2.3.1. Direct numerical simulation: finite-difference Navier–Stokes solver, CFDSU

The fully compressible Navier–Stokes equations (2.1) are solved using a finite differencing code, CFDSU, originally written by Nagarajan, Lele & Ferziger (Reference Nagarajan, Lele and Ferziger2003). It is a Fortran-based code that is continuously being developed at Purdue University. The code CFDSU has been used previously for various types of flow applications, which include large-eddy simulation modelling (Chapelier, Wasistho & Scalo Reference Chapelier, Wasistho and Scalo2018); Chen & Scalo (Reference Chen and Scalo2021), vortex dynamics (Chapelier, Wasistho & Scalo Reference Chapelier, Wasistho and Scalo2019) and hypersonic transitional flow (Sousa et al. Reference Sousa, Patel, Chapelier, Wartemann, Wagner and Scalo2019, Reference Sousa, Wartemann, Wagner and Scalo2024).

The governing equations are solved on a structured curvilinear grid in a staggered arrangement. Thermodynamic variables, including density $\rho$ , temperature $T$ and pressure $p$ , are stored at the cell centres, while the other conservative variables involving velocities are stored at cell faces. Temporal integration has been performed using a fourth-order accurate, six-stage Runge–Kutta scheme (Allampalli et al. Reference Allampalli, Hixon, Nallasamy and Sawyer2009). Spatial discretization is done using the compact sixth-order scheme (C6) by Visbal & Gaitonde (Reference Visbal and Gaitonde2002). This numerical strategy is expected to achieve spectral-like numerical accuracy with less numerical dissipation. This is desirable for resolving the relevant length and time scales involved in a hypersonic BL transition problem. A grid transformation is done to map the physical domain in Cartesian coordinates to a cylindrical coordinate system, where the governing equations are numerically solved. The curved grid in the cylindrical coordinate system (physical space) is further mapped to an equispaced orthogonal grid (computational space) using a curvilinear grid transformation suitable for applying the structured spatial scheme. The relevant equations for grid transformations involved are given in Nagarajan, Lele & Ferziger (Reference Nagarajan, Lele and Ferziger2007).

2.3.2. Linear stability theory: Laguerre–Galerkin spectral linear stability solver

The stability of the BL flow has been analysed using a spectral LST solver, employing the Laguerre functions as the orthogonal basis. The linearized perturbation equations, obtained by linearizing the governing equations (2.1) about a steady base flow, can be posed as an eigenvalue problem. This requires applying the normal mode approximation and the locally parallel flow approximation, which imparts two types of eigenvalue problems: temporal and spatial. The spatial eigenvalue problem has been previously solved using the Chebyshev spectral collocation method (Malik Reference Malik1990). Recently, Sousa et al. (Reference Sousa, Wartemann, Wagner and Scalo2023) showed the suitability of the Laguerre–Galerkin spectral method for open BL stability problems. The Laguerre functions are defined on the positive real axis $[0,\infty )$ , and their natural property of decaying exponentially away from zero allows the boundary conditions of open shear flows to be automatically satisfied without a hard Dirichlet imposition. In the present work, the methodology outlined in Sousa et al. (Reference Sousa, Wartemann, Wagner and Scalo2023) has been used to develop an in-house stability solver to solve the spatial eigenvalue problem.

3.2.1. Blowing/suction wall forcing

A blowing/suction type wall excitation was first proposed by Gaster & Grant (Reference Gaster and Grant1975) for low-speed transition, which was used later on by Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2014) and Sousa et al.2023 for high-speed transition simulations. The resulting wall-normal velocity fluctuations read

(3.1) \begin{equation} v'_w (x, y=0, t) = A_0 \cos ^3 \left ( \pi \xi \right ) \sin \left (2 \pi f_d t \right ), \quad \forall t \in [0,1/f_d], \end{equation}

where $A_0$ is the maximum amplitude of the pulse and $\xi = ( x - x_d )/w_d$ , such that $\xi \in [ -1/2,1/2 ]$ , where $x_d$ and $w_d$ are portrayed in figure 1.

A finite pulse triggers a wide spectrum of frequencies with no direct control over the frequency spectrum. In the following section, a second forcing method is developed where pseudorandom noise is projected on a spectral basis and filtered, guaranteeing grid independence and control over the spectral distribution.

3.2.2. Legendre-space filtered wall forcing

A univariate pseudorandom noise field $\phi (x_i)$ is sampled on a uniformly spaced grid $x_i$ in the streamwise direction and then projected onto the Legendre spectral space. Therein, the high-wavenumber modes that are unresolvable on the coarsest grid considered in this work are removed (see figure 2). The thus-obtained signal comprising a collection of Legendre modes is transformed back to the physical space with the same analytical expression regardless of grid refinement.

Figure 2. Schematic illustration of the Legendre-space filtered forcing strategy. (a) Sharp filtering of the sampled noise in the Legendre space. The horizontal axis denotes the Legendre mode number $n$ . (b) The effect of the filtering process in the physical domain. The initially sampled noise is shown in black, while the analytical noise obtained upon filtering is displayed in red. The black circles denote the Legendre–Gauss nodes where the discrete noise is initially sampled.

