Theoretical investigation of alteration and radiation of large-scale structures due to jet impingement

Abstract

Jet flows impinge on launch pad structures and aircraft carrier deck blast deflectors. Turbulent structures are deformed and acoustic radiation is reflected by the deflector. The coupling of reflected acoustic waves with the instability waves of the jet turbulence increases their amplitude and causes a feedback loop. Resultant far-field acoustic radiation is amplified. This amplification results in additional tones with significant spectral broadening occurring at frequencies corresponding to the constructive interference. We present a simple prediction methodology in the form of an acoustic analogy. The analogy accounts for reflected acoustic waves through a tailored Green’s function and models the large-scale structures as spatially and temporarily growing and decaying instability waves. The predictions are compared with two experimental datasets. Predictions compare favorably with measured frequencies and spectral broadening in the far-field.

Keywords

Jet impingement feedback large-scale noise reflection scattering

Introduction

High-speed off-design jet flows from military engines and rockets continue to cause harmful noise radiation. Additional noise sources form when jet flows impinge on blast deflectors. For example, acoustic loading on the propulsion system is quantified by Eldred¹ and noise on aircraft carriers is quantified by Aubert and McKinley.² Radiation reflected or created by the blast deflector alters the turbulent structure within the jet. At particular frequencies, the large-scale turbulence within the jet is amplified and coupled with the feedback from the impingement surface. This feedback process causes the far-field radiated noise from large-scale structures to be amplified. In this paper, we propose a simple model to predict the spectral broadening and amplification of the radiated noise from large-scale coherent turbulence in the presence of structures.

Noise radiation from jet flows is complex and contains many sources that include mixing and shock-associated noise. The shock-associated noise is both tonal and broadband in nature, where the broadband component is often called broadband shock-associated noise (BBSAN). Mixing noise is divided into contributions from the relatively spatially incoherent turbulence that is locally isotropic, the large-scale turbulent structures that are highly spatially coherent and anisotropic, and the Mach wave radiation. In this paper, a jet operating in an environment without interference of its development is a “free-jet”. Additional noise sources occur when disturbances or solid structures are present. For example, when a jet interacts with a plate that is aligned with the flow (a ground surface) or impinges on a surface (a deflector), then additional sources are created, which are called jet structure interaction noise (JSIN), and the free-jet sources are altered.

Figures 1 and ² show an isometric and side view of the jet impingement problem analyzed in this paper. In Figure 1, the observer and source vectors are defined as X and Y, respectively. The location of the trailing edge of the deflector is defined by x_p and y_p. Figure 2 shows the inclination angle α of the impinging plate, which is the angle of the deflector relative to the jet axis. Figure 2 illustrates large-scale structures of the jet as well as the acoustic waves scattered by the deflector edge. Instability waves generated from the nozzle will propagate downstream, deflect from the plate, and propagate back upstream and interact with the jet.

Figure 1.

Isometric view of the problem setup.

Figure 2.

Side view of the problem setup with large-scale structures and acoustic waves.

The measured spectra from an off-design jet is shown in Figure 3, where both the free-jet of the small-hot jet acoustic rig (SHJAR)³ and installed jet of the jet structure interaction test (JSIT)⁴ are shown as lines with symbols. The JSIT contained a large flat plate near the jet flow. Predictions and measurements of the free and installed jet correspond to the SHJAR Small Metal Chevron 16 (SMC016) nozzle operating at $M_{d} = 1.5, M_{j} = 1.29$ , TTR = 1, $x_{p} D^{- 1} = 20, y_{p} D^{- 1} = - 2$ , at $R D^{- 1} = 100$ , and Ψ = 50º. Here, M_d is the design Mach number, M_j is the fully-expanded Mach number, TTR is the total temperature ratio, x is the distance from the nozzle exit to the observer, and $Ψ$ is the observer angle. Predictions for each noise component are performed individually and summed to create total predictions using the methods of Miller.^5–7

Figure 3.

Predictions and measurement of the free-jet and installed jet corresponding to the SMC016 nozzle operating at $M_{d} = 1.5, M_{j} = 1.29$ , TTR = 1, $x_{p} / D = 20, y_{p} / D = - 2$ , at $R / D = 100$ , and Ψ=50°.

When jets operate in the vicinity of structures, the incident acoustic waves may reflect back into the flow-field. These acoustic waves alter the aerodynamic noise sources and a coupling feedback loop is created. This feedback mechanism was documented using ultra-high frame rate schlieren in an impinging underexpanded supersonic jet by Mitchell et al.⁸ Figure 4 shows one frame of the schlieren of Mitchell et al.,⁸ where the off-design jet flows from top to bottom. The jet impinges on a blast deflector that is normal to the flow ( $α = π / 2$ ). One acoustic wave is labeled by Mitchell et al.,⁸ which has propagated upstream and originated from the impingement plate. This acoustic wave amplifies new instability waves or large-scale turbulent structures. One such large-scale structure is identified in another frame of the high-speed schlieren system of Mitchell et al.⁸ (shown in Figure 5). The process is cyclic and repeats at specific frequencies. These amplified instability waves radiate noise. The noise appears as tones with significant spectral broadening in the large-scale mixing noise spectra in the jet far-field.

