Mechanisms of noise reduction by corner rounding in turbulent flow over forward-facing steps

Abstract

The aeroacoustic effects of rounding a forward-facing step in a low-Mach-number turbulent boundary layer are investigated using large-eddy simulation and Lighthill’s theory. The step height is $26\%$ of the thickness of the unperturbed boundary layer at $R{e}_{\theta }=4755$ , and the rounding radius relative to the step height ranges from 0 to $100\%$ . Consistent with previous experimental findings, step rounding is shown to cause reduced flow separation and increased peak pressure fluctuations on the step upper-surface except in the $100\%$ rounding case. The acoustic radiation is significantly weakened by step rounding, and the noise reduction increases with increasing rounding radius and frequency. The acoustic source fields for different rounding radii are analyzed in conjunction with tailored, acoustically compact Green’s functions to investigate the mechanisms for noise reduction. It is found that while step rounding affects both turbulence production and acoustic diffraction by the step, noise reduction is primarily due to the reduced surface diffraction effect. Additionally, a boundary-element analysis shows that the effect of step rounding on the acoustic directivity is insignificant for acoustically compact steps, but grows with increasing acoustic noncompactness of the step height.

Introduction

Surface discontinuities can perturb the turbulent boundary layer on the exteriors of road, air and marine vehicles and generate noise. They also cause elevated levels of surface-pressure fluctuations, leading to structural vibrations and additional acoustic radiation. Steps and gaps are canonical examples of surface discontinuities whose aeroacoustic effects have received significant attention. ¹

Farabee and Zoccola ² made the first measurements of the sound from flow over steps. At free-stream Mach numbers of 0.075 and 0.12, step heights of 0.76 and 1.4 cm, and boundary layer thicknesses approximately equal to twice the step heights, they found that the noise from forward-facing steps was generally 5 dB above the facility background level, whereas the noise from backward-facing steps was indiscernible from the background noise. It was therefore concluded that the forward step was a significantly stronger noise source. This conclusion was confirmed in a numerical study by Ji and Wang ³ using large-eddy simulation (LES) and Lighthill’s aeroacoustic theory. ⁴ They computed the forward- and backward-step noise from turbulent boundary layers at momentum-thickness Reynolds number $R{e}_{\theta }=4755$ for four step heights ranging from $0.83\%$ to $53\%$ of the boundary-layer thickness. At low Mach numbers, the steps were considered acoustically compact, allowing the use of the approximate tailored Green’s function of Howe. ⁵ Consistent with Farabee and Zoccola, ² the numerical results showed that a forward step was noisier than a backward step of the same height, and provided a physical explanation through an analysis of the Lighthill stress and the step Green’s function: a backward step generates sound primarily through a diffraction mechanism, whereas a forward step generates sound through a combination of diffraction and turbulence generation by the step. The difference between the sound pressure levels from forward and backward steps was found to decrease with decreasing step height as the forward step produced less turbulence in the boundary layer.

The use of the acoustically compact Green’s function, which leads to a streamwise dipole field, is reasonable for flows at very low Mach numbers such as those encountered in underwater applications. However, the compact-step assumptions can be violated in airflows at not-so-low Mach numbers and/or high frequencies. Wind-tunnel tests (e.g., Refs 6 and 7) have revealed some dipole-like features of step noise but also significant departures from dipole sound depending on the frequency and flow Mach number. Catlett et al. ⁶ measured the sound from steps in an anechoic wall-jet facility under similar conditions as considered by Ji and Wang. ³ They also confirmed that forward step noise is stronger than backward step noise, and noted difference as much as 10 dB. Dipole-like directivity was observed only at low frequencies, and strong noncompactness effects including local spectral dips were found at higher frequencies for larger steps. Hao et al. ⁸ extended the LES study of Ji and Wang ³ to acoustically noncompact forward steps. Instead of using the tailored Green’s function for noncompact steps, ⁹ they employed a boundary-element method (BEM) for computational efficiency. The Lighthill integral equation was solved by adapting the BEM procedure proposed by Khalighi et al. ¹⁰ for $R{e}_{\theta }=4755$ , step heights of $13\%$ and $53\%$ of the boundary-layer thickness, and free-stream Mach numbers of 0.01, 0.1 and 0.2. Their results demonstrated that with decreasing acoustic wavelength relative to the step height, the acoustic directivity evolved from a simple streamwise-dipole field to asymmetrically distorted dipole-like fields, and eventually to multi-lobed, complex edge-scattering patterns as in the experiment of Catlett et al. ⁶

