this article was carried out using this grid scheme. The maximum grid spacing between nodes within the cavity was $Δ y_{\max} = 2.2 \times 10^{- 5} m$ , and the maximum grid spacing between nodes of the other region was $Δ y_{\max} = 2.0 \times 10^{- 3} m$ . For the calculation of the cavity aerodynamic noise, the maximum frequency was 10 kHz, so the maximum grid spacing of the calculation area satisfied $Δ y_{\max} < λ / 12$ , where $λ$ is the minimum wavelength of the sound waves to be calculated. The time step of the model was set to be $Δ t = 0.8 \times Δ y_{\min} / c = 1 \times 10^{- 7} s$ . The Reynolds number R_e was 12,185 and the Mach number M was 0.2.

Figure 3.

The sound pressure in the cavity calculation area at different times $t U_{0} / D$ is a dimensionless quantity.

A vortex was shed from the cavity front edge and moved to the downstream until it impinged onto the trailing edge of the cavity, generating an acoustic pressure wave, which traveled upstream and led to instabilities in the shear layer and to the shedding of a new vortex. Rossiter¹⁴ developed an empirical formula to predict the resulting oscillation frequencies, which was based on previous studies on edge tones

f = \frac{U_{0} (m - γ)}{L (M + \frac{1}{k_{c}})}

(20)

where f is the frequency, U₀ is the velocity of the flow, m is the integer mode number, L is the cavity length, M is the Mach number, 1/k_c is the ratio between convection velocity of the vortices and the free-stream velocity, and γ is a factor accounting for the lag time between the passage of a vortex and the emission of a sound pulse at the trailing edge of the cavity. A Fourier analysis of the pressure near the corner of the trailing edge is given in Figure 4. It can be seen that the dominant frequency was obvious. Table 1 shows the comparison of the calculated spectral frequencies and the Rossiter formula. Only the first two modes are shown. The calculation errors were within the acceptable range compared to the results estimated by Rossiter formula.

Figure 4.

Fourier analysis of the pressure near the corner of the trailing edge.

Table 1.

Comparison of the calculated spectral frequencies and the estimated results by Rossiter formula.

Mode	Spectral frequencies (Hz)		Error (%)
Mode	Calculated results	Estimated by Rossiter formula	Error (%)
First	905	870	4.02
Second	1949	2030	3.99

The cavity unsteady flow field

Figure 5 shows the generation and propagation of the vortexes in the cavity at different times. At $t U_{0} / D = 6.0$ , the shear boundary layer separated from the cavity front edge and formed two vortexes, marking them to be E₁ and E₂, respectively. The formed vortexes moved to the downstream and merged with the surrounding small-scale vortexes. At $t U_{0} / D = 8.0$ , the vortexes E₁ and E₂ evolved into two large-scale vortexes. It can be seen that many vortex filaments split from the edge of the vortex core continuously. At $t U_{0} / D = 10.0$ , the vortex E₂ rotated clockwise to the trailing edge of the cavity, while the vortex E₁ impacted the trailing edge of the cavity and ruptured to be two parts. One part moved downstream along the trailing edge of the cavity and the boundary layer thickness increased. The other part was involved into the cavity under the effect of cavity flow, and as a result, the flow form changed by the interaction with the rest vortex. At $t U_{0} / D = 12.0$ , the vortexes on the front edge of the cavity shed periodically and complex vortexes structure was formed in the cavity.

Figure 5.

Vorticity at different times: (a) $t U_{0} / D = 6.0$ , (b) $t U_{0} / D = 8.0$ , (c) $t U_{0} / D = 10.0$ , and (d) $t U_{0} / D = 12.0$ .

