2.1. Simulation strategy and initial condition

The simulation strategy for the simulations presented consists of three steps (see figure 3). First, in a precursor calculation (step 1), using a finite-volume code developed in our CFD laboratory by Gross & Fasel (Reference Gross and Fasel2008, Reference Gross and Fasel2010), the steady undisturbed basic flow around the entire axisymmetric cone geometry is computed using a second-order-accurate discretization (see figure 3). Due to the low-order accuracy this code is too diffusive to directly perform stability and transition simulations with a reasonable number of grid points. Therefore, in the second step, the steady basic flow obtained by the precursor calculation is used as an initial condition for calculating a highly accurate basic flow using the high-order-accurate finite-difference code, which was also developed in our laboratory by Laible, Mayer & Fasel (Reference Laible, Mayer and Fasel2008, Reference Laible, Mayer and Fasel2009). The computational domain (see figure 3) for the high-order-accurate basic flow computation (in step 2) does not cover the entire flow around the cone because it is beneficial to focus all computational resources into the region of interest, i.e. by starting the computational domain downstream of the nose tip. Thus, in the high-order-accurate finite-difference code, the nose tip is not included. However, the oblique shock emanating from the nose tip is included by placing the free-stream boundary above the shock (see § 2.3). The basic flow (initialized using the precursor results) has to be converged again, since the underlying numerical schemes of the finite-volume code and the high-order-accurate finite-difference code have different truncation errors. Finally, in the third step, the newly converged basic flow serves as an initial undisturbed state for the unsteady simulations with a specified disturbance input (see § 3) using the same high-order-accurate code. In the following only the high-order-accurate finite-difference code used for steps 2 and 3 will be discussed. For details regarding the finite-volume code used in step 1, see Gross & Fasel (Reference Gross and Fasel2008, Reference Gross and Fasel2010).

Figure 3.

Schematic of the strategy used for the simulations: (a) precursor calculation; (b) high-order-accuracy DNS. Note that the computational domain for the high-order simulation does not cover the entire flow. The contours shown here are for density

For stability and transition investigations it is imperative to have an accurate undisturbed basic (undisturbed) flow field. For validation purposes, in figures 4 and 5, the streamwise velocity and temperature profiles obtained from the precursor calculation using the finite-volume code and the high-order-accurate finite-difference code are compared to Mangler-transformed (Mangler Reference Mangler1948) profiles calculated from a flat-plate similarity profile. The profiles, plotted here for four different streamwise locations, show excellent agreement. Note that the finite-volume code used for the precursor calculation predicts the boundary layer profiles accurately and provides a high quality initial condition for the simulations performed using the high-order finite-difference code.

Figure 4.

Comparison of streamwise velocity profiles computed by the precursor finite-volume code, the high-order code and the Mangler-transformed flat-plate similarity profile at several streamwise locations: (a) $x^{\ast }=0.100~\text{m}$ ; (b) $x^{\ast }=0.200~\text{m}$ ; (c) $x^{\ast }=0.300~\text{m}$ and (d) $x^{\ast }=0.400~\text{m}$ .

Figure 5.

Comparison of temperature profiles computed by the precursor finite-volume code, the high-order code and the Mangler-transformed flat-plate similarity profile at several streamwise locations: (a) $x^{\ast }=0.100~\text{m}$ ; (b) $x^{\ast }=0.200~\text{m}$ ; (c) $x^{\ast }=0.300~\text{m}$ and (d) $x^{\ast }=0.400~\text{m}$ .

2.2. Numerical method

The details of the high-order-accurate finite-difference code used here, along with its continuing development, have been described in Laible et al. (Reference Laible, Mayer and Fasel2008, Reference Laible, Mayer and Fasel2009), Sivasubramanian et al. (Reference Sivasubramanian, Mayer, Laible and Fasel2009), Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2010, Reference Sivasubramanian and Fasel2011) and Laible (Reference Laible2011). Therefore, only a short overview will be given here. The numerical method is based on the three-dimensional Navier–Stokes equations, the continuity equation and the energy equation for compressible flows in conical coordinates. For the full set of governing equations in conical coordinates the reader is referred to e.g. Sivasubramanian & Fasel (Reference Sivasubramanian and Fasel2012b ). The so-called ‘spatial model’ is employed, so that the disturbance waves can grow or decay in the downstream direction. This is in contrast to the so-called ‘temporal model’, where the disturbances grow in time. For a detailed discussion of the spatial versus temporal model for stability and transition investigation see Fasel (Reference Fasel, Arnal and Michel1990).

The governing equations are integrated in time using the standard explicit fourth-order Runge–Kutta method (Ferziger Reference Ferziger1998). The spatial discretization is mainly based on high-order-accurate standard central finite differences. In particular, sixth-order central finite differences were used in the streamwise direction and fourth-order central finite differences were used in the wall-normal direction to discretize the derivatives of the viscous terms and the source term. 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 differences (Zhong Reference Zhong1998) with ninth-order accuracy are applied to evaluate the derivatives for these fluxes. In the azimuthal direction, a pseudo-spectral discretization using fast Fourier transforms is employed. Since high-order-accurate boundary closures may develop oscillations and, hence, are usually unstable, special attention was given to the wall-next boundary stencils. The present code employs a method suggested by Zhong & Tatineni (Reference Zhong and Tatineni2003). For further details see Laible (Reference Laible2011).

Note that if the high-order-accurate finite differences as described above were used for the calculation of the shock, strong oscillations would be introduced into the computational domain. Therefore, before the simulation, the shock position is detected using the smoothness estimater for the pressure as suggested by Balsara & Shu (Reference Balsara and Shu2000). Also the order of accuracy of the wall-normal upwind-difference stencils is decreased from ninth-order to first-order in the near-shock region in order to avoid numerical oscillations. Since the shock is multiple boundary layer thicknesses away from the wall and the gradients in the inviscid region are rather small, the higher truncation error of the first-order stencils has no significant effect on the overall accuracy of the scheme and the results. Hence, the instability modes are also not affected by the inclusion of the shock.

