# Noise emissions from excited jets

Osama Marzouk

► To cite this version:

Osama Marzouk. Noise emissions from excited jets. 22nd National Conference on Noise Control Engineering (NOISE-CON 2007), Institute of Noise Control Engineering of the United States of America (INCE-USA), Oct 2007, Reno, Nevada, United States. pp.1034-1045. hal-04887157

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# Noise emissions from excited jets

Osama Marzouk<sup>a</sup>

Department of Engineering Science and Mechanics, MC 0219,
Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061

## ABSTRACT

Exhaust gases emitted from breathing and non-breathing engines are examples of free jet flows, which are encountered in different industrial applications, such as gas turbine engines and rocket motors. Better performance in these applications is associated with higher jet speeds, which implies more noise emissions. This motivated a lot of research to predict and mitigate noise emissions from free jets.

Computational aeroacoustics simulations typically involve issues that might not be significant in traditional computational fluid dynamics. Special attention should be paid to the numerical implementation as we are interested in the noise emission rather than the flow motion or the thermal field. This study numerically investigates the generated noise and perturbations from excited high-speed jets, which are modeled by the linearized Euler equations. We use a fourth-order numerical method, which eliminates any spatial bias in the calculations. The mean flow is prescribed by experimental-analytical approach. Harmonic hydrodynamic excitation is applied at the nozzle exit. The propagation of this excitation is investigated for the pressure (acoustic), velocity, and density waves. We investigate the characteristics of these waves and the differences between them, specially the saturation and decay phenomena. Some features of the noise emission pattern are identified and discussed.

## 1. INTRODUCTION

Aeroacoustics is concerned with the sound generated by the motion of gases. This motion induces pressure perturbations which are heard if they are strong enough and fall into the audible human spectrum. Accurate prediction of noise is a building block in the process of noise control or the design of devices with low noise. Computational aeroacoustics (CAA) has proven to be a reliable tool to predict and simulate noise generation for different classes of problems. Computational aeroacoustics problems differ from classical computational fluid dynamics (CFD) ones in various aspects. This is ascribed to the unique nature of CAA which attempt to capture unsteady low-amplitude waves, whereas industrial CFD commonly attempts to capture steady fields or large-amplitude variations. In addition to that, CAA is concerned with acoustic propagation in the far field thus requires large computational domains. The CFD analysis usually focuses on the near-wall region in order to calculate the forces exerted by the flow field on the structures in contact with the fluid. Also, with the wide range of scales in the audible frequencies (20 Hz to 20 kHz), it becomes necessary to extend the simulation for long times to capture the low-frequency waves which might have relatively very long periods. These issues motivated researchers to develop special techniques that suit CAA problems more than classical CFD counterparts.

<footnote><sup>a</sup> Email address: omarzouk@vt.edu</footnote>
There are different approaches used in CAA, with different levels of accuracy and computational cost. The direct numerical simulation (DNS) approach integrates the compressible Navier-Stokes equations to resolve the flow quantities directly without any modeling or domain decomposition. This method is limited to a certain range of problems due to the tremendous computational time and power needed. It is more suitable as a validation tool of other simplified approaches or to study the underlying physics of the problem. Freund⁴ is one of the few researchers who utilized the DNS approach in CAA. He computed the flow field and the associated acoustic field of subsonic jet, Mach 0.9. The calculated pressure field was used to compute the Lighthill acoustic source. Good agreement was reported between the Lighthill's analogy and the DNS method. However, the difference in the computations amount between the Lighthill analytical approach and the DNS numerical approach is substantial.

Large eddy simulation (LES) is a simplified level of the Navier-Stokes approach, where the flow field is decomposed into large scales and small, subgrid, scales. Goldstein⁵ combined this approach, which is still too expensive for many industrial applications, with another more simplified approach. The latter is the Reynolds-averaged Navier-Stokes equations (RANS) model, which is based on temporal decomposition in contrast to the spatial decomposition of LES. In the RANS approach, the flow field is decomposed into time-averaged and turbulent fields. The time-averaged field is resolved directly, whereas the turbulent one is modeled. In the proposed hybrid approach, the transition occurs from RANS-based modeling near the nozzle lip, to LES-based modeling downstream. The low-frequency sound waves are directly resolved by the hybrid approach, whereas the residual small-scale high-frequency waves are computed through modeling the source terms via linear equations. This approach was motivated by the fact that acoustic contribution from the subgrid scales in LES are not accounted for. Decreasing the size of these scales, by refining the grid, solves this problem but incurs heavy demands for computational resources.