Radiated acoustic in near field of the cavities

According to the calculated noise source, the sound pressure of the unsteady flow in the cavity was simulated. Figure 6 shows the instantaneous pressure perturbation contours in near field of the cavity. It can be seen that the acoustic wave was corrugated. The shed vortexes moved downstream along with the flow, and radiated acoustic would generate when the vortexes impact the trailing edge of the cavity. The acoustic wave propagated to the upstream and caused the boundary layer to be unsteady. There were strong convection and surface scattering in the front edge of the cavity. The radiated acoustic produced by the boundary layer separation and vortex impact cavity wall intervened, and an obvious interference phenomenon appeared. This was in good agreement with the experimental results carried out by Karamcheti,¹⁵ as shown in Figure 7. Obvious directivity can be seen in the near acoustic field. It was revealed that the radiated acoustic in the area is located at 135°–180° interval and near the front edge of the cavity if taking the z axis to be the rotation axis.

Figure 6.

Instantaneous pressure perturbation contours.

Figure 7.

Acoustic interference observed in the experiment.

Radiated acoustic in far field of the cavities

In order to analyze the acoustic characteristic in the far field of the cavity, several observation points were set on the arcs of R = 20D and R = 40D with an interval of 15°. The sound pressure level curves of each observation point were obtained by analyzing the detected sound pressure signals, as shown in Figure 8. It can be seen that the sound pressure level of R = 20D was apparently higher than the sound pressure level of R = 40D. It meant that the sound wave attenuated gradually with the increase in the propagation distance. The radiated acoustic in the far field of the cavities also showed obvious directivity as in the near field. The sound pressure in the area located at 45°–135° interval was much higher, and the directivity of the acoustic field had no change with the increase in the observation distance within a certain spread range.

Figure 8.

Sound pressure level of different observation angles in far field.

Numerical simulation of the jet noise

The hybrid algorithm was applied to study the effects of nozzle structure on the radiation noise. Two nozzles were designed: an elliptical nozzle and a rectangular nozzle, as shown in Figure 9. The two nozzles were of the same export area and the ratio of long to short half axises of the elliptical nozzle was 1.5:1. Figure 10 shows the grid structure of the model. The model was meshed by the hexahedral grid, and the grids in the nozzle exit and the jet boundary layer whose velocity gradient was great were refined to guarantee high computational accuracy. A damp truncation region was set in the export region near the outlet boundary to reduce the influence of the outlet boundary on the calculation of the sound field. The total grid number was 1,668,864.

Figure 9.

The structure of the nozzles.

Figure 10.

The grid structure of the model.

The calculation region and the boundary conditions of the model are shown in Figure 11. The environment boundary was set to be the pressure outlet. The inlet velocity of the nozzle was 60 m/s and the corresponding Reynolds number was 200,000. The flow temperature and the ambient temperature were both set to be 300 K. The ambient pressure was set to be 101,325 Pa. The impact of air from the environment in the simulation was ignored and the nozzle wall was set to be standard wall condition. Along the axis of the nozzle center, several observation points were set on the arcs of R = 40D_j with an interval of 30°. Besides, the radial section of R = 80D_j at L = 20D_j away from the nozzle exit was set to be monitoring surface.

Figure 11.

The schematic of the calculation region.

Analysis of the flow field

When the airflow jet into the environment by the gradual change nozzle, the jet flow intensity attenuated as the strong airflow mixing was generated nearby the border of the jet. Figure 12 shows the velocity distribution of the two different nozzles along the axis of the nozzle center. It can be seen that the flow velocity was maximum near the nozzle exit and the jet was complete turbulence. Comparison of the velocity distribution revealed that the axial velocity of the elliptical nozzle at about L = 8D_o away from the nozzle exit showed rapid attenuation which was much higher than that of the rectangular nozzle.

Figure 12.

Velocity distribution along the axis of the nozzle center.