The aim of the modulation operation is to obtain an analytically filtered noise $\bar {\phi }(x)$ that can be evaluated at any point in space and stays invariant with grid refinement. The bar signifies the modulated function in the Legendre space throughout the following formulation. The Legendre-space modulation operator $\mathcal {M}_L$ is given as

(3.2) \begin{equation} \bar {\phi }(x)= \mathcal {M}_L\left (\sigma ^L_n, \phi \right ) = \mathcal {L}^{-1} \left [ \sigma ^L_n \mathcal {L}(\phi (x_i)) \right ], \quad \sigma ^L_n = \begin{cases} 1,\quad \forall n \leq N_{cL}\\ 0,\quad \text {otherwise} \end{cases}, \end{equation}

which will be evaluated at the DNS grid points $x=x_i^g$ for different grid refinement levels. These can be different from the sampled grid points $x_i$ . Here $\sigma ^L_n$ denotes a sharp Legendre-space filter, removing the noise content above a user-defined cutoff mode number $N_{cL}$ . Here $\mathcal {L}$ and $\mathcal {L}^{-1}$ denote the forward and backward discrete Legendre transform operations, respectively,

(3.3) \begin{equation} \tilde {\phi }_n = \mathcal {L}\left [\phi (x_i)\right ] = \frac {1}{\gamma _n}\sum _{i=0}^{N_x} \phi (x_i)L_n(x_i)w_i, \quad \bar {\phi }(x) = \mathcal {L}^{-1}\left [\tilde {\phi }_n\right ] = \sum _{n=0}^{N_{cL}} \tilde {\phi }_n L_n(x), \end{equation}

where, $\gamma _n= ({2}/{2n+1})$ , and $w_i={2}/{ ((1-x_i^2 ) ( L'_{N_x+1}(x_i) )^2)}$ denotes the nodal weights for Legendre–Gauss quadrature (Shen, Tang & Wang Reference Shen, Tang and Wang2011). While $N_x$ varies depending on the grid size, $N_{cL}$ is kept constant to achieve grid independence.

Modulation can be explored in other spectral spaces, like Chebyshev, Hermite, etc. However, in this work, a Legendre-space modulation has been chosen due to the more uniform distribution of Legendre–Gauss quadrature nodes over the excitation region. The Chebyshev–Gauss quadrature nodes tend to be more aggressively concentrated at the ends.

Modulation in time is achieved via a series of $m_f$ discrete frequencies that are intended to be introduced into the BL. For each frequency, a different noise field $\bar {\phi }$ is created. The final wall-normal velocity noise signal is a summation of all the $m_f$ reconstructed fields and has the form

(3.4) \begin{equation} v'_w (x, y=0, t) = A_0 \sum _{m=0}^{m_f} c(f_m) \cdot \Re \left ( \bar {\phi }(x, f_m) e^{-2\pi i f_m t} \right ), \end{equation}

where $c(f_m)$ is a frequency-dependent amplitude modulation that can be used to decide the colour of the noise, for e.g., pink, white, brown, etc. Pink noise, which is generally accepted to mimic ambient noises, has been used in this work. Here ‘ $A_0$ ’ is the amplitude parameter that indicates the strength of the introduced signal and is used for all the analyses in this work. The forcing has been applied over the time span $t \in [0,n_{c}/f_0 ]$ in the simulations, where $f_0$ is the lowest frequency and $n_{c}$ represents the number of periodic cycles in time.

4.1.1. Kinetic energy

For a rigid body of fixed mass $m$ , acted upon by a force $F_i=m \textrm{d}u_i/\textrm{d}t$ displacing it by $\textrm{d}x_i=u_i \textrm{d}t$ , the total work done is $\int \textrm{d}W=\int mdu_i/\textrm{d}t \cdot u_i \textrm{d}t=\int _0^{u_i} mu_i \,\textrm{d}u_i$ , which gets stored as KE.

The KE (per unit volume) of the perturbed flow can be similarly defined on a fluid element as the work done by the force (per unit volume) $\textrm{d}(\rho u_i)/\textrm{d}t$ to displace such element by $\textrm{d}x_i=u_i \textrm{d}t$ . Thus, the total KE of the flow (per unit volume) can be given as

(4.12) \begin{equation} \begin{aligned} E_k &= \frac {1}{2} \rho u_i^2 = \frac {1}{2} \left ( \rho _0+\rho ' \right ) \left ( u_{0i} + u'_{\!i} \right )^2 \\ &= \underbrace {\frac {1}{2} \rho _0 u_{0i}^2}_{\textit {O}^{(0)}} + \underbrace {\rho _0 u_{0i} u'_{\!i} + \frac {1}{2} \rho ' u_{0i}^2}_{\textit {O}^{(1)}} + \underbrace {\frac {1}{2} \rho _0 u_i^{\prime 2} + \rho ' u_{0i} u'_{\!i}}_{\textit {O}^{(2)}} + \underbrace {\frac {1}{2} \rho ' u_i^{\prime 2}}_{\textit {O}^{(3)}}. \end{aligned} \end{equation}

The zeroth-order term represents the base flow KE, while the following two terms constitute the first-order fluctuation energy, comprising the perturbation momentum carried by the non-zero mean flow. Finally, the second-order terms obtained here form the KE component of the disturbance energy expression (4.16).

