4 Results

The random forcing approach introduced above was applied to a high-resolution DNS to initiate laminar–turbulent transition. The computational domain with an extent of $0.15~\text{m}\leqslant x\leqslant 0.51~\text{m}$ , $0~\text{m}\leqslant \unicode[STIX]{x1D702}\leqslant 0.09~\text{m}~({\approx}100\unicode[STIX]{x1D6FF}_{99,outflow})$ and $0~\text{rad}\leqslant \unicode[STIX]{x1D711}\leqslant \unicode[STIX]{x03C0}/40~\text{rad}$ was resolved with $n_{x}=4750$ , $n_{\unicode[STIX]{x1D702}}=350$ and $n_{\unicode[STIX]{x1D711}}=129$ points in the streamwise, wall-normal and azimuthal directions, respectively. Here $\unicode[STIX]{x1D6FF}_{99}$ denotes the boundary layer thickness as the location where 99% of the edge velocity are reached. This setup is similar to the ones used for the ‘controlled’ breakdown simulations by Hader & Fasel (Reference Hader and Fasel2016, Reference Hader and Fasel2017). The details of the DNS code used here, along with its continuing development, have been described in Laible, Mayer & Fasel (Reference Laible, Mayer and Fasel2008, Reference Laible, Mayer and Fasel2009), Laible (Reference Laible2011), Laible & Fasel (Reference Laible and Fasel2011). For brevity only a short summary will be given. The numerical method is based on the three-dimensional Navier–Stokes equations, the continuity equation, and the energy equation for a compressible fluid in generalized conical coordinates. The so-called spatial model is applied, where all three spatial dimensions and time are discretized without any modelling assumptions. High-order accurate, non-compact, central finite differences are used for discretization of the wall-normal (fourth-order) and the streamwise direction (sixth-order). In the azimuthal direction, a pseudo-spectral discretization using fast Fourier transforms is employed. The inviscid fluxes are separated into an upwind flux and a downwind flux using van Leer’s splitting (van Leer Reference van Leer1982). Then, grid centred upwind biased differences (Zhong Reference Zhong1998) with ninth-order accuracy are applied to evaluate the derivatives for these fluxes. Special attention is given to the wall-next boundary stencils. Since high-order accurate boundary closures are typically unstable, the order of the stencil is successively reduced from ninth order to third order at the wall. The wall is assumed to be isothermal for the base flow and will remain so even in the presence of the disturbances. Consequently, the temperature disturbances at the wall vanish. The isothermal wall assumption with respect to the disturbances is justified because the wall temperature of the model remains nearly constant in the experiments due to the very short run times of the BAM6QT (see Ward Reference Ward2010; Chynoweth Reference Chynoweth2015). At the outflow boundary, a buffer is employed (see Meitz Reference Meitz1996) that effectively relaminarizes the flow. Here, the buffer is applied over 150 grid points from $4600\leqslant i\leqslant 4750$ . At the inflow and the free-stream boundary, five grid points are used to enforce the laminar base flow values obtained from a precursor calculation using a lower-order finite-volume code (for details see Gross & Fasel Reference Gross and Fasel2008). Periodicity and symmetry conditions are imposed at the azimuthal boundaries, allowing a reduction of the simulation domain by a factor of two. These boundary conditions were selected because they have been successfully employed in previous high-resolution DNS to study laminar–turbulent transition.