If viscous effects are insignificant, the viscous terms in the Navier-Stokes equations can be dropped, which results in the Euler equations. Then, the flow field can be split into mean and perturbation fields. Provided that the mean flow is known, the perturbation field will be governed by the linearized form of Euler equations. The present study uses the linearized Euler equations to model and compute the perturbation field associated with supersonic axisymmetric jet flow. Harmonic excitation of hydrodynamic form is applied in the potential core of the mean flow which amplifies downstream, saturates, and then decays. We investigate the acoustic field in this problem. We also study the similarities and differences between the pressure (acoustic) waves and the velocity or density waves.

## 2. LINEARIZATION OF EULER EQUATIONS

Navier-Stokes equations fully model the jet motion. By dropping the viscous terms in these equations, we get the Euler equations. In the considered free shear flows, large-scale dynamics are the main sound source in supersonic jets. Viscous effects for these scales are negligible. So, an Euler-based method is a good candidate for noise prediction in this type of problems. Although Euler equations are simplified form of the Navier-Stokes counterparts, both contain the nonlinear (inertia) terms. Linearizing these equations is another simplification that remarkably reduces the computational work needed to integrate the resulting governing system (linearized Euler equations, LEE). It is important to mention here that these simplifications restrict the applicability of the simplified model. However, if the implied conditions are met, the simplified model should be capable of capturing most of the features captured by the more expensive models. For acoustic simulations, the perturbations amplitudes are typically much smaller than mean quantities. This renders linearized models a feasible simulation approach. In fact, the
linearization technique is used in many fields even outside fluid mechanics and aeroacoustics. For example, it is used to reduce complicated structural models into simple linear ones.

Given a general flow variable, $b$, we can represent it as the sum of two quantities: the mean (time averaged), $B$, and the perturbation (fluctuating), $b'$ as follows

$$b = B + b' \qquad (1)$$

where $B$ is assumed to be known and the goal is to solve for $b'$. After solving for the perturbation quantity, we can add it to the mean one to find the total quantity. However, in the present study, we are merely interested in the perturbation field since the pressure perturbations are the source of the noise emissions.

Upon incorporating the above splitting into the nonlinear equations, and then subtracting the terms containing only the mean quantities, we get the linearized system that governs the perturbations. It worth mentioning here that the mean field can be time varying by itself, as in the case of a flow field around oscillating structures. Since we are considering axisymmetric jet, the equations are solved in an axisymmetric coordinate system $(x, r)$, where $x$ is the axial coordinate (aligned with the jet and nozzle centerline) and $r$ is the radial coordinate. The linearized Euler equations in conservative vector from are

$$\frac{\partial Q'}{\partial t} + \frac{\partial F'}{\partial x} + \frac{1}{r} \frac{\partial (r G')}{\partial r} = \frac{1}{r} S' \qquad (2)$$

where $Q'$ is the vector of the conserved perturbation variables

$$Q': \left[ \rho' (\rho u)' (\rho v)' (\rho e)' \right]^T \qquad (3a)$$

and $S'$ is the source term vector

$$S' : [0 \quad 0 \quad p' \quad 0]^T \qquad (3b)$$

The symbols $\rho$, $u$, $v$, $e$, and $p$ are the density, axial velocity component, radial velocity component, total (internal plus kinetic) specific energy, and pressure; respectively. Equation (2) represents a linear system governing the perturbation variables, with coefficients containing the mean quantities.