2.3. Boundary conditions

The inflow (see figure 2) for hypersonic boundary layer simulations is separated into two regions: a subsonic region ( $M<1$ ) close to the wall and a supersonic/hypersonic region ( $M>1$ ) away from the wall. In the supersonic/hypersonic region, Dirichlet conditions for $u,v,w,T,p$ and ${\it\rho}$ are specified (obtained in our case from a precursor calculation, see § 2.1). For the subsonic region in the boundary layer, a special treatment is occasionally necessary for some applications to avoid undesired reflections, e.g. non-reflecting boundary condition are used as suggested by Poinsot & Lele (Reference Poinsot and Lele1992). However, for the simulations presented here, there was no evidence of reflections at the inflow boundary, even when the Dirichlet conditions were also applied for the subsonic region. Therefore, for the simulations presented in this paper, Dirichlet conditions were prescribed at the inflow over the entire domain height. On the cone surface, no-penetration ( $v=0$ ) and no-slip ( $u=0;w=0$ ) conditions were enforced. The wall is set to be isothermal with $T_{w}^{\ast }=300~\text{K}$ for the steady base-flow calculations. The wall temperature $T_{w}^{\ast }=300~\text{K}$ is slightly lower than the laminar recovery temperature. For the unsteady simulations the temperature fluctuations were set to zero at the wall. The value of the pressure at the wall boundary is obtained from the $y$ -momentum equation. Finally, the density at the wall is computed using the equation of state. At the outflow boundary, the second derivatives of the primitive variables $u$ , $v$ , $w$ , $T$ , and $p$ are set to zero: $\partial ^{2}u/\partial x^{2}=0,\partial ^{2}v/\partial x^{2}=0,\partial ^{2}w/\partial x^{2}=0,\partial ^{2}T/\partial x^{2}=0,\partial ^{2}p/\partial x^{2}=0$ . The density was then determined from the temperature and the pressure by using the equation of state. For simulations of the nonlinear breakdown a buffer domain technique is applied, where finite-amplitude disturbances are ramped down to zero at the outflow (see Meitz & Fasel Reference Meitz and Fasel2000). Since for all simulations presented here the free stream is located above the oblique shock emanating from the nose of the cone, Dirichlet conditions ( $u,v,w,T,p,{\it\rho}$ prescribed) can be prescribed at this boundary.

In the azimuthal direction a pseudo-spectral approach with Fourier modes (Canuto et al. Reference Canuto, Hussaini, Quateroni and Zang1988) was used to calculate the derivatives. In order to save computational resources, symmetric transformations (cosine) are considered for all variables except the azimuthal velocity component, for which anti-symmetric (sine) transformations are used. The adoption of the symmetric and anti-symmetric transformations automatically enforces symmetry conditions on the azimuthal boundaries. Therefore, the azimuthal domain contains only half of the azimuthal wavelength of the primary wave for the fundamental breakdown simulations presented in this paper.

4.1. Linear regime

The linear transition regime is studied using both LST and DNS. While in a DNS the complete set of nonlinear governing equations is solved without any simplification, in LST these equations are linearized assuming very small disturbance amplitudes. Furthermore, in LST it is assumed that disturbances ${\it\phi}^{\prime }$ follow the wave ansatz

The complex eigenfunction $\hat{{\it\phi}}$ is only dependent on the wall-normal direction $y$ . For the spatial model considered here, ${\it\alpha}$ is complex, while ${\it\beta}$ and ${\it\omega}$ are real. The wave angle of a wave with respect to the streamwise direction is given by

where

are the streamwise and spanwise (or azimuthal) wavenumbers, respectively. The streamwise wavenumber ${\it\alpha}_{r}$ is the real part of the complex streamwise wavenumber ${\it\alpha}$ in (4.1) ( ${\it\alpha}={\it\alpha}_{r}+\text{i}{\it\alpha}_{i}$ ), and ${\it\omega}$ is the angular frequency. For negative values of ${\it\alpha}_{i}$ , disturbances are amplified in the streamwise direction whereas for positive values they are damped. The wave ansatz from (4.1) with the appropriate boundary conditions reduces the linear system of partial differential equations to an eighth-order system of ordinary differential equations, which in fact reduces to an eigenvalue problem, which can be solved efficiently.

For the cone geometry, the decreasing spanwise curvature and the decreasing wave angle of the disturbances as they travel downstream play an important role in both the linear and nonlinear development of the transition process. The decreasing wave angle is a consequence of the body divergence due to the cone geometry, which causes the azimuthal wavelength of a specific wave to increase in the downstream direction. The azimuthal wavelength at the streamwise position $x$ is defined as

The cone radius $r$ is a function of the streamwise direction $x$ and $k_{c}$ represents the azimuthal mode number. Here the assumption is made that the ratio of the boundary layer thickness and the cone radius is small and that the azimuthal wavelength is equal to the arc length on the cone surface. In this paper, a specific wave is defined by its azimuthal mode number, $k_{c}$ , and a certain reduced frequency, $F$ . For example, a wave with azimuthal mode number $k_{c}=1$ has an azimuthal wavelength equal to the circumference of the cone cross-section, a wave with azimuthal mode number $k_{c}=2$ has an azimuthal wavelength of half the circumference, etc. Note that for a cone the azimuthal mode number, $k_{c}$ , has to be an integer and consequently only a discrete set of physically possible azimuthal wavelengths can exist. This is in contrast to a flat plate for which the spanwise wavelength can take on any value in a continuous fashion. The non-dimensional frequency, $F$ , and the local Reynolds number, $R_{x}$ , are defined as

respectively. Here, subscript $e$ indicates the boundary layer edge values.

In order to solve the LST eigenvalue problem for a compressible boundary layer, the linear stability solver by Mack (Reference Mack, Alder, Fernbach and Rotenberg1965, Reference Mack1969, Reference Mack1987) was employed. Mack’s linear stability solver is based on the flat-plate equations and does not account for transverse curvature and body divergence. However, it accounts for the change in azimuthal wavelength with the streamwise location. For the linear stability analysis, self-similar compressible boundary layer profiles were used as base flow by employing Mangler transformation (Mangler Reference Mangler1948). Contours of constant amplification rate (stability diagram) for axisymmetric disturbances ( $k_{c}=0$ ) are presented in figure 7. The first- and second-mode unstable frequency bands are clearly distinguishable as they are separated by a stable region. The beginning and the end of the computational domain is also indicated in the stability diagram in figure 7.

Figure 7.

Contours of constant amplification rate ${\it\alpha}_{i}$ for axisymmetric disturbances ( $k_{c}=0$ ). Generated using Mack’s LST solver (Mack Reference Mack, Alder, Fernbach and Rotenberg1965, Reference Mack1987). The low-frequency band in the diagram corresponds to the first-mode unstable region and the high-frequency band represents the second-mode unstable region. The vertical dashed lines indicate the beginning and the end of the computational domain used in the DNS.