In addition to steps, Catlett et al. ⁶ also measured the sound from small gaps composed of a backward-facing step of height 1.17 cm followed by forward-facing steps with different heights and separations, at several free-stream Mach numbers of up to 0.062. Motivated by this experiment, Hao et al. ¹¹ conducted a numerical investigation with the backward-step height fixed at $13\%$ of the boundary-layer thickness, three forward-step heights, and three separation distances. Both experimental and numerical results show that the acoustic field is dominated by the forward-facing step if it is exposed to the incoming turbulent flow. If the forward-facing step is shielded from the incoming turbulence, which happens if it is shorter than the backward step in narrow gaps, the acoustic field resembles that of a backward-step field.

Since a forward step generates strong flow-noise through diffraction and turbulence production, noise mitigation can be achieved by modifying the exterior corner of the step and thereby both mechanisms. In a series of experiments, Awasthi et al.^12–15 investigated the effect of step rounding on the radiated sound and surface pressure fluctuations at low Mach numbers. Measurements were made in an anechoic wall-jet facility^12,15 at relatively low Reynolds numbers (step-height Reynolds numbers $R{e}_h$ from 8,300 to 16,800) and in an anechoic wind tunnel^13,14 at high Reynolds numbers ( $R{e}_h=$ 35,000 and 104,000). The steps were rounded with radii equal to $0\%$ , $6.25\%$ , $12.5\%$ , $25\%$ , $50\%$ and $100\%$ of the step height, although only the first four cases yielded reliable acoustic data. These measurements showed that rounding the step led to smaller separation bubble size on the upper surface, faster decay of wall-pressure fluctuations, and lower sound pressure level. The noise reduction was small with $6.25\%$ rounding but grew progressively larger with increasing rounding radius, without significant changes in the spectral shape or directivity. Significant effects of step noncompactness were also observed. To facilitate noise prediction for acoustically noncompact steps, Glegg et al. ¹⁶ developed a theory based on the half-space Green’s function and the assumption that turbulence causes equal net unsteady loading on both sides of the step upper corner. Using this theory and experimental data, Awasthi et al.^14,15 developed a prediction model and applied it to unrounded and rounded steps with some success.

In the present study, the noise from rounded forward-facing steps in turbulent boundary layers is investigated numerically, with a focus on elucidating the mechanisms of how step rounding reduces noise generation. The flow configuration is illustrated schematically in Figure 1. The height of the step $h$ is fixed at $26\%$ of the unperturbed boundary-layer thickness $\delta$ at the step location, where $R{e}_{\theta }=4755$ . The Reynolds number based on the step height is $R{e}_h=$ 12,960. Five different rounding radii, $r/h=0\%$ , $12.5\%$ , $25\%$ , $50\%$ and $100\%$ , are considered as in the experiment of Awasthi et al.^12,15 Using the techniques established in previous step- and gap-flow studies,^3,8,17 LES is employed in combination with the Lighthill theory to predict the radiated noise, and the Lighthill stress field is analyzed in relation to the step Green’s function to illuminate the noise-reduction mechanisms. Compared to source analysis based on the fluctuating surface pressure, which is itself caused by the turbulent flow field, the present analysis has the advantage that it directly examines turbulent velocity fluctuations as the fundamental acoustic source and the effect of the step geometry on its radiation efficiency.

Figure 1

. Schematic of turbulent boundary-layer flow over a rounded forward-facing step.

Flow field

Computational method

The turbulent boundary-layer flows over the forward steps are computed using a finite-volume LES code for incompressible flow.^18,19 The spatially filtered Navier-Stokes equations are solved on a staggered grid with the dynamic Smagorinsky subgrid-scale stress model.^20,21 The energy-conservative, non-dissipative numerical scheme is based on second-order central differencing in space and a fractional-step time-advancement method with the third-order Runge-Kutta scheme for the convective terms and second-order Crank-Nicolson scheme for the viscous and subgrid-scale stress terms. The Poisson equation for pressure is solved using a combination of Fourier collocation method in the spanwise direction and multigrid iterative method in the other two directions.