When the airflow jet into the environment through the nozzle, large-scale vortexes were formed. In order to have an intuitive understanding of the generation and propagation of the vortexes jet from different nozzles in the turbulent jet flow field, the Q vortex criterion was used to define the vortex structure of the flow field, Q was expressed by

Q = \frac{1}{2} (Ω_{ij} Ω_{ij} - S_{ij} S_{ij}) = - \frac{1}{2} \frac{\partial u_{i}}{d x_{j}} \frac{\partial u_{j}}{d x_{i}}

(21)

where $Ω_{ij}$ is the asymmetric part of velocity gradient tensor, $Ω_{ij} = 0 . 5^{*} (\partial u_{i} / d x_{j} - \partial u_{j} / d x_{i})$ , $S_{ij}$ is the symmetric part of the velocity gradient tensor, $S_{ij} = 0 . 5^{*} (\partial u_{i} / d x_{j} + \partial u_{j} / d x_{i})$ . In this article, the dimensionless quantity $Q^{*} = Q / (U_{j} / D_{j})^{2}$ was adopted to describe the vortex structure.

Figure 13 shows the iso-surface distribution of $Q^{*} = 0.5$ of the two nozzles. The emergence, mergence, and submergence of the vortexes were accompanied by the jet process. In the region of L = 8D_o away from the nozzle exit, the mixture of the environmental fluid into the turbulent jet fluid was mainly influenced by the large-scale vertical motion. The vortexes near the jet boundary layer wrapped the ambient fluid into the jet region, and the jet intension attenuated accordingly. Thus, it can be seen that the emergence and evolution of vortex structures were the main factors of the turbulent jet attenuation. Figure 13 also revealed that the distribution region, the scale, and intensity of the vortexes generated by the elliptical nozzle in the near field of the nozzle exit were larger than those of the rectangular nozzle. Thus, more ambient fluids were swirled into the jet region for the elliptical nozzle, and the decay rate of the jet velocity was also higher.

Figure 13.

The iso-surface distribution of $Q^{*} = 0.5$ of the nozzles.

Analysis of the sound field

On the basis of the flow field study, the characteristics of the radiated sound were analyzed. Figure 14 shows the sound pressure spectrum curves of the two nozzles in the far field. It can be seen that the sound pressure level of the rectangular nozzle was higher than that of the ellipse nozzle within the entire spectrum. The sound pressure attenuated obviously in the spectrum around 4 kHz. One reason for this was that the small-scale vortexes which would lead to high-frequency sound were ignored in the calculation of the flow field.

Figure 14.

The sound pressure spectrum curves of the two nozzles in the far field.

The variation of the nozzle structure not only affected the vortex distribution in the jet field but also affected the characteristics of the radiated sound. Figures 15 and 16 show the sound pressure distributions of the axial cross section and radial section in the far field. It can be seen that the sound pressure of the rectangular nozzle was higher than that of the ellipse nozzle. Combining with the jet flow field analysis results, it can be concluded that the nozzle structure had great influence on the vortex motion and distribution in the jet flow and led to the variation of the jet flow intensity and scope. As revealed in Figure 15, the sound pressure of the observation points in the 30°–65° fan area of the axial cross section at R = 40D_j was little greater than that of the other observation points. Thus, it can be seen that the radiated jet noise was mainly concentrated in the downstream of the jet. Figure 16 shows the sound pressure of the observation points in the radial section of R = 80D_j at L = 20D_j away from the nozzle exit. The sound pressure distributions of the two nozzles presented obvious directivity. And the sound pressure in the short-axis direction of the nozzle section was higher than that in the long-axis direction.

Figure 15.

The sound pressure level of the axial cross section at R = 40D.

Figure 16.

The sound pressure level in the radial section of R = 80D_j at L = 20D_j away from the nozzle exit.