Prior to conducting the fundamental breakdown simulations, DNS with very low amplitudes were performed such that the linear regime was maintained throughout the entire computational domain. These simulations allow a direct comparison of the DNS results with LST. In figure 8 the streamwise wavenumber ${\it\alpha}$ from the DNS is compared with that from LST for axisymmetric disturbance waves with three different frequencies of $F=1.1312\times 10^{-4}$ , $F=1.1731\times 10^{-4}$ and $F=1.2318\times 10^{-4}$ . In the DNS the complete Navier–Stokes equations are employed and therefore the non-parallel effects are also included (Gaster Reference Gaster1974; Saric & Nayfeh Reference Saric and Nayfeh1975; Fasel & Konzelmann Reference Fasel and Konzelmann1990). Hence, the spatial growth rate, ${\it\alpha}_{i}$ , depends on the criterion used. Here, the spatial growth rate and the streamwise wavenumbers were calculated based on the wall pressure disturbance as follows:

Here $A$ represents the amplitude and ${\it\theta}$ the phase of the wall pressure disturbance. The streamwise distributions of amplitude $A(x)$ and phase ${\it\theta}(x)$ were obtained by performing Fourier transformations of the time signal of wall pressure disturbance. There is very good agreement between the growth rates and streamwise wavenumbers from DNS and LST (generally the streamwise wavenumber is less sensitive to the criterion used and therefore non-parallel effects are less obvious). Close to the disturbance forcing location, however, the streamwise wavenumber ${\it\alpha}_{r}$ and the spatial growth rate ${\it\alpha}_{i}$ as calculated from the DNS data, are modulated by the superposition of damped waves. This modulation is more pronounced for the spatial growth rate ${\it\alpha}_{i}$ than for the streamwise wavenumber ${\it\alpha}_{r}$ . Close to the location where the waves exhibit maximum growth rate there is good agreement between the spatial amplification rate ${\it\alpha}_{i}$ from DNS and LST. The reason for the small differences in spatial growth rate ${\it\alpha}_{i}$ at other locations is most likely due to non-parallel effects.

Figure 8.

Downstream development of the complex streamwise wavenumber ${\it\alpha}$ obtained by LST (symbols) and DNS (using low-forcing amplitude) for axisymmetric waves ( $k_{c}=0$ ) with three different frequencies: $F=1.1312\times 10^{-4}$ (——, ○), $F=1.1731\times 10^{-4}$ (- - - -, $\Box$ ) and $F=1.2318\times 10^{-4}$ (– $\cdot$ – $\cdot$ –, ♢). (a) ${\it\alpha}_{r}$ (streamwise wavenumber) and (b) ${\it\alpha}_{i}$ (spatial amplification rate). LST results were computed using Mack’s solver (Mack Reference Mack, Alder, Fernbach and Rotenberg1965, Reference Mack1987).

In figure 9 the wall-normal amplitude and phase distributions for the streamwise velocity component, temperature and pressure from the DNS are compared at $R_{x}=2501$ to results obtained by LST using Mack’s stability solver for axisymmetric disturbance waves with frequency $F=1.1312\times 10^{-4}$ . The amplitude distributions from both linear theory and DNS are normalized by their respective maximum values. The excellent agreement between both results substantiates that the linear eigenbehaviour of the unstable mode is correctly reproduced in the DNS. The agreement with theory confirms that the disturbances introduced via the blowing and suction slot indeed initiate the desired physically relevant instability waves.

Figure 9.

Comparison of wall-normal amplitude (a–c) and phase distribution (d–f) of the streamwise velocity, temperature and pressure disturbance to theoretical predictions from LST for frequency $F=1.1312\times 10^{-4}$ and azimuthal mode number $k_{c}=0$ (axisymmetric disturbance wave) at $Re_{x}=2501$ . LST results were computed using Mack’s solver (Mack Reference Mack, Alder, Fernbach and Rotenberg1965, Reference Mack1987).

4.2. Parameter study

Fundamental resonance or (K- or Klebanoff-type) breakdown (Herbert Reference Herbert1988; Kachanov Reference Kachanov1994) is due to a secondary instability mechanism involving a two-dimensional (here an axisymmetric) primary wave ( $1,0$ ) and a symmetric pair of oblique secondary waves (( $1,1$ ) and ( $1,-1$ )) of the same frequency. Note that in the present paper, a wave is denoted by ( $n,k$ ), where $n$ represents the frequency and $k$ the azimuthal wavenumber, both normalized by the corresponding values of the secondary pair of oblique waves. For the incompressible case of fundamental resonance, for a flat plate, the strength of the fundamental resonance is strongly influenced by the amplitude of the primary wave ( $1,0$ ) and the wave angle of the secondary oblique wave pair. For a cone this issue is even more complicated than for a flat plate, because the wave angle of a disturbance wave changes in the downstream direction. Therefore, we first performed a parameter study to find the most strongly resonating oblique wave pair. Then, we performed a set of highly resolved controlled fundamental breakdown simulations using the most strongly resonating oblique wave pair as secondary waves.

Fundamental resonance is initiated by forcing the primary wave ( $1,0$ ) at a large amplitude and the secondary disturbance waves ( $1,\pm 1$ ) at a low amplitude. The dominant (linearly most amplified) axisymmetric wave was chosen as the primary wave, which was found from comparing the $N$ -factors for axisymmetric waves for various frequencies. The streamwise development of the $N$ -factors is plotted in figure 10 for axisymmetric disturbance waves ( $k_{c}=0$ ) as obtained from a low-amplitude wavepacket simulation (see for example Sivasubramanian & Fasel Reference Sivasubramanian and Fasel2014). The $N$ -factor of an instability wave is defined as

where the subscript $n$ and $k_{c}$ denote the frequency and azimuthal wavenumber of the wave, respectively; $A_{n,k_{c}}(x)$ is the amplitude of the disturbance wave at a certain downstream position and $A_{n,k_{c},0}$ is the amplitude of the disturbance wave at its lower neutral point. The amplitudes of the disturbance waves were obtained by performing Fourier transformations of the time signal of the disturbance waves. In figure 10 the $N$ -factor reached by the most amplified axisymmetric wave ( $k_{c}=0$ ) is approximately 8.0 with a corresponding non-dimensional frequency of $F=1.1312\times 10^{-4}$ (  $f^{\ast }\approx 210~\text{kHz}$ ). It was therefore chosen as the primary wave ( $1,0$ ) for a parameter study to determine the most strongly resonating oblique secondary wave pair ( $1,\pm 1$ ).