4.1.2. Acoustic and thermal potential energy

Due to the presence of a disturbance, the mechanical work done on a fluid element to compress it gets stored as its PE, referred to as the acoustic PE in the literature (Kinsler et al. Reference Kinsler, Frey, Coppens and Sanders2000). In a constant-volume fluid element, the compression/expansion work can be quantified by the enthalpy fluctuation when no heat exchange occurs (isentropic). The second-order acoustic PE (per unit volume) then reads

(4.13) \begin{equation} E_a= \int (\rho ^{\prime} dh^{\prime})_s = \int \left (\rho ^{\prime} \frac {\partial h}{\partial p}\right )_s \textrm{d}{p}^{\prime} = \int _0^{p^\prime} \frac {p^\prime}{\gamma p_0} \,\textrm{d}{p^\prime} = \frac {1}{2} \frac {p^{\prime 2}}{\gamma p_0}, \end{equation}

where the expression $ (\rho^{\prime}/\rho _0=p^{\prime}/\gamma p_0 - s^{\prime}/C_p )$ is used to replace $\rho '$ . The above equation indicates that if a control element is compressed in the presence of a propagating disturbance, its acoustic PE increases while it decreases during an expansion. This can be alternatively derived considering a Lagrangian control-mass element, whose volume changes as it moves with the flow. The isentropic $pdV$ work on such an element reads

(4.14) \begin{equation} {E_a} = - \int _{V_0}^V \frac {p^{\prime}}{V} \,\textrm{d}V = \int _0^{\rho ^{\prime}} \frac {p^{\prime}}{\rho } \,\textrm{d}\rho ^{\prime} = \int _0^{p^{\prime}} \frac {p^{\prime}}{\rho } \,\frac {\textrm{d} p^{\prime}}{c^2} = \frac {1}{2}\frac {p^{\prime 2}}{\gamma p_0}, \end{equation}

which is equivalent to (4.13). The negative sign above indicates a PE increase (work done on the element) during compression caused by a positive acoustic pressure $p^{\prime}$ (Kinsler et al. Reference Kinsler, Frey, Coppens and Sanders2000).

Similarly, the heat added to a fluid element gets stored as the thermal PE. On a constant volume element, this heat addition can be quantified by the enthalpy fluctuation when no compression/expansion work is done to cause pressure change (isobaric). This gives the second-order thermal PE as

(4.15) \begin{equation} E_t = -\int (\rho^{\prime} dh^{\prime})_p = -\int \left (\rho^{\prime} \frac {\partial h}{\partial s}\right )_p \textrm{d}s^{\prime} = \int _0^{s^{\prime}} \frac {\rho _0 T_0}{C_p} s^{\prime} \,\textrm{d}s^{\prime} = \frac {\rho _0 T_0}{2C_p}s^{\prime 2}. \end{equation}

The negative sign is needed to associate a density decrease ( $\rho^{\prime}\lt 0$ ) at constant pressure with a heat gain, which indicates an increase in thermal PE. Heat gain at constant pressure occurs due to the influx of high-temperature ( $T^{\prime}\gt 0$ ) fluid into the control volume.

4.1.3. Deriving the energy equation

Finally, combining all of the aforementioned components that make up the total energy of a perturbation to the second order, yields

(4.16) \begin{equation} E_2 = E_t + E_k + E_a = \underbrace {\frac {1}{2} \frac {\rho _0 T_0}{C_p}s^{\prime 2}}_{i} + \underbrace {\rho^{\prime} u_{0i}u_i^{\prime}}_{ii} + \underbrace {\frac {1}{2} \rho _0 u_i^{\prime 2}}_{iii}+ \underbrace {\frac {1}{2}\frac {p^{\prime 2}}{\gamma p_0}}_{iv}. \end{equation}

Throughout this manuscript, each term in the above expression is referred to by the numerals given below. Correctly deriving the disturbance energy equation entails choosing an appropriate combination of (4.6c ), (4.10) and (4.11) to obtain $E_2$ , which reads

(4.17)

The result of this combination contains perturbation terms up to the fifth order of nonlinearity because of (4.6c ) and (4.11). By truncating up to the second order, the desired perturbation energy conservation equation can be recovered. Contrary to Myers’ naming convention, it will be denoted as the second-order energy equation hereafter.

The multipliers in (4.17) are required to achieve the specific perturbation energy $(E_2)$ chosen in the current work (4.16). While its expression is justified by the considerations in §§ 4.1.1 and 4.1.2, we note that one might choose to use a different expression for the perturbation energy. For example, George & Sujith (Reference George and Sujith2012) offers an expression for a family of allowable perturbation norms. The question of the uniqueness of the perturbation energy of a fluctuating thermofluid-dynamic field remains a topic for continued academic investigation.