Figure 4

Schlieren of a jet impinging on a blast deflector. Flow is from top to bottom and an impingement plate is at the bottom. An acoustic wave is traveling upstream and alters the instability wave. Used with permission of Mitchell et al.⁸

Figure 5.

A schlieren of a jet moving left to right where a large instability wave is circled. Used by permission of Mitchell et al.⁸

Experimental research

Several experimental investigations on jet impingement have demonstrated the effects of a feedback loop generated by acoustic waves interacting with instability waves. Powell⁹ was likely the first to examine the interaction of acoustic waves within a supersonic off-design jet flow. Powell⁹ placed a reflector near the flow and also introduced an edge in the flow to examine this effect. Glass¹⁰ created shadowgraphs of the jet when there was a reflecting surface, with and without fiberglass, near the nozzle. The results showed that the jet decayed more rapidly when the reflecting surface was exposed and, subsequently, a dominant tone was present in the cases of more rapid jet decay. This lead to the notion of an acoustic feedback mechanism disrupting the flow. Weightman et al.¹¹ employed ultra-high-speed schlieren images to analyze a periodic transient shock in the wall jet region of an underexpanded jet impingement. Weightman et al.¹¹ argued the phase lag between the shock formation and acoustic wave formation causes a phase lag in the acoustic feedback process. Henderson et al.¹² investigated the structure of a supersonic jet impinging on a flat plate at various M_j and $x_{p} D^{- 1}$ . They determined that the production of tones occurred when the Mach number upstream of the stand-off shock is greater than M_j. However, when the stand-off shock was near the plate, no tones were produced.

A similar experiment of Akamine et al.¹³ examined acoustic waves generated from the impingement region of the jet with a flat plate at a 45º. They determined that two types of acoustic waves were generated in this impingement region, with propagation directions of 30º and 75º from the plane of the plate. Ito et al.¹⁴ classified a particular jet impingement flow as “Type 1 with Bubble”, which occurs when the plate angle is greater than 50º and $x_{p} D^{- 1}$ is approximately 0.5–1.5. Krothapalli et al.¹⁵ investigated the role that large-scale structures and the feedback mechanism play in a loss of lift phenomena, which may occur for vertical take-off and landing aircraft near the ground. Their experiment included a nozzle and lift plate positioned above a large flat plate. They showed that as the nozzle gets closer to the flat plate, a “suckdown” force of as much as 60% of the jet thrust can occur, and suggests that eliminating the oscillations within the jet can reduce the amount of lift lost. Worden et al.¹⁶ focused on addressing the specific issue of jet impingement on blast deflectors. In their experiment, a nozzle was placed at a distance of $x_{p} D^{- 1} \approx 6$ to 12 from a blast deflector. Worden et al.¹⁶ found that noise in the upstream direction is largely dominated by impingement noise and results in an increase in sound pressure level (SPL) of up to 12 dB.

Computational and theoretical research

Several studies involving large eddy simulation (LES) and detached eddy simulation (DES) have been performed to study the structure and acoustic feedback mechanism in impinging jets. Brown and Frendi¹⁷ used DES to study the near-field acoustic spectra of a converging-diverging nozzle exhausting on a flat plate and showed that smaller plate to nozzle distances resulted in increased near-field SPL. Gojon et al.¹⁸ and Gojon and Bogey¹⁹ investigated acoustic properties of planar and round jets impinging on a flat plate. In both cases, near-field pressure data was sampled and Fourier decomposition was performed to extract the amplitude and phase of dominant tones. Uzun et al.²⁰ used LES to investigate tonal production from jet impingement on a flat plate during isothermal and heated conditions. They performed dynamic mode decomposition and showed that the most dominant mode of the flow-field had a temporal frequency related to the impingement tone frequency. The results of Uzun et al.²⁰ show the relationship between the acoustic feedback mechanism and tonal noise produced by the jet. Golubev et al.²¹ used an LES with a wall-model to capture the acoustic feedback mechanism to study the variation of α and the directivity of the acoustic field.

A number of other computational investigations have been performed without the use of LES and DES. Goto et al.²² performed Reynolds-averaged Navier–Stokes (RANS) simulations to investigate the shock structure of jets impinging on a plate at different angles. Using an implicit LES alongside the RANS simulation, they found that treble points from the jet shock, plate shock, and tail shock move intensely along the plate. Honda et al.,²³ using a modified weighted nonlinear scheme, observed acoustic waves emerging from two different sources near the plate. These two sources are the interaction of the plate shock and shear layer and the interaction of the bubble shock and shear layer. For a more comprehensive review of numerical studies of the jet impingement flow-field.^24–26

Several other computational investigations have taken different approaches to analyzing the jet impingement problem. Hong and Jeon²⁷ and Tsutsumi et al.²⁸ implemented a Roe flux splitting scheme and a hybrid LES/RANS scheme, respectively, to resolve the impingement flow-field. McIlroy and Fujii²⁹ approached the jet impingement problem using a simple high-resolution upwind scheme for the convective terms. This method enabled McIlroy and Fujii²⁹ to capture the structure of the flow such as shocks and pressure peaks on the plate.