The computational domain size for all simulations is $L_x=12.8\delta$ , $L_y=6.4\delta$ and $L_z=2.1\delta$ in the streamwise, wall-normal and spanwise directions, respectively. In terms of the step height $h$ , it is $48h\times 24h\times 8.0h$ with $20h$ upstream of the step face and $28h$ downstream of the step face. The origin of the coordinate system is set to be the bottom corner of the step at mid-span. The total number of grid points ranges from $1.87\times 10^7$ for the $0\%$ rounded step to $2.76\times 10^7$ for the $100\%$ rounded step. The body-fitted grid is curvilinear in the plane normal to the step and uniform Cartesian in the spanwise direction. The streamwise grid spacing $\mathrm{\Delta }{x}^{+}$ , scaled by the wall unit at the computational inlet, ranges from 5 at the step face to 50. The wall-normal grid spacing $\mathrm{\Delta }{y}^{+}$ ranges from 2 at the surfaces to 50 at the edge of the boundary layer. The spanwise grid spacing $\mathrm{\Delta }{z}^{+}$ is 25.

Turbulent inflow at $R{e}_{\theta }=4100$ is supplied by an auxiliary LES of a flat-plate turbulent boundary layer using the rescale-and-recycle technique of Lund et al. ²² The Dirichlet boundary condition $(u,v,w)=(U_0,\mathrm{0,0})$ , where $U_0$ is the free-stream velocity, is applied at the top of the domain, whereas on the step surfaces $(u,v,w)=(\mathrm{0,0,0})$ . The convective outflow boundary condition is used at the outlet, and the periodic boundary condition is used in the spanwise direction. The Courant-Friedrichs-Lewy number is fixed at one in all simulations.

The accuracy of the flow-simulation approach described above has been established in prior studies of flows over steps^3,17 and gaps ¹¹ with sharp edges through comparisons with experimental results^23,24 and grid-convergence studies. ¹¹

Flow characteristics

The mean streamwise velocity, $U/U_0$ , and streamlines near the five steps of different rounding radii are shown in Figure 2. The boundary layer is deflected upward as the flow approaches the step, forming recirculation zones at the lower corner and above the upper surface. The recirculation bubble on the upper surface is very sensitive to the upper-corner rounding. The bubble size decreases dramatically with increasing rounding radius. For the $100\%$ rounding case, the bubble is hardly visible. In comparison, the size of the lower-corner recirculation zone is less sensitive to the step rounding radius.

Figure 2.

Isocontours of mean streamwise velocity $U/U_0$ and streamlines near steps with different rounding radii $r/h$ : (a) $0\%$ ; (b) $12.5\%$ ; (c) $25\%$ ; (d) $50\%$ ; (e) $100\%$ .

Figure 3 shows instantaneous spanwise vorticity fields in the mid-span plane for the five steps. As the flow approaches the step, the near-wall vortical structures are lifted up following the mean streamlines and severely distorted. These vorticies then interact with the separated shear layer developed from the step upper surface, causing intense mixing, breakdown of the shear layer and generation of small-scale turbulence. Increasing rounding radius leads to delayed and weaker interactions because the separation is weaker and occurs farther downstream.

Figure 3.

Instantaneous spanwise vorticity ${\omega }_{z}h/U_0$ near steps in the mid-span plane for different rounding radii $r/h$ : (a) $0\%$ ; (b) $12.5\%$ ; (c) $25\%$ ; (d) $50\%$ ; (e) $100\%$ . 12 contour levels from −5 (blue) to 5 (red) are shown.

To quantify the recirculation region in Figure 2, the wall skin-friction coefficient, $C_f={\tau }_w/(\rho U_0^2/2)$ , is shown in Figure 4. The lengths of the recirculation bubbles at the lower corner judged by negative $C_f$ are very similar for the four rounded steps and somewhat smaller than that of the sharp-edged step. This is consistent with the experimental observations of Awasthi et al.^12,13 that step rounding has little impact on the upstream separation location except for a slight decrease in the bubble size. Around the rounded edges strong flow acceleration leads to very large $C_f$ values compared to the sharp-edged case and delayed boundary-layer separation. The flow separates from the top surface for all steps, but the separation point moves downstream with increasing rounding radius. The mean bubble lengths, $s_b/h$ , and reattachment locations, $x_r/h$ , are listed in Table 1 along with the mean reattachment locations measured in the wall-jet boundary layers of Awasthi et al. ¹² There is good agreement between the computational and experimental mean reattachment lengths except for the $100\%$ rounding case, considering the different configuration. The large discrepancy in the $100\%$ rounding case is due to the fact that the separation bubble is very weak and sensitive to flow conditions, making it difficult to compute or measure accurately. The computed reattachment lengths also show a similar trend as those from the wind-tunnel measurements of Awasthi et al. ¹³ at much higher Reynolds numbers (see Figure 5 in Ref 13).