### 3. MEAN-FLOW QUANTITIES

As mentioned in the previous section, the mean (time-averaged) quantities of the flow are needed in order to solve for the perturbation (fluctuating) quantities. For this purpose, we use a mean flow based on the measurements of Troutt and McLaughlin¹⁵ for a near isentropic supersonic cold jet at Mach number 2.1 (based on the conditions of the jet at the nozzle exit). A glow-discharge device located at a distance of 0.4 nozzle radius ($R_j$) after the nozzle was used to excite the flow. Tam and Burton¹³ proposed continuous functions and relations which accurately fit these measurements. We are using their technique to represent the needed mean-flow quantities. The mean-flow solution for other jets can be obtained by solving the boundary layer equations for wakes¹², which yields a reasonable mean-flow profile. This step will be minor part of the overall process; the major part will be computing the perturbation field since high accuracy is required in that step. Figure 1 shows a schematic sketch of supersonic free jet, with the three main regions: the potential core, transitional region, and fully developed region. The potential core is a near-conical region dominated by potential flow. It extends to $x=10R_j$ after the nozzle. The mean axial velocity is represented by the following half-Gaussian profile
$$ U = \begin{cases} U_j & r < h(x) \\ U_j \exp\left[-\ln(2)\left(\frac{r-h(x)}{b(x)}\right)^2\right] & r \ge h(x) \end{cases} \qquad (4) $$

where $U_j$ is the jet exit velocity, $h$ is the radius of the potential core, $b$ is the half-width of the mixing layer, whose outer radius, $r(0.5)$, is located at $U = 1/2 U_c$, where $U_c$ is the mean centerline velocity. Inside the potential core, we have $U_c = U_j$.

The transitional region is the zone surrounding the potential core, extending to $x=16R_j$. The flow transforms from the uniform potential pattern to the fully-developed viscous one. The following profile is used for the mean axial velocity in the transitional region

$$ U = \begin{cases} U_c(x) & r < h(x) \\ U_c(x) \exp\left[-\ln(2)\left(\frac{r-h(x)}{b(x)}\right)^2\right] & r \ge h(x) \end{cases} \qquad (5) $$

The fully-developed region starts after the transitional region. The velocity profile is characterized by a similarity rule. The following profile is used for the mean axial velocity in the fully-developed region

$$ U = U_c(x) \exp\left[-\ln(2)\left(\frac{r-h(x)}{b(x)}\right)^2\right] \qquad (6) $$

Careful attention is paid while calculating $U_c$, $h$, and $b$ so that the axial momentum is conserved. Other expressions exist for the rest of the mean-flow quantities (i.e., radial velocity, density, temperature, and pressure) based on mass and energy conservation as well as the equation of state. Figure 2 gives the axial distribution of the calculated $h$, $r(0.5)$, and $b$. The same figure shows good agreement of $b$ with the measurements. Figure 3 compares the calculated and measured normalized centerline axial velocity ($U_c/U_j$), and demonstrates the high capability of the analytical technique to represent the measured data. The potential core can be identified in this figure by having $U_c/U_j=1$.

# 4. PERTURBATION QUANTITIES

## A. Primitive Variables

The perturbation field is governed by the system in Equation (2). This system is integrated numerically in time and the development of the perturbation quantities is obtained. The conservative system in Equation (2) yields the perturbed density, $\rho'$, and the conservative perturbation quantities: $(\rho u)'$, $(\rho v)'$, and $(\rho e)'$. They are used along with the mean-flow quantities to extract the primitive perturbed variables: $u'$, $v'$, and $p'$. We have

$$ (\rho u)' = \rho' U + \bar{\rho} u' \qquad (7a) $$

$$ (\rho v)' = \rho' V + \bar{\rho} v' \qquad (7b) $$
$$ (\rho e)' = \frac{p'}{\gamma - 1} + \langle (\rho u)' U + (\rho v)' V \rangle - \frac{1}{2} \rho' (U^2 + V^2) \quad (7c) $$

where $\bar{\rho}$ is the mean density.

## B. Scheme

Computational aeroacoustics analysis requires very accurate numerical schemes and boundary treatments since it inherently tries to capture small-amplitudes fields. So, a classical scheme used for fluid dynamics problems might be unsatisfactory for the present problem. In particular, dissipation and dispersion properties of the numerical scheme need to be very weak. These properties tend to damp the acoustic oscillations or to generate nonphysical oscillations that might affect the acoustic field significantly.

We use a version of the McCormack predictor-corrector scheme that is second order in time and fourth order in space. The differential operator in Equation (2) is split into x-operator and r-operator. The sequence of integrating these operators in one time step is reversed in the following time step. Therefore, a complete integration cycle is performed over two time steps rather than one. This reduces any directional bias in the solution. The time step is constrained by resolution requirement more than stability requirements; it is small enough to capture the anticipated waves. Another difference between classical fluid dynamics problems and aeroacoustics ones is the domain size. The computational domain here extends to $x = 70 R_j$ in the axial direction and $r = 32 R_j$ in the radial direction. This is large enough to include the far-field propagation of perturbation waves.

