Mechanisms and control of secondary-instability-induced-transition in a supersonic boundary layer are studied numerically via direct numerical simulation. The aim is to investigate and compare the transition mechanisms of fundamental, subharmonic, asymmetric subharmonic, and detuned resonances, and to control these secondary instabilities using a local wall cooling strip. The results indicate that the nonlinear interaction between the high-amplitude primary mode and low-amplitude secondary modes is the main contributor to transition. The mutual- and self-interactions of the primary and secondary modes generate other harmonic modes with laminar breakdown soon appearing. The asymmetric subharmonic resonance induces the earliest transition, while the fundamental subharmonic has the latest. Wall cooling effects are also studied. The results show that a lower wall temperature significantly suppresses the secondary instabilities, and steady modes become dominant and lead to obvious streamwise vortexes. Numerical data demonstrate that all secondary-instability-induced transitions result in fully developed turbulent boundary layers, as supported by the skin friction and scaled velocity profiles. The transition control cases indicate that the local wall cooling strip can significantly delay the transition by suppressing the growth of the primary mode. An upstream control strip is found to have a more obvious suppression effect. The fundamental and asymmetric subharmonic resonances are sensitive to the location of the local wall cooling strip and show a stronger transition delaying effect.

Laminar–turbulent transition has attracted increased attention in designing supersonic aircraft as it significantly influences the friction drag and heat transfer.1–5 Thus, understanding the physical mechanisms of transition is essential in fluid dynamics. In actual flight environments, the natural transition is a common path to turbulence. Therein, different instability modes first experience linear amplification.6 In incompressible boundary layers, the most amplified disturbance is the two-dimensional (2D) Tollmien–Schlichting wave.7 In supersonic flow, the three-dimensional (3D) first modes generally have the largest growth rate.8 The 2D second mode becomes important when the Mach number exceeds approximately four.8 When these disturbance modes grow linearly to a sufficient amplitude, the nonlinear effects become more important, and nonlinear growth occurs. This nonlinear growth finally causes a breakdown to turbulence.9 

Various nonlinear transition mechanisms have been discovered. One example is the oblique-type breakdown,10 in which two symmetric oblique modes generate a steady vortex mode with a fast growth rate. The wave-vortex triad between the oblique modes and the streamwise vortex generates many more harmonic modes and causes a breakdown to turbulence. In addition to oblique-type breakdown, secondary instability11 is another potential nonlinear transition path. This mechanism generally contains a high-amplitude primary mode and two low-amplitude secondary modes. The primary and secondary modes first experience their linear development process. Once the primary mode is amplified to a sufficient amplitude, secondary instabilities are excited and increase much quicker. Secondary instability mechanisms can be divided into fundamental, subharmonic, and detuned resonances.11 For the first two, secondary modes have the same and a half frequency as the primary mode. In these secondary instabilities, the primary mode is usually 2D. Thus, the two secondary modes have equal but opposite wave angles. However, there exists another type of potential secondary instability, namely, the asymmetric subharmonic resonance.12,13 In this scenario, a fundamental oblique wave composes a wave triad with two subharmonic and asymmetric oblique modes. For supersonic boundary layers, asymmetric subharmonic resonance may be more important in realistic scenarios because oblique modes are much more unstable than 2D ones.

Several studies have considered secondary instabilities. Ng and Erlebacher14 found that subharmonic resonance dominates in subsonic and low-Mach-number supersonic flows for low-disturbance environments. In contrast, the fundamental resonance is important when the primary mode has a high amplitude. Masad and Nayfeh15,16 and El-Hady17 made a series of Floquet analyses on secondary instability in low-speed and supersonic boundary layers. They found that suction and a higher Mach number stabilize the primary, fundamental, and subharmonic waves. Chang and Malik18 made nonlinear stability analyses for a supersonic boundary layer and found that the dominant secondary instability shifts from subharmonic to fundamental for stronger primary modes. The asymmetric subharmonic resonance was first reported through the experimental result of Kosinov et al.12 Later, Fezer and Kloker19 used DNS to study interactions between the fundamental oblique mode and the subharmonic oblique mode. However, no obvious secondary growth occurred. The growth rate of the subharmonic oblique mode was less than that of the fundamental oblique mode, indicating the transition is induced primarily by the latter, and the former only accelerates the transition. Recently, secondary instability in hypersonic flows was studied using high-resolution DNS and the high-efficiency nonlinear parabolized stability equation (NPSE). Franko and Lele20,21 and Leinemann et al.22 used DNS to study different nonlinear transition mechanisms. Their results show that second-mode fundamental resonance is the dominant secondary instability in hypersonic flows, while the first-mode oblique breakdown may cause earlier transitions. Xu and Liu23 studied the temperature effect on the secondary instability of Mack modes. They pointed out that the growth rates of secondary instability modes are proportional to the stagnation temperature and inversely proportional to the wall temperature.