Figure 4.

Skin-friction coefficient near the step for different rounding radii $r/h$ : , $0\%$ ; , $12.5\%$ ; , $25\%$ ; , $50\%$ ; , $100\%$ .

Table 1.

Mean separation bubble sizes and reattachment locations compared with experimental data of Awasthi et al. ¹²

Case	Computation		Experiment
$r/h$	Bubble size $s_b/h$	Reattachment $x_r/h$	Reattachment $x_r/h$
$0\%$	2.44	2.44	2.69
$12.5\%$	1.33	1.41	1.36
$25\%$	0.86	1.05	1.19
$50\%$	0.55	1.02	0.97
$100\%$	0.47	1.50	0.98

Figure 5.

Turbulent kinetic energy $(\overline{u^{\prime 2}}+\overline{v^{\prime 2}}+\overline{w^{\prime 2}})/(2U_0^2)$ near the steps of different rounding radii $r/h$ : (a) $0\%$ ; (b) $12.5\%$ ; (c) $25\%$ ; (d) $50\%$ ; (e) $100\%$ . 11 contour levels from 0 (blue) to 0.09 (red) are shown. Maximum levels are: (a) 0.088; (b) 0.096; (c) 0.094; (d) 0.068; (e) 0.041.

The distributions of turbulent kinetic energy, which is related to acoustic sources, are shown in Figure 5. In each case, the high-energy region is associated with the separated shear layer and shrinks in size with increasing rounding radius as the separation bubble becomes smaller. The maximum kinetic energy levels are comparable for the first three steps with small rounding and no rounding. It decreases modestly in the $50\%$ and $100\%$ rounding cases due to increasingly weaker separation bubbles. As the bubble size decreases with increasing rounding, the peak of the turbulent kinetic energy moves closer to the surface, causing stronger surface pressure fluctuations.

The streamwise variations of the root-mean-square (r.m.s.) of surface pressure fluctuations for the five steps are shown in Figure 6. The top plot shows the $p_{\mathit{rms}}$ distribution over the entire domain length, whereas the bottom plot provides a close-up view near the step face. As can be seen, with the exception of the $100\%$ rounding case, edge rounding leads to stronger pressure fluctuations in a narrower region and a quicker return of the fluctuation level to the equilibrium-boundary-layer value in the downstream. The flows for all rounded steps nearly recover the equilibrium turbulent boundary layer at $x/h=20$ . This is consistent with the conclusions of Ji and Wang ¹⁷ and Hao et al. ¹¹ that the recovery distance scales inversely with the separation bubble size. Near the step face the $p_{\mathit{rms}}$ distributions exhibit complex, non-monotonous behavior. The first spike in the vicinity of $x/h=0$ is due to the switch from the lower surface to the upper surface of the step and the turbulence distortions associated with strong flow acceleration. The subsequent variations are due to the complex flow topology in the separated regions. The global peak $p_{\mathit{rms}}$ occurs near the downstream reattachment point, which is again consistent with previous results for sharp-edged steps and gaps.^3,11,17

Figure 6.

Root-mean-square of wall pressure fluctuations for different rounding radii $r/h$ : , $0\%$ ; , $12.5\%$ ; , $25\%$ ; , $50\%$ ; , $100\%$ .

Acoustic field

Following the previous studies^3,8,11 of noise from sharp-edged steps and gaps, the acoustic calculations are based on the Lighthill equation in an integral form, ²⁵

$\alpha \widehat{p}(\mathbf{x},\omega )={\int }_S{n}_i\widehat{p}(\mathbf{y},\omega )\frac{\partial G\left(\mathbf{x}|\mathbf{y},\omega \right)}{\partial y_i}{\mathrm{d}}^2\mathbf{y}+{\int }_V{\widehat{T}}_{ij}(\mathbf{y},\omega )\frac{{\partial }^{2}G\left(\mathbf{x}|\mathbf{y},\omega \right)}{\partial y_i\partial y_j}{\mathrm{d}}^3\mathbf{y},$

