# Investigation of Strouhal number effect on acoustic fields

Osama Marzouk

► To cite this version:

Osama Marzouk. Investigation of Strouhal number effect on acoustic fields. 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.1056-1067. hal-04887169

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# Investigation of Strouhal number effect on acoustic fields

Osama Marzouk<sup>a</sup>

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

## ABSTRACT

Understanding how jet flows respond to aerodynamic excitation helps constructing better understanding of the physical aspects of the problem. Also, this might lead to ideas and active techniques to control the induced noise. There are several approaches for noise computations, with different levels of accuracy and computational cost. In this study, we numerically simulate the acoustic field generated from supersonic axisymmetric jet flow. Aerodynamic excitation, or forced oscillatory flow signal, is introduced around the nozzle exit with different Strouhal numbers (nondimensional frequencies) below and above the critical value. Then, the acoustic fields are predicted at these different excitations by integrating the governing equations of the fluctuations. The adopted approach is very efficient in suppressing any spurious waves, except little oscillations during the transient development of the acoustic field at low Strouhal number. The effect of the Strouhal number on the acoustic field is investigated. Big differences in the acoustic fields exist between these cases. These differences are highlighted and discussed.

## 1. INTRODUCTION

Assessing audible fields generated due to compression-expansion pressure waves induced by jet flows has been a problem of interest for long time. For example, Morley<sup>8</sup> measured the intensity of far-field noise from turbulent air jets in 1939. Researchers have used experimental, analytical, and numerical approaches for this purpose. One of the numerical approaches utilizes the linear stability theory to capture the generation and propagation of waves, including acoustic waves. In some research cases; such as those done by Tam and Morris<sup>13</sup>, Tam and Burton<sup>12</sup>, and Owis<sup>9</sup>; the physical features of jet noise were studied using the stability-wave equations. In this approach, the sound source is represented by an instability wave assuming normal mode decomposition for the perturbations in the axial direction. Although this approach can be helpful in analytically modeling various aspects of the problem, such as the wave speed and length, it cannot be viewed as a generic approach applicable to a wide range of problems since it has several limitations. It requires normal mode decomposition and allows for discrete frequencies only. In the present study, we use a more generic approach, which is the linearized Euler equations model. Similar to the linear stability theory, this approach is linear and requires a prescribed mean flow. However, it does not require normal mode decomposition. It simultaneously describes both the sound generation and the propagation. Thus, it handles the problem of matching the near-field sound source to the far-field radiated sound properly. The linearized Euler equations approach has been used by several researchers; such as Sreenivas and Whitfield<sup>10</sup>, Bailly and Juve<sup>1</sup>, and Elshamy<sup>4</sup>; to capture both the flow and acoustic perturbations.

<footnote><sup>a</sup> Email address: omarzouk@vt.edu</footnote>
For subsonic and supersonic jet flows, there is a critical Strouhal number, $St$ (nondimensional frequency), where the core region becomes very responsive to any excitation waves. Instability waves grow to higher levels and the acoustic field becomes more intensive. This critical Strouhal number was found to be around 0.3 as indicated in some studies, such as Crow and Champagne³ and Troutt and McLaughlin¹⁶. In the present study, a supersonic axisymmetric jet is excited by introducing a harmonic wave around the nozzle exit. The frequency of the excitation has substantial effects on the pattern of the coherent acoustic field. These effects are related to both the acoustic generation, such as the location of the virtual sound source, and the acoustic propagation, such as the pattern of traveling waves. Several frequencies around the critical value are considered; namely Strouhal numbers of 0.1, 0.4, and 0.6.

## 2. COMPUTATIONAL APPROACH

### A. Mean Flow

Supersonic free jets are characterized by a converging region of uniform velocity (potential core) that starts from the nozzle exit, and a region of fully-developed diverging shear (or mixing) layer. Figure 1 shows a schematic sketch of the problem addressed in the present study. The coordinates are relative to the jet radius at nozzle exit ($R_j$).

The mean, or base, flow is a steady-state flow field, whose profile is known by some arbitrary approach, such as the steady Navier-Stokes equations or another simplified method. In the adopted approach of computing the acoustic field, the flow is linearized around this mean profile, which is for a supersonic isentropic circular jet. The Reynolds number is 70,000, which is a moderate value for this class of problems. There are experimental studies¹⁶ that considered this jet flow. Tam and Burton¹² developed analytical approach with continuous distributions of the measured data that fit the measurements very well and also satisfy the conservation laws. Contours of the mean axial velocity relative to the jet exit velocity (half domain only shown) are given in Figure 2. Table 1 lists the main properties of this jet.