Figure 10.

$N$ -factor curves for axisymmetric waves ( $k_{c}=0$ ) obtained from a low-amplitude wavepacket simulation.

Towards this end a series of low-resolution simulations was performed for various azimuthal wavenumbers ( $k_{c}$ ). The results from this parameter study are presented in figures 11 and 12. In figure 11, the downstream amplitude development of selected modes is shown for the low-resolution simulation performed for azimuthal mode numbers $k_{c}=60$ , $80$ , $100$ and $120$ . These curves were obtained from the wall-normal maximum of the streamwise velocity disturbances. As can be observed in figure 11, when the axisymmetric primary wave ( $1,0$ ) exceeds a certain amplitude, the secondary oblique wave pair ( $1,\pm 1$ ) and the steady longitudinal vortex mode ( $0,1$ ) experience strong secondary growth. Figure 12 shows the growth rate ( ${\it\sigma}$ ) of the secondary oblique wave pair ( $1,\pm 1$ ) as a function of azimuthal mode number ( $k_{c}$ ). The growth rates ( ${\it\sigma}$ ) were extracted at the streamwise position $x^{\ast }=0.46~\text{m}~(R_{x}=2501)$ . A broad band of azimuthal modes experiences resonant secondary growth, but according to figure 12, waves with $k_{c}=100$ exhibit the strongest secondary growth rate.

Figure 11.

Maximum of streamwise velocity disturbance versus downstream distance for the cases with azimuthal wavenumber (a) $k_{c}=60$ , (b) $k_{c}=80$ , (c) $k_{c}=100$ and (d) $k_{c}=120$ . Shown are selected modes, which play an important role in the early nonlinear stage of fundamental resonance.

Figure 12.

Fundamental secondary growth rate ( ${\it\sigma}$ ) as a function of azimuthal mode number ( $k_{c}$ ).

4.3. High-resolution fundamental breakdown simulations

In the parameter study discussed above for fundamental resonance, oblique waves with azimuthal mode number $k_{c}=100$ were identified as the most strongly resonating oblique waves. Therefore, a highly resolved fundamental breakdown simulation (CFUND1) was performed with azimuthal mode $k_{c}=100$ for the secondary wave pair.  In these simulations, the axisymmetric primary wave ( $1,0$ ) with a frequency of $f^{\ast }\approx 210~\text{kHz}$ was forced with a large amplitude ( $A_{1,0}=4.0\,\%$ of the free-stream velocity) and the oblique secondary waves ( $1,\pm 1$ ) were forced at a low amplitude ( $A_{1,\pm 1}=0.1\,\%$ of the free-stream velocity). Here each of the two modes ( $1,1$ ) and ( $1,-1$ ) are forced with the same amplitude of 0.1 % of the free-stream velocity.

The downstream development of the wall-normal amplitude maximum of the streamwise velocity disturbance is presented in figure 13. Selected modes that play an important role in the nonlinear stages of the fundamental breakdown are shown. As before, when the axisymmetric primary wave ( $1,0$ ) reaches a certain amplitude (at $x^{\ast }\sim 0.4~\text{m}$ ), resonance sets in and the secondary mode ( $1,1$ ) and the steady longitudinal vortex mode ( $0,1$ ) start to grow much faster than the primary wave and eventually reach the same amplitude levels as the primary mode ( $1,0$ ) at $x^{\ast }\sim 0.45~\text{m}$ . As the steady longitudinal vortex mode ( $0,1$ ) starts to grow, it also produces mean flow distortion (mode $0,0$ ). Higher harmonic modes ( $2,0$ ), ( $2,1$ ) and ( $1,2$ ) are also generated. When the amplitudes of modes ( $1,1$ ) and ( $0,1$ ) approach the amplitude of the primary mode ( $1,0$ ) all higher modes experience rapid streamwise growth and the transition process becomes strongly nonlinear, which is an indication of the onset of the final breakdown to turbulence. As more and more higher steady modes (( $0,2$ ), ( $0,3$ ), ( $0,4$ ), ( $0,5$ ) etc.) are generated, the mean flow deformation (mode $0,0$ ) increases. Note that in the nonlinear region, the steady longitudinal vortex mode ( $0,1$ ) has the highest amplitude ( $0.45~\text{m}<x^{\ast }<0.57~\text{m}$ ). Close to the end of the computational domain the steady longitudinal vortex mode ( $0,2$ ) reaches higher amplitudes than ( $0,1$ ). However, the steady mode ( $0,0$ ) has the highest amplitude, indicating a strong mean flow distortion due to transition to turbulence. A more detailed presentation of the downstream development of the wall-normal amplitude maximum of the streamwise velocity disturbance is provided in figure 14. Modes with the fundamental frequency ( $1,k$ ), two-dimensional modes ( $h,0$ ) and three-dimensional modes ( $h,1$ ) are presented in figures 14(a), 14(b) and 14(c), respectively. They provide a detailed view of how the spectrum broadens due to nonlinear interactions, which produce higher modes as the flow transitions from the laminar to the turbulent state.

Figure 13.

Streamwise development of the maximum $u$ -velocity disturbance amplitude obtained from the fundamental breakdown simulation. Note that when the axisymmetric primary wave ( $1,0$ ) reaches a certain amplitude, the secondary mode ( $1,1$ ) and the steady longitudinal vortex mode ( $0,1$ ) grow faster than the primary wave and eventually reach the same amplitude as the primary mode. Higher harmonics, modes ( $2,0$ ) and ( $2,1$ ), are also generated, and when modes ( $1,1$ ) and ( $0,1$ ) approach the amplitude of the primary mode ( $1,0$ ) all higher modes experience rapid streamwise growth. In the nonlinear region the steady longitudinal vortex mode ( $0,1$ ) has the highest amplitude. Close to the end of the computational domain, however, the steady mode ( $0,2$ ) reaches higher amplitudes than ( $0,1$ ).

Figure 14.

Detailed presentation of the streamwise development of the maximum $u$ -velocity disturbance amplitude obtained from the fundamental breakdown simulation. (a) Modes with fundamental frequency ( $1,k$ ); (b) two-dimensional modes ( $h,0$ ) and (c) three-dimensional modes ( $h,1$ ).

Figure 15.

