Silent inflow condition for turbulent boundary layers

The generation of a turbulent inflow is a tricky problem. In the framework of aeroacoustics, another important constraint is that the numerical strategy used to reach a turbulent state induces a spurious noise which is lower than the acoustic field of interest. For the study of noise radiated directly by a turbulent boundary layer on a flat plate, this constraint is severe since wall turbulence is a very inefficient source. That is why a method based on a transition by modal interaction using a base flow with an inflection point is proposed to cope with that. The base flow must be a solution of the equations so we use a profile behind a backward-facing step representative of experimental trip bands. A triad of resonant waves is selected by a local stability analysis of the linearized compressible equations and is added with a weak amplitude in the inlet plane. The compressible stability calculation allows the specification of the thermodynamic quantities at the inlet, which turns out to be fundamental to ensure a quiet inflow. A smooth transition is achieved with the rapid formation of $\mathrm{\Lambda }$-shape vortices in a staggered organization as in subharmonic transition. The dominance of oblique waves promotes a rapid breakdown by the liftup mechanism of low-speed streaks. The quality of the fully turbulent state is assessed and the direct noise radiation from a turbulent boundary layer at Mach 0.5 is obtained with a very low level of spurious noise.

Figure 1

Spurious fluctuating pressure due to the introduction of a vortex. Three successive instants are shown by using the inflow condition (2) for (a) $u^{\prime },v^{\prime }$ only (levels $\pm 10$ Pa); (b) $u^{\prime },v^{\prime },p^{\prime }$ on three points and update every Runge-Kutta substep (levels $\pm 1$ Pa); (c) $u^{\prime },v^{\prime },p^{\prime }$ on five points and update every physical time step (levels $\pm 1$ Pa); (d) $u^{\prime },v^{\prime },p^{\prime }$ on five points and update every Runge-Kutta substep (levels $\pm 1$ Pa).

Figure 2

Attachment length $x_a$ for the 2D base flow ($\blacksquare$) compared to channel flow experiments (step height $h=0.9423$ channel height) by Armaly et al. [82] ($\circ \circ$) and incompressible channel flow simulations (same conditions) by Biswas et al. [83] ($▵▵$).

Figure 3

Base flow obtained seven points downstream of the step at $M=0.5,{\text{Re}}_h=462$. From left to right, streamwise velocity, vertical velocity, pressure, and density.

Figure 4

Choice of a Craik's triad: on the left, dispersion curves for the wave number ${\alpha }_r$ as function of negative wave angle $\mathrm{\Psi },$ and on the right, amplification factor ${\alpha }_i$ as function of positive wave angle $\mathrm{\Psi }$. Red and blue isolines are associated to the 2D fundamental wave ($•$) and 2D fundamental wave ($•$), respectively. The magenta isoline in the left part represents the phase velocity, identical for the three waves.

Figure 5

Eigenfunctions from the local stability analysis for the 2D fundamental [(a)–(d)] and the 3D subharmonic [(e)–(h)]: $----$ modulus; $-\; -\; -$, real part; and $-·-·$, imaginary part.

Figure 6

Instantaneous views in the streamwise-spanwise plane at $y/h=0.61$ of the vertical velocity for different forcing amplitudes: (a) ${\varepsilon }_{2\mathrm{D}}={\varepsilon }_{3\mathrm{D}}=4\times {10}^{-4}$; (b) ${\varepsilon }_{2\mathrm{D}}={\varepsilon }_{3\mathrm{D}}=6\times {10}^{-4}$; (c) ${\varepsilon }_{2\mathrm{D}}={\varepsilon }_{3\mathrm{D}}=8\times {10}^{-4}$; (d) ${\varepsilon }_{2\mathrm{D}}={\varepsilon }_{3\mathrm{D}}=1\times {10}^{-3}$; (e) ${\varepsilon }_{2\mathrm{D}}=0$ and ${\varepsilon }_{3\mathrm{D}}=8\times {10}^{-4}$; (f) ${\varepsilon }_{2\mathrm{D}}=0$ and ${\varepsilon }_{3\mathrm{D}}=1\times {10}^{-3}$; (g) ${\varepsilon }_{2\mathrm{D}}=1\times {10}^{-3}$ and ${\varepsilon }_{3\mathrm{D}}=0$. Same color scale $\pm 1$ m/s for all figures.

Figure 7

Evolution of the integrated modal energy for $\varepsilon =6\times {10}^{-4}$ generated by Fourier mode: (1,1), $----$; (1,3), $----$; (2,0), $-·-·$; (0,2), $-\; -\; -$; (0,4), $-\; -\; -$; and (0,6), $......$.

Figure 8

Frequency wave number spectra at three successive locations: $x/h=15$ [(a), (d)], 30 [(b), (e)], and 60 [(c), (f)] for the transition from a triad of modes [(a)–(c)] and from a pair of oblique modes [(d)–(f)]. Open black circles with decreasing size are markers of oblique modes (1,1), (1,3), and (1,5). Open black squares with decreasing size are for streak modes (0,2), (0,4), and (0,6). The red lozenge marks plane mode (2,0).