A number of theoretical studies examined jet impingement and the acoustic feedback phenomena. Messersmith³⁰ employed a convection-entrainment model, similar to the one developed by Nosseir and Ho,³¹ to predict the impingement tones produced by jet impingement. The model provides a simple means of predicting impingement tones and requires only the nozzle pressure ratio (NPR), TTR, and x_p. Papamoschou³² and Papamoschou et al.³³ constructed a wavepacket model and calibrated it with far-field acoustic measurements. The model used steady RANS simulations and a scattering technique to predict installed jet noise. Cavalieri et al.³⁴ computed a scattered sound field around a semi-infinite flat plate using a tailored Green’s function and compared the results to experimental data. They showed that the plate-jet interaction causes an amplification of low frequency noise, which alters the far-field measurements at low frequencies. Huber et al.³⁵ investigated the flight effect on noise scattered by the trailing edge of a flat plate. They modeled the flat plate using a tailored Green’s function, assuming the plate was semi-infinite, and a boundary element method formulation of the Helmholtz equation. Both the tailored Green’s function approach and the boundary element solution agreed with measurement. This comparison provides evidence that for a sufficiently large plate a tailored Green’s function can be used in place of a boundary element method formulation.

Present approach

In this paper, we predict the interference pattern of the radiated acoustic waves from the large-scale turbulence or instability waves within an off-design supersonic jet. The scattered waves (due to impingement) interact with the model of the instability wave basis function and amplify or destruct it through the effect of interference. The scattered portion of the problem is governed by the tailored Green’s function of a semi-infinite flat plate. The source depends on the scattered acoustic field, which results in a model equation that is not truly closed form. For this reason, we evaluate the equation recursively to obtain an amplified instability wave, and then predict the far-field acoustic pressure. The far-field acoustic pressure, due to the amplified instability wave basis function, is compared with measurement. The initial model is calibrated with jets operating without impingement.

The next section of this paper presents the mathematical model for the prediction of the interference pattern of radiated acoustic pressure spectra. Next, the results are presented for the tailored Green’s function, the basis function of the instability wave, static jets without impingement, and the interference pattern due to jet impingement. Finally, a summary of the findings are presented.

Mathematical model

The governing equations are the Navier–Stokes equations. We initially follow Lighthill³⁶ and define the continuity equation as

\frac{\partial ρ}{\partial t} + \frac{\partial ρ u_{i}}{\partial x_{i}} = 0

(1)

and the momentum equation as

\frac{\partial ρ u_{i}}{\partial t} + \frac{\partial ρ u_{i} u_{j}}{\partial x_{j}} = \frac{\partial τ_{i j}}{\partial x_{j}}

(2)

where

τ_{i j} = - p δ_{i j} + μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) - \frac{2}{3} μ \frac{\partial u_{k}}{\partial x_{k}} δ_{i j}

(3)

p is the pressure, t is time, u is the velocity vector, x is the spatially independent variable, δ_ij is the Kronecker delta function, ρ is the density, and μ is the viscosity. Terms where the same subscript resides imply summation (standard Einstein notation). We apply the partial derivative operator on equation (1) with respect to time and separately apply the divergence operator on equation (2). Subtracting the modified continuity equation from the divergence of the momentum equation, then adding

(\frac{\partial^{2} p}{\partial x_{i} \partial x_{j}} - c_{\infty}^{2} \frac{\partial^{2} ρ}{\partial x_{i} \partial x_{j}}) δ_{i j}

(4)

to both sides of the resulting equation, and simplifying yields Lighthill’s acoustic analogy

\frac{\partial^{2} ρ}{\partial t^{2}} - c_{\infty}^{2} \frac{\partial^{2} ρ}{\partial x_{i} \partial x_{i}} = \frac{\partial^{2} T_{i j}}{\partial x_{i} \partial x_{j}}

(5)

where

T_{i j}

is the Lighthill stress tensor (see Lighthill³⁶ for details and discussion). We have now obtained a wave equation with a right-hand-side equivalent source. The Green’s function of equation (5) is governed by

\frac{\partial^{2} g}{\partial t^{2}} - c_{\infty}^{2} \frac{\partial^{2} g}{\partial x_{i} \partial x_{i}} = δ (x - y) δ (t - τ)

(6)

where

g (x; y, t; τ)

is the Green’s function, x is an observer location, y is a source location, t is observer time, τ is source time, and δ is the Dirac delta function. Using the Green’s function and a convolution integral, assuming that

p - p_{\infty} = c_{\infty}^{2} (ρ - ρ_{\infty})

for small values of p and p, the general solution of equation (5) is

p - p_{\infty} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{\partial^{2} T_{i j}}{\partial x_{i} \partial x_{j}} g d y

(7)

We must now find the free-space Green’s function and tailored Green’s function. It is well known that the solution of equation (6) in three-dimensional space is

g (x; y, t; τ) = \frac{δ (t - τ - | x - y | c_{\infty}^{- 1})}{4 π | x - y |}

(8)