(1)

where $p$ is the fluctuating pressure which is equal to the acoustic pressure outside the nonlinear flow region, the caret represents temporal Fourier transform, $\omega$ is the angular frequency, $\mathbf{x}$ and $\mathbf{y}$ are position vectors for the observer and source, respectively, $n_i$ is the unit normal to the solid surface $S$ (into the fluid), $V$ is the fluid region external to the solid surface, and $G(\mathbf{x}|\mathbf{y},\omega )$ is an acoustic Green’s function. The geometric parameter $\alpha$ is equal to $1/2$ for $\mathbf{x}\in S$ , 1 for $\mathbf{x}\in V$ , and 0 elsewhere. The Lighthill stress tensor is approximated as $T_{ij}\approx {\rho }_0(u_i-U_i)(u_j-U_j)$ , where ${\rho }_0$ is the free-stream density, $u_i$ are instantaneous velocity components, and $U_i={\delta }_{i1}{U}_0$ is the free-stream velocity.

With the choice of a hard-wall Green’s function, the surface integral in equation (1) vanishes. At very low Mach numbers typical of underwater applications $(M\sim 0.01)$ , the step height is acoustically compact over most frequencies of interest, and the approximate Green’s function derived by Howe ⁵ can be employed, leading to a streamwise-dipole field for the far-field acoustic pressure (see Ref 3 for details),

$\widehat{p}(\mathbf{x},\omega )=-\frac{ik}{2\pi }\frac{e^{ik|\mathbf{x}|}\cos \theta }{|\mathbf{x}|}\widehat{D}(\omega ),$

(2)

where $\cos \theta =x_1/|\mathbf{x}|$ defines the observer angle measured counter-clockwise from the downstream direction, $k=\omega /c_0$ is the acoustic wavenumber, and $\widehat{D}(\omega )$ is the Fourier transform of

$D(t)=\int \int \frac{{\partial }^2{Y}_1}{\partial y_i\partial y_j}{Q}_{ij}(y_1,y_2,t)\mathrm{d}{y}_2\mathrm{d}{y}_1,$

(3)

with

$Q_{ij}(y_1,y_2,t)=\int T_{ij}(\mathbf{y},t)\mathrm{d}{y}_3$

(4)

being the Lighthill stress tensor integrated over the span of the computational domain. The quantity $Y_1(y_1,y_2)$ is the part of the Green’s function that is dependent on source coordinates under the acoustic-compactness approximation. It satisfies the Laplace equation with $\partial Y_1/\partial n=0$ on the surfaces. This equation is solved using the Schwarz-Christoffel transformation for the sharp-edged step. For rounded steps it is solved numerically on the same grid used for flow-field LES. From equation (2), the far-field sound pressure spectrum can be expressed as

${\mathrm{\Phi }}_{pp}(\omega )=\frac{k^2{\cos }^2\theta }{4{\pi }^2|\mathbf{x}{|}^2}{\mathrm{\Phi }}_{DD}(\omega )$

(5)

where ${\mathrm{\Phi }}_{DD}(\omega )$ is the spectral density of $D(t)$ .

Figure 7 compares the sound pressure spectra calculated from equations (3)–(5) for steps with different rounding radii. These spectra are normalized based on the compact dipole scaling given by equation (5), and symbols $P_0$ and $\gamma$ in the normalization are the free-stream pressure and ratio of specific heats, respectively. The shapes of the broadband spectra are similar for all steps but the spectral level decreases with increasing rounding radius. Compared with the sharp-edged case, the peak spectral level is decreased by 2.4 dB, 4.4 dB, 7.7 dB and 10.1 dB for steps with $12.5\%$ , $25\%$ , $50\%$ and $100\%$ rounding, respectively. At higher frequencies the spectral level declines even more with increasing edge rounding. For example, at $f\delta /U_0=8.0$ it is lowered by 5.5 dB, 7.3 dB, 19.3 dB and 31.0 dB with $12.5\%$ , $25\%$ , $50\%$ and $100\%$ rounding, respectively. This largely agrees with the experimental findings of Awasthi et al.^12,15 Although it is difficult to make a quantitative comparison with their wall-jet experiment because of different flow configurations, the numerical and experimental results show qualitative agreement in terms of noise reduction by step rounding and its dependence on rounding radius and frequency.

Figure 7.

Sound pressure spectra for acoustically compact steps with different rounding radii $r/h$ : , $0\%$ ; , $12.5\%$ ; , $25\%$ ; , $50\%$ ; , $100\%$ .

