Two-point radiation statistics from large-scale turbulent structures within supersonic jets

Abstract

The noise from large-scale coherent turbulent structures within jets remains the dominant source. For the purpose of developing future control systems for the large-scale noise source, we investigate the statistics between upstream and downstream radiating waves. We investigate two off-design supersonic jet flows with instability theory and associated noise radiation, large-eddy simulation (LES), and experiments. We compare the auto-correlation, cross-correlation, coherence, and other statistics predicted by aeroacoustic instability theory. As instability waves are closely connected with the formation of large-scale turbulent structures, they yield insight into large-scale noise statistics. We investigate two nozzles at two supersonic off-design conditions. The first is a biconic nozzle operating at an unheated condition, and the second is a NASA nozzle operating at a heated condition. We find that for these jets, the noise from instability waves is coherent between 0.40 to 0.70 at large-scale radiation frequencies between the downstream and upstream radiation directions.

Keywords

Jet aeroacoustics experiment large-eddy simulation computation fluid dynamics instability theory coherence

Introduction

Jet noise from fighter aircraft remains one of the major sources of noise during aircraft carrier operations, near military airfields, and consistently contributes to hearing loss even with the most advanced hearing protection. Large-scale coherent structures within jet turbulence^1–3 cause the dominant noise, which radiates in the downstream direction. Noise from large-scale turbulent structures also radiates in the upstream direction, however, it is dominated by the noise from fine-scale turbulent mixing noise⁴ and broadband shock-associated noise (BBSAN).⁵ Reduction of noise from large-scale turbulent structures without affecting aircraft performance remains one of the most difficult challenges in practical jet engine development.

We examine the statistical relation between downstream and upstream radiating noise from large-scale turbulent structures within jets. By understanding the statistics of large-scale radiation, we hope to create the possibility of one-day designing a feedback control system for large-scale jet noise. This approach will rely on the ability to extract the upstream propagating noise from the large-scale structures from within a signal that is dominated by fine-scale mixing noise. Such sensors might be mounted on the aircraft wing. Wing mounted microphones for jet noise measurement were pioneered in World War II by the Germans on the Messerschmitt Me 262.⁶ The control signal after extraction would be used to feed an actuator within the nozzle system of the aircraft.

Here, we focus on the difficult task of understanding the statistics of upstream traveling waves from large-scale turbulent structures. We seek to understand their relation with downstream radiation and how far below the fine-scale mixing noise they reside. To accomplish this task, we performed a combined theoretical, experimental, and computational based program. We modeled the large-scale structures using the established instability wave theory of Tam et al.^7[ Also, we performed large-eddy simulations (LES) of the corresponding jet flows with our newly established solver.¹⁰ The Kirchhoff surface (KS)^11,12 approach is used to compute the radiated far-field noise based on the LES. Both the model and LES are validated with experiments performed at University of Florida and NASA.¹³ Furthermore, the experimental measurements are examined to understand these statistical properties. We then perform a statistical separation of the modes to match the statistics of the large-scale structures found from the aeroacoustic model and experiments. This is possible because the large-scale mixing noise turbulence has a unique statistical nature,¹⁴ which is presented in this paper.

The paper is presented as follows. First a background and motivation review is presented. Our methodology is presented in terms of experiments, LES, and instability predictions. Results of instability predictions, LES, and experimental statistics are then presented. Finally, we summarize our program and suggest a way forward.

Background review

Figures 1 and 2 show acoustic spectra from a high-speed off-design supersonic heated jet. The y-axis is sound pressure level (SPL) and the x-axis is non-dimensional frequency, $St = f D / u_{j}$ . Here, f is frequency, D is the nozzle exit diameter, and u_j is the fully-expanded velocity. The experiment is shown as a red line and the other lines represent the prediction of mixing noise, BBSAN, and the total noise, as predicted by Miller.¹⁵ Note that the mixing noise prediction is a summation of the fine-scale and large-scale mixing noise. In the downstream direction, the mixing noise is dominant relative to all other noise sources. In the upstream direction (as measured and shown in Figure 1) the BBSAN is dominant, but at mid- to low-frequencies mixing noise is dominant. The mixing noise in the upstream direction (Figure 1) has unique statistics that are likely separable (see sources of jet noise by Tam et al.¹⁴). The statistics of the large-scale mixing noise, as measured in the upstream direction, are highly correlated with the statistics of the large-scale mixing noise that dominates the downstream direction.

Significant research has been conducted on the large-scale instability waves (that cause the dominant radiation) and control.^16gni For low Reynolds number flow, the noise generation predominately occurs from instability waves, as shown by Morrison and McLaughlin.²¹ Morrison and McLaughlin²¹ used a low-level excitation within the upstream portion of the flow-field that caused the majority of sound to be generated near the potential core, where the instability waves are the highest. Papamoschou and Roshko²² examined the growth rate and turbulent structures of a compressible plane shear layer in a range of Mach numbers, M, from 0.2 to 4 using schlieren and Pitot probes. Suzuki and Colonius^23,24 examined instability waves within subsonic jets using a near-field phased array of microphones. Measurements were performed to ascertain instability waves through comparisons with eigenfunction reference solutions derived from linear theory. Suzuki and Colonius²³ showed that the instability waves evolved in the mean flow from the nozzle exit to the end of the potential core. Noise was shown to be highly correlated to phased-array measurements and instability waves (see Suzuki and Colonius²⁴). In a recent and similar approach, Zaitsev et al.²⁵ examined instability waves in supersonic jet flows using an azimuthal decomposition method.