Based on the understanding of the nonlinear instability mechanisms, transition control has attracted considerable interest in the research community. Many methods can be used for controlling transition, such as wall blowing,24,25 secondary recirculation jet,26–33 and wall cooling and heating.34 In addition, streaks have been applied successfully to control transition in both supersonic35 and hypersonic flows.36 Among these, the local wall cooling or heating is often mentioned because wall heating/cooling has significant influence on the development of disturbance modes in both the supersonic and hypersonic flows.

Based on previous studies, secondary instabilities have been suggested as a possible breakdown mechanism in both supersonic and hypersonic boundary layers. However, most related DNS results are for hypersonic flows, where full breakdown and complete turbulence are achieved. In supersonic flows, secondary instabilities are generally studied via stability theory, either via Floquet analysis or NPSE. Therein, only the linear and weak nonlinear evolution processes are investigated, while strong nonlinear and turbulent regions are not included. Therefore, we focus on providing clear evidence of the four different types of secondary-instability-induced transition in supersonic flows, including fundamental, subharmonic, asymmetric subharmonic, and detuned resonances, and comparing their inner mechanisms. At the same time, the wall cooling effects for the four secondary-instability-induced transitions are also investigated. The linear growth behaviors are first studied via LST. The DNS computations are then performed to show the entire process of transition to turbulence as well as the turbulence mean results. In addition, most transition control studies focus on the oblique-mode breakdown transition process, the control of secondary-instability-induced transitions in supersonic flows is less discussed. Therefore, we make transition control studies for these scenarios using a local wall cooling strip, and the different strip locations are compared.

The detailed computational setup, including the numerical methods and descriptions of the physical problem, is presented in Sec. II. The numerical results are shown in Secs. III and IV. Section V summarizes the main conclusions.

The simulations are based on the unsteady compressible Navier–Stokes equations:
Q t + F j c x j = F j ν x j ,
(1)
where t is the time and xj is the Cartesian coordinate in the j direction. Q , F j c, and F j ν represent the vectors of the conserved variables, convective fluxes, and viscous fluxes, respectively. These are defined as
Q = [ ρ ρ u i ρ e ] , F j c = [ ρ u j ρ u i u j + p δ i j ( ρ e + p ) u j ] , F j ν = [ 0 σ i j σ i j u i + k T x j ]
(2)
with
e = p ( γ 1 ) ρ + u i 2 2 , σ i j = μ ( u j x i + u i x j ) 2 3 μ δ i j · u .
(3)
Here, δij is the Kronecker delta, k is the thermal conductivity, u = [ u , v , w ] T is the velocity vector, p is the pressure, and ρ is the density. The viscosity μ is calculated using Sutherland's law37 with T s = 110.4 K as
μ = μ ref ( T T ref ) 3 / 2 T ref + T s T + T s .
(4)
The governing equations are closed by the state equation of a perfect gas. The specific heat ratio γ, gas constant R, and Prandtl number Pr are set to 1.4, 287, and 0.72, respectively. The three velocity components are nondimensionalized by the streamwise freestream velocity u , and other flow variables are nondimensionalized by their freestream values. The reference length is L ref = 0.001 m.

The governing equation is solved on a body-fitted grid using the high-order finite-difference DNS code OpenCFD as developed by Li et al.38,39 A third-order Runger–Kutta method40 is used for time advancing. For spatial discretization, the convective flux vectors are computed with a seventh-order WENO-SYMBO method41 utilizing Steger-Warming splitting.42 An eighth-order central-differencing scheme discretizes the viscous flux vectors.

The considered physical problem is a supersonic flat-plate boundary layer with an isothermal wall. The flow conditions follow the NPSE study of Chang and Malik18 and are listed in Table I, where M , p , T , and R e are the freestream Mach number, pressure, temperature, and unit Reynolds number, respectively. Figure 1 shows the detailed configuration of the computational domain and the boundary condition. The inflow position is placed at x in = 5 mm with a boundary layer thickness of δ in = 8.636 × 10 2 mm. The computational domain is L x × L y × L z = 85 × 5 × 2.42 mm3. A 15-mm long sponge region is placed at x > 70 mm. In the sponge region, the computational grid size in the streamwise direction is quickly amplified to avoid disturbance reflections from the outlet domain. The computational grid is sized at n x × n y × n z = 3163 × 109 × 269 through a grid independence study shown in the  Appendix.