5.1.1. Grid sensitivity

The results of grid sensitivity analysis with the forcing type BL5 are also included in figure 5. Three grid refinement levels are considered in this work, the details of which are given in table 4. Results for the grids $9216\times 384$ (dashed) and $12\,288\times 512$ (solid) are in good agreement, as the coarse grid $6144\times 256$ (dotted) differs slightly. The DNS results from the grid $9216\times 384$ have been used for all succeeding analyses.

5.1.2. Behaviour of power spectra at higher amplitudes

Figure 6. Power spectra evolution in the streamwise direction is shown for DNS runs on the grid $9216\times 384$ with forcing type BL5 (described in table 3). The PSDs are shown for streamwise locations in the range x = 35 () to 95 cm (), using a sequential colour map (arrows designate increasing $x$ ). The forcing amplitude $A_0$ increases row-wise. A temporary attenuation effect in the PSD growth, observed at higher fluctuation levels, is highlighted with dashed boxes.

Figure 6 shows a more detailed streamwise evolution of the DNS power spectra. The PSDs are shown at uniformly spaced locations in the range $x=35-95\,\textrm{cm}$ at intervals of $5\,\textrm{cm}$ . The figure has been broken down into three rows to avoid plot overcrowding. The PSD evolution of the linear amplitude case is omitted since it is qualitatively similar to the weakly nonlinear amplitude. The spatial PSD evolution can be divided into two distinct stages. In the first stage, immediately downstream of the forcing region between $35$ and $50\,\textrm{cm}$ , the spectral content at all frequencies increases monotonically without any frequency shift. In the next stage, after $50\,\textrm{cm}$ , the second modes start to amplify with a steady frequency shift. The forcing region having ended at $x=30.9\,\textrm{cm}$ , the unstable modes have not modulated with the BL height in the first phase and cannot sustain standing wave resonance yet. Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2014) points out that just downstream of a localized forcing, the wave packets are predominantly three-dimensional, and the axisymmetric second modes do not start to destabilize readily. The steady frequency shift and amplification observed in the second stage is a characteristic of second-mode evolution and is known to occur due to the increasing BL height.

However, the PSD evolution deviates from this trend at higher forcing amplitudes as a temporary nonlinear attenuation is observed, which has been marked in figure 6 with dashed rectangles. The first occurrence of this for $A_0=1\,\textrm{m}\,\textrm{s}^-{^1}$ is at $x=55\,\textrm{cm}$ after the forcing mechanism stabilizes and continues to occur periodically downstream. For the nonlinear amplitude $A_0=0.1\,\textrm{m}\,\textrm{s}^-{^1}$ , this behaviour is only seen temporarily at $x\approx 0.6\,\textrm{m}$ . This effect was observed in the experiments conducted by Casper et al. (Reference Casper, Beresh, Henfling, Spillers, Pruett and Schneider2016) at Sandia’s HWT-8 wind tunnel facility on a $7^\circ$ cone. Schlieren imaging showed signs of intermittent turbulence in the BL, which were reasoned to be the culprit for the abrupt attenuation. Chynoweth et al. (Reference Chynoweth, Schneider, Hader, Fasel, Batista, Kuehl, Juliano and Wheaton2019) also observed a similar effect on a Mach 6 flared cone as second-mode pressure fluctuations were seen to drop with simultaneous spectrum broadening. This was similarly concluded to be a sign of intermittency and ensuing breakdown. It is to be noted that these attenuations of the power spectrum are also observed in the axisymmetric DNS of the current work, where three-dimensional turbulence/intermittency effects are not accounted for, but nonlinear mechanisms are captured. Succeeding the PSD dip, the power spectra amplification of the primary mode starts to saturate, while the higher-frequency modes are seen to amplify more aggressively. In § 6, the perturbation energy corollary is used to investigate this effect further. However, before moving to analyse the nonlinear behaviour, the linear behaviour is first validated against locally parallel LST predictions.

5.2.1. DNS versus LST pressure spectra

Linear stability theory serves two purposes in this work. Firstly, the DNS results for the linear forcing cases are validated against LST results. Secondly, any deviation of the DNS results from LST predictions helps identify the onset of nonlinear effects. Instead of comparing the $N$ -factors obtained from LST, they are used to predict the pressure spectra for direct comparison with DNS, as shown in figure 7. Using the phasor representation of a harmonic wave $e^{i (\alpha x - \omega t )}$ , the $N$ -factor for the spatial stability theory can be given as

(5.1) \begin{equation} N=\ln \left (\frac {\hat {p}(\omega ,x)}{\hat {p}_0(\omega ,x_0)}\right )=\int _{x_0}^x -\alpha _i (\omega ,x) \,\textrm{d}x, \end{equation}

where $\hat {p}_0(\omega ,x_0)$ denotes the Fourier spectrum of the wall pressure fluctuations at an arbitrarily chosen location $x_0$ . For the results shown, the DNS spectrum at $x_0=0.6\,\textrm{m}$ was chosen, which is a location where the applied forcing is deemed to be settled.