Figure 3 shows a large-scale structure from the measurement of Mitchell et al.²⁶ using a high-speed schlieren within an impinging jet. Here, the flow moves from left to right and it is centered about the jet shear layer. Mitchell et al.²⁶ circled one of the large-scale structures within the jet shear layer. It is highly anisotropic (turbulent statistics depend on rotation of the coordinate frame), coherent spatially, coherent temporarily, and is responsible for the dominant noise source and most energetic noise source. These structures might be controlled through actuation systems on the nozzle.^20,27,28

Previous approaches – Instability theory and prediction

The instability wave theory used to compute the noise radiated from the large-scale turbulent structures in a plane turbulent shear layer was developed by Tam and Morris.²⁹ Tam and Burton^7,8 extended this method by separating the flow-field into the inner and outer region, and applied the matched asymptotic expansion. The far-field pressure was predicted and validated by an axisymmetric jet at jet Mach number, M_j = 2. Dahl³⁰ adopted this method for supersonic coaxial jets, and the computed results (directivity pattern) agreed well with the measurements. Kelvin-Helmholtz instability waves are responsible for the formulation of large-scale turbulent structures, and have been known since the nineteen sixties.³¹ Tam and Hu³² analyzed the effects of three families of instability waves for supersonic jets based on experimental results and showed theoretically that the Kelvin-Helmholtz instability waves are important in relation to the mechanisms of noise generation. Tam and Chen⁹ quantified the effects of the whole instability wave by applying a stochastic turbulence theory.³³ Mohseni³⁴ verified the assumption of instability wave theory of Tam by comparing direct numerical simulation (DNS), linear Navier-Stokes (LNS) simulation, and instability wave solutions. The amplitude of the instability wave was scaled with the simulation results, and the comparisons were used to illustrate the sound generation mechanism for Mach wave radiation. Recently, the bi-orthogonal and proper orthogonal decomposition (POD) methods were used to extract the instability waves from direct numerical simulation (DNS), LES, or experiments to explain the contributions of the instability waves. These methods were used to determine the amplitude of the instability wave from linear stability theory or parabolized stability equations (PSE) for subsonic, transonic, and supersonic jets.^35–37

Methodology

Experiment

Anechoic wind tunnel facility

The experiments were conducted in the University of Florida Anechoic Wind Tunnel facility, which has a cut-off frequency of approximately 100 Hz and test section dimensions of 5.5 m long by 5 m wide by 2.3 m high. The axisymmetric supersonic jet nozzle used in this study is located in the center of the UF Anechoic Wind Tunnel inlet and is aligned so that the shear layer growth does not extend beyond the jet diffuser. The jet nozzle is flanged to a stagnation plenum with six circumferential ports for total pressure and temperature measurements. A ceramic substrate with a mesh density of 400 cells per square inch is used inside the stagnation plenum to condition the incoming air. The stagnation plenum is flanged onto a steel pipe which extends to the air delivery system outside the room, where the jet is supplied with clean dry air from a Sullair LS-20T compressor at up to 200 psi. A flow silencer is included between the piping and stagnation chamber to reduce flow noise generated by the valves and compressors, which feed the compressed air to the jet. During experiments the air from the jet is exhausted through the wind tunnel flanechoically treated diffuser to the outdoor ambient environment. For the experiments conducted in this study, the tunnel is operated without wind tunnel base flow. The tunnel inlet serves to provide adequate air for jet entrainment. Figure 4 shows the UF Anechoic Wind Tunnel Test Chamber with the supersonic nozzle installed, more details of the UF Anechoic Wind Tunnel can be found in Mathew.³⁸

Axisymmetric supersonic jet design

A biconic nozzle that is similar to one developed by Siener¹⁷ was developed for this test. This axisymmetric supersonic jet has a design Mach number M_d of 1.76. The interior profile of the jet nozzle is a matched 3rd order to 5th order polynomial between the exit of the stagnation chamber, which has a diameter of 0.152 m. The straightening section of the convergent-divergent nozzle has a diameter of 0.0571 m. The area ratio is 1.40, between the nozzle throat and the nozzle exit, which have diameters of 0.0429 m and 0.0508 m, respectively. Four 5.08 mm holes are machined into the nozzle with a 90-degree separation at 70% of the nozzle diverging section for future flow control applications. In the current study, these holes are plugged with Aluminum 6061 cylinders that are machined to match the interior surface of the nozzle. The interior surface of the jet nozzle is machined with a 0.8 μm root-mean-square surface roughness to ensure a smooth surface for the flow. A three-dimensional rendering of the nozzle and the nozzle placed in the anechoic facility can be seen in Figure 5.

Instrumentation and jet control

The stagnation pressure was measured using an Omega PX309-100A5V pressure transducer with a 0.25% static accuracy based on the full-scale pressure of 100 psia. Ambient air measurements were taken using a Mensor Series 6100 Digital Pressure transducer with a maximum error of 0.16 psi. The stagnation temperature was measured with an Omega TJ36-CPSS thermocouple, which has a maximum error of 0.75%. This thermocouple was placed inside of the stagnation chamber of the jet plenum for initial stagnation temperature measurements and then removed when acoustic data was acquired. The temperature ratio of the jet was found to be nominally equal to one for all experiments.

In order to maintain a constant exit Mach number, the jet flow rate was regulated using a pneumatic Fisher valve in conjunction with a ControlAir E/P Transducer. A LabVIEW proportional–integral–derivative (PID) controller was implemented to continuously send voltage signals to adjust the fisher valve position through an NI-9263 4-channel voltage output module. The PID controller was tuned such that the overall root-mean-square fluctuations of the calculated exit Mach number are less than 1%. The real-time Mach number was calculated based on the isentropic relations using the measured stagnation pressure and ambient pressure.

Experimental acoustic measurements

The experiments were performed with an over-expanded supersonic jet with M_j, of 1.3 and M_d of 1.76. The nozzle pressure ratio and nozzle temperature ratio were 2.77 and 1.00, respectively. The experiments were performed in a blow down manner, and each run was 45 seconds to ensure that Mach number deviations are less than 1%. Acoustic measurements were taken with two GRAS 46-BE 1/4-inch CCP free-field microphones, which have a flat free field frequency response of 10 Hz to 80 kHz and dynamic range of 35 to 160 dB (ref. 20 μPa). The sampling frequency for all acoustic measurements was 102 kHz with both channels sampled simultaneously through an NI PXI-4472 sound and vibration module. The microphones were held by microphone stands with a three-dimensional printed part that is fit into copper piping to hold the microphone and distance it from the support. The stands are wrapped with acoustic damping material to minimize reflections.