<table><caption>**Table 1:** Jet properties at the nozzle exit</caption><thead><tr><td>Property</td><td>value</td></tr></thead><tbody><tr><td>Jet radius (mm)</td><td>5</td></tr><tr><td>Reynolds Number</td><td>70,000</td></tr><tr><td>Mach number</td><td>2.1</td></tr><tr><td>Jet Speed (m/s)</td><td>525</td></tr><tr><td>Static Pressure (N/m²)</td><td>5,060</td></tr><tr><td>Total Pressure (N/m²)</td><td>46,270</td></tr><tr><td>Total Temperature (°K)</td><td>294</td></tr></tbody></table>

### B. Perturbed Flow

The axisymmetric augmented (including mass and energy conservation equations as well as momentum conservation equations) Euler equations, in vector form, are
$$ \frac{\partial Q}{\partial t} + \frac{\partial F}{\partial x} + \frac{1}{r} \frac{\partial (rG)}{\partial r} = \frac{1}{r} S \quad (1) $$

where $Q: [\rho \ \rho u \ \rho v \ \rho e]^T$ is the a vector of the resolved conservative variables and $S: [0 \ 0 \ p \ 0]^T$ is a source terms that arises from transforming the equations from the Cartesian form to the axisymmetric form. There is no source term in the Cartesian form. The axial and radial momentum and energy fluxes are $F$ and $G$, respectively. Also, $p$ is the pressure, $\rho$ is the density, $u$ is the velocity in the axial direction, $v$ is the velocity in the radial direction velocity, and $e$ is the total energy per unit mass.

Upon rewriting each variable as the sum of mean and perturbation quantities and then substituting into Equation (1), we get two sets of equations that look similar to Equation (1). The first set governs the mean flow. We do not need this one since the mean flow is prescribed in alternative approach that allows for capturing the viscous and turbulent effects. The second set of equations governs the perturbation field. In fact, they can be written in the same form of Equation (1), but the expressions of the flux vectors will be different.

The governing equations for the perturbation field are discretized and integrated numerically using 2-4 McCormack scheme. We used this scheme for its relative simplicity and high accuracy. Implicit schemes are commonly favored over explicit ones in computational fluid dynamics problems because they allow using large time steps, although they involve complicated mathematical operations and calculations. However, in the present problem, we use explicit scheme since a very small time step is needed anyway due to the resolution requirement so that high frequency acoustic components are satisfactorily captured.

Figure 3 illustrates the computational grid used in the simulation (half domain only shown). The axial spacing is uniform with $\Delta x = R_j/6$. This permits good spatial wave resolution. In supersonic flows, it was found that the spatial wave number and the angular frequency are related with approximately linear relation (Troutt and McLaughlin¹⁶, Tam¹¹, and Owis⁹). For example, a wave with angular frequency of $0.15\pi$ in a supersonic flow has a wave number $\approx 0.353\pi$, which corresponds to a wavelength of $5.67R_j$. Thus there are 34 points along such wave, which is satisfactory. More points are placed near the centerline, where the radial gradients are steeper than those regions near or outside the edge of the mixing layer. Nonreflecting boundary conditions; based on the method of characteristics (Thompson¹⁵), acoustic radiation, and asymptotic analysis (Tam and Webb¹⁴); are used at the entrainment and outflow regions to prevent spurious waves. Figure 4 shows these boundary conditions and the locations where they are applied (half domain only shown).

### C. Input Excitation

At the inflow boundary, where the characteristics-based boundary condition is used, we add an excitation in terms of forced oscillations in the density, axial velocity, and pressure. These excitation components have Gaussian distribution with different amplitudes but same frequency, which is related to the Strouhal number. For example, the pressure excitation has the following form

$$ p'_{\text{INPUT}} = -100 \ln(2) (r - 0.78 R_j)^2 e^{(-i 2\pi \text{St} t)} \quad (2) $$

where

$$ \text{Strouhal number: } \text{St} = \frac{(\text{frequency})(2R_j)}{(\text{jet exit speed})} \quad (3) $$
There is a value (or narrow range) of St around 0.3, where the potential core becomes very responsive to excitations. The input excitation waves grow and saturate at relatively high amplitudes. We consider here cases of St below and above the critical, and investigate the acoustic field in each case and compare them to each other.