TABLE I.

Flow conditions.

M p (Pa) T (K) R e (m−1)
1.6  77 366  300  2.704 × 10 7 
M p (Pa) T (K) R e (m−1)
1.6  77 366  300  2.704 × 10 7 
FIG. 1.

Computational domain used in DNS simulations.

FIG. 1.

Computational domain used in DNS simulations.

Close modal
A laminar boundary layer profile is imposed at the inlet boundary. This profile is obtained by solving the following similarity equations for a compressible laminar boundary layer:7,
{ d d η ( ρ u ρ u d 2 f d η 2 ) + f d 2 f d η 2 = 0 d d η ( ρ u ρ u d g d η ) + Pr f d g d η + Pr ( γ 1 ) M 2 ρ u ρ u ( d 2 f d η 2 ) 2 = 0
(5)
with
d f d η = u u , g = T T , η = u 2 ξ 0 y ρ d y , ξ = 0 x ρ u μ d x
(6)
and
f ( 0 ) = d f d η | η = 0 = 0 , g ( 0 ) = T w T , lim η + d f d η = lim η + g = 1 .
(7)
Periodic conditions are used for the lateral boundary. Non-reflection43 and pressure outlet44 conditions are applied to the far-field boundary and the outflow domain, respectively.
A blow and suction strip is used at the wall to generate boundary layer disturbances. This blow and suction strip extends from x start = 10 mm to x end = 12.7 mm and takes the following form:
ρ v = ρ u o ( x ) A ( h , k ) sin ( k β 0 z h ω 0 t ) ,
(8)
where o(x) defines the variation of the blow and suction in the x direction:
o ( x ) = 4 27 sin ( 2 π x x start x end x start ) [ 1 cos ( 2 π x x start x end x start ) ] .
(9)
In Eq. (8), the subscript (h, k) represents a disturbance mode with frequency h ω 0 and spanwise wavenumber k β 0, while A ( h , k ) denotes the forcing amplitude. The fundamental spanwise wavenumber is calculated as β 0 = 2 π / λ z = 2596.36 m−1, and λ z = L z = 2.42 × 10 3 m is the fundamental spanwise wavelength. The fundamental angular frequency is ω 0 = 2 π f 0 = 600.80 kHz, and the fundamental physical frequency is f 0 = 95.62 kHz.

For the fundamental resonance, one primary mode (1, 0) and two secondary modes (1, ±1) are excited. The primary mode is still (1, 0) for subharmonic resonance, but the two secondary modes are (1/2, ±1). For asymmetric subharmonic resonance, an oblique wave (1, 1) is the primary mode, and (1/2, 2) and (1/2, −1) are secondary modes. For detuned resonance, (1, 0) is the primary mode, and (1/4, 1) and (3/4, −1) are secondary modes. The forcing amplitude of the primary mode is 5 × 10 3 except for asymmetric subharmonic resonance cases, in which the value is set as 1 × 10 3. For the secondary modes, their initial amplitudes are set to 1 × 10 5. Table II lists detailed information of all cases, including the wall temperature Tw, the cooling temperature Tc, and the location of the cooling strip. The naming scheme for the cases is as follows. The first string represents the scenario as F (fundamental), S (subharmonic), A (asymmetric-subharmonic), and D (detuned). The second string C represents the wall cooling, followed by a subscript indicating the location of the control strip. Here, Tw and Tc are represented by their ratio with respect to the turbulent adiabatic wall temperature, which is computed as T a w = T [ 1 + Pr 1 / 2 M 2 ( γ 1 ) / 2 ] 429 K.

TABLE II.

Details of all different cases.