All acoustic measurements and predictions were made relative to the jet upstream direction (e.g. 45 deg. is in the upstream direction relative to the nozzle exit). Far-field measurements were taken at a radius of 30 D as measured from the jet nozzle exit to minimize reflections from the diffuser and sidewalls. This was performed at polar angles from $50 °$ to $150 °$ in $10 °$ increments as measured from the nozzle inlet. Near-field measurements were also taken at a radius of 15 D for similar polar angles. A total of three different experimental campaigns were conducted for simultaneous far-field measurements, simultaneous near-field measurements, and simultaneous near-field versus far-field measurements. A total of 55 experiments were performed for the far-field measurements to compare the noise statistics of all possible angle combinations. Similar experiments were performed for near-field measurements to make comparisons between noise statistics at different angles. Experiments were also performed with a near-field microphone located at $50 °$ . The far-field microphone traversed from $150 °$ to $60 °$ to avoid contamination from reflections when both microphones were placed in the same polar angle plane. These measurements were performed to compare near-field versus far-field noise statistics.

Computational fluid dynamics

An open-source LES code called High-Fidelity Large-Eddy Simulation (HiFiLES)³⁹ was modified to simulate supersonic jet flows. The code uses the Energy Stable Flux Reconstruction (ESFR) scheme⁴⁰ to achieve high-order of accuracy on unstructured grids by using piece-wise high-order polynomials to approximate the solution inside each element. The high-order FEM framework allows us to obtain reasonably accurate unsteady data on a relatively coarse computational grid. A correction function was chosen to account for the common interface flux and to reconstruct the continuous flux function inside the element. By carefully choosing the correction function, the scheme can recover many existing high-order schemes such as the discontinuous Galerkin (DG) scheme⁴¹ and the spectral difference (SD) scheme⁴² with a simplified implementation. On the interfaces of the elements, an approximate Riemann solver is used to compute the common numerical flux. Details of the numerical methods can be found within Huynh,^43,44 Vincent et al.,⁴⁰ and Castonguay et al.⁴⁵ The time integration scheme used is Runge-Kutta (RK) scheme.¹⁰ Different RK schemes are applied depending on the requirements of accuracy.

Due to the low dissipation and low dispersion nature of the numerical scheme, explicit LES can be used to model the unresolved scales of turbulence. Here, we used the wall-adapting local eddy-viscosity (WALE) model by Nicoud and Ducros⁴⁶ because it can correctly model the sub-grid scale flux in transitional and near wall flows in the jet simulations. In order to solve the numerical issues introduced by the simulation of supersonic jet flows and to acquire the data for acoustic analysis, new features were implemented in the modified code. The shock detection algorithm of Persson⁴⁷ is used in conjunction with a modal exponential filter to stabilize the solution on a per element basis. A compressible wall model based on a single layer wall function and Van Driest transformation of velocity⁴⁸ was used to approximate the wall stress and allows for a coarser mesh near the wall. A er element baobe” was implemented¹⁰ to sample time-resolved data efficiently in the computational domain for instability analysis (mean-flow) and acoustic analysis.

Kirchhoff surface method

The acoustic disturbances can be described by the linearized, inviscid wave equation when the viscous attenuation and non-linear effects are minimal and can be neglected, and the mean velocity and its gradients are sufficiently small. The wave equation for a pressure fluctuation within a quiescent medium is

[\frac{1}{c_{\infty}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \nabla^{2}] p' = 0,

(1)

where

c_{\infty}

is the ambient speed of sound,

p'

is the fluctuating pressure, and t is time. We apply the method from Freund¹¹ and Bodony and Lele⁴⁹ to solve this equation via transforming coordinates for a KS (S) of radius R_s along the jet axis. The KS, as in Figure 6, must be far away enough from the hydrodynamic region of jets and enclose all the nonlinear effects so that the generating noise can be described by the linear wave equation.

The partially transformed form of equation (1) outside of the surface S is

(\frac{d^{2}}{d r^{2}} + \frac{1}{r} \frac{d^{2}}{d r} + \frac{ω_{n}^{2}}{c_{\infty}^{2}} - k_{x}^{2} - \frac{m^{2}}{r^{2}}) {\hat{\hat{p}}}_{m n} (k_{x}, r) = 0,

(2)

where k_x is the wavenumber along the streamwise direction and

{\hat{\hat{p}}}_{m n} (k_{x}, r)

is the fluctuating pressure with Fourier transform in x, θ, and t.

The exact solution of equation (2)¹² in cylindrical coordinates is

\begin{array}{l} p' (x, r, θ, t) = \frac{1}{2 π} \int_{- \infty}^{\infty} \sum_{n = - N_{t} / 2}^{N_{t} / 2 - 1} \sum_{m = - N_{θ} / 2}^{N_{θ} / 2 - 1} {\hat{\hat{p}}}_{m n} (k_{x}, R_{s}) T_{m}^{(j)} \\ exp (i (k_{x} x + m θ - ω_{n} t)) d k_{x}, \end{array}

(3)

where

T_{m}^{(j)}

depends on the value of

ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2}

and m.

T_{m}^{(j)}

can be expressed as

T_{m}^{(j)} = {\begin{array}{l} \frac{H_{m}^{(j)} (r \sqrt{ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2}})}{H_{m}^{(j)} (R_{s} \sqrt{ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2}})} ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2} > 0 \\ \frac{R_{s}^{| m |}}{r^{| m |}} ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2} = 0 and m \neq 0 \\ 1 ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2} = 0 and m = 0 \\ \frac{K_{m} (r \sqrt{ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2}})}{K_{m} (R_{s} \sqrt{ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2}})} ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2} < 0, \end{array}

(4)

where j = 1 for

ω_{n} > 0

and j = 2 for

ω_{n} < 0

to enforce the outgoing acoustic wave.