(a) Time- and azimuthal-averaged skin friction coefficient and (b) streamwise development of the maximum $u$ -velocity disturbance amplitude for the fundamental breakdown simulation. The initial rise in skin friction is caused by the large-amplitude primary wave ( $1,0$ ). This is followed by a dip caused by the nonlinear saturation of the primary wave ( $1,0$ ). Then a steeper rise in skin friction occurs when all higher modes experience nonlinear growth.

The downstream development of the temporally and azimuthally averaged skin friction coefficient, $C_{f}$ , is shown in figure 15(a). The skin friction coefficient is calculated as

The Reynolds number, $\mathit{Re}$ , is based on the reference length, $L^{\ast }$ , and the flow quantities in the free stream. Note that symbols with an overbar, $\overline{{\it\phi}}$ , represent Reynolds-averaged, i.e. time- and spanwise-averaged, flow quantities. The skin friction curves obtained from the simulations were compared with a correlation for a turbulent boundary layer from White (Reference White2006),

Here, the subscript $e$ refers to boundary layer edge conditions and the subscript $w$ refers to wall conditions. The factor $S$ can be obtained from

with $A$ and $B$ defined as

where $a$ and $b$ are given by

respectively. Note that $\overline{T}_{aw}$ denotes the adiabatic wall temperature for a turbulent boundary layer at the same flow conditions. Since this value is not known, it has to be estimated using the turbulent recovery factor (White Reference White2006; Roy & Blottner Reference Roy and Blottner2006),

The above relationship used for estimating the skin friction is only valid for a turbulent flat-plate boundary layer. Therefore, a correction for the cone geometry was employed as suggested by van Driest (Reference van Driest1952) and White (Reference White2006) based on the von Kármán momentum integral: $C_{f,cone}=G\times C_{f,plate}$ , where $1.1<G<1.15$ . Note that the theoretical turbulent skin friction curve in figure 15(a) is calculated with $G=1.1$ . The skin friction plotted in figure 15(a) initially follows the laminar curve up to $x^{\ast }\sim 0.38~\text{m}$ , then it increases from the laminar curve towards the turbulent curve. However, farther downstream it unexpectedly drops sharply to almost the laminar value before rising again steeply and eventually overshooting the turbulent skin friction curve. The skin friction curve suggests that the boundary layer is close to a turbulent boundary layer at the end of the computational domain.

A close examination of the results shown in figure 15(a,b) reveals that the initial deviation of the skin friction from its laminar value is due to the large amplitude reached by the primary wave ( $1,0$ ). The first peak in the skin friction corresponds to the streamwise location where the primary wave ( $1,0$ ) seems to ‘saturate’ ( $x^{\ast }\sim 0.45~\text{m}$ ). This peak in skin friction also roughly coincides with the location where the secondary disturbances (e.g. modes ( $1,\pm 1$ ) and ( $0,1$ )) reach their largest amplitude levels, which in turn cause a mean flow deformation ( $0,0$ ). Due to the mean flow deformation the primary wave starts to decay following its ‘nonlinear saturation’ and the skin friction dips almost to the laminar value. Together with the primary wave, higher modes also starts to decay (seen clearly in figure 14 b,c at $x^{\ast }\sim 0.46~\text{m}$ ). Following the dip, a much steeper rise in the skin friction occurs as all higher modes experience very strong nonlinear amplification ( $x^{\ast }\sim 0.52~\text{m}$ ). As a result of this steep rise, the skin friction overshoots the theoretical turbulent skin friction estimate.

Note that this particular development of the skin friction is probably due to the ‘controlled’ transition scenario where only a few selected waves are forced initially. In a ‘natural’ transition scenario this development may be less pronounced or even completely absent. Also note that the mode ( $0,0$ ) is the only mode that directly affects the development of the temporally and azimuthally averaged skin friction coefficient. All other modes influence the skin friction development indirectly through their nonlinear effect on the development of mode ( $0,0$ ). More importantly, the maximum amplitude of mode ( $0,0$ ) does not seem to decay noticeably here. Therefore, it is the change in the shape of mode ( $0,0$ ) that influences the skin friction development and causes the dip following the first peak. Hence, the change in the shape of mode ( $0,0$ ) may very well be connected to the decay of the primary mode ( $1,0$ ) and other higher modes after the first peak in skin friction. These results raise the interesting question of whether the first peak in the skin friction could be influenced by the amplitude of the oblique secondary wave pair ( $1,\pm 1$ ).

In order to answer this question, we performed two more ‘controlled’ fundamental breakdown simulations, CFUND 2 and CFUND 3, where the forcing amplitudes for the oblique secondary waves were changed to 0.01 % and 1.0 % of the freestream velocity, respectively. The forcing amplitude for the primary wave was kept the same as in CFUND 1 (4.0 % of the free-stream velocity). A summary of the forcing parameters used in these fundamental breakdown simulations is provided in table 3. For selected modes, the streamwise velocity disturbances development from CFUND 2 and CFUND 3 are plotted in figure 16. Overall, the downstream amplitude development looks very similar to CFUND 1. However, the time- and azimuthally averaged skin friction coefficients from CFUND 2 and CFUND 3 as plotted in figure 18 look slightly different. As the forcing amplitude of the oblique secondary waves is decreased (CFUND 2) the first peak in the skin friction distribution shifts slightly downstream. This is due to the fact that the secondary growth of the oblique wave starts at a lower amplitude level and therefore reaches nonlinear amplitudes slightly downstream. However, when the forcing amplitude of the oblique secondary waves is increased (CFUND 3), the first peak shifts slightly upstream. Here, it is due to the fact that the secondary growth of the oblique wave starts at a higher amplitude level and consequently reaches nonlinear amplitudes slightly upstream. As a result, the onset of the rapid streamwise growth of higher modes also shifts slightly downstream in CFUND 2 and shifts slightly upstream in CFUND 3 with increasing forcing amplitude of secondary waves (see figure 16 a,b). In CFUND 2 and CFUND 3, as the primary wave is forced with the same amplitude as in CFUND 1, the location of transition onset remains the same for all three cases. This confirms that the initial rise in skin friction (onset of transition) is due to the large amplitude of the primary wave ( $1,0$ ). A comparison of the streamwise development of the key modes between CFUND 1, CFUND 2 and CFUND 3 is provided in figure 17. Figure 17(a) clearly presents that the secondary wave ( $1,1$ ) reaches nonlinear amplitudes downstream and upstream (compared to CFUND 1) in CFUND 2 and CFUND 3, respectively. Similarly, the steady mode ( $0,1$ ) reaches nonlinear amplitudes downstream and upstream compared to CFUND 1 and, as a consequence, induces mean flow deformation either downstream or upstream (see figure 17 b).