Figure 7. Comparison of DNS (solid lines) and LST predictions (lines with markers) of the power spectra of wall pressure fluctuations for frequencies ranging from $110-130\,\textrm{kHz}$ at intervals of $5\,\textrm{kHz}$ . This frequency span covers the typical range of second-mode frequencies for the base flow considered in this work.

In the comparisons shown in figure 7, only the linear ( $A_0=10^{-3}\,\textrm{m}\,\textrm{s}^-{^1}$ ) and the weakly nonlinear ( $A_0=10^{-2}\,\textrm{m}\, \textrm{s}^-{^1}$ ) cases are shown since the highly nonlinear cases deviate significantly from the linear theory predictions. The figure shows the comparison for forcing types BL5, L1 and C1. For type BL5, the linear amplitude forcing matched well with LST predictions, while the weakly nonlinear amplitude shows deviations for frequencies $110\,\textrm{kHz}$ and $115\,\textrm{kHz}$ (highlighted in figure 7 biii). On the other hand, forcing type L1 showed no such deviation. As given in table 3, the forcing was applied for one cycle of $120\,\textrm{kHz}$ frequency for type L1, while it was done for a range of frequencies over five cycles for type BL5. This shows that the onset of nonlinear effects is dependent not only on the forcing amplitude but also on the frequency spectrum and the time period over which the forcing is applied. The blowing/suction forcing (case C1) followed a similar trend as type L1. This suggests that the transition induced through forcing types L1 and C1 are predominantly in the linear regime. Additionally, figure 7 shows that the spectral energy for the blowing/suction pulse is much lower than that for forcing type L1, even though the forcing time periods, frequency content and amplitudes are identical for both. This higher receptivity of the harmonic forcing over the blowing/suction type is due to the richer spatial features in the former compared with the smoother cosine-cubed function used in the latter (3.1). Achieving a similar fluctuation level through the blowing/suction pulse requires the imposition of exceedingly high amplitudes, leading to significant mean flow distortions.

5.2.2. Phase speeds and growth rates of unstable modes: wave steepening

The pressure spectra deviation observed at modes $110\,\textrm{kHz}$ in the previous section is now investigated further. Comparison is made with the behaviour at $120\,\textrm{kHz}$ , which follows the linear theory predictions, as shown in figure 7. The DNS and LST predictions of the growth rates and phase speeds of the two frequencies are evaluated to aid this analysis. Using the phasor representation of wall pressure fluctuations and neglecting the base pressure gradient (Balakumar & Kegerise Reference Balakumar and Kegerise2015), it can be seen that

(5.2) \begin{equation} \alpha (\omega ,x)=\alpha _r+j\alpha _i = \frac {1}{j\hat {p}(\omega ,x)}\frac {\partial \hat {p}(\omega ,x)}{\partial x}, \quad c_r(\omega ,x)=\textrm {Re} \left \{\frac {\omega }{\alpha (\omega ,x)}\right \}, \end{equation}

where, $\alpha$ denotes the complex wavenumber in the streamwise direction and $c_r$ denotes the streamwise phase speed. The Fourier spectra, available at a set of discrete frequencies, are interpolated using a piecewise cubic Hermite interpolating polynomial to obtain the pressure spectra at any given frequency. The spatial derivative of the spectra, $\partial \hat {p}/\partial x$ , is then calculated using the same C6 scheme as in DNS.

Figure 8. Phase speeds (left-hand vertical axis, black) and growth rates (right-hand vertical axis, red) for $f=110$ (a) and $120\,\textrm{kHz}$ (b) are shown. Two forcing types, L1 and BL5 (see table 3) are compared, with initial amplitudes $A_0=10^{-3}\,\textrm{m}\,\textrm{s}^-{^1}$ (dashed) and $A_0=10^{-2}\,\textrm{m} \, \textrm{s}^-{^1}$ (solid). The linear theory (LST) predictions are shown as circles. Mode F (S) denotes the fast (slow) acoustic modes, which propagate supersonically (subsonically) relative to the BL edge velocity $U_e$ .

The comparison of growth rates and phase speeds for forcing cases BL5 and L1 have been shown in figure 8. Non-dimensional phase speeds of the fast (Mode F) and the slow acoustic modes (Mode S) are also shown, which are $1+1/M_e$ and $1-1/M_e$ , respectively. The second-mode instabilities are seen to be propagating subsonically relative to the BL edge convective velocity $U_e$ at all times. The DNS growth rates and phase speeds for the linear amplitude agree with the LST predictions for both forcing types. The pressure spectra at 110 kHz are susceptible to low signal-to-noise ratios (SNR), as the spectrum at 110 kHz is typically six orders of magnitude lower than the peak amplitude in the upstream region (until $x\lt 0.7\,\textrm{m}$ ). This is why the phase speed and growth rate extraction from DNS results via a derivative calculation (5.2) are susceptible to the irregularities observed in figure 8. The SNR at 120 kHz is relatively higher, resulting in a smoother prediction. Farther downstream, as the spectra content at the respective frequencies increases, the behaviour of the power spectra derivatives is numerically more stable.