For the far-field, only those waves that satisfy $ω_{n}^{2} / c_{\infty}^{2} - k_{x}^{2} > 0$ radiate as noise. Therefore, the stationary phase method⁵⁰ is applied to equation (3) to compute the noise more readily. Finally, the solution in the spherical coordinate system is

\begin{array}{l} p' (R, θ, ψ, t) = \sum_{m = - N}^{N} (\sum_{n = 0}^{N_{t} / 2 - 1} {\hat{\hat{p}}}_{m n} (k_{n} cos ψ, R_{s}) \frac{e^{i k_{n} R {(- i)}^{m + 1}}}{π R H_{m}^{(1)} (R_{s} k_{n} sin ψ)} exp (i (m θ - ω_{n} t)) \\ + \sum_{n = - N_{t} / 2}^{0} {\hat{\hat{p}}}_{m n} (k_{n} cos ψ, R_{s}) \frac{e^{i k_{n} R {(i)}^{m + 1}}}{π R H_{m}^{(2)} (- k_{n} R_{s} sin ψ)} exp (i (m θ - ω_{n} t)), \end{array}

(5)

where the stationary phase is

k_{x} = ω_{n} / c_{\infty} cos ψ, k_{n} = ω_{n} c_{\infty}

, and

N \approx k_{x} R_{s} sin ψ

. Only selected wavenumbers can propagate to the far-field, and they are dependent on the radial frequency, radiation angle from the jet exit, and azimuthal number. The radiation conditions are

k_{x} = ω_{n} / c_{\infty} cos ψ

and

k_{x} R_{s} sin ψ \geq m

Instability theory

The small perturbation equations that govern the instability waves are derived from the linearized compressible inviscid equations of motion in non-dimensional form. The length, velocity, time, density, and pressure scales are the radius of the nozzle R_j, jet fully-expanded exit velocity u_j, $R_{j} / u_{j}$ , the jet fully-expanded density at the nozzle exit ρ_j, and $ρ_{j} u_{j}^{2}$ , respectively. The governing equations are

\frac{\partial ρ'}{\partial t} + \bar{V} \cdot \nabla ρ' + V' \cdot \nabla \bar{ρ} + \bar{ρ} \nabla \cdot V' + ρ' \nabla \cdot \bar{V} = 0,

(6)

\frac{\partial V'}{\partial t} + \bar{V} \cdot \nabla V' + V' \cdot \nabla \bar{V} + \frac{ρ'}{\bar{ρ}} (\bar{V} \nabla \cdot \bar{V}) = - \frac{1}{\bar{ρ}} \nabla p',

(7)

and

\frac{\partial p'}{\partial t} + \bar{V} \cdot \nabla p' + γ \bar{p} \nabla \cdot V' + γ p' \nabla \cdot \bar{V} = 0,

(8)

where the prime denotes the perturbation or a small-amplitude instability wave,

\bar{V}

is the mean velocity vector, and

V'

is the velocity perturbation vector.

The cylindrical coordinate system, ( $r, x, θ$ ), with its origin at the center of the nozzle exit are used. All the physical variables can be represented as $p' (x, r, θ, t) = Re [p (x, r) exp (i m θ - i ω t)]$ , where Re represents the real part. The term $exp (i m θ - i ω t)$ can be separated because of the axial symmetry of the mean flow (azimuthal harmonics within round jets), and the assumption that the instability waves are generated by an external harmonic in time.⁵¹ Equations (6) through (8) are now

- i ω u + \bar{v} \frac{\partial u}{\partial r} + \bar{u} \frac{\partial u}{\partial x} + v \frac{\partial \bar{u}}{\partial r} + u \frac{\partial \bar{u}}{\partial x} = - \frac{1}{\bar{ρ}} \frac{\partial p}{\partial x},

(9)

- i ω v + \bar{v} \frac{\partial v}{\partial r} + \bar{u} \frac{\partial v}{\partial x} + v \frac{\partial \bar{v}}{\partial r} + u \frac{\partial \bar{v}}{\partial x} = - \frac{1}{\bar{ρ}} \frac{\partial \bar{p}}{\partial r},

(10)

- i ω w + \bar{v} \frac{\partial w}{\partial r} + \bar{u} \frac{\partial w}{\partial x} + 2 \frac{\bar{v} w}{r} = - \frac{1}{r^{2}} \frac{i m}{\bar{ρ}} p,

(11)

and

\begin{array}{l} - i ω p + \bar{v} \frac{\partial p}{\partial r} + \bar{u} \frac{\partial p}{\partial z} + γ \bar{p} (\frac{1}{r} \frac{\partial v r}{\partial r} + \frac{i m}{r} w + \frac{\partial u}{\partial x}) \\ + γ p (\frac{1}{r} \frac{\partial \bar{v} r}{\partial r} + \frac{\partial \bar{u}}{\partial x}) = 0. \end{array}

(12)

Tam and Burton^7,8 separated the jet flow into the inner and outer region to solve this problem. The solution of equations (9) to (12) can be expressed as an asymptotic series of waves traveling through a non-uniform medium, such as

p (r, s) = \sum_{n = 0}^{\infty} δ_{n} (ϵ) {\hat{p}}_{n} (r, s) exp [i \int_{0}^{s} α (s) d s],

(13)

where

δ_{n} (ϵ)

is the gauge function of the asymptotic expansion, the subscript n is order of the flow properties,

{\hat{p}}_{n} (s, r)

is the shape function, and α is the local wavenumber.