A tailored Green’s function, g_t, is required for the solution in the presence of an impingement plate. We adopt the approach of Ffowcs-Williams and Hall,³⁷ where Lighthill’s equation is written as the inhomogeneous Helmholtz equation. The acoustic velocity normal to the semi-infinite plane is zero. Ffowcs-Williams and Hall³⁷ apply Gauss’ theorem to convert the volume integrals into surface integrals. These surface integrals vanish because of the velocity normal boundary condition. They showed that the tailored Green’s function of the Helmholtz equation in the far-field for the semi-infinite plate is described by equation (9(in the frequency domain. The inverse Fourier transform of g_t will then be taken to obtain a time dependent function that is integrated in equation (22)

g_{t} (x; y, x_{p}, y_{p}, α, ω) = \frac{e^{(1 / 4) κ π}}{\sqrt{π}} (\frac{e^{- i κ R}}{4 π R} \int_{- \infty}^{u_{R}} e^{- i u^{2}} d u + \frac{e^{- i κ R'}}{4 π R'} \int_{- \infty}^{u_{R'}} e^{- i u^{2}} d u)

(9)

where u_R and

u_{R'}

are

u_{R} = 2 \sqrt{\frac{k r_{o} r_{s}}{B + R}} \cos (\frac{β_{o} - β_{s}}{2})

(10)

and

u_{R^{'}} = 2 \sqrt{\frac{k r_{o} r_{s}}{B + R'}} \cos (\frac{β_{o} + β_{s}}{2})

(11)

Here, R and R′ are the distances between the observer and the source, as well as the observer and image source, respectively. The image source is defined as the mirror image of the source across the plane of the impingement plate. The coordinate system (r, β, z) originates on the trailing edge of the plate with the z-axis parallel to the edge and β = 0 on the flat plate. We can define the values for r, β and z for some observer or source A = (a₁, a₂, a₃) as

r = \sqrt{{(a_{2} - y_{p})}^{2} + {(a_{1} - x_{p})}^{2}}

(12)

β = π - \cos - 1 (\frac{x_{p} (a_{1} - x_{p}) + y_{p} (a_{2} - y_{p})}{\sqrt{{(a_{2} - y_{p})}^{2} + {(a_{1} - x_{p})}^{2}} \sqrt{{(y_{p})}^{2} + {(x_{p})}^{2}}})

(13)

and

z = a_{3}

(14)

The variable κ is the wavenumber defined as $ω c_{\infty}^{- 1}$ and B is

B = \sqrt{{(r_{o} + r_{s})}^{2} + {(z_{o} - z_{s})}^{2}}

(15)

We evaluate the integrals of equation (9) through the complex exponential in the integrand with use of Euler’s equation in equation (16) and obtain

\int_{- \infty}^{u_{R}} e^{- i u^{2}} d u = \int_{- \infty}^{u_{R}} \cos (u^{2}) - i \sin (u^{2}) d u

(16)

We substitute $u = 2^{- 1 / 2} π^{1 / 2} t$ and make use of the symmetric relations of Fresnel integrals (see Zhang et al.³⁸ for details). We obtain expressions for both integrals which are

\int_{- \infty}^{u_{R}} e^{- i u^{2}} d u = \frac{1}{2} \sqrt{\frac{π}{2}} [2 C (\sqrt{\frac{2}{π}} u_{R}) + 1] - i \frac{1}{2} \sqrt{\frac{π}{2}} [2 S (\sqrt{\frac{2}{π}} u_{R}) + 1]

(17)

and

\int_{- \infty}^{u_{R'}} e^{- i u^{2}} d u = \frac{1}{2} \sqrt{\frac{π}{2}} [2 C (\sqrt{\frac{2}{π}} u_{R'}) + 1] - i \frac{1}{2} \sqrt{\frac{π}{2}} [2 S (\sqrt{\frac{2}{π}} u_{R'}) + 1]

(18)

where C(x) and S(x) represent the Fresnel cosine and sine integrals with upper bound x.

We now model the source term within equation (7), which is the double divergence of the Lighthill stress tensor. Recall that the Lighthill stress tensor is $T_{i j} = ρ u_{i} u_{j} + (p - c_{\infty}^{2} ρ) δ_{i j} - τ_{i j}$ . We propose an equivalent source model of the form

T_{i j} \approx ({\bar{ρ}}_{j} + \hat{ρ}) {({\bar{u}}_{j} + {\hat{u}}_{j})}^{2} P

(19)

where the overbar represents an average, the hat represents the fluctuation due to the waves reflected by the plate, and

P

represents the normalized large-scale structure. We make use of the relation of

u' = p' ρ_{\infty}^{- 1} c_{\infty}^{- 1}

and

p' = c_{\infty}^{2} ρ'

when finding

\hat{u}

and

\hat{ρ}

. In this model,

\bar{u}, \hat{u}, \bar{ρ}

, and

\hat{ρ}

alter the magnitude of the equivalent source. The variables with hat operators are found from iterative acoustic calculations involving the tailored Green’s function of the wave equation with an impingement plate. This approach violates the idea that the acoustic observer (for calculation of

\hat{ρ}

and

\hat{u}

) cannot be at the source; however, we overcome this through the use of a numerical tailored Green’s function. When finding the far-field acoustic pressure, the observer is no longer at a source location, and this mathematical problem no longer occurs. The choice of using a scaling term like