Figure 9

(a) Displacement thickness ($-\; -\; -$) and momentum thickness ($----$); (b) evolution of the Reynolds number based on momentum thickness ${\text{Re}}_{\theta }$ along the plate: $----$), present LES; $\blacksquare$), DNS of FST transition of Wu and Moin [91]; $▴$, DNS of $K$ type and $▾, H$-type transitions of Sayadi et al. [93]; $-·-·$, Blasius solution.

Figure 10

(a) Shape factor $H$ as function of ${\text{Re}}_{\theta }$: $----$, present LES; $\blacksquare$, FST [91]; $▴, K$-type [93]; $▾, H$-type [93] transitions; $•$, high-Re DNS [94]; $\ast$, Blasius solution; $+$, momentum integral estimate [27]. (b) Skin-friction coefficient $C_f$ as function of ${\text{Re}}_{\theta }$: $----$, present LES; $\blacksquare$, FST [91]; $▴, K$-type [93]; $▾, H$-type [93] transitions; $•$, high-Re DNS [94].

Figure 11

(a) Profiles of the mean longitudinal velocity in wall units for ${\text{Re}}_{\theta }$=1551 ($----$) compared with DNS of Jimenez et al. [95] at ${\text{Re}}_{\theta }$ = 1551 ($○$) and for ${\text{Re}}_{\theta }$ = 2300 ($-\; -\; -$) compared with DNS of Schlatter and Örlü [94] at ${\text{Re}}_{\theta }$ = 2540 ($○$). (b) Turbulent intensities normalized by $u_{\tau }$ at ${\text{Re}}_{\theta }$ = 2300: Present LES: $----, u_{\text{rms}}$; $-·-·, v_{\text{rms}}$; $-\; -\; -, w_{\text{rms}}$. DNS of Jimenez at ${\text{Re}}_{\theta }$ = 1968: $○, u_{\text{rms}}$; $+, v_{\text{rms}}$; $\square , w_{\text{rms}}$. DNS of Schlatter at ${\text{Re}}_{\theta }$ = 2540: $○, u_{\text{rms}}$; $+, v_{\text{rms}}$; $\square , w_{\text{rms}}$.

Figure 12

Locations of the extraction planes ($-\; -\; -$) for turbulent spectra: $x/h=100,200,300,400,500,1000,1500,2000$, superimposed on the instantaneous streamwise velocity field.

Figure 13

One-dimensional velocity spectra for the three velocity components at $y/h\approx 0.6$: (a) $u$; (b) $v$; (c) $w$ for the successive streamwise locations depicted in Fig. 12. The red dashed line indicates the $-5/3$ slope.

Figure 14

Instantaneous snapshots of the pressure fluctuations $p^{\prime }$ in the median plane (levels $\pm 3$ Pa) at four instants.

Figure 15

Power spectral densities of the pressure perturbations $\tilde{\varphi }\left(\omega \right)=\varphi \left(\omega \right)U_{\infty }/\left({\rho }^2{u}_{\tau }^4{M}^2{{\delta }^{*}}^3\right)$ in the acoustic field as a function of the Strouhal number $\omega {\delta }_{\text{ref}}^{*}/U_{\infty }$. (a) LES with the new inflow condition at $y/h=250$, and for various streamwise locations: ($----▿$) $x/h=455$, ($----○$) $x/h=850$, ($----*$) $x/h=1244$, ($----+$) $x/h=1638$, and ($----▵$) $x/h=2032$. (b) Previous LES results [4] at $y/{\delta }_{\text{ref}}=9.6$, and for various streamwise locations: ($----▿$) $x/{\delta }_{\text{ref}}=6.6$, ($----○$) $x/{\delta }_{\text{ref}}=19.8$, ($----*$) $x/{\delta }_{\text{ref}}=33.1$, ($----+$) $x/{\delta }_{\text{ref}}=46.4$, ($----▵$) $x/{\delta }_{\text{ref}}=59.7$, and ($----\square$) $x/{\delta }_{\text{ref}}=72.9$, where subscript ${•}_{\text{ref}}$ denotes a median boundary-layer thickness (for comparison, $y/h$ = 250 in subplot (a) corresponds to $y/{\delta }_{\text{ref}}=8.8$).

Figure 16

Instantaneous plots of the streamwise velocity in a streamwise-spanwise plane at $y$ = $0.77{\delta }_0^{*}$ for increasing values of the forcing amplitude: ${\varepsilon }_{3\mathrm{D}}=8\times {10}^{-3}$ (a), $2\times {10}^{-2}$ (b), and $3\times {10}^{-2}$ (c).

Figure 17

Amplitude growth of streamwise velocity modes for an oblique-wave pair with amplitude $\varepsilon =8\times {10}^{-3}$ (a). Lines represent the present computation, open symbols represent the DNS of Joslin et al., and filled symbols represent the PSE of Joslin et al.: mode (1,1) ($----,\square ,\blacksquare$), mode (1,3) ($----,\square ,\blacksquare$), mode (0,2) ($-\; -\; -,○,•$), and mode (0,4) ($-\; -\; -,○,•$). Maps of the modes (1,1) (b), (1,3) (c), (0,2) (d), and (0,4) (e). Color levels of $|\widehat{u}|$ are between 0 and 0.03 for panels (b) and (c) and between 0 and 0.3 for panels (d) and (e).