The random forcing described by (3.1) was applied at $6\leqslant i\leqslant 10$ , $1\leqslant j\leqslant 345$ and $k=1$ , as discussed above, with a noise level of $A_{noise}=10^{-2}$ . This noise level was selected based on the results from preliminary axisymmetric calculations, where the random forcing was applied at the same streamwise and wall-normal locations with varying values of $A_{noise}$ . The noise level was chosen such that the resulting saturation location of a primary wave was in the neighbourhood of that observed for the ‘controlled’ breakdown DNS to ensure that the nonlinear breakdown would occur in the computational domain. The choice to use $x=0.15$  m as the inflow boundary was based on the results of the linear stability regime (Hader & Fasel Reference Hader and Fasel2016), which showed that, at this location, the second mode is already amplified. This avoids the need to compute parts of the cone geometry where the disturbances are decaying, and therefore are not of interest for a nonlinear breakdown investigation. The domain height was designed such that the free-stream boundary is approximately 100 laminar boundary-layer thicknesses (calculated at the outflow) above the wall to minimize the effects of possible reflections from the free-stream boundary. The grid resolution was based on our previous experience with the controlled breakdown simulations (Hader & Fasel Reference Hader and Fasel2016, Reference Hader and Fasel2017). In these cases, the wall units (grid line spacing rescaled with the friction velocity) were $\unicode[STIX]{x0394}x_{max}^{+}\approx 6$ in the streamwise direction, $\unicode[STIX]{x0394}\unicode[STIX]{x1D702}_{max}^{+}\approx 1$ in the wall-normal direction, and $\unicode[STIX]{x0394}\unicode[STIX]{x1D701}_{max}^{+}\approx 2.3$ in the azimuthal direction, respectively. Consequently, the wall-normal grid line spacing is adequate to resolve the viscous sublayer and the streamwise and azimuthal wall units are comparable to the values reported in Mayer, Von Terzi & Fasel (Reference Mayer, Von Terzi and Fasel2011), who computed a complete breakdown all the way to turbulence for a flat plate at Mach 3.

Note that the extent of the computational domain in the azimuthal direction determines the smallest azimuthal wavenumber that is physically resolved in the simulation. The domain size of $\unicode[STIX]{x1D711}_{simulation}=\unicode[STIX]{x03C0}/40$ along with the symmetry conditions that are employed on the azimuthal boundaries limit the smallest azimuthal wavenumber that is contained in the simulation to $k_{c}=40$ . Consequently, only integer multiples of $k_{c}=40$ are resolved in the presented random forcing DNS.

The entire flow field was sampled for a duration of $T_{sampling}=2\times 10^{-4}$  s with a sampling frequency of $f_{sampling}=6$  MHz to accurately resolve frequencies up to 3 MHz with a frequency interval of $\unicode[STIX]{x0394}f=5$  kHz. With these sampling parameters, the most amplified second mode frequency band for the primary instability (the largest $N$ -factors were obtained for 300–350 kHz, see Hader & Fasel Reference Hader and Fasel2016) and the higher harmonics are adequately resolved. The flow was averaged (arithmetic average) for a duration of $T_{average}=2\times 10^{-3}$  s to obtain a value on the order of the tunnel run time at the BAM6QT facility. The wall pressure disturbance at the wall was sampled with $f_{sampling}=10$  MHz for a duration of $T_{sampling}=2\times 10^{-3}$  s allowing for a reasonable comparison to the experimental data that was recorded for $T_{sampling,experiment}=9.3\times 10^{-3}$  s at a rate of $f_{sampling,experiment}=5$  MHz (Chynoweth, B. 2017, private communication).

The normalized pressure disturbance contours are shown in figure 4 (top) along with the corresponding normalized pressure signals at the symmetry line ( $\unicode[STIX]{x1D711}=0$ , $k=1$ ) in figure 4 (bottom) at the same time instant ( $t=1$  ms) when the initial transient has left the computational domain. The contours for the unrolled cone surface ( $\unicode[STIX]{x1D701}$ , see figure 2 b) are shown in figure 4(a) (top). As can be observed, at the wall the random disturbances quickly contaminate the entire spanwise domain and then begin to form a coherent two-dimensional wave packet after a short downstream distance. The two-dimensional waves are the dominant second mode waves that are most unstable in high-speed boundary layers (Mack Reference Mack1969). This shows that the boundary layer naturally ‘selects’ the most unstable frequencies from the broadband spectrum introduced by the random forcing after a very short downstream distance. In contrast, the normalized pressure disturbance contours in the unrolled surface approximately two boundary-layer thicknesses away from the cone wall in figure 4(b) indicate the presence of random 3D patterns, which are equivalent to the acoustic noise in the free stream. The normalized pressure disturbance signals extracted at the symmetry line show that in the boundary layer, (figure 4 a, bottom) the randomly forced disturbances quickly form wave packets, whereas in the free-stream, (figure 4 b, bottom) random pressure fluctuations without a preferred wavelength are observed.

Figure 4.

Pressure disturbance contours (top) and pressure disturbance signal at $\unicode[STIX]{x1D711}=0$  rad (bottom) at the same time instant ( $t=1$  ms). (a) For the unrolled cone surface and (b) in an unrolled surface at a distance $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FF}_{99}=2.2$ from the cone wall.