Table 3.

Forcing parameters for the fundamental breakdown simulations. Note that the notation $[n,k]$ is used in order to identify a particular wave according to its frequency $n$ and its azimuthal wavenumber $k;n$ denotes multiples of the fundamental forcing frequency $f^{\ast }=210~\text{kHz}$ and $k$ multiples of the azimuthal wavenumber.

Figure 16.

Streamwise development of the maximum $u$ -velocity disturbance amplitude from CFUND 2 (a) and CFUND 3 (b). Shown are selected modes that play an important role in the early nonlinear stage of fundamental resonance.

Figure 17.

Comparison of the streamwise development of the maximum $u$ -velocity disturbance amplitude of key modes from CFUND 1, CFUND 2 and CFUND 3.

Figure 18.

Time- and azimuthal-averaged skin friction coefficient from CFUND 1, CFUND 2 and CFUND 3.

Figure 19.

Time- and azimuthal-averaged Stanton number obtained from CFUND 1.

The downstream development of the time- and azimuthal-averaged Stanton number, $\mathit{St}$ , is presented in figure 19. The Stanton number distribution follows the laminar curve up to $x^{\ast }\sim 0.38~\text{m}$ and then increases towards the turbulent curve. Eventually it overshoots the turbulent curve before dropping almost down to the laminar value. However, downstream it steeply increases again towards the turbulent curve and stays above the turbulent value curve. The stronger overshoots observed here, even in the transitional stages, at the streamwise location where the first peak appears for skin friction coefficient is unique and is a direct result of stronger temperature gradients produced by the transition process. Such overshoots during the transitional stages as observed here could lead to localized regions of very large heat transfer. The downstream development of the time-averaged skin friction and Stanton number along the streak is plotted in figure 20. The skin friction and Stanton number along the streak have much larger values compared to their azimuthal-averaged values and exhibit massive overshoots over the turbulent values both in the early and late transition region. Note that the particular development of the Stanton number observed here is strongly affected by the isothermal wall boundary condition used in the simulations presented in this paper.

Figure 20.

Time- and azimuthal-averaged (a) skin friction and (b) Stanton number obtained from CFUND 1.

Isocontours of the time-averaged skin friction and wall-normal temperature gradient ( $\text{d}T/\text{d}y$ ) at the wall are shown in figures 21(a) and 21(b). Remarkable are the streamwise aligned ‘hot’ streaks, which are a consequence of the large amplitudes reached by the steady longitudinal vortex modes. The steady mode ( $0,1$ ) is the first to reach the highest amplitude and it is, therefore, responsible for the first upstream appearing streaks. Downstream, however, these streaks get weaker. As the steady mode ( $0,2$ ) reaches the highest amplitude close to the end of the computational domain, there, the azimuthal spacing of the streaks is related to that mode. These streaks look qualitatively similar to the streamwise pattern observed in the Purdue experiments (under quiet-flow conditions) for a flared cone using temperature-sensitive paint (see figure 7b in Berridge et al. Reference Berridge, Chou, Ward, Steen, Gilbert, Juliano, Schneider and Gronvall2010 and figure 18 in Ward et al. Reference Ward, Wheaton, Chou, Berridge, Letterman, Luersen and Schneider2012). The associated streamwise temperature distribution obtained using temperature-sensitive-paint data first increases from the laminar reference value, then decreases downstream of the first peak and finally increases again as the flow transitions to turbulence (see figure 11a in Berridge et al. Reference Berridge, Chou, Ward, Steen, Gilbert, Juliano, Schneider and Gronvall2010).

This behaviour is similar to the trend observed in heat transfer distributions obtained from the second-mode fundamental breakdown simulations (see figure 19). Under noisy-flow conditions, however, the streamwise streaks were not observed in the Purdue experiments (the Purdue tunnel can be run either as a quiet-flow tunnel or as a conventional noisy-flow tunnel). The streamwise temperature distribution measured under noisy-flow conditions (see figure 11b in Berridge et al. Reference Berridge, Chou, Ward, Steen, Gilbert, Juliano, Schneider and Gronvall2010) was also different from the distribution measured under quiet-flow conditions: it increased gradually from the laminar value to the turbulent value (as in Horvath et al. Reference Horvath, Berry, Hollis, Chang and Singer2002). It is not understood why the streaks appear under quiet-flow and not under noisy-flow. Thus, the streamwise streaks and the streamwise temperature distribution measured in the Purdue experiments under quiet-flow conditions are similar to the streaks and the heat transfer curves obtained from the second-mode fundamental (K-type) breakdown simulations. Therefore, we conjecture that second-mode fundamental breakdown mechanism may have played a role in the natural transition process in the Purdue experiments under quiet-flow conditions. Overall, the simulations presented in this paper demonstrate that second-mode fundamental breakdown is a viable route to turbulence in hypersonic cone boundary layers at Mach 6 and confirms the findings of our previous research in which we investigated the ‘natural’ transition process in hypersonic cone boundary layers using wavepackets and identified second-mode fundamental breakdown to be a relevant mechanism for transition to turbulence in Mach 6 cone boundary layers.

Figure 21.

Time-averaged (a) skin friction and (b) wall-normal temperature gradient ( $\text{d}T/\text{d}y$ ) at the wall obtained from the fundamental breakdown simulation CFUND 1. The streamwise aligned ‘hot’ streaks look qualitatively similar to the streamwise streaks observed in the Purdue experiments using temperature-sensitive paint for a flared cone.

4.4. Flow structures

A close observation of the dominant flow structures and their streamwise development can provide further insight into the underlying physical mechanisms of the transition process. In figure 22(a) the flow structures from the fundamental breakdown simulation are plotted using the $Q$ criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). The flow structures reveal the typical evolution of a K-type breakdown: during the initial stage of the transition process the dominant wave is axisymmetric. Once nonlinear interactions cause the oblique secondary waves to amplify rapidly, the dominant axisymmetric waves become modulated (peak–valley splitting) in the circumferential direction (see figures 22 b and 23 a). Eventually, the interaction of the dominant axisymmetric wave and finite-amplitude oblique waves leads to the formation of ${\it\Lambda}$ -vortices. The ${\it\Lambda}$ -vortices appear in an aligned pattern because the primary axisymmetric wave and the secondary oblique waves have the same frequency. The evolution of the flow structures during the final stage of transition can be better observed in the close-up view of figures 23 and 24. The aligned arrangement of the ${\it\Lambda}$ -vortices can be seen clearly. The tips of the ${\it\Lambda}$ -vortices lift away from the surface while the legs remain close to the wall. Hairpin-shaped vortices start to appear on the tips of the ${\it\Lambda}$ -vortices. Eventually these structures break down to smaller scales as the flow starts to become turbulent.