ρ u^{2}

will result in the jet noise scaling as a quadrupole source that is consistent with the Lighthill

u_{j}^{8}

law. We model the growing and decaying spatial and temporal waves within the shear layer as

P = Re exp [- \frac{{(l_{x} - y_{1})}^{2}}{l_{x}^{2}}] \exp [- \frac{{(D_{j} / 2 - {(y_{2}^{2} + y_{3}^{2})}^{1 / 2})}^{2}}{l_{r}^{2}}] \cos [m ψ] \times \exp [i (k x - k {\bar{u}}_{c} τ)]

(20)

where l_x is the axial length scale at wavenumber κ, l_r is a radial length scale at wavenumber κ, m is the azimuthal mode number,

{\bar{u}}_{c}

is the convection velocity, κ is the wavenumber, and

ϕ

is the azimuthal angle. The fully expanded diameter, D_j, is the diameter that represents the equivalent exit diameter for fully-expanded jet flow

\frac{D_{j}}{D} = {[\frac{1 + \frac{γ - 1}{2} M_{j}^{2}}{1 + \frac{γ - 1}{2} M_{d}^{2}}]}^{\frac{γ + 1}{4 (γ - 1)}} {(\frac{M_{d}}{M_{j}})}^{1 / 2}

(21)

where D is the nozzle diameter and γ is the ratio of specific heats. Note that D_j is equal to D for subsonic jets.

We drop the higher order term within equation (19). The total or actual instability wave is a summation over all possible modes and wavenumbers. We adopt our models of equations (19) and (20) and substitute them into equation (7). The resultant prediction equation is

\begin{matrix} p' (x, t) = \int_{- \infty}^{\infty} … \int_{- \infty}^{\infty} g (x; y, t; τ) \sum_{m = - \infty}^{\infty} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} ({\bar{ρ}}_{j} + \hat{ρ}) ({\bar{u}}_{j} + \hat{u}) 2 \exp [- \frac{{(l_{x} - y_{1})}^{2}}{l_{x}^{2}}] \\ \times \exp [- \frac{{(D_{j} / 2 - {(y_{2}^{2} + y_{3}^{2})}^{1 / 2})}^{2}}{l_{r}^{2}}] \cos [m ψ] \cos [(κ y_{1} - κ {\bar{u}}_{c} τ)] d κ d y \end{matrix}

(22)

We now seek the pressure field from equation (22) in the far-field without any impingement plate nearby. Integrations of the double divergence of $T_{i j}$ are shown in Appendix 1. We adopt the length scale models of Miller,³⁹ where $l_{x} = (1.07 / D_{j}) (0.1028 M_{j} + 0.0654) y_{1} D_{j}$ and $l_{r} = 0.33 l_{x}$ . We model the convection velocity as, ${\bar{u}}_{c} \approx 0.7 \bar{u}$ , where

\bar{u} = {\begin{array}{l} u_{j} & if y_{1} < y_{c} \\ (u_{j} - u_{\infty}) (1 - \exp [\frac{1.35}{1 - y_{1} / y_{c}}]) + u_{\infty} & if y_{1} \geq y_{c} \end{array}

(23)

and y_c is the jet core length. The core length, y_c is

y_{c} = {\begin{array}{l} D_{j} (4.3 + 1.2 M_{j}^{2} + 1.2 (1 - T_{j} / T_{\infty})) & if T_{j} < T_{\infty} \\ D_{j} (4.3 + 1.2 M_{j}^{2}) & if T_{j} \geq T_{\infty} \end{array}

(24)

Evaluation of the model equation

We now discuss the numerical evaluation of equation (22). Equation (22) is initially evaluated with a given plate impingement angle α, where α is zero for a ground surface and $π / 2$ for normal impingement. The initial evaluation uses the tailored Green’s function of equation (9). The observer vector is set at x for every source position y. We are able to then estimate the values of $\hat{ρ}$ and $\hat{u}$ . We continually repeat the evaluation of equation (22) using updated values of $\hat{ρ}$ and $\hat{u}$ . Each evaluation of the model equation uses numerical integration over the volume of the jet using the trapezoidal rule. Generally, small scale laboratory jet flows are restricted to lower modes (for one of the many examples see Malla et al.⁴⁰). For this reason, we restrict our modal content to m = 0 and $m = \pm 1$ . The final time equation (22) is evaluated, $x$ is set at the observer positions corresponding to the measurement microphone locations for comparison with experiment. Retarded time, τ, is computed using the propagation vector x−y and $c_{\infty}$ . This process is repeated for each observer time of interest, t.

Results

Scattering

The tailored Green’s function of Ffowcs-Williams and Hall³⁷ governs the scattering of sound about a semi-infinite flat plate and was presented in equation (18). Here, we validate g_t to demonstrate that its implementation is correct and to assess error relative to another prediction approach and measurement.

We first examine the case where the source point within equation (18) lies upon the boundary condition where the normal velocity is zero. In this scenario, the resultant incident field will be equal to that of the free-space Green’s function (see Ffowcs-Williams and Hall³⁷ for details). Figures 6 and 7 show a contour plot of a source in a free-field and a source placed on the semi-infinite flat plate, respectively. Both contour plots have a radiating acoustic field with equal source magnitude. The tailored Green’s function (Figure 7) reduces to the free-space Green’s function correctly in our numerical implementation.