The phase speed for the weakly nonlinear amplitude imposed with forcing type BL5 peaks at the same downstream location ( $x \approx 0.8-0.9\,\textrm{m}$ ), where the pressure amplitude dropped (denoted as nonlinear deviation in figure 7). This indicates the pressure drop to be a consequence of the sudden decrease in growth rate as the phase speed increases (figure 8). An increase/decrease in the phase speed of a mode is an indication of wave steepening. As the phase speed of a mode increases or decreases, it leads to dispersion of the instability wave packet and creates wave steepening effects. This effect is observed to be significant for the more nonlinear forcing type BL5 compared with type L1 with the same amplitude. Thus, the initial forcing characteristics play a major role in initiating the steepening effect. This steepening phenomenon is a precursor to nonlinear breakdown and is an indication of the onset of nonlinearities in the transition process.

5.2.3. Streamwise evolution of growth rates: signs of nonlinearities

Figure 9. Spatial evolution of the modal growth rates $-\alpha _i$ of the disturbance predicted using DNS (contour) and LST (overlaying white isolines). The four forcing types (see table 3) are shown in each row with the same amplitude level $A_0=10^{-2}\,\textrm{m}\, \textrm{s}^-{^1}$ (weakly nonlinear). The development of nonlinear effects can be seen at the localized hot spots as the growth rate spectrum broadens (highlighted with red arrow), which is visible for all the forcing types except the blowing/suction pulse (C1).

The growth rate, $-\alpha _i (\omega ,x)$ , is now presented as a neutral stability curve in figure 9, with the LST predictions overlaid on it. These plots show the streamwise evolution of growth rates at different frequencies and help identify the nonlinear deviations over a broader spectral range. As in figure 8, spurious growth rates appear at low frequencies until $x\lt 0.7\,\textrm{m}$ due to the low SNR of the pressure spectra. A $\tanh$ -smoothening has been enforced along $x$ to remove the spurious artefacts while taking care that the spectral broadening later downstream is not affected. The smoothening is only done for frequencies below the neutral curve.

Local hot spots of the growth rate can be identified (lower right-hand side), after which the disturbance growth is observed to saturate. All four forcing types for the weakly nonlinear forcing ( $A_0=10^{-2}\,\textrm{m}\, \textrm{s}^-{^1}$ ) are now presented together. The blowing/suction pulse (case C1) shows no evidence of nonlinear action, as the DNS and LST neutral curves coincide with each other. Case L1 $(n_{c}=1)$ showed signs of nonlinearity at $x\approx 0.9\,\textrm{m}$ , whereas case L5 $(n_{c}=5)$ behaves similarly to case BL5 $(n_{c}=5)$ , with the spectral broadening occurring earlier at $x\approx 0.8\,\textrm{m}$ . A broad-band forcing frequency content promotes mode-to-mode interactions, while a higher forcing time increases the second-mode amplification rate, triggering nonlinear mechanisms. Due to this nonlinear action, frequencies below the LST lower neutral branch are observed to amplify farther downstream. The presence of such lower frequency content has been observed by Chynoweth et al. (Reference Chynoweth, Schneider, Hader, Fasel, Batista, Kuehl, Juliano and Wheaton2019) on a flared cone and compared with nonlinear PSE calculations using finite bandwidth initial disturbances. The DNS results in the current work employ a similar broad-band forcing, demonstrating that apart from the spectral bandwidth of the initial disturbance, the forcing time also causes an aggravated broadening effect.

The preceding sections show that the DNS results agree well with the LST predictions for lower amplitude forcing. The DNS captures the full nonlinear dynamics of the BL and is used to identify deviations from the linear theory predictions at the weakly nonlinear forcing. At such forcing levels, the linear regime persists much longer than the shorter nonlinear regime. The streamwise evolution of the modal growth rates is used to identify the linear-to-nonlinear regime transition, as indications of wave steepening and spectral broadening are captured. At higher amplitudes, the growth of the power spectra of wall pressure fluctuations is not monotonic. It is observed to attenuate periodically in between successive growth phases. This is believed to be an indication of nonlinear effects and is analysed in the following section using the disturbance energy corollary.

6.1.1. Control volume for energy budget calculations

The integral limits $\Omega _{V,x}$ and $\Omega _{V,y}$ in the streamwise and wall-normal directions, respectively, define the control volume or observation window $\Omega _V$ . Keeping the wall-normal span constant at $\triangle y=7\,\textrm{mm}$ , three different streamwise spans have been used (given in table 5). All domains start at $x=0.33\,\textrm{m}$ (downstream of the forcing region) and at $y=0$ (cone surface). The three $\Omega _V$ roughly capture the locations where disturbances of different amplitude levels start to deviate from linear behaviour. The axisymmetricity of the control volume is accounted for analytically using a constant azimuthal span. The time span for the integral has been adequately chosen for each control volume $\Omega _V$ such that the second-mode instabilities do not cross the boundaries of the $\Omega _V$ . This eliminates the energy flux term $I_{2j}$ at the side and the top boundaries by Gauss’ divergence theorem. The energy flux at the cone surface evaluates to zero due to the no-slip boundary condition for the base flow and homogeneous boundary conditions for the velocity fluctuations. This allows the disturbance power $\frac {\partial E_2}{\partial t}$ to be equated with the source terms $D_2$ directly. Note that $E_2$ is energy density; hence, the integral evaluated in (6.1) has the SI units of power and has not been specified in the following figures. For all analyses in this section, DNS results of the broad-band Legendre forcing (type BL5) have been used.