Case T w / T a w T c / T w Location (mm)
F, S, A, D  1.0  ⋯  ⋯ 
FC, SC, AC, DC  0.84  ⋯  ⋯ 
FC1, FC2, FC3  1.0  0.5  20–26, 26–32, 32–38 
SC1, SC2, SC3  1.0  0.5  16–22, 22–28, 28–34 
AC1, AC2, AC3  1.0  0.5  17.5–23.5, 23.5–29.5, 29.5–35.5 
DC1, DC2, DC3  1.0  0.5  16–22, 22–28, 28–34 
Case T w / T a w T c / T w Location (mm)
F, S, A, D  1.0  ⋯  ⋯ 
FC, SC, AC, DC  0.84  ⋯  ⋯ 
FC1, FC2, FC3  1.0  0.5  20–26, 26–32, 32–38 
SC1, SC2, SC3  1.0  0.5  16–22, 22–28, 28–34 
AC1, AC2, AC3  1.0  0.5  17.5–23.5, 23.5–29.5, 29.5–35.5 
DC1, DC2, DC3  1.0  0.5  16–22, 22–28, 28–34 
First, LST is used to investigate the linear behavior of the primary and secondary instability modes. In LST, the disturbance is expressed as45 
q ( x , y , z , t ) = q ̂ ( y ) exp [ i ( α x + β z ω t ) ] , q = [ ρ , u , v , w , T ] T ,
(10)
where α = α r + i α i, αr is the streamwise wavenumber, and αi is the spatial growth rate. A negative value of αi represents an amplified mode.

Figure 2 presents the dimensionless growth rate α i L LST ( L LST = x / R e ) and the amplification factor N for modes with different spanwise wavenumbers. For 2D modes in Fig. 2(a), the maximum growth rate is only about 0.001, and the maximum amplification factor is approximately 1. For 3D modes with β = 2596.36 m−1 (k = 1), the neutral curve expands and the growth rate becomes large, thereby leading to a much stronger amplification factor. When the spanwise wavenumber increases to β = 5192.72 m−1 (k = 2), the growth rate is stronger than the 2D modes but the unstable region shrinks, and the growth distance shortens. This significantly decreases the amplification factor. Generally, modes with k = 1 experience stronger linear growth than other 3D modes.

FIG. 2.

Linear stability results for (a) 2D modes, (b) 3D modes with β = 2596.36 m−1, and (c) 3D modes with β = 5192.72 m−1. The contours represent the growth rate α i, and the black solid lines are isolines of the amplification factor N.

FIG. 2.

Linear stability results for (a) 2D modes, (b) 3D modes with β = 2596.36 m−1, and (c) 3D modes with β = 5192.72 m−1. The contours represent the growth rate α i, and the black solid lines are isolines of the amplification factor N.

Close modal

The flow features and detailed nonlinear breakdown process for different secondary-instability-induced transitions are discussed. Figure 3 shows the streamwise velocity at y / δ in = 0.5 for cases at the adiabatic wall temperature. Figure 3(a) presents the fundamental resonance case. Upstream of the blow and suction strip, the streamwise velocity is steady laminar flow. Downstream of the blow and suction strip and in front of the breakdown to turbulence, the flow field has typical 2D characteristics with periodic patterns in the streamwise direction because of the high-amplitude forced 2D mode (1, 0). When the flow develops to x 40 mm, the 2D strip bends, and weak 3D patterns appear. Further downstream, the three-dimensionality of the flow becomes much higher, and the flow finally breaks down to turbulence. Counters of the streamwise velocity for the subharmonic resonance case are shown in Fig. 3(b). The flow field first presents a laminar state, and the 3D effect appears at x 37 mm and becomes much stronger at x 40 mm. Unlike the aligned Λ-type patterns in the fundamental resonance cases, the streamwise velocity in the subharmonic resonance case shows an X-type pattern. Thus, the disturbance development process in subharmonic resonances has obvious differences from the fundamental resonances. Figure 3(c) shows the streamwise velocity for the asymmetric subharmonic resonance case. The flow shows a typical oblique wave pattern in the upstream region caused by the 3D primary mode (1, 1). This wave pattern is periodic in both the streamwise and spanwise directions. Small and messy 3D structures appear at x 32 mm before breakdown occurs. However, we can find that asymmetric subharmonic cases present asymmetric 3D patterns in front of the breakdown, which differs significantly from the symmetric patterns in fundamental and subharmonic resonance cases. Figure 3(d) shows the streamwise velocity for the detuned resonance case. The 3D patterns appear at x 38 mm and become much stronger behind x 40 mm. Different from fundamental resonance and subharmonic resonance cases, the 3D pattern is not symmetric in the spanwise direction.

FIG. 3.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case F, (b) case S, (c) case A, and (d) case D.

FIG. 3.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case F, (b) case S, (c) case A, and (d) case D.