A comparison of the time-averaged Stanton number contours on the surface of the cone between the results from the DNS of the ‘controlled’ fundamental breakdown (Hader & Fasel Reference Hader and Fasel2016) and the DNS results with random forcing is provided in figure 5(a). For the ‘controlled’ fundamental breakdown a large-amplitude two-dimensional primary wave ( $f=300$  kHz) was forced together with a pair of small-amplitude three-dimensional secondary waves at the same frequency using wall-normal velocity disturbances through a wall blowing/suction slot. For the secondary wave, the azimuthal wavenumber that led to the largest secondary growth rate after resonance onset was chosen ( $k_{c}=80$ ). Compared to the ‘controlled’ fundamental breakdown, the random forcing approach does not favour any particular disturbance waves, both with respect to frequency and azimuthal wavenumbers (that are resolved in the computational domain), but rather introduces a wide range of frequencies and azimuthal wavenumbers at the inflow boundary of the computational domain (figure 3 b). As can be observed in figure 5 for both the ‘controlled’ and random forcing simulations, ‘primary’ streaks appear for a limited downstream extent until ‘secondary’ streaks appear farther downstream where the flow has already progressed far into the highly nonlinear breakdown regime (figure 5 a). Although the ‘primary’ streaks and the ‘secondary’ streaks do not appear at exactly the same downstream locations for the two cases, it is remarkable that the random forcing approach leads to a pattern of streamwise streaks as observed in the ‘controlled’ breakdown simulation and in the BAM6QT experiments. A close-up of the ‘primary’ streak region (unrolled in azimuthal direction) is provided in figure 5(b). In the top half, the ‘primary’ streak region obtained for the ‘controlled’ fundamental breakdown is shown and compared to the ‘primary’ streaks observed for the random forcing breakdown on the bottom half. As shown by Laible (Reference Laible2011), and Laible & Fasel (Reference Laible and Fasel2011), all other conditions being identical, reducing the initial primary disturbance amplitude slightly would shift the streak onset farther downstream. As a result, both the onset location and the downstream extent of the streaks observed in the ‘controlled’ breakdown simulations depend on the initial amplitudes of the primary and secondary disturbance waves.

Figure 5.

(a) Time-averaged Stanton number contours on the surface of the cone (left: ‘controlled’ fundamental breakdown; right: random forcing breakdown) and (b) a close-up of the ‘primary streak’ region (unrolled in azimuthal direction) for the ‘controlled’ fundamental breakdown (top) and the random forcing breakdown (bottom).

The azimuthal spacing of the ‘primary’ streaks obtained with the ‘controlled’ breakdown DNS, the random forcing DNS and the experiments at BAM6QT ( $p_{0}=131.9$  psi, $T_{0}=432.6$  K, $Re_{1}=9.7\times 10^{6}~\text{m}^{-1}$ , Chynoweth, B. 2017, private communication) is compared in figure 6(a). For the DNS results a total of 80 ‘primary’ streaks around the circumference of the cone can be observed. The distance in the azimuthal direction between the ‘primary’ streaks obtained in the experiments is not constant but is in the same range as for the DNS. The cause for the misalignment of the streaks between experimental and numerical results is not yet known and will be subject to future investigations. Whereas for the ‘controlled’ fundamental breakdown case the secondary wave was forced with an azimuthal wavenumber of $k_{c}=80$ (Hader & Fasel Reference Hader and Fasel2016, Reference Hader and Fasel2017), no particular azimuthal wavenumber was favoured in the random forcing breakdown case (except that due to the limited azimuthal domain extent only azimuthal wavenumbers which are an integer multiple of $k_{c}=40$ are resolved in the simulation). This indicates that the assumption made for the ‘controlled’ breakdown simulations, namely that in a ‘natural’ breakdown scenario the secondary wave with the largest integrated growth rate ( $N$ -factor) after the secondary resonance onset is the 3D wave that will most likely prevail and dominate the nonlinear regime and thus the ‘primary’ streak development, is valid (provided of course that the fundamental resonance scenario is indeed the route to transition for the given flow conditions and geometry of the Purdue experiments). Therefore, the results of the random forcing DNS are providing further (almost conclusive) evidence that the observed streaks in the Purdue experiment are indeed due to a fundamental resonance. Further support for this can be obtained by comparing the downstream development of the time-averaged Stanton number between the ‘controlled’ fundamental breakdown, the random forcing case and the experiments (Chynoweth, B. 2017, private communication) as provided in figure 6(b). The Stanton number development for the base flow as well as for the laminar and turbulent estimate for a straight cone (White Reference White2006) are provided for reference. For all cases, after an initial departure from the laminar curve, the Stanton number increases slowly at first before rising rapidly when the ‘primary’ streaks begin to form, resulting in a massive overshoot far beyond the turbulent estimates. When the ‘primary’ streaks disappear, the time-averaged Stanton number drops almost back to the laminar base flow value. When the ‘secondary’ streaks appear, the time-averaged Stanton number rises again; however, the peak values are lower than for the ‘primary’ streaks. The Stanton number development in the downstream direction of the random forcing DNS and the experimental results show remarkable agreement.