6.1.2. Behaviour of the energy budget

Figure 10 compares the time evolution of the rate of change of disturbance energy (solid lines) and the energy sources (markers). In each of the panels of figure 10, the DNS schlieren is shown in the top row, designating the second-mode position at the end of the temporal span used for the energy budget calculations shown in the following row. As can be observed in the first observation window $\Omega _{V_1}$ , the source term for the strongly nonlinear forcing starts to deviate, whereas the energy budgets for the three lower amplitudes close well (figure 10 a). For the nonlinear amplitude, the budget starts to diverge similarly in $\Omega _{V_2}$ (shown in figure 10 b). The larger control volume $(\Omega _{V_3})$ shows the disturbance energy for the strongly nonlinear forcing ( $A_0=1\,\textrm{m}\, \textrm{s}^-{^1}$ ) to show temporary attenuations periodically between successive growth phases (figure 10 c), while the energy growth persists following a period of attenuation for the nonlinear forcing ( $A_0=10^{-1}\,\textrm{m}\, \textrm{s}^-{^1}$ ).

For the lower amplitude forcings, $A_0=10^{-3}$ and $10^{-2}\,\textrm{m}\, \textrm{s}^-{^1}$ , the disturbance power increases exponentially, closely matching the source. In contrast, the growth is not monotonically increasing at higher amplitudes, as the disturbance energy is observed to saturate. This deviation from the monotonic growth overlaps with the periodic dips seen in the pressure power spectra at higher amplitudes (figure 6). This behaviour of the disturbance energy occurs periodically for the strongly nonlinear case while only occurring temporarily for the nonlinear case. Power spectra in figure 6 confirm the same behaviour, suggesting a common mechanism responsible for this effect. The axisymmetric nature of the simulations dismisses turbulence as a dissipation mechanism. This indicates the presence of a saturation mechanism that causes the disturbance energy growth to show repeating cycles of amplification and attenuation at higher forcing amplitudes.

6.2.1. Analysis of second-order disturbance energy, $E_2$

The disturbance energy comprises the thermal PE (term $i$ ), mean flow energy (term $ii$ ), KE (term $iii$ ) and acoustic PE due to compression/rarefaction (term $iv$ ) (Chu Reference Chu1965). Thermal energy is observed to dominate the energy content in all cases. Terms $iii$ and $iv$ , which constitute the total acoustic energy in the absence of mean flow, increase monotonically in time as the second modes propagate downstream. The behaviour of term $ii$ , which signifies the acoustic energy carried by the mean flow, is irregular.

Five lines that characterize the BL physics (Knisely & Zhong Reference Knisely and Zhong2019) are marked in figure 11, which are:

1. (i) the BL edge $ (\delta _{BL} )$ ;

2. (ii) the generalized inflection line– where $\frac {\partial }{\partial y} \left(\rho \frac {\partial u_0}{\partial y} \right)=0$ ;

3. (iii) the critical layer– where relative Mach number $\overline {M}(y)=\frac {u_0(y)-c_{r}}{a(y)}=0$ ;

4. (iv) the relative sonic line– where relative Mach number $\overline {M}(y)=-1$ ;

5. (v) the local sonic line– where local Mach number $M(y)=\frac {u_0(y)}{a(y)}=1$ .

The critical layer and the relative sonic line are calculated using the second mode phase speed $(c_r)$ evaluated earlier in § 5.2.2.

The acoustic PE (term $iv$ ) dominates the disturbance energy content $E_2$ near the wall as the pressure fluctuations are maximum at the wall. In the neighbourhood of the relative sonic line, the KE and mean flow energy dominate the disturbance energy content. The relative sonic line marks the region below which the instability propagates supersonically relative to the mean flow (Fedorov Reference Fedorov2011). The acoustic and thermal instabilities can only interact with each other above the relative sonic line when their respective phase speeds synchronize. The acoustic instabilities cannot interact with the base flow above the relative sonic line and consequently propagate as trapped waves beneath it. Farther above, the thermal energy dominates the acoustic energy, as the temperature and density fluctuations reach a maximum near the critical layer $(\approx 0.8\delta _{BL})$ . In contrast to all the other energies, the mean flow acoustic energy can be seen to change signs and reach a minimum at the inflection line (see figure 11). This negative value of the mean flow energy occurs due to its definition, as the density and velocity fluctuations are out of phase with each other at the generalized inflection line. Appendix B.1 explores the positive definiteness of this term in the present context.