Figure 22.

Visualization of flow structures using isosurface of $Q$ criterion ( $Q=20\,000$ ) obtained from the fundamental breakdown simulation CFUND 1. (a) Full view of the cone and (b) last downstream part of the cone. The isosurfaces are coloured by the streamwise velocity magnitude.

Figure 23.

Visualization of flow structures using isosurface of $Q$ criterion ( $Q=20\,000$ ) obtained from the fundamental breakdown simulation CFUND 1: (a) close-up view of a small section of the cone; (b) close-up view of the last downstream part. The isosurfaces are coloured by the streamwise velocity magnitude.

Figure 24.

Visualization of flow structures using isosurface of $Q$ criterion ( $Q=20\,000$ ) obtained from the fundamental breakdown simulation CFUND 1. Close-up view of the last downstream part of the cone. The isosurfaces are coloured by the streamwise velocity magnitude.

Instantaneous streamwise velocity contours are presented in the $x{-}{\it\varphi}$ plane for three wall-normal positions inside the boundary layer (see figure 25). This figure illustrates various flow features such as the transition onset, the breakup region and the early turbulent region close to the wall (figure 25 a) and farther away (figure 25 c). Dark regions denote low-velocity flow and brighter regions correspond to high-velocity flow. Two-dimensional structures seem to appear upstream in figure 25 for all three wall-normal positions. These structures correspond to the second-mode axisymmetric wave (primary wave) forced at a high amplitude. Farther downstream these two-dimensional structures appear modulated in the azimuthal direction due to nonlinear interactions between the axisymmetric primary wave and the secondary oblique waves. The azimuthal modulation becomes stronger downstream, indicating the presence of strong oblique waves (generated by nonlinear interactions). Furthermore, the aligned arrangements characteristic of K-type transition are clearly visible (see for example figure 25 b). Eventually the flow breaks up into small-scale structures as the flow transitions to turbulence.

Figure 25.

Contours of the instantaneous streamwise velocity in the $x{-}{\it\varphi}$ plane for three wall-normal positions from CFUND 1: (a) $y^{\ast }=0.35~\text{mm}$ ; (b) $y^{\ast }=1.20~\text{mm}$ ; and (c) $y^{\ast }=2.0~\text{mm}$ .

To gain more information about the dynamics of the flow field during the transition process, contours of the instantaneous spanwise vorticity component are plotted in the $x{-}y$ plane for various azimuthal positions (see figure 26). The five selected azimuthal positions are (a) ${\it\varphi}=0~(0^{\circ })$ , (b) ${\it\varphi}=0.0078~(0.45^{\circ })$ , (c) ${\it\varphi}=0.0157~(0.9^{\circ })$ , (d) ${\it\varphi}=0.0235~(1.35^{\circ })$ and (e) ${\it\varphi}=0.0314~(1.8^{\circ })$ . Here dark contour regions represent negative spanwise vorticity and white regions represent positive spanwise vorticity. Figure 26 provides a detailed view of the breakdown region and the downstream development of the small-scale structures. The plane at ${\it\varphi}=0$ is located in the spanwise valley location and the plane at ${\it\varphi}=0.0314$ is in the spanwise peak location. In the plane at ${\it\varphi}=0.0314$ (spanwise peak), prior to breakdown (up to $x^{\ast }\sim 0.51~\text{m}$ ) a strong concentration of spanwise vorticity can be observed away from the wall. Downstream of this location ( $x^{\ast }\sim 0.51~\text{m}$ ), for all azimuthal positions, the flow is dominated by small-scale structures.

Figure 26.

Contours of the instantaneous spanwise vorticity in the $x{-}y$ plane for five azimuthal positions from CFUND 1. (a) ${\it\varphi}=0$ ; (b) ${\it\varphi}=0.0078$ ; (c) ${\it\varphi}=0.0157$ ; (d) ${\it\varphi}=0.0235$ and (e) ${\it\varphi}=0.0314$ (from bottom to top). All plots have the same contour levels.

Contours of the instantaneous wall-normal density gradient are shown in figure 27 for several azimuthal positions. The selected azimuthal positions are the same as in figure 26. As before, wall-normal density gradient contours are shown in the $x{-}y$ plane for each azimuthal position. This type of flow visualization could be used for comparison with schlieren photographs which would reveal flow structures in high-speed experiments. At all azimuthal positions ‘rope-like’ structures appear close to the boundary layer edge up to the location where the flow starts to break down. Such structures have been observed in hypersonic boundary layer flows by several experimentalists (see for example Lachowicz et al. Reference Lachowicz, Chokani and Wilkinson1996). Flow visualization from the numerical simulations by Pruett & Chang (Reference Pruett and Chang1995) and Pruett & Chang (Reference Pruett and Chang1998) also clearly reveals instability waves of rope-like appearance. The structures observed in our simulation are qualitatively similar in appearance to the schlieren image obtained by Lachowicz et al. (Reference Lachowicz, Chokani and Wilkinson1996) and to the flow visualization by Pruett & Chang (Reference Pruett and Chang1998). The wavelength of these structures is approximately twice the boundary layer thickness, which is consistent with the nature of the second-mode instability waves. Furthermore, as the boundary layer starts to break down, acoustic waves ‘radiating’ away from the boundary layer and towards the free stream are also observed for all azimuthal positions (see figure 27).

Figure 27.

Contours of the instantaneous wall normal density gradient in the $x{-}y$ plane for five azimuthal positions from CFUND 1. (a) ${\it\varphi}=0$ , (b) ${\it\varphi}=0.0078$ , (c) ${\it\varphi}=0.0157$ , (d) ${\it\varphi}=0.0235$ and (e) ${\it\varphi}=0.0314$ (from bottom to top). All plots have the same contour levels.