Figure 6.

Comparison of the time-averaged Stanton number for the fundamental breakdown DNS (Hader & Fasel Reference Hader and Fasel2016), the random forcing DNS (for $p_{0}=140$  psi,  $T_{0}=420$  K,  $Re_{1}=10.8\times 10^{6}~\text{m}^{-1}$ ) and the experimental results (for $p_{0}=131.9$  psi,  $T_{0}=432.6$  K,  $Re_{1}=9.7\times 10^{6}~\text{m}^{-1}$ , courtesy of Chynoweth, B. 2017, private communication) in (a) the azimuthal direction and (b) the downstream direction at $\unicode[STIX]{x1D711}=0$  rad.

The downstream development of the maximum temperature amplitudes of various disturbance components for the ‘controlled’ fundamental resonance case is compared to the random forcing case in figure 7. With the random forcing (figure 7 b), all two- and three-dimensional disturbance modes (resolved in the computational domain) at a wide range of frequencies (see figure 7 b) are introduced at approximately the same amplitude level, whereas for the fundamental breakdown (figure 7 a) only the primary wave and the secondary wave are forced, and are introduced at much larger amplitudes compared to the numerical background noise caused by the discretization error. Based on a low-amplitude axisymmetric pulse simulation (Hader & Fasel Reference Hader and Fasel2016), the primary wave frequency to be used for the ‘controlled’ fundamental breakdown simulation was $f=300$  kHz, as for this frequency the largest $N$ -factor was obtained. For the random forcing case, no single frequency is favoured by the broadband forcing, and the individual modes are allowed to develop naturally in the downstream direction. In this scenario, the two-dimensional second mode with a frequency of $f=295$  kHz appears to dominate over the other frequency modes, which is in remarkable agreement with the $f=300$  kHz of the primary wave used for the ‘controlled’ fundamental breakdown case. Based on our investigations of the primary instability (see Hader & Fasel Reference Hader and Fasel2016) the branch I neutral point for both frequencies is $x_{0,I}\approx 0.085$  m and the spatial growth rates vary in the range $60~\text{m}^{-1}\leqslant -\unicode[STIX]{x1D6FC}_{i}\leqslant 70~\text{m}^{-1}$ . Compared to the ‘controlled’ fundamental breakdown in figure 7(a), for the random forcing case (figure 7 b) the three-dimensional disturbance waves with the various spanwise wavenumbers are all ‘starting’ at approximately the same amplitude levels. None of the 3D modes are initially amplified until they experience strong growth once the two-dimensional waves start to saturate close to $x\approx 0.35$  m. Consistent with the fundamental resonance study by Hader & Fasel (Reference Hader and Fasel2016), the disturbance wave with an azimuthal wavenumber of $k_{c}=80$ grows most rapidly, and nonlinearly generates the steady longitudinal mode with the same azimuthal wavenumber together with the two-dimensional wave of the same frequency: $(f=295~\text{kHz},k_{c}=0)\pm (f=295~\text{kHz},k_{c}=80)\rightarrow (f=0~\text{kHz},k_{c}=80)$ . The ( $f=0$  kHz, $k_{c}=80$ ) mode is the dominant mode when the ‘primary’ streaks appear, and reaches its maximum when the time-averaged Stanton number becomes maximal. The azimuthal wavenumber of this dominant longitudinal mode determines the ‘primary’ azimuthal streak count. The ‘primary’ streaks disappear because the streamwise vortices that are responsible for their formation are lifted away from the wall. Further downstream these vortices are pushed back towards the wall, leading to the ‘secondary’ streaks (see Hader & Fasel Reference Hader and Fasel2016). At this point, the ‘controlled’ as well as the random forcing DNS results show that higher harmonics ( $k_{c}=160$ , $k_{c}=240$ ) of the dominant steady streamwise mode ( $k_{c}=80$ ) have reached large amplitudes (figure 7), causing an increased streak count, as observed by Hader & Fasel (Reference Hader and Fasel2016) for the ‘controlled’ breakdown simulation.