In the inner region, the multiple scales analysis is used. A slow variable $s = ϵ x$ is introduced as in equation (13). Equations (9) through (12) are arranged according to order n by substituting $ρ, u, v, w,$ and p in the form of equation (13), combining the mean flow in the inner region ( $\bar{V} = (\bar{u} (x, r), ϵ \bar{v} (s, r), 0)$ ), transforming the coordinate system (s, r), eliminating the term of $exp [i \int_{0}^{s} α (s) d s]$ , rearranging equations according to the order n, and eliminating other dependent variables. Finally, the equation in terms of ${\hat{p}}_{n}$ for the $n^{th}$ order is

\begin{array}{l} \frac{\partial^{2} {\hat{p}}_{n}}{\partial r^{2}} + [\frac{1}{r} + \frac{2}{\bar{ω}} \frac{\partial \bar{u}}{\partial r} - \frac{1}{\bar{ρ}} \frac{\partial \bar{ρ}}{\partial r}] \frac{\partial {\hat{p}}_{n}}{\partial r} \\ + [\bar{ρ} M_{j}^{2} {\bar{ω}}^{2} - \frac{m^{2}}{r^{2}} - α^{2}] {\hat{p}}_{n} = G_{n} (r, s), \end{array}

(14)

where

\bar{ω} = ω - α \bar{u}

. When n = 0, the term G_n at the right side equals zero, namely, the equation becomes homogeneous and is called the Rayleigh equation.

The solution of the Rayleigh equation can be expressed via the sum of two independent linear solutions, $ζ_{1}^{p} (s, r)$ and $ζ_{2}^{p} (s, r)$ , as

{\hat{p}}_{0} (r, s) = A_{0} (s) ζ_{1}^{p} (r, s) + B_{0} (s) ζ_{2}^{p} (r, s),

(15)

where

A_{0} (s)

and

B_{0} (s)

are the amplitude functions varying in the streamwise direction.

In the outer region, an outer variable, $\bar{r} = ϵ r$ , is selected to ensure the same scale with the variable s due to the uniform condition assumption of zero mean velocity, where the mean flow are $\bar{V} = (0, ϵ \bar{v} = ϵ {\bar{v}}_{\infty} / r, 0)$ . Equations (9) through (12) are arranged with a similar procedure in the inner region via performing the Fourier transform on the variable s. Finally, a second ordinary differential equation can be obtained (see Tam and Burton⁸). The solution in terms of pressure in the outer region is

\begin{array}{l} p (\bar{r}, s) = \int_{- \infty}^{\infty} g (k, ϵ) [1 + i \frac{ϵ^{3}}{{\bar{ω}}_{k}} \frac{{\bar{v}}_{\infty}}{\bar{r}} \frac{\partial}{\partial \bar{r}}] {\times {({\bar{r}}^{2} - ϵ^{4} {\bar{ρ}}_{\infty}^{2} M^{2} ω^{2} {\bar{v}}_{\infty}^{2})}^{- 1 / 2 i ϵ^{2} {\bar{ρ}}_{\infty} M_{j}^{2} {\bar{ω}}_{k} {\bar{v}}_{\infty}} \\ \times H_{q}^{1} (\frac{1}{i ϵ} {(ϵ^{2} k^{2} - {\bar{ρ}}_{\infty} M_{j}^{2} {\bar{ω}}_{k}^{2})}^{1 / 2}) \times ({\bar{r}}^{2} - ϵ^{4} {\bar{ρ}}_{\infty}^{2} M_{j}^{2} {\bar{v}}_{\infty}^{2})} e^{i k s} d k, \end{array}

(16)

where

g (k, ϵ) = \frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{A} (s) e^{i \int_{0}^{x} α (x; ω, m) d x - iks} d k

and

\tilde{A} (s)

is the amplitude function.

The method of matched asymptotic expansions is applied^8,30 to determine the three unknown amplitude functions. The matching principle and procedure of Dahl³⁰ is used. The relations and values of the amplitude function are derived, ${\tilde{A}}_{0} (s) = A_{0} (s), B_{0} (s) = 0$ , and $A_{0} (s) = {\hat{A}}_{0} \exp [- β (s)]$ . $β (s) = \int_{0}^{s} I_{2} / I_{1} d s$ is the non-parallel flow correction factor, I₁ and I₂ are two complex integral terms, and ${\hat{A}}_{0}$ is the amplitude of the instability wave at the nozzle exit plane.

To summarize, the inner solution and outer solution of zeroth order, in terms of pressure p, at the given radial frequency and azimuthal mode number are

p_{0}^{inner} (x, r) = A_{0} (x) ζ_{1}^{p} (r, x) \exp [i \int_{0}^{x} α (x) d x]

(17)

and

p_{0}^{outer} (x, r) = \int_{- \infty}^{\infty} {\hat{A}}_{0} \hat{g} (η) H_{m}^{(1)} [i λ (η) r] \exp (i η x) d η .

(18)

To obtain an estimate of the sound radiated to the far-field, equation (18) can be easily computed via transforming cylindrical to spherical coordinates and applying the method of stationary phase (see Dahl³⁰ for details). The final solution is

p (R, ψ) = \frac{R}{2} \hat{A_{0}} g (\bar{η}) exp [i (Φ (\bar{η}) - \frac{π}{2} (m + 1))],

(19)

where

\bar{η}

is the stationary phase and

Φ (\bar{η})

is a function of

\bar{η}

Correlation and coherence

Correlation analysis is a useful tool to examine the relation of source and radiation statistics. Previously, experiments have used correlation analysis to relate pressure-field measurements to source phenomena for characterizing the source.^14,20,52,53 The correlation coefficient between two waveforms is defined as

R_{m n} (τ) = \frac{〈 p_{m} (t) p_{n} (t + τ) 〉}{{〈 p_{m}^{2} (t) 〉}^{1 / 2} {〈 p_{n}^{2} (t) 〉}^{1 / 2}},

(20)

where

p_{m} (t)

is the pressure time signal of the m^th microphone,

p_{n} (t)

is the pressure time signal of the n^th microphone, τ is the retarded time, and

〈 〉

denotes the time-averaged quantity.