Close modal

Modal decomposition is performed to further analyze the role of different disturbance modes in the nonlinear breakdown process. Snapshots of the flow field are sampled within two time periods of the primary mode (1, 0). Fast Fourier transformed is then applied in both time and the spanwise direction to extract modes with different frequencies and spanwise wavenumbers. Note that the amplitude of each mode is represented by the maximum streamwise velocity disturbance.

The streamwise developments of selected modes in the fundamental resonance cases are shown in Fig. 4(a). Modes (1, 0) and (1, 1) first experience linear growth. Then, mode (1, 1) begins to drop at x 18 mm. Further downstream, mode (1, 1) begins to grow again at x 20 mm at a rate much greater than the LST results. This indicates that secondary growth occurs. At the same time, the development of mode (1, 0) gradually departs from and is lower than the LST result. The nonlinear interactions between modes (1, 0) and (1, 1) lead to the generation of the steady mode (0, 1), which has a similar growth rate as mode (1, 1). The nonlinear interactions between the two secondary modes (1, 1) and (1, −1) generate the steady mode (0, 2). Mode (1, 2) comes from two paths: (1, 0) + (0, 2) and (1, 1) + (0, 1). As modes (0, 2) and (1, 2) have nearly the same growth rate, the first path may be dominant. After x 37 mm, the amplitudes of modes (1, 1), (0, 1), (0, 2), (1, 2), and other high-spanwise-wavenumber modes gradually overtake that of mode (1, 0), leading to the 3D flow part in Fig. 3(a). Further downstream, the transition occurs. Figure 4 presents streamwise developments of selected modes in the subharmonic resonance case. The result of case S5 is plotted in Fig. 4(a). The amplitude of the secondary mode (1/2, 1) first shows a short decrease before x 15 mm. Further downstream, both the DNS and LST results show that mode (1/2, 1) begins to grow, but the DNS result shows a much larger growth rate that significantly departs from the LST results. The interactions between modes (1, 0) and (1/2, 1) generate mode (3/2, 1), and the self-interaction of mode (1/2, 1) generates mode (1, 2). Mode (0, 2) comes from two paths: the interactions between modes (1, 1) and (1, −1) and the fundamental resonance between modes (1, 0) and (1, 2). The two paths lead to fast growth in mode (0, 2). At x 35 mm, the amplitude of mode (1/2, 1) [as well as mode (1/2, −1)] exceeds that of mode (1, 0), leading to a weak 3D effect in Fig. 3(a). Because the frequency of the subharmonic mode is half the fundamental mode, the streamwise wavenumbers are related by about 0.5, which causes staggered wave patterns in front of the breakdown. Further downstream, the amplitudes of different modes gradually saturate. Figure 4(c) presents the modal decomposition results for the asymmetric subharmonic resonance case. The primary mode (1, 1) first experiences linear growth. The secondary mode (1/2, −1) first decreases until secondary growth occurs at x 23 mm. A more complex situation is observed for the secondary mode (1/2, 2). The DNS results show that it first experiences a sharp decrease. At x 15 mm, mode (1/2, 2) begins to grow quickly. The secondary growth position of mode (1/2, 2) is much further ahead than that of mode (1/2, −1). In addition to the three initially forced modes, we see that more modes are generated by the nonlinear interaction processes and experience fast growth. For example, (3/2, 0) = (1, 1) + (1/2, −1), (3/2, 3) = (1, 1) + (1/2, 2), and (0, 3) = (1/2, 2) − (1/2, −1). These modes start to saturate at R e x 9.4 × 10 5, and transition finally occurs. Figure 4(d) presents streamwise developments of selected modes in the detuned resonance case. The two secondary modes first begin to grow at x  15 mm. Then, when the flow develops to x 23 mm, the growth rate of mode (3/4, −1) increases obviously. At the same time, mode (7/4, −1) begins to grow. In addition, modes (1/2, 2) and (1/2, −2) are generated by nonlinear interactions and have almost the same growth rate and amplitude. At x 37 mm, the amplitude of the two secondary modes (1/4, 1) and (3/4, −1) exceeds the primary mode (1, 0), and the amplitude of mode (1/4, 1) has an obvious peak at x 45 mm. Behind x 50 mm, all the modes reach a saturation condition, which corresponds to the fully developed turbulence in Fig. 3(d).

FIG. 4.

Development of the maximum streamwise velocity disturbances for different modes: (a) case F, (b) case S, (c) case A, and (d) case D.

FIG. 4.

Development of the maximum streamwise velocity disturbances for different modes: (a) case F, (b) case S, (c) case A, and (d) case D.