## 3. RESULTS

For the considered supersonic axisymmetric free jet, input excitation, in terms of forced oscillations of flow variables, is applied around the nozzle exit. Numerical simulations were performed for different low- and high-frequency excitations. The Strouhal number values considered are below the critical range (subcritical: St 0.1), and above the critical range (supercritical: St 0.4 and 0.6). The St 0.4 value is the closest to the critical range, and can be considered part of it. The acoustic field induced in each case was examined from different perspectives. The acoustic intensity (in decibels, dB) was calculated from the root-mean-square of the pressure fluctuations by

$$ \text{Acoustic dB} = 20 \log(P'_{\text{rms}}/P'_{\text{scale}}) \qquad (4) $$

where $P'_{\text{scale}}$ is related to the human threshold of hearing.

The presented perturbed pressure results are made relative to the product of the mean jet density and the mean jet velocity squared at the nozzle exit. Figure 5 shows flooded contours of the root-mean-square of the perturbed pressure fields at different Strouhal numbers. This figure is very helpful in demonstrating the differences in the acoustic fields between these cases. In the case of St 0.1 (low frequency), the virtual sound source that corresponds to the region of peak pressure fluctuations is located at about $38R_j$. In addition, the acoustic field is confined and decays rapidly. There are oblique and axial directions of propagation. However, the axial propagation is stronger. For St 0.4 (slightly above the critical), two changes occurs in the virtual sound source. It become flattened and elongated axially. Also, it is located closer to the nozzle exit at about $13R_j$; that is one-third the observed location for the St 0.1 case. There are still oblique and axial directions of propagation, but the intensity along the oblique direction increased and became comparable to the axial one. Increasing the Strouhal numbers further to 0.6, which is roughly twice the critical value, results in big changes in the acoustic field. The virtual sound source reaches the nozzle exit. It has the form of two sources along the axial direction. The axial propagation is attenuated to large extent. Additional oblique propagation direction appears which is more inclined toward the radial direction.

Figure 6 compares the patterns of the perturbed pressure fields at the different Strouhal numbers. It can be easily noticed that the large-scale waves for the low-St case is characterized by low-frequency pressure waves, and vice versa. In the St 0.1 case, the oblique propagation is minimal, as also indicated in Figure 5. The case of St 0.6 has very little axial propagation as compared to the oblique ones. Another observation is that at low St, each pressure wave (compression or expansion) has a relatively large region of concentration near the centerline, which decays rapidly in the radial direction. At the higher Strouhal number, such as St 0.4, these regions of concentrations become smaller, and the radial decay rate decreases.

Figures 7 is intended to illustrate the initial and steady-state acoustic propagation process, where the perturbed pressure field (half domain only shown) for St 0.1 is given in two cases: during the transient mode and after the steady-state mode is reached. Similar illustrations are given in Figures 8 and 9 for St 0.4 and St 0.6, respectively. There are some oscillations in the wave front in the case of St 0.1. However, the other cases have almost no oscillations during the transient propagation. This is an indication of the good quality of the numerical simulation and the adopted boundary conditions.
## 4. CONCLUSIONS

Numerical simulation was employed to solve the system governing the perturbations and acoustic fields for supersonic circular jet subject to input harmonic excitation (or forced aerodynamics oscillations). The governing equations were integrated numerically in the time domain. The employed numerical approach and boundary conditions were able to minimize possible spurious waves due to reflections at the boundaries. The effect of the Strouhal number, $St$, (or nondimensional frequency) of the excitation on the acoustic field was examined. The transient and steady-state patterns of acoustic propagations were generated at different Strouhal numbers below and above the observed critical value of about 0.3.

Big differences between the resolved acoustic fields at different Strouhal numbers were observed. Increasing the Strouhal number results in shifting the virtual sound source, where the pressure fluctuations reach their maximum values, upstream toward the nozzle exit. Another effect of increasing the Strouhal number is attenuating the axial propagation of acoustic waves and intensifying the oblique one. At the low Strouhal number, the pressure waves have relatively large regions of concentrations near the centerline, with rapid radial decay. At high Strouhal numbers, these concentration regions become smaller and the radial decay decreases as well.