Correlation can be used more easily to identify waveform periodicity and to obtain spatio-temporal length scales and phase speeds. On the other hand, coherence (normalized form of square cross-spectrum)^53,54 is useful for extracting the spatial phase relations of the field as a function of frequency. The coherence is defined as

γ_{m n}^{2} (f) = \frac{| G_{m n} (f) |^{2}}{G_{m m} (f) G_{n n} (f)},

(21)

where

G_{m n} (f)

is the cross-spectral density between the

m^{th}

and

n^{th}

microphones, and

G_{m m} (f)

and

G_{n n} (f)

are the auto-spectra density of the

m^{th}

and

n^{th}

microphone.

Results and analysis

In this section, the flow-field and jet aeroacoustics of two jet cases are discussed, and the focus is on the noise generated from large-scale turbulent structures. The first case is an unheated over-expanded supersonic jet that matches the experiment conducted at the University of Florida Anechoic Jet Facility. The nozzle used in this case is the bi-conic (SERDP) nozzle. The second case is a heated under-expanded supersonic jet from the SHJAR database.¹³ The nozzle geometry is NASA's baseline small metal chevron converging nozzle (SMC000). Both nozzles have a D=0.0508 m exit diameter. The bi-conic nozzle has a design Mach number $M_{d} = 1.764$ and lip thickness $h_{lip} = 0.01 D$ . The SMC000 nozzle has a design Mach number $M_{d} = 1.0$ and lip thickness $h_{lip} = 0.025 D$ . Table 1 presents the operating conditions of the jet cases, where Re _D is the Reynolds number based on the nozzle diameter, the subscript o represents total conditions, and the subscript $\infty$ represents ambient conditions. First, the simulation setup and results of the two jet cases are presented. We then examine the jet noise predicted by instability wave theory.

Table 1.

Operating conditions.

Nozzle	$p_{o} / p_{\infty}$	$T_{o} / T_{\infty}$	T_j K	M_d	M_j	u_j m/s	Re _D
SERDP	2.770	1.00	218.9	1.76	1.30	386	$1.60 \times 10^{6}$
SMC000	3.514	3.20	670.4	1.00	1.47	762	$6.26 \times 10^{5}$

The KS method is used to compute the far-field noise in order to calibrate the amplitudes of the instability wave models. The time series of pressure perturbations from the instability wave models are constructed via equation (19). Finally, we compare the auto-correlation, cross-correlation, coherence, and other statistics predicted by instability theory with LES and experiment.

Results of computational fluid dynamics

The computational domain of both simulations are a conical frustum located from $x / D = - 6$ to $x / D = 60$ , with radius of $r / D = 30$ and $r / D = 37.25$ , respectively. The origin of the domain is located at the center of the nozzle exit plane and the nozzle geometry is included in the domain. The meshes are only refined in the jet exhaust and acoustic near-field region to save computational power and to resolve the turbulent motions and acoustic waves. The mesh for the SERDP case and the SMC000 case have approximately 3.2 million and 2.5 million tetrahedral elements, respectively. The detailed description of the mesh and elements can be found in Shen and Miller.¹⁰ For both simulations, the internal and exterior walls of the nozzle are adiabatic non-slip walls. An anisotropic wall layer with initial layer thickness equivalent to $y^{+} = 225$ is used in the SERDP case with the compressible wall model to capture the turbulent boundary layer inside of the nozzle. In the SMC000 case, we use $y^{+} = 280$ for the initial layer thickness. A characteristic based pressure inlet boundary condition is used at the nozzle inlet with prescribed total pressure and temperature. A Riemann invariant far-field boundary with ambient atmospheric parameters are used for all external boundaries of the flow in both simulations. The time integration schemes are the low-storage strong stability-preserving (SSP) third order four stage Runge-Kutta (RK34(2N $^{*}$ )) scheme by Ketcheson⁵⁵ for SERDP case and the low-storage fourth order five stage Runge-Kutta scheme (LSRK45(2N)) scheme by Carpenter and Kennegy⁵⁶ for SMC000 case, respectively. Details of the simulations setup can be found in Shen and Miller.¹⁰

The simulation results are validated by comparing the far-field noise spectra predicted by the FWH acoustic solver with measurements. The noise spectra of the SERDP case at all the observer angles agree favorably with the experimental data with a maximum error of 3 dB. However, the SMC000 case over-predicted the noise by up to 5 dB in the sideline direction. This error is mainly due to the transitional shear layer at the nozzle outlet in the simulation, which radiates excessive noise through vortex pairing not observed in the experiment.⁵⁷ Full details of the validation of these cases using LES are shown in Shen and Miller.¹⁰

Figure 7 shows the schlieren images of the SERDP jet. The experimental schlieren is averaged over one thousand times sampled at 8810 Hz. The numerical schlieren is an instantaneous snapshot, due to the high computational cost of LES. Two images are aligned to compare the shock positions. At the nozzle outlet, a barrel shock is formed due to the strong pressure mismatch, and a complex shock system forms inside the bi-conic nozzle. The LES result accurately captures the location of the barrel shock. Downstream from the barrel shock, periodic shock cells are also accurately captured by the simulation relative to the experiment. These shocks are much weaker than the initial barrel shock and are more oscillatory.