## C. Boundary Conditions

Similar to the integration scheme, the boundary conditions treatment is a big issue when simulating noise emissions. We use radiation and outflow conditions that avoid nonphysical oscillations and allow acoustic and perturbation waves to leave the computational domain without reflection as in the realistic situation.

At the inflow boundary, and from the centerline to $r = 4 R_j$, an input hydrodynamic (non-propagating) periodic excitation is applied to reproduce the excitation in the experimental setup and to increase the level of coherence in the propagation field. This excitation has the following form

$$ [\rho' \ u' \ v' \ p']^T = \delta \text{Real} \ \{[\bar{\rho}(r) \ \hat{u}(r) \ 0 \ \hat{p}(r)]^T \exp(-i\omega t)\} \quad (8) $$

where $\delta$ is a scaling parameter chosen as 0.03 so that calculated sound pressure levels match the measured ones; $\bar{\rho}(r)$, $\hat{u}(r)$, and $\hat{p}(r)$ have Gaussian profile; and $\omega = 0.1 \pi$. Although this excitation is expressed in terms of the primitive variables, it is transformed and expressed in terms of the conservative variables using Equation (7) before incorporating this excitation into the integration scheme.

# 5. RESULTS

The conservative perturbation variables are directly resolved by integrating Equation (2) and then the primitive perturbation variables are evaluated in a post-processing step. The mean flow is steady and time independent, whereas the perturbation field is, by definition, time dependent. As the simulation starts, perturbation waves travel from the near field to the far field, and after a while, the large-scale waves take a periodic pattern. The frequency of this pattern is the frequency of the applied excitation.
To validate the adopted technique, the perturbed pressure field is used to compute the acoustic field in terms of the sound pressure levels: $20 \log(P'_{rms}/P'_{scale})$, and the results are compared with available measurements¹⁵ in Figure 4. Good agreement is observed. The scaling perturbation pressure, $P'_{scale}$, is $20^{-6} \text{ N/m}^2$.

For the rest of the figures, the results are normalized with respect to reference values, which are the mean jet density and velocity at the nozzle exit: $\bar{\rho}$ and $U_j$. This leads to better feeling of the order of magnitude of the perturbed solution. In Figure 5, the evolution of the perturbed pressure at different points located in the near and far fields is compared. The perturbed signals settle quickly to a periodic function with a period of 20 time units (which is the period of the applied excitation). The amplitude of the perturbation decays as we go way in the radial direction. There is a phase shift between the perturbed pressure signals at these points as a result of the distances between them, which are traveled by the wave in finite times. Figure 6 illustrates the propagation process of the perturbed pressure waves (acoustic waves), where snapshots of the perturbed pressure field are given at successive instants until the waves traverse the entire domain. This figure indicates directional pattern of propagation. Figure 7 shows the axial distribution of the RMS perturbed pressure at different radial distances. Near the centerline, there is a region of concentrated perturbation about $25 R_j$ after the nozzle. Figures 8-11 show the axial distributions of the perturbed pressure, axial velocity component, radial velocity component, and density; respectively. Comparing these figures shows that the location of the amplitude peaks, or saturation, in the velocity and density perturbations occurs at about twice the distance of the peak pressure perturbation. This is a remarkable difference in the propagation of these waves from the acoustic propagation.

## 6. CONCLUSIONS

The development of the perturbation field for a supersonic free jet subject to hydrodynamic excitation was modeled by the linearized version of the Euler equations. A prescribed mean-flow pattern was used to integrate the equations governing the perturbation field. The perturbation field was computed and validated by comparing the calculated acoustic field to available measurements. Then, the temporal development of the acoustic waves was investigated in the near and far fields. The acoustic waves are emitted from the region of saturated pressure perturbation following a preferred directional pattern. All perturbed variables exhibit periodic pattern after the initial transience is passed. Also, they all decay in the radial direction. However, the saturation point of the pressure (acoustic) waves occurs at an axial distance that is half those observed for the velocities and the density.