Conclusion

This article presented a hybrid method to provide a potential prediction tool for numerical simulations of aerodynamic sound problems. The radiated near-field sound was calculated using the LEE with acoustic source terms. The acoustic source terms were extracted from the calculations of the unsteady Navier–Stokes equations using the LES technique. The FW-H equation was adopted to calculate the sound characteristics of the far field. First, the characteristics of the unsteady flow around the cavity had been investigated. The simulation results showed that radiated acoustic was generated along with the formation and breaking of vortexes in the cavity. The radiated acoustic produced by the boundary layer separation and vortex impact cavity wall intervened, and an obvious interference phenomenon appeared. The far-field radiated noise generated by the cavities presented obvious directivity, and the directivity of the acoustic field had no change with the increase in the observation distance within a certain spread range. Then, the hybrid method was used to simulate the unsteady turbulence jet radiated noise of the elliptical and rectangular nozzles. It was revealed that the distribution region, the scale, and intensity of the vortexes generated by the elliptical nozzle in the near field of the nozzle exit were larger than those of the rectangular nozzle. Thus, more ambient fluids were swirled into the jet region for the elliptical nozzle, and the decay rate of the jet velocity was also higher. The sound pressure distributions of the two nozzles presented obvious directivity. The sound pressure in the short-axis direction of the nozzle section was higher than that in the long-axis direction.

Footnotes

Academic Editor: Jiin-Yuh Jang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by the National Natural Science Foundation of China (nos 51306166 and 51206101), the Scientific Research Foundation of Shandong Province Outstanding Young Scientist Award (no. BS2013NJ017), the Shandong Province Natural Science Foundation (no. ZR2012EEQ012), and the Qingdao Science and Technology Program of Basic Research Projects (nos. 13-1-4-210-jch and 13-1-4-248-jch).

References

Howe

MS.

Contributions to the theory of aerodynamic sound with application to excess jet noise and the theory of the flute. J Fluid Mech 1975; 71: 625–673.

Lighthill

. On sound generated aero-dynamically. I. General theory. Proc R Soc Lon Ser A 1952; 211: 564–587.

Freund

Lele

Moin

Direct numerical simulation of a Mach 1.92 turbulent jet and its sound field. AIAA J 2000; 38: 2023–2031.

Freund

. Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9. J Fluid Mech 2001; 438: 277–305.

Terracol

Manoha

Herrero

. Hybrid method for airframe noise numerical prediction. Theor Comp Fluid Dyn 2005; 19: 197–227.

Bailly

Bogey

Gloerfelt

Some useful hybrid approaches for predicting aerodynamic noise. CR Mecanique 2005; 333: 666–675.

Ewert

Schroderb

On the simulation of trailing edge noise with a hybrid LES/APE method. J Sound Vib 2004; 270: 509–524.

Bechara

Bailly

Lafon

Stochastic approach to noise modeling for free turbulent flows. J Acoust Soc Am 1994; 97: 3518–3531.

Wang

Moin

Computation of trailing-edge flow and noise using large-eddy simulation. AIAA J 2000; 38: 2201–2209.

10.

Zhao

Frankel

Mongeau

Large eddy simulation of sound radiation from subsonic turbulent jets. AIAA J 2001; 39: 1469–1477.

11.

Ribaul

Sarkar

Stanley

Large eddy simulation of a plane jet. Phys Fluids 1999; 11: 3069–3083.

12.

McMullan

Pokora

Page

. Large eddy simulation of a high Reynolds number subsonic turbulent jet for acoustic source capture. In: 14th AIAA/CAES aeroacoustics conference, Vancouver, BC, Canada, 5–7 May 2008. Reston, VA: AIAA.

13.

Ffowcs Williams

Hawkings

. Sound generation by turbulence and surface in arbitrary motion. Philos T Roy Soc B 1969; 264: 321–342.

14.

Rossiter

JE.

Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Royal Aircraft Establishment ARC R. &M. No. 3438, 1964, http://naca.central.cranfield.ac.uk/reports/arc/rm/3438.pdf

15.

Karamcheti

Acoustic radiation from two-dimensional rectangular cutouts in aerodynamic surfaces. Technical note 3487, 33 pp., 1955, NACA, http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930084329.pdf

Simulations of cavity flow noise and turbulent jet noise using a hybrid method