Acoustics

Far-field noise via Kirchhoff surface method

The far-field noise for each observation angle is predicted by using 2230 time samples on the KS with radius $R_{s} = 3.5 D$ and length $x_{s} = 35 D$ , with twelve evenly spaced azimuthal locations. Each spectra are averaged at the azimuthal locations to obtain the final spectra. Figure 8 shows comparisons of acoustic predictions relative to measurement at one upstream angle ( $60 °$ ), one sideline angle ( $90 °$ ), and one downstream angle ( $140 °$ ) at $R / D = 30$ . The prediction methods contain far-field assumptions, while measurements are conducted at $R / D = 30$ due to the size of the facility. We expect there to be some mismatch between our predictions and measurements for this reason. It can be seen that BBSAN dominates near $St \sim 1$ at the sideline and upstream directions, and turbulent mixing noise is dominant in the downstream direction. For the KS approach, the predicted noise decays rapidly for $St \geq 1$ . Based on grid resolution of the LES, ${St}_{\max}$ is approximately 1.48 computed with ${St}_{\max} = \frac{(P + 1) c_{\infty} D}{8 Δ x u_{j}}$ , where $P = 3$ is the order of polynomial and $Δ x = 0.3 D$ is the point spacing of the CFD simulation.⁵⁸ In addition, the specific decay frequency also depends on the observation angle.¹² Therefore, the results are reasonable. There is an over-prediction at $60 °$ and $90 °$ when $St \leq 0.1$ . We believe there are two reasons causing this. The first is the open-shell of the KS, and the second is the statistical outlier values of pressure fluctuations on the KS, as shown in Figure 9(a). These values cause spurious spectral contributions at low wavenumbers. It can be seen that the noise spectrum is improved in Figure 8 when we remove these abnormal values of pressure fluctuations by setting them to zero as shown in Figure 9(b).

Far-field noise from instability wave models

The amplitudes of the instability waves are determined using equation (19) to reconstruct the time series of pressure perturbations. The amplitudes of the instability wave models depend on frequency and azimuthal mode. Unfortunately, there is insufficient measurement data in the azimuthal direction to calibrate the models. Therefore, we calibrated our models with the KS predictions, which contain azimuthal variation. We apply the Fourier transform of the time series of pressure fluctuations at the far-field locations obtained from the KS method in terms of time and azimuthal direction. The amplitudes are obtained via matching the predictions from the instability wave models with associated Fourier transforms in the downstream direction, where the large-scale turbulent structures are dominant. In this paper, we choose the dominant direction via the overall sound pressure level (OASPL) (Figures 10 and 11) within the range of $0 < St \leq 0.7$ , which is the range of validity of instability waves for the present calculation. To clarify, we use predictions to represent the instability model predictions in the latter sections. Therefore, we choose $ψ = 150 °$ for SERDP case and $ψ = 140 °$ for SMC000 case as the angle to calibrate the amplitudes of instability waves. The time series of pressure perturbations from the instability wave models are then obtained with the inverse Fourier transform.

Figure 12 shows the sound spectra at $ψ = 60 °, 90 °$ , and $150 °$ for SERDP at $R / D = 100$ . The acoustic predictions are presented within the range of $St < 1$ . It can be seen that the value of SPL of the predictions from the instability wave modes are lower than the experiment and KS method at lower $St < 0.1$ , and higher at St $\geq$ 0.5. The value of SPL at the intermediate $St$ range are close to the experiment and KS method. The near-field sound spectrum ( $ψ = 50 °$ and $R / D = 15$ ) is also predicted with the instability wave model and the comparison with the experimental measurement is shown in Figure 13. The figure shows that there is an over-prediction for the KS method at lower frequencies, which is similar to the far-field results. The prediction from instability wave models show poor agreement with the experiment. But this large discrepancy is expected at the sideline and upstream directions, where the noise from fine-scale and BBSAN dominants.

Figure 14 shows the sound spectra at $ψ = 60 °, 90 °$ , and $140 °$ for SMC000. The acoustic predictions are presented within the range of $St < 1$ . There are screech tones at $St \sim 0.2$ and corresponding harmonic frequencies. BBSAN dominates in the sideline and upstream directions, while turbulent mixing noise dominates in the downstream direction. The value of SPL of the predictions from the instability wave modes are much lower than the experiment and KS method, especially for $St > 0.3$ at the sideline and upstream directions. This is likely due to the screech tones.

Correlation and coherence analysis

In this section, the auto-correlation and cross-correlation of predictions and measurements are compared for the SERDP case. We show these analyses for the SMC000 case with respect to the KS predictions because the time-domain experimental data is not available. The noise source of the bi-conic free jet are analyzed based on the correlation results. In addition, the coherence analyses are used to understand the noise sources in greater detail in the frequency domain.

Auto-correlation

Figure 15 shows a comparison among prediction, experiment, and KS method auto-correlations for the SERDP case ( $M_{j} = 1.3$ ) from $ψ = 150 °$ to $60 °$ . The sampling time for the instability wave predictions is computed via $Δ t = D / (2 u_{j} {St}_{\max})$ , St $_{\max} = 0.7$ , and $Δ t$ is $9.4 \times 10^{- 5}$ s. The length of the time series is 0.26 s, which is approximately 2802 time steps and contains 10 time periods. The sampling time for LES is $Δ t = 1 \times 10^{- 5}$ s. There are distinct shapes from experimental measurements, which represent different noise sources as described by previous investigations for different jets flow with or without shocks.^14,59 The wide peak and large negative peak at $ψ = 150 °$ and $140 °$ demonstrate the character of the noise from large-scale turbulent structures based on the explanation of Tam et al.¹⁴ When $ψ \leq 110 °$ , there is a peak and clear f the noise from large-scale turbuleriment, which implies the combination of noise from fine-scale turbulent structures and BBSAN.⁵⁹ Conversely, the results of predictions show the wide peak, which provide evidence that the noise from instability waves is similar to that of large-scale turbulent structures. Furthermore, the amplitude of negative peak becomes smaller, which shows that the instability waves become weaker in the upstream directions. Instability waves can be used to approximate part of the noise from large-scale turbulent structures.