Figure 6.

Contour plot of the magnitude of the free-field Green’s function with a single source at origin.

Figure 7.

Contour plot of the magnitude of the tailored Green’s function with a single source placed on the semi-infinite plate.

Dunn and Tinetti⁴¹ developed the fast scattering code (FSC) and compared their numerical predictions to an experiment (NASA TP 1004) conducted by Ahtye and Mcculley,⁴¹ which measured the ΔdB reduction due to a source of unit strength shielded by the presence of a plate with square edges. Figure 8 shows the experimental setup of Ahtye and Mcculley.⁴² In the experiment, the plate was placed 0.25 m from a monopole source of unit strength. Measurements of SPL were obtained on a linear microphone array that was five meters parallel to the surface of the plate on the shielded side. SPL was also measured along the microphone array from the same unit source without the presence of the plate to serve as a baseline. We refer to the ΔdB difference between free-field and shielded predictions or measurements as the “shielded SPL”. In order to analyze the problem of a finite plate shielding a point source, four semi-infinite flat plates are considered, with each edge corresponding to an edge of the finite plate. The four solutions for scattering around each edge are then summed together to obtain the solution for a finite plate. Note that the tailored Green’s function proposed by Ffowcs-Williams and Hall³⁷ does not incorporate thickness or the type of edge of the plate.

Figure 8.

Diagram of NASA TP 1004 experiment and fast scattering code computational analysis.

Figures 9 to 12 show a comparison of shielding SPL predictions and measurements for different frequencies plotted against θ, which is defined Figure 8. The tailored Green’s function captures the mean shielding SPL compared to the experimental results when θ is less than 35º. When θ is larger than 35º, the tailored Green’s function prediction loses accuracy as it slightly over predicts SPL relative to the experimental results. Diffraction patterns are not predicted by g_t because they are essentially canceled out through interference in the semi-infinite plate solution. Diffraction patterns are however observed in measurement and the predictions of FSC. Figures 9 to 12 also show that as the frequency increases so does the absolute shielding SPL. Both the evaluation of g_t and the FSC code are able to capture the trend of increasing shielding SPL observed in experiment. The ability of g_t to capture the mean shielding SPL and the trend of increasing shielding SPL with frequency supports the method for use in our scattered jet noise predictions.

Figure 9.

Comparison of plate shielding results with fast scattering code⁴¹ and NASA Experiment TP 1004⁴³ for f = 1000 Hz.

Figure 10.

Comparison of plate shielding results with fast scattering code⁴¹ and NASA Experiment TP 1004⁴³ for f = 2000 Hz.

Figure 11.

Comparison of plate shielding results with fast scattering code⁴¹ and NASA Experiment TP 1004⁴³ for f = 4000 Hz.

Figure 12.

Comparison of plate shielding results with fast scattering code⁴¹ and NASA Experiment TP 1004⁴³ for f = 8000 Hz.

Wave model

A central point of this research is the use of a basis function of the instability wave shown in equation (20) for $P$ . We graph the behavior of this equation for typical values within a turbulent flow. Figure 13 shows the normalized instability wave basis function, P, at various times, t, on the x-axis. Arguments of equation (20) are $κ = 10.0, {\bar{u}}_{c} = 0.50, l_{x} = 1.0, l_{r} = 1.0$ , and $τ_{c} = 1.0$ . The arguments are held constant while t varies from the time which the instability wave is at its largest amplitude (t = 0) through its decay at t = 1. The envelope represents the maximum bound of the instability wave in space-time. Figure 13 shows that as time increases the instability wave decreases, but the wavenumber and propagation speed are altered. In the context of this model, instability waves reside everywhere within the turbulent flow and their amplitudes, wavenumbers, and propagation speeds vary.

Figure 13.

Normalized large-scale structure function centered at x = 0 at various times t constrained by an envelope.

We now examine the normalized instability wave basis function presented in equation (20). For demonstration purposes, we set D = 1 m, ${\bar{u}}_{c} = 10$ m/s, and m = 3. The length scales of the model are l_c = 4, l_r = l_y, $l_{x} = 2.5, l_{y} = 0.25, l_{z} = 0.25$ , and $τ_{c} = 0.5$ . The ambient speed of sound is 343 m/s, f = 0.75 Hz, and wavenumber $κ = 2 π f$ . Contours of the model in the x–y plane and the y–z plane are shown in Figure 14 for modes 0 to 3. The nozzle exit resides on the left side between y of −0.5 and 0.5. The lip line of the nozzle, within the cross-stream plots, resides on the circle centered about the origin with radius 0.5. The centers of the wave are on the nozzle lip-line, located l_c = 4, downstream from the nozzle exit. In the streamwise direction, the waves propagate at speed ${\bar{u}}_{c}$ and have wavelength dependent on κ. The contour plot of the cross-stream direction shows that the waves decay as the distance from the nozzle lip line increases. We assume that the instability or large-scale turbulent structures are a summation of the individual instability wave model over m and κ.

Figure 14.