4.5. Quantitative analysis

For quantitative global analyses, the time- and azimuthal-averaged streamwise velocity and temperature profiles shown in figures 28 and 29 are inspected. The time interval over which the relevant data were extracted and averaged is equal to four forcing periods of the primary wave. Figure 28(a,b) shows isocontours of streamwise velocity and temperature in the $x{-}y$ plane, whereas figure 29(a,b) shows wall-normal profiles extracted at different downstream positions (rescaled with the local boundary layer thickness). The objective here is to show how these profiles develop during transition and how they converge to a turbulent profile. It is clear from these plots that with increasing downstream distance, the boundary layer thickness is increasing and the mean streamwise velocity and temperature profiles are becoming ‘fuller’, which is a consequence of the transition to turbulence.

Figure 28.

Contours of (a) the mean (in time and azimuthal direction) streamwise velocity and (b) the mean temperature from CFUND 1.

Figure 29.

Selected wall-normal mean (a) streamwise velocity and (b) temperature profiles from CFUND 1.

The downstream development of the shape factor,

and the Reynolds number based on the momentum thickness,

are shown in figure 30. Note that the momentum thickness and the displacement thickness considered in the above formulae are calculated from ${\it\theta}=\int _{y_{w}}^{y_{e}}({\it\rho}u)/({\it\rho}_{e}u_{e})(1-u/u_{e})\text{d}y$ , and ${\it\delta}^{\ast }=\int _{y_{w}}^{y_{e}}1-({\it\rho}u)/({\it\rho}_{e}u_{e})\text{d}y$ , respectively. The shape factor development in figure 30(a) shows similar features as the skin friction coefficient and Stanton number. It first decreases from the laminar value and then rises towards the laminar value before dropping again to a lower value as the boundary layer transitions to turbulence. The shape factor curves from all three cases can be seen to feature similar characteristics. The distribution of Reynolds number based on the momentum thickness (see figure 30 b), however, does not exhibit the trend observed for the shape factor (and also for skin friction and Stanton number). It initially follows the laminar curve up to $x^{\ast }\sim 0.42~\text{m}$ and then increases monotonically to its highest value of $\mathit{Re}_{{\it\theta}}\sim 2500$ close to the end of the computational domain.

Figure 30.

Downstream development of (a) the shape factor and (b) the Reynolds number based on momentum thickness.

More insight into how far transition has progressed can be gained from the time-averaged streamwise velocity profiles plotted in wall coordinates. Here, the streamwise velocity is rescaled with the friction velocity,

and the wall-normal coordinate is renormalized by

Furthermore, for compressible flow the velocity is typically transformed using the van Driest transformation,

The transformed velocity profiles corresponding to several streamwise positions are shown in figure 31. Note that the dotted lines represent the theoretical curves for the linear sublayer and the logarithmic overlap region,

where ${\it\kappa}=0.41$ and $C=5.2$ . For a location upstream of the transition onset location ( $x^{\ast }\sim 0.35~\text{m}$ ), the velocity profile looks like a laminar profile. For locations close to the peak in the skin friction ( $x^{\ast }=0.54~\text{m}$ ), the turbulent log-layer shape is approached. This indicates that the flow is starting to become fully turbulent.

Figure 31.

van-Driest-transformed streamwise velocity profiles normalized with the friction velocity for different streamwise positions obtained from the fundamental breakdown simulation CFUND 1. For streamwise position close to the peak in the skin friction ( $x^{\ast }=0.54~\text{m}$ ), the van-Driest-transformed velocity approaches the theoretical curves. The dotted lines indicate the linear sublayer and the law of the wall.

Figure 32.

Energy spectra versus $k_{{\it\varphi}}$ for streamwise velocity at (a) $x^{\ast }=0.52~\text{m}$ ; (b) $x^{\ast }=0.54~\text{m}$ and (c) $x^{\ast }=0.56~\text{m}$ from CFUND 1. The red and blue lines indicate the theoretical estimate for the decay of the energy spectrum: $E\sim k_{{\it\varphi}}^{-5/3}$ (red line) and $E\sim k_{{\it\varphi}}^{-7}$ (blue line).

Figure 33.

Energy spectra versus ${\it\omega}$ for streamwise velocity at (a) $x^{\ast }=0.52~\text{m}$ ; (b) $x^{\ast }=0.54~\text{m}$ and (c) $x^{\ast }=0.56~\text{m}$ from CFUND 1. The red and blue lines indicate the theoretical estimate for the decay of the energy spectrum: $E\sim {\it\omega}^{-5/3}$ (red line) and $E\sim {\it\omega}^{-7}$ (blue line).

The state of the transition process can also be estimated using the energy spectra. For a spatial simulation typically both the time-averaged azimuthal wavenumber spectra and the azimuthal-averaged frequency spectra for different downstream positions are employed (see figures 32 and 33),

where ${\it\phi}$ is the place holder for the velocity components. The indices $j$ and $k$ represent a grid point in the wall-normal and azimuthal direction, whereas $i$ is the index for the time steps over which the average is performed; $k_{{\it\varphi}}$ and ${\it\omega}$ denote the azimuthal wavenumber and the frequency, respectively. Furthermore, $N_{t}$ and $N_{{\it\varphi}}$ are the number of time steps and points over which the average is performed. The wall-normal position where the spectra are extracted is $y^{\ast }=1.0~\text{mm}$ . The rate of the energy decay for the range of wavenumbers in which neither the large eddies nor the small eddies have an influence on the spectrum (inertial subrange) can be predicted on theoretical grounds as $E\sim k_{{\it\varphi}}^{-5/3}$ (Weizäcker Reference Weizäcker1948). This theoretical behaviour is indicated by the red dashed lines in figure 32. Also indicated with blue dash-dotted lines is the decay according to $E\sim k_{{\it\varphi}}^{-7}$ at the high-wavenumber end of the spectrum as predicted by Heisenberg (Reference Heisenberg1948). In figure 32, the time-averaged energy spectra are plotted versus the azimuthal wavenumber for streamwise velocity. For all three streamwise positions considered, the spectra agree well with the theoretically predicted decay rates for the inertial subrange and the high-wavenumber end of the spectra. As the flow transitions from laminar to transitional to turbulent, energy is transported from the base flow to large scales and then to smaller and smaller scales. This trend can be observed in the energy spectra presented in figure 32. In figure 33, the azimuthal-averaged energy spectra are plotted versus the frequency. For all three streamwise positions, the azimuthally averaged spectrum experience a decay as predicted by theory. Therefore, based on these spectra, one can conclude that the boundary layer is close to fully turbulent at the end of the computational domain.