Noise-reduction mechanisms

The formulation described in the preceding section provides not only a means to compute the flow-induced acoustic field but also a rigorous framework for analyzing the acoustic source mechanisms. Through equations (1)–(4), the acoustic pressure is directly related to the turbulent flow-field with account for the effect of the step geometry via a tailored Green’s function. A step amplifies sound generation in a boundary layer by promoting turbulence production (larger $T_{ij}$ ) and efficient conversion of flow energy to acoustic energy (larger ${\partial }^{2}G/\partial y_i\partial y_j$ , or ${\partial }^2{Y}_1/\partial y_i\partial y_j$ for acoustically compact steps). These two effects are known as turbulence-generation and diffraction mechanisms, respectively.^1,3 By examining how the step geometry modifies the relative importance of these two effects, the mechanisms for noise reduction by step-rounding can be elucidated.

The spatial distributions of the second derivatives of the compact Green’s functions, $\left|{\partial }^2{Y}_1/\partial y_1^2\right|h$ and $\left|{\partial }^2{Y}_1/\partial y_1\partial y_2\right|h$ , for the sharp-edged step and $50\%$ rounded step are contrasted in Figure 8. The largest values of the Green’s function derivatives are located near the step upper corner. Rounding the step by $50\%$ is found to reduce the maximum values by two decades and at the same time expand the high-value region along the rounded surface, which makes the important acoustic source regions less localized. The step-rounding effects for the other steps are qualitatively similar, increasing in severity with increasing rounding radius. For the same source field, smaller Green’s function derivatives lead to lower sound emission as suggested by equation (3).

Figure 8.

Magnitude of the second derivatives of $Y_1$ for steps with (a,b) zero rounding and (c,d) $r/h=50\%$ rounding: (a,c), $\left|\frac{{\partial }^2{Y}_1}{\partial y_1^2}\right|h$ ; (b,d), $\left|\frac{{\partial }^2{Y}_1}{\partial y_1\partial y_2}\right|h$ . Logarithmic scale with 9 contour levels from $10^{-2}$ (blue) to $10^1$ (red).

The previous investigation of Ji and Wang ³ revealed that turbulence generation by a forward step plays an important role in sound generation, which, in combination with step diffraction, makes the forward-step sound significantly louder than the backward-step sound. The latter is dominated by diffraction. Edge rounding changes the intensity and spatial distributions of turbulent kinetic energy (cf. Figure 5) and the second derivatives of Green’s function, and therefore alters both the diffraction and turbulence generation mechanisms. To analyze quantitatively the noise-reduction mechanisms, the spatial distributions of Lighthill source functions in the frequency domain are examined with and without accounting for the Green’s function effect. Figure 9 compares the sum of the power spectral densities of Lighthill stresses, $({\mathrm{\Phi }}_{Q_{11}}+2{\mathrm{\Phi }}_{Q_{12}}+{\mathrm{\Phi }}_{Q_{22}})U_0/{({\rho }_0{U}_0^{2}h)}^{2}h$ , for the sharp-edged and $50\%$ rounded steps at three different frequencies, $f\delta /U_0=0.5$ , 2.0 and 8.0. Similar to previous step and gap observations,^3,8,11 the Lighthill source field becomes increasingly localized at higher frequencies. With edge rounding the intense-source region is smaller and closer to the top surface due to the smaller separation bubble size. However, the maximum source strength does not show a consistent trend. It is decreased at the low frequency, increased slightly at the intermediate frequency, and increased by a decade at the high frequency. Overall, the differences in maximum source strengths between the two steps are much smaller than the two-decade differences in their Green’s function derivatives shown in Figure 8.

Figure 9.

Power spectral density of Lighthill stresses, $({\mathrm{\Phi }}_{Q_{11}}+2{\mathrm{\Phi }}_{Q_{12}}+{\mathrm{\Phi }}_{Q_{22}})U_0/{({\rho }_0{U}_0^{2}h)}^{2}h$ , for steps with (a–c) zero rounding and (d–f) $r/h=50\%$ rounding, at three different frequencies, $f\delta /U_0=$ : (a,d) 0.5; (b,e) 2.0; (c,f) 8.0. Linear scale with 11 contour levels from 0 to (a) $4.9\times 10^{-3}$ ; (b) $3.4\times 10^{-4}$ ; (c) $5.0\times 10^{-5}$ ; (d) $1.8\times 10^{-3}$ ; (e) $3.9\times 10^{-4}$ ; (f) $4.4\times 10^{-4}$ .