Figure 12. Time evolution of individual terms in the disturbance energy $E_2$ (a) and source terms $D_2$ (b), for initial forcing amplitudes $A_0=10^{-2},\,10^{-1},\,1\,\textrm{m}\, \textrm{s}^-{^1}$ . The colour codes are given in tables 6 and 7. In panel (b), the dotted lines indicate the dissipation terms $(-)$ , while the solid lines denote production terms $(+)$ . The balance between these terms dictates the growth/attenuation of the instabilities.

6.2.2. Analysis of second-order disturbance sources, $D_2$

The interplay between the thermal and acoustic fluctuations is now analysed using the source terms (4.20). The dissipation terms in $D_2$ are term III due to heat diffusion and term V due to viscous action. The second modes are known to be inviscid-type instabilities (Mack Reference Mack1969). Accordingly, term V is observed to not play a major role until farther downstream when the fluctuation levels become significant. On the other hand, thermal diffusion is active even at lower amplitudes. The strongly nonlinear forcing (figure 12 a iii, b iii) shows that the valleys in the total disturbance source $(D_2)$ correspond with the peaks in the viscous (V) and thermal diffusion (III) terms. This suggests that at sufficiently high fluctuation levels, the diffusive effects $(-)$ overpower the energy production $(+)$ periodically in between intervals of growth. These diffusion effects are seen to be significant at the wall and the critical layer (figure 11).

Figure 12 also demonstrates that the thermal diffusion (term III) increases in magnitude or becomes more negative in response to the growth of term IV. At lower (linear) amplitudes, term IV is always higher than term III, as the magnitude of both increases exponentially. Whereas, at higher (nonlinear) amplitudes, term III $(-)$ periodically surpasses term IV $(+)$ , balancing the total disturbance energy production.

Term IV represents the energy production occurring due to the entropy fluctuations advecting through a base entropy gradient. For a hypersonic BL with zero pressure gradient, the base entropy gradient is proportional to the base temperature gradient, i.e., $\partial s_0/\partial y \approx C_p(T_0)^{-1}\partial T_0/\partial y$ . Term IV is dominated primarily by the term $-p_0 s^{\prime} (R C_p )^{-1} v^{\prime} \partial s_0/\partial y\,(+)$ , with a relatively lower contribution from $-p^{\prime} v^{\prime} (C_p)^{-1} \partial s_0/\partial y\,(+)$ . The importance of term IV suggests that the interaction of the fluctuations with the base temperature gradient of the BL is one of the primary mechanisms of second-mode growth. On the other hand, thermal diffusion (term III) is dominated by $s^{\prime}(Pr)^{-1} \partial ^2 T^{\prime}/\partial y^2\,(-)$ near the BL edge, with a lower contribution from $R\mu _0 p^{\prime} (p_0Pr)^{-1} \partial ^2 T^{\prime}/\partial y^2\,(-)$ at the wall. This suggests that the heat generated by the advecting perturbation near the critical layer is diffused away from it. The direction of this diffusion is dictated by the fluctuating temperature field, as is evident from the presence of the term $\partial ^2 T^{\prime}/\partial y^2$ . This energy production near the BL edge can interact with the compression/rarefaction waves below, creating a feedback mechanism that promotes the growth of the acoustic-type second-mode waves.

This interplay is evident from the behaviour of terms I and II (figure 12), which are seen to balance each other irrespective of the amplitude levels. Term I facilitates the acoustic–thermal energy coupling and is dominated by the term $-p^{\prime}u_0 (C_p)^{-1} \partial s^{\prime}/\partial x\,(-)$ . This feeds the terms $-\rho^{\prime} u_0 v^{\prime} \partial u_0/\partial y\,(+)$ and $-(\rho^{\prime} u_0)^2 (\rho _0)^{-1} \partial u_0/\partial x\,(+)$ in term II, causing the disturbance KE (term $iii$ in $E_2$ ) to increase. This is verified from figure 11 as both terms $iii$ of $E_2$ and II of $D_2$ increase at the neighbourhood of the relative sonic line. The disturbance energy is further augmented by term VII, which is observed to only activate when the fluctuation levels are high. This is the only term exhibiting disturbance energy production below the relative sonic line and is dominated by the term $\rho _0 u_0 v^{\prime} \partial u^{\prime}/\partial y\,(+)$ , suggesting the presence of vorticity effects at high fluctuation levels. The viscous diffusion at the wall (term V) is observed to grow in response to the evolution of term VII.

To summarise, the primary source for second-mode energy growth is identified to be the entropy/velocity fluctuations interacting with the base entropy (temperature) gradient. At higher amplitudes, the disturbance energy growth encounters periodic attenuation due to thermoviscous diffusion effects that counteract the production mechanisms. The balance between the terms V $(-)$ and VII $(+)$ near the wall and of terms III $(-)$ and IV $(+)$ near the critical layer is collectively responsible for this effect that leads to saturation of the disturbance energy growth. The periodic dips in the power spectra of the wall pressure fluctuations, discussed in § 5.1.2, can be attributed to this attenuation effect. Whereas at lower amplitudes, the contributions from the production terms IV and VII are always higher than the dissipation effects, facilitating the monotonic growth of the second-mode waves in the linear regime.