Contours of the normalized equivalent wave for modes 0 through 3 in the x–y and x–z planes: (a) m = 0 in x–y plane; (b) m = 0 in y–z plane; (c) m = 1 in x–y plane; (d) m = 1 in y–z plane; (e) m = 2 in x–y plane; (f) m = 2 in y–z plane; (g) m = 3 in x–y plane; (h) m = 3 in y–z plane.

Free-field jet aeroacoustic predictions

We now turn our attention to acoustic predictions. For the acoustic predictions, the jet operating conditions are used to estimate the jet velocity and density using the empirical models in equations (23) and (24). We examine the far-field prediction of the model against a static jet without any reflections and multiple predictions of jet impingement in the near-field. Equation (22) is evaluated and calibrated with two conditions of the SHJAR experimental database of NASA Glenn Research Center (see Bridges and Brown³ for details). These predictions and measurements are shown in Figures 15 and ¹⁶. In the first comparison of Figure 15, a jet from the SMC000 nozzle operates at M_d = 1, $M_{j} = 1.0$ , and TTR = 1.00. This nozzle is convergent. The second comparison of Figure 16 shows spectra from the SMC016 nozzle operating at $M_{d} = 1.5, M_{j} = 1.5$ , and TTR = 1.00. This nozzle is designed with the method of characteristics and operates predominantly shock free. In both cases, the nozzle has an exit diameter of 0.0508 m and the observer is located at 100 nozzle diameters from the nozzle exit at an angle of ψ = 130º.

Figure 15.

Prediction compared to measurement of SHJAR of a $M_{d} = 1.0, M_{j} = 1.0$ , D = 0.0508 m, TTR = 1.00 jet at $R / D = 100$ .

Figure 16.

Prediction compared to measurement of SHJAR of a $M_{d} = 1.5, M_{j} = 1.5$ , D = 0.0508 m, TTR = 1.00 jet at $R / D = 100$ .

The predictions are very similar to the large-scale similarity spectra proposed by Tam et al.⁴³ This is expected because the noise predicted is from a basis function designed to mimic measurements of large-scale structures or instability waves within the jet. This approach does not represent a total noise prediction method capable of predicting the total spectra accurately because it lacks a fine-scale mixing noise source. The purpose of the free-field predictions is to calibrate the prediction method for subsequent examination of the constructive interference pattern due to the acoustic-instability wave feedback mechanism. We have calibrated the coefficients in our model through comparison with the method of the large-scale similarity spectra of Tam et al.⁴³ and associated SHJAR experiments. The calibration is performed by adjusting the coefficients for the M_d = 1, $M_{j} = 1.0$ , and TTR = 1.00 operating condition to match the highest magnitude and peak frequency of the spectra at $R / D = 100$ and ψ = 130º. This optimization is performed through trial and error by altering the coefficients and evaluating the model. Once the coefficients are set against this condition they are never altered.

Jet aeroacoustic impingement predictions compared with Messersmith

We now examine the experiment of Messersmith³⁰ and make predictions for comparison. Messersmith³⁰ conducted experiments of jets impinging on large flat plates at relatively short $x_{p} D^{- 1}$ distances with α = 90º. The nozzles of Messersmith³⁰ are convergent and convergent-divergent at M_d = 1 and $M_{d} = 1.5$ , respectively. Both nozzle exit diameters are D = 0.00635 m and operate at $M_{j} = 1.358$ and $M_{j} = 1.559$ with TTR = 1. Microphones are placed very close to the jet flow-field in the near-field at the sideline direction just outside the jet shear layer. The acoustic measurement spectra of pressure is shown for four cases in Figure 17, where the y-axis is SPL per unit $f D c_{\infty}^{- 1}$ and the x-axis is $f D c_{\infty}^{- 1}$ . This x-axis unit is not typically used but repeated here following the work of Messersmith.³⁰ Figure 17(a) contains conditions M_d = 1, $M_{j} = 1.358$ , and $x_{p} D^{- 1} = 3$ , Figure 17(b) contains conditions $M_{d} = 1.5, M_{j} = 1.358$ , and $x_{p} D^{- 1} = 4$ , Figure 17(c) contains conditions M_d = 1, $M_{j} = 1.559$ , and $x_{p} D^{- 1} = 3$ , and Figure 17(c) contains conditions $M_{d} = 1.5, M_{j} = 1.559$ , and $x_{p} D^{- 1} = 4$ . The constructive interference pattern due to jet impingement is observed as tones with significant spectral broadening.

Figure 17.

Predictions compared to measurements of Messersmith.³⁰ Impingement angle $α = π / 2$ and TTR = 1: (a) M_d = 1, M_j = 1.358, x_p/D=3; (b) M_d = 1.5, M_j = 1.358, x_p/D = 4; (c) M_d = 1, M_j = 1.559, x_p/D = 3; (d) M_d = 1.5, M_j = 1.559, x_p/D = 4.