## REFERENCES

<footnote n="1">¹ Alvin Bayliss and Eli Turkel, “Far field boundary conditions for compressible flows,” *J. Comp. Phys.* **48**(2), 182-199 (1982).</footnote>

<footnote n="2">² Daniel J. Bodony and Sanjiva K. Lele, “Review of the current status of jet noise predictions using large-eddy simulation,” *Proc. 44th Aerospace Science Meeting and Exhibit*, Reno, Nevada, January 9-12, AIAA 2006-0468 (2006).</footnote>

<footnote n="3">³ Björn Engquist and Andrew Majda, “Absorbing boundary conditions for the numerical simulation of waves,” *Math. Comput.* **31**(139), 629-651 (1977).</footnote>

<footnote n="4">⁴ Jonathan B. Freund, “Direct numerical simulation of the noise from a Mach 0.9 jet,” *Proc. 3rd ASME/JSME Joint Fluids Engineering Conference*, San Francisco, California, July 18-23, FEDSM’99-7251 (1999).</footnote>

<footnote n="5">⁵ M. E. Goldstein, “A hybrid RANS/LES approach for predicting jet noise,” *Proc. 12th AIAA/CEAS Aeroacoustics Conference*, Cambridge, Massachusetts, May 8-10, AIAA 2006-2631 (2006), and in *AIAA Journal* **44**(12), 3136-3142 (2006).</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1:** A schematic sketch of supersonic free jet.</figcaption></figure>

<footnote n="6">⁶ David Gottlib and Eli Turkel, "Dissipative two-four method for time dependent problems," *Math. Comput.* **30**(136), 703-723 (1976).</footnote>

<footnote n="7">⁷ Klaus A. Hoffmann and Steve T. Chiang, *Computational Fluid Dynamics for Engineers, vol. 2* (Engineering Education System, Austin, Texas, 1989).</footnote>

<footnote n="8">⁸ Moumen M. Idres, *Computational Modeling of Airborne Noise Demonstrated via Benchmarks, Supersonic Jet, and Railway Barrier* (Ph.D. Dissertation, Department of Aerospace Engineering, Old Dominion University, Norfolk, Virginia, 1999).</footnote>

<footnote n="9">⁹ Michael J. Lighthill, "On sound generated aerodynamically: I- general theory," *P. Roy. Soc. Lond. A Mat.* **211**(1107), 564-587 (1952).</footnote>

<footnote n="10">¹⁰ Michael J. Lighthill, "On sound generated aerodynamically: II- turbulence as a source of sound," *P. Roy. Soc. Lond. A Mat.* **222**(1148), 1-32 (1954).</footnote>

<footnote n="11">¹¹ B. Rembold, J. B. Freund, and M. Wang, "An evaluation of LES for jet noise prediction," *Proc. Summer Program*, Center for Turbulence Research, Stanford University (2002).</footnote>

<footnote n="12">¹² Joseph C. Schetz, "Free Shear Flows," Chap. 9 in *Boundary Layer Analysis*, edited by Joseph C. Schetz (Prentice Hall, New Jersey, 1993).</footnote>

<footnote n="13">¹³ Christopher K. W. Tam and Dale E. Burton, "Sound Generated by instability waves of supersonic flows. Part2. axisymmetric jets," *J. Fluid Mech.* **138**, 273-295 (1984).</footnote>

<footnote n="14">¹⁴ Kevin W. Thompson, "Time-dependent boundary conditions for hyperbolic systems II," *J. Comp. Phys.* **89**(2), 439-461 (1990).</footnote>

<footnote n="15">¹⁵ T. R. Troutt and D. K. McLaughlin, "Experiments on the flow and acoustic properties of a moderate-Reynolds-number supersonic jet," *J. Fluid Mech.* **116**, 123-156 (1982).</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Axial distribution of the mean-flow profile parameters, *h*(*x*) and *b*(*x*).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3:** Axial distribution of the normalized centerline axial velocity.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 4:** Contours of the sound pressure levels compared with measurements.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5:** Time-histories of the normalized perturbed pressure at a radius of
a) $R_j$
b) $5R_j$.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 6:** Snapshots of the normalized perturbed pressure field at successive instants.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 7:** Axial distributions of the RMS normalized pressure perturbation.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 8:** Axial distributions of the normalized pressure perturbation.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 9:** Axial distributions of the normalized axial velocity perturbation.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 10:** Axial distributions of the normalized radial velocity perturbation.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 11:** Axial distributions of the normalized density perturbation.</figcaption></figure>