Figure 16 shows predictions of auto-correlations via instability theory compared with KS predictions for the SMC000 nozzle from $ψ = 160 °$ to $50 °$ . The sampling time $Δ t$ for the instability wave is around $4.75 \times 10^{- 5}$ s. The length of the time series is 0.07 s, which is around 1400 time steps and contains 5 time periods. The sampling time for LES is $5.5 \times 10^{- 6}$ s. It can be seen that the shape of the two predictions are similar when $ψ \geq 110 °$ (downstream direction). When the observation angles become smaller, the KS predictions are lower than that of the instability wave predictions, which means that the effects of large-scale turbulent structures are weaker. There are also sharp peak and a clear ‘ringing’ similar with the SERDP experiment from $ψ = 100^{°}$ to $50 °$ , however, the intensity of S predict and variation are larger due to the screech tone in the SMC000 jet flow. The prediction of acoustic waves from the instabilities have a relatively wide peak with smaller associated negative peaks, which means that the effects of large-scale turbulent structures become weaker in the upstream direction.

Cross-correlations

Figure 17 shows a comparison of maximum normalized cross-correlations from the SERDP nozzle. Comparisons are made between the predictions, KS results, and experiments, where the $ψ = 150 °$ microphone is the fixed microphone. The other microphone is moved along an arc centered on the nozzle exit. For the cross-correlation of $140 °$ , the instability wave prediction and experiment is close, which indicates that the dominate noise at the downstream direction is from the large-scale turbulent structures. The cross-correlation coefficient of the instability predictions at other observer angles is much larger than experiment. This shows that the noise source in the upstream and sideline directions is very different from the downstream direction, which is related to the instability model.

Next, we study the correlation between the near- and far-field acoustic pressure of predictions, experimental measurement, and KS method. We choose the near-field microphone ( $R / D = 15$ and $ψ = 50 °$ ) as the fixed reference microphone. Other microphones are at $R / D = 30$ from $ψ = 150 °$ to $60 °$ . For the calculation of the time delay (τ), the near-field acoustic pressure has an approximate time advancement $d τ = (| y_{n} | - | y_{f} |) / c_{\infty}$ , where the subscript n and f denote the near- and far-field, and y is the distance between the noise source and observer locations.²⁰ The maximum normalized cross-correlation between the near- and far-field is shown in Figure 18. It can be seen that the correlation coefficients of experimental measurements are less than 0.1 from $ψ = 150 °$ to $90 °$ , while there is an incremental change for $ψ = 80 °$ to $60 °$ . These results imply that the near-field upstream and far-field downstream radiation directions are uncorrelated, and the sound sources are distinct. The correlation coefficients of prediction results vary in the range between 0.2 to 0.6.

Figure 19 shows the comparison of the maximum normalized cross-correlation of the SMC000 nozzle between the predictions and the KS method results. Here, the $ψ = 140 °$ microphone is fixed, and the other microphone is positioned on an arc centered about the nozzle exit. It can be seen that the results of predictions are around 0.2 to 0.4 for all the angles, which show that the noise source at different angles are similar with the reference angles (large-scale turbulent structures). The results of KS method are higher than 0.2 when $ψ \geq 110 °$ , and then become lower.

Coherence

An alternative method for ascertaining the level of correlation throughout the radiated noise field is formed via the coherence spectra computed via equation (21). We show the coherence at several St and observation angles. Figure 20 shows the coherence of far-field acoustic pressure for the SERDP case, where the reference microphone is located at $ψ = 150 °$ . The variation of coherence at distinct St in Figure 20 shows a lower coherence value than 0.3 for the chosen St (Figure 20(b)), while the coherence value of predictions are in the range of 0.4 to 1 as shown in Figure 20(a). These results show that significant coherence is present according to the instability wave prediction alone compared to experimental measurements. The latter has dominant sources at varying observer angles, and therefore coherence is not expected to be large over a wide range of observer angles.

Next, we present the coherence between near- and far-field acoustic pressures in Figure 21, where the reference microphone is at the near-field upstream position ( $ψ = 50 °$ and $R / D = 15$ ). The coherence from instability waves varies within the range of 0.4 to 0.7 at all St, while the experimental results have coherence values less than a maximum of 0.30 and are nearly zero through the entire St investigated.

Figure 22 shows comparisons between the coherence of the far-field acoustic pressure of the instability wave model and KS predictions for the SMC000 case. The reference microphone is located at $ψ = 140 °$ in the far-field. We examine coherence at distinct $St = 0.1$ , 0.2, and 0.4 for the SMC000 case. The coherence values based upon predictions from instability wave models are relatively higher than KS methods for different angles at each St. There is a decrease after $ψ = 150 °$ for $St = 0.1$ and 0.4, and it remains at a lower value around 0.2 from the KS method.

Conclusion

We have examined radiation statistics from two supersonic jet flows through the use of instability theory, LES, and experimental measurements. Our LES and newly implemented Kirchhoff surface prediction scheme have been validated against experiments for both the flow-field and acoustic spectra. We have compared statistics of near-field upstream radiation with downstream far-field radiation at ψ = 50 deg. Coherence between the upstream and downstream directions calculated from experimental measurements have a lower value relative to the predictions of instability waves. As expected, there is high coherence from instability waves in the upstream and downstream directions that reside between 0.40 and 0.70 for the St we investigated. The noise from large-scale structures and their origin (the instability waves) is dominated by fine-scale noise in the upstream direction. A most difficult task remains, which is to extract the noise from large-scale structures in the upstream direction. These unique statistics of large-scale structures are masked by at least 10 dB of fine-scale mixing noise. If they were to be extracted, then it might be possible to integrate them into a control system that has a sensor on the aircraft fuselage, which is in the upstream direction relative to the exhaust flow.

Footnotes

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 author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article This program is supported by SERDP WP19-1014 SERDP/ESTCP under program manager Robin Nissan. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of SERDP/ESTCP.

ORCID iDs

Jianhui Cheng

Steven AE Miller

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