Predictions of the location of amplification, using the approach of Messersmith,³⁰ are shown as circles. The predictions of Messersmith³⁰ do not include amplitude, and we have selected their amplitude to be 110 dB in each case for comparison purposes only. For this case, we evaluate g_t using the mirror source technique, where the source is placed on the opposite side of a fictitious plane that is centered on the nozzle axis and at the corresponding distanced $x_{p} D^{- 1}$ . Predictions using the newly developed approach of equation (22) are shown as solid lines. In Figure 17(a), the constructive interference pattern is at relatively discrete frequencies and separated by an almost constant $Δ f D c_{\infty}^{- 1}$ of approximately 0.45. The prediction has captured this relatively constant frequency spacing, but has not correctly predicted the relative amplitudes between measurement SPL. Figure 17(b) is unique compared to the other predictions because the constructive amplification only occurs at two discrete frequencies of $f D c_{\infty}^{- 1} \approx 0.425$ and 0.850. Here, the prediction of Messersmith³⁰ is slightly higher at two discrete frequencies while the prediction of equation (22) is slightly lower. Improvements of the newly developed model, relative to the predictions of Messersmith, are observed through the inclusion of spectral widths and the predicted amplitudes. Figure 17(c) and (d) shows similar predictions to that of Figure 17(a). We have generally captured the correct $Δ f D c_{\infty}^{- 1}$ between amplification peaks. Unfortunately, the model does not capture some of the large overpressures compared to the broadband spectrum. Perhaps, this is partly due to the screeching nature of these off-design supersonic unheated jets which alter modal content.

Jet aeroacoustic impingement predictions compared with Worden et al.

We now examine the recent experiment of Worden et al.,¹⁶ who conducted jet impingement studies at the Anechoic High-Temperature Jet Facility at Florida State University. A convergent-divergent nozzle with $M_{d} = 1.5$ was used and operated on-design at $M_{j} = 1.5$ and NPR $\approx$ 3.7. An impingement plate was positioned downstream from the nozzle exit at locations of $x_{p} D^{- 1} = 6, 8, 10$ , and 12 with constant angle of α = 50º. A ground plane is present at $y_{p} D^{- 1} = - 3.6$ . A linear microphone array was used at a sideline distance of $z_{p} D^{- 1} = 15$ . We elect to examine data from the microphone located at 61º upstream from the deflector ( $x_{p} D^{- 1} = 8.9$ upstream from the plate impingement point). The height of the linear microphone array is the same as the nozzle height at 3.6D. The TTRs are prescribed as 1 and 2.81, respectively.

Equation (22) is evaluated through iterations involving g_t and updating $\hat{ρ}$ and $\hat{u}$ , which alters the equivalent source during each iteration. At the end of each iteration, we examine the prediction’s percent change in SPL relative to the previous prediction. We show this convergence property of the model in Figure 18 for the predictions corresponding to the case of Worden et al.¹⁶ with observer location $x_{p} D^{- 1} = 8$ . After approximately 40 iterations the percent change in SPL relative to the previous iteration is approximately 2%. We find similar convergence properties of the model for other observer angles and for the predictions created for comparison with Messersmith.³⁰

Figure 18.

Prediction model convergence in terms of percentage change in ΔSPL versus previous iteration.

Measurements of Worden et al.¹⁶ are shown in Figures 19 (unheated) and 20 (heated), respectively. The y-axis is SPL per unit St and the x-axis is St. Measurements are shown as lines with symbols and predictions using equation (22) are shown as solid lines of corresponding color. We predict the amplified pattern in the region of maximum SPL. Here, g_t is calculated through the evaluation of equation (9), where a single edge is modeled that corresponds to the edge of the experiment. Note that at this radiation angle, the fine-scale mixing noise makes a contribution to the total noise, thus it appears that the model underpredicts the measured spectra. Predictions capture the Δf for both the unheated and heated jet amplification pattern. Predictions shown in Figure 20 over estimate the primary and secondary harmonic interference patterns and spectral widths.

Figure 19.

Predictions compared to measurements of Worden et al.¹⁶ of an unheated on-design jet.

Figure 20.

Predictions compared to measurements of Worden et al.¹⁶ of a heated on-design jet.

Summary and conclusion

Jet structure interaction often involves the exhaust impinging on a blast deflector. Such flows are often present during rocket launch, impingement of engine exhaust on aircraft pylons or flaps, or aircraft take-off during carrier operations. The instability waves in conjunction with reflected waves can constructively interfere to create a “ripple” or constructive interference pattern of the acoustic pressure spectra. This feedback mechanism is modeled through construction of an instability wave basis function or large-scale structure model for an equivalent source of an acoustic analogy. A tailored Green’s function for a semi-infinite flat plate is used to model the deflector. The model is calibrated against static jets without reflections and then compared to two experimental datasets with reflections. The model is initially evaluated recursively to find the correct amplification of the instability wave basis function and then reevaluated to find the far-field acoustic pressure spectrum. Predictions compare favorably, often predict the correct frequencies of constructive interference, and often predict the correct spectral broadening near each interference frequency. The large-scale structure or instability model must be made more general, put on a firmer theoretical foundation, and linked to the underlying governing equations of fluid motion.

Footnotes

Authors’ note

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 the Office of Naval Research.

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: S. A. E. Miller was supported by Office of Naval Research Grant N00014-17–1-2583 “Turbulent Statistics and Associated Acoustic Sources Near Shock Wave Shear Layer Interactions”, under the advisement of Steven Martens, John Spyropoulos, and Russell Powers. Alexander N Carr was supported by the University of Florida Herbert Wertheim College of Engineering Graduate School Preeminence Award.

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