## REFERENCES

¹ Christophe Bailly and Daniel Juve “Numerical solution of acoustic propagation problems using linearized Euler equations,” *Proc. 4th AIAA/CEAS Aeroacoustics Conference*, Toulouse, France, June 2-4, AIAA 1998-2267 (1998), and in *AIAA Journal* **38**(1), 22-29 (2000).

² Alvin Bayliss, Lucio Maestrello, Paresh Parikh, and Eli Turkel, “A fourth order scheme for the unsteady compressible Navier-Stokes equations,” *Proc. 18th Fluid Dynamics and Plasmodynamics and Lasers Conference*, Cincinnati, Ohio, July 16-18, AIAA 1985-1694 (1985).

³ S. C. Crow and F. H. Champagne, “Orderly structure in jet turbulence,” *J. Fluid Mech.* **48**, 547-591 (1971).

⁴ Omar Elshamy, *Flow Field and Acoustics of Subsonic Impinging Jets* (M.Sc. Thesis, Mechanical Power Department, Faculty of Engineering, Cairo University, 2002).

⁵ David Gottlib and Eli Turkel, “Dissipative two-four method for time dependent problems,” *Math. Comput.* **30**(136), 703-723 (1976).

⁶ Ray Hixon and Eli Turkel, “High-accuracy compact McCormack-type schemes for computational aeroacoustics,” *Proc. 13th Aerospace Sciences Meeting and Exhibit*, Reno, Nevada, January 12-15, AIAA 1998-0365 (1998).

⁷ Cheolwan Kim, Philip L. Roe, and Jeffrey P. Thomas, “Accurate schemes for advection and aeroacoustics,” *Proc. 13th AIAA Computational Fluid Dynamics Conference*, Snowmass Village, Colorado, June 29-July 2, AIAA 1997-2091 (1997).

⁸ A. W. Morley, “Estimation of aeroplane noise level: some empirical laws with an account of the present experiments on which they are based,” *Airc. Eng. Aerosp. Tec.* **11**(123), 187-189 (1939).

⁹ Farouk Owis, *Jet Stability and Noise Computations Using Direct Numerical Simulation* (Ph.D. Dissertation, Department of Aerospace Engineering, Old Dominion University, Norfolk, Virginia, 1999).

¹⁰ Kidambi Sreenivas and David L. Whitfield, 1998, “Time- and frequency-domain numerical simulation of linearized Euler equations,” *AIAA Journal* **36**(6), 968-975 (1998).

¹¹ Christopher K. W. Tam, “On the noise of a nearly ideally expanded supersonic jet,” *J. Fluid Mech.* **51**, 69-95 (1972).

¹² 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).

¹³ Christopher K. W. Tam and Philip J. Morris, “The radiation of sound by the instability waves of a compressible plane turbulent shear layer,” *J. Fluid Mech.* **98**(2), 349-381 (1980).

¹⁴ Christopher K. W. Tam and Jay C. Webb, “Dispersion-relation-preserving finite difference schemes for computational acoustics,” *J. Comp. Phys.* **107**(2), 262-281 (1993).

¹⁵ Kevin W. Thompson, “Time-dependent boundary conditions for hyperbolic systems II,” *J. Comp. Phys.* **89**(2), 439-461 (1990).

¹⁶ 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).
<figure><img src="image_2.png" /><figcaption>**Figure 1:** Width of the core and mixing layer.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 1:** Contours of the relative axial velocity.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Figure 3:** Computational grid (1/5 resolution in both x and r only shown).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4:** Computational domain and boundary conditions.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 5:** Root-mean-square of the relative perturbed pressure fields at different Strouhal numbers.</figcaption></figure>
<figure><img src="image_7.png" /></figure>

St =0.1

<figure><img src="image_8.png" /></figure>

St =0.4

<figure><img src="image_9.png" /></figure>

St =0.6

<caption>Figure 6: Relative perturbed pressure fields at different Strouhal numbers.</caption>
<figure><img src="image_10.png" /><figcaption>**Figure 7:** Development of the relative perturbed pressure field at Strouhal number 0.1.</figcaption></figure>

Top: transient

Bottom: steady state
<figure><img src="image_11.png" /><figcaption>**Figure 8:** Development of the relative perturbed pressure field at Strouhal number 0.4.</figcaption></figure>

Top: transient Bottom: steady state
<figure><img src="image_12.png" /><figcaption>**Figure 9:** Development of the relative perturbed pressure field at Strouhal number 0.6.</figcaption></figure>

Top: transient Bottom: steady state