To evaluate the combined acoustic source and Green’s function effect, the products of the Lighthill stresses and the Green’s function derivatives,

$R_{ij}(y_1,y_2,t)=\frac{{\partial }^2{Y}_1}{\partial y_i\partial y_j}{Q}_{ij}(y_1,y_2,t),$

(6)

(no summation over repeated indices implied here) are calculated. The sum of their power spectral densities, $({\mathrm{\Phi }}_{R_{11}}+2{\mathrm{\Phi }}_{R_{12}}+{\mathrm{\Phi }}_{R_{22}})U_0/{({\rho }_0{U}_0^2)}^{2}h$ , is shown in Figure 10 for the two steps at the same three frequencies. At all frequencies, step rounding leads to a larger effective source region, but the maximum values (see the caption) are reduced by two decades, which is consistent with the observations made in Figure 8 about the Green’s function derivatives. The much weaker Green’s function-weighted source far outweighs the larger effective source region, leading to the large reduction in radiated sound seen in Figure 7.

Figure 10.

Power spectral density of Green’s function-weighted Lighthill stresses, $({\mathrm{\Phi }}_{R_{11}}+2{\mathrm{\Phi }}_{R_{12}}+{\mathrm{\Phi }}_{R_{22}})U_0/{({\rho }_0{U}_0^2)}^{2}h$ , for steps with (a–c) zero rounding and (d–f) $r/h=50\%$ rounding at three different frequencies, $f\delta /U_0=$ : (a,d) 0.5; (b,e) 2.0; (c,f) 8.0. Logarithmic scale with 11 contour levels from (a) $3.0\times 10^{-1}$ to $3.0\times 10^{-6}$ ; (b) $3.0\times 10^{-2}$ to $3.0\times 10^{-7}$ ; (c) $2.0\times 10^{-4}$ to $2.0\times 10^{-9}$ ; (d) $2.0\times 10^{-3}$ to $2.0\times 10^{-8}$ ; (e) $3.0\times 10^{-4}$ to $3.0\times 10^{-9}$ ; (f) $4.0\times 10^{-6}$ to $4.0\times 10^{-11}$ .

In conclusion, the analysis in this section demonstrates that the noise reduction by rounding the upper corner of a forward-facing step in a turbulent boundary layer is mainly caused by a drastic reduction of acoustic diffraction by the step surface. Although the step shape also has a strong effect on the acoustic source strength and distribution, its impact on noise generation is less significant in comparison. It should be noted that the analysis has been facilitated by the use of the approximate Green’s functions for compact steps; however, the conclusions are expected to be valid, at least qualitatively, for steps that are acoustically noncompact due to either large step heights or high frequencies. The noise reduction by rounding acoustically-noncompact steps is discussed in the next section.

Noncompact steps

When the flow Mach number is not very low, as in most wind-tunnel experiments, the step height may not be acoustically compact. In this case the BEM approach proposed by Khalighi et al. ¹⁰ can be adapted to obtain acoustic solutions. Using the half-space Green’s function, equation (1) is first solved as an integral equation to obtain the fluctuating pressure, including the acoustic contribution, on the surface $S$ . This surface pressure is then used in the surface integral to calculate the far-field acoustic pressure. Details of the solution procedure can be found in Ref 8.

The effect of step rounding on the sound field with account for step noncompactness is illustrated in Figures 11 and 12 for $M=0.1$ . In Figure 11 the sound pressure spectra, ${\mathrm{\Phi }}_{pp}{U}_0/(P_0^2\delta )$ , are plotted versus observer angle $\theta$ (deg.) at $|\mathbf{x}|/h=1000$ for four frequencies, $f\delta /U_0=0.5$ , 2.0, 4.0 and 8.0, corresponding to $kh\approx 0.082$ , 0.33, 0.65 and 1.31, or $\lambda /h\approx 77$ , 19, 9.6 and 4.8, respectively. At the lowest frequency, the steps are acoustically compact, and therefore the directivity of a streamwise-dipole field is observed for all steps with the spectral level decreasing with increasing rounding radius. At $f\delta /U_0=2.0$ , the spectra still show dipole-like directivity but there is significant sound radiation in the wall-normal direction. At $f\delta /U_0=4.0$ , the steps become mildly noncompact, and the sound directivities deviate more from that of a compact dipole. There is a marked asymmetry relative to the $90°$ angle, and the directivities upstream of the steps differ much less from the ideal dipole directivity than those downstream of the steps. At the highest frequency plotted, $f\delta /U_0=8.0$ , the steps are highly noncompact, and the directivities are drastically different from the dipole directivity.