Figure 7.

Downstream development of the maximum temperature amplitude for various disturbance components: (a) ‘controlled’ fundamental breakdown, (b) random forcing DNS.

The amplitude development comparison in figure 7 clearly indicates that even for the random forcing, which does not favour any particular frequencies or azimuthal wavenumbers (in contrast to the ‘controlled’ forcing case), the fundamental resonance is indeed the relevant mechanism for the investigated geometry and flow conditions.

Figure 8.

Comparison of experimental measurements at the BAM6QT ( $p_{0}=131.9$  psi, $Re_{1}=9.7\times 10^{6}~\text{m}^{-1}$ ) to DNS results ( $p_{0}=140.0$  psi,  $Re_{1}=10.8\times 10^{6}~\text{m}^{-1}$ ): (a) normalized wall pressure disturbance over time and (b) corresponding PSD spectra.

A comparison of simulation data with measurements from the BAM6QT is provided in figure 8. The experimental data are for $p_{0}=131.9$  psi, $Re_{1}=9.7\times 10^{6}~\text{m}^{-1}$ and were provided by (Chynoweth, B. 2017, private communication). Although the provided experimental data are for conditions slightly different than those used in the simulation ( $p_{0}=140$  psi, $Re_{1}=10.8\times 10^{6}~\text{m}^{-1}$ ), a direct comparison nevertheless provides valuable insights. The experimental time signals of the normalized pressure disturbance at the wall at the various pressure sensor locations (figure 8 a) show good qualitative agreement with results from the random forcing DNS. The experiment and simulation time signals in figure 8(a) show ‘wave packet like intermittency’, with sequences of wave packets that are ‘chasing’ each other. Despite the slightly different flow conditions between experiment and DNS, the main trends are consistent with each other. The amplitudes of the normalized pressure disturbances obtained with the random forcing DNS are of the same order of magnitude as the experimental measurements (figure 8 a). The amplitude of the experimental signal at $x=0.441$  m has dropped significantly compared to the amplitude at $x=0.390$  m due to energy being distributed over a wider range of frequencies and azimuthal wavenumbers, whereas the amplitude of the simulation data shows a less drastic drop when comparing these two locations. This is most likely caused by the fact that the experiment may already have progressed further into the breakdown regime compared to the simulation at this particular downstream location. At $x=0.492$  m the experimental time signal as well as the simulation data show random fluctuations, and the wave packet character of the time signals as seen at the upstream locations has disappeared. The corresponding PSD spectra for both experimental and simulation data have been calculated using the post-processing script provided by (Chynoweth, B. 2017, private communication). The comparison in figure 8(b) shows remarkable agreement between experiments and the random forcing DNS. The dominant amplified frequency bands from the experimental data are well captured in the simulations. The PSD amplitudes are in reasonable agreement (computed amplitudes are two to three times larger compared to the measured amplitudes) at $x=0.390$  m and $x=0.492$  m, considering the differences in flow conditions and the limitations of the simulation regarding the azimuthal domain width, which only allows for azimuthal wavenumbers of $k_{c}=40$ and its higher harmonics compared to the experiments, where no such limitations are present. The differences at the other locations can be attributed to the difference in flow conditions as well as the fact that the ‘primary’ streaks and the ‘secondary’ streaks appear further downstream in the DNS compared to the experimental data. As discussed already in connection with the results shown in figure 5, the streak onset can be moved in the upstream or downstream directions by increasing or decreasing, respectively, the initial forcing amplitude.