Figure 11.

Acoustic directivity for flow over rounded forward steps at $M=0.1$ and four frequencies: (a) $f\delta /U_0=0.5$ $(kh\approx 0.082)$ ; (b) $f\delta /U_0=2.0$ $(kh\approx 0.33)$ ; (c) $f\delta /U_0=4.0$ $(kh\approx 0.65)$ ; (d) $f\delta /U_0=8.0$ $(kh\approx 1.31)$ . Shown are the sound-pressure spectra ${\mathrm{\Phi }}_{pp}{U}_0/(P_0^2\delta )$ versus observer angle $\theta$ (deg.) at $|\mathbf{x}|/h=1000$ for steps with different rounding radii $r/h$ : , $0\%$ ; , $12.5\%$ ; , $25\%$ ; , $50\%$ ; , $100\%$ .

Figure 12.

Acoustic directivity in terms of sound pressure spectra normalized by their respective mean values over $0°$ to $180°$ angles for flow over rounded forward steps at $M=0.1$ and four frequencies. Other parameters and definitions are the same as in Figure 11.

The directivity curves in Figure 11 differ in both magnitude and angular dependence for the different steps. To quantify the reduction in sound pressure level, the sound-pressure spectra in Figure 11 are averaged over observer angles from $0°$ to $180°$ and the results are compared. At $f\delta /U_0=0.5$ , the averaged spectral levels in the order of increasing rounding radius are 1.8, 3.9, 7.3 and 9.0 dB lower than that of the sharp-edged step, which is close to the differences seen in Figure 7 for acoustically compact steps. At $f\delta /U_0=8.0$ , the averaged spectral levels for the corresponding rounded steps are 3.9, 6.7, 17.4 and 25.2 dB lower. Therefore the effect of step rounding on the sound pressure level is larger at higher frequencies, again consistent with Figure 7.

To highlight the effect of the step-rounding radius on directivity, the curves in Figure 11 are normalized by their $\theta$ -averaged values and replotted in Figure 12. At $f\delta /U_0=0.5$ , or equivalently $kh\approx 0.082$ , the steps are acoustically very compact, and all configurations exhibit nearly identical directivity patterns of a streamwise dipole. Step rounding has an appreciable influence on the directivity at $f\delta /U_0=2.0$ $(kh\approx 0.33)$ and $f\delta /U_0=4.0$ $(kh\approx 0.65)$ , although the overall patterns remain similar across different rounding radii. When the steps are acoustically highly noncompact, as in the case of $f\delta /U_0=8.0$ $(kh\approx 1.31)$ , the directivity becomes much more sensitive to the rounding radius. These results illustrate that the acoustic directivity is primarily determined by the parameter $kh$ unless the step is acoustically highly noncompact, as anticipated from theoretical considerations. Numerical predictions have also been compared with the experimental results of Awasthi et al. ¹² at the same $kh$ values, and similar directivities were observed despite differences in flow parameters and flow configurations.

Conclusions

In this study, the noise from rounded forward-facing steps in a low-Mach-number turbulent boundary layer has been investigated using large-eddy simulation and Lighthill’s theory, with the objective of quantifying the aeroacoustic effects of step rounding and elucidating its noise-reduction mechanisms. Five steps with rounding radii equal to $0\%$ , $12.5\%$ , $25\%$ , $50\%$ and $100\%$ of the step height are considered. The step height is $26\%$ of the thickness of the unperturbed boundary layer at a momentum-thickness Reynolds number of 4755. Consistent with previous experimental results, step rounding is found to cause a significant reduction in the radiated sound pressure level, and the size of reduction increases with increasing rounding radius and frequency. The acoustic source fields for different rounding radii have been analyzed in conjunction with tailored, acoustically compact Green’s functions. The results indicate that, while step rounding affects both turbulence production and acoustic diffraction by the step, noise reduction is primarily due to the weaker diffraction effect. Step rounding also affects the acoustic directivity for acoustically noncompact steps, as demonstrated by BEM calculations. In addition, it leads to weaker flow separation and quicker recovery towards equilibrium boundary layer judged by surface-pressure fluctuations. The peak level of surface-pressure fluctuations is amplified by step rounding for up to $50\%$ rounding but is reduced for the case of $100\%$ rounding.