Close modal

The wall temperature is an important influencing factor to instabilities and transitions. For high-speed flows, the solid wall typically has a lower temperature than the adiabatic wall temperature. As Tw decreases, the growth rate of first-mode disturbance will decrease. To further examine the wall cooling effect on secondary-instability-induced transitions, four additional simulations were undertaken at Tw = 360 K, and these cases are named as cooling cases.

Figure 5 shows the streamwise velocity at y / δ in = 0.5 for cases with wall cooling. For the fundamental resonance case, it can be seen that with a lower wall temperature, the flow remains fully laminar, and no 3D effects are observed. For the subharmonic resonance case, the X-type pattern appears at x 45 mm, which shows an obvious delay compared to that in the adiabatic wall temperature case. In addition, in the region where 50`1 mm < x < 56 mm, two high-speed streamwise streaks are observed. For the asymmetric subharmonic case, a lower wall temperature also shows an obvious suppression effect to transition, and an obvious streamwise vortex becomes obvious in the wall cooling case. In the detuned resonance case, wall cooling significantly delays the transition onset position from x 40 to 60 mm. Overall, the delaying effect in the transition from strong to weak is ordered as fundamental resonance, detuned resonance, asymmetry subharmonic resonance, and subharmonic resonance.

FIG. 5.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case FC, (b) case SC, (c) case AC, and (d) case DC.

FIG. 5.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case FC, (b) case SC, (c) case AC, and (d) case DC.

Close modal

Figure 6 plots the streamwise developments of selected modes in the four wall cooling cases. For the fundamental resonance case in Fig. 6(a), modes (1, 0), (1, 1), and (0, 1) are shown. Obviously, with a lower wall temperature, the primary mode (1, 0) has a lower growth rate in the region where 13 < x < 17. Further downstream, the amplitude of mode (1, 0) begins to decrease. At the same time, no obvious secondary growth of mode (1, 1) is observed, and the transition does not occur. For the subharmonic resonance case in Fig. 6(b), although the primary mode (1, 0) is strongly suppressed by a cooling wall, secondary growth of mode (1/2, 1) still occurs, but the secondary growth rate is lower. In addition, at x 48 mm, steady mode (0, 2) exceeds other modes and has its largest amplitude at x 54 mm. This strong steady mode (0, 2) results in the two high-speed streamwise streaks in Fig. 6(b). For the asymmetric subharmonic resonance case in Fig. 6(c), obvious secondary instabilities are observed; the secondary growth position moves downstream and the secondary growth rate becomes lower. Another main difference in the asymmetric subharmonic cases is the evolution of the steady mode (0, 1). In the adiabatic wall temperature case, although mode (0, 1) overtakes mode (1, 1) at x 35 mm, the amplitudes of the two modes are similar. Thus, no obvious streamwise vortex pattern is observed [see Fig. 3(c)]. In the wall cooling case, mode (0, 1) overtakes other modes at x 43 mm and has a much larger amplitude. Therefore, this large-amplitude steady streamwise vortex mode (0, 1) causes the high-speed streak in Fig. 6(c). For the detuned resonance case in Fig. 6(d), secondary instabilities still occur, and the secondary growth position moves downstream and the secondary growth rate becomes lower. The main difference between the adiabatic and cooling wall cases is that the amplitudes of modes (1/4, 1) and (1/2, −2) are almost the same. Therefore, the 3D wave patterns in the spanwise direction in Fig. 6(d) are not very regular.

FIG. 6.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases F and FC, (b) cases S and SC, (c) cases A and AC, and (d) cases D and DC.

FIG. 6.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases F and FC, (b) cases S and SC, (c) cases A and AC, and (d) cases D and DC.

Close modal

This section provides turbulent mean and fluctuation results for adiabatic and cooling wall cases. Figure 7 plots the streamwise developments of the span- and time-averaged skin friction coefficient Cf, momentum thickness Reynolds number R e θ, and the shape factor H 12 = δ * / δ. Here, θ and δ * are the momentum thickness and displacement thickness, respectively. The turbulence reference values obtained from the Van Driest II correlation46 are also shown for comparison. For the fundamental resonance, the Cf begins to increase sharply at x 38 mm in the adiabatic wall temperature case, which indicates the transition occurs; in the wall cooling case, the Cf keeps at a low value, which indicates a fully laminar flow. For the subharmonic resonance, the transition occurs at x 35 mm at the adiabatic wall temperature, and it moves to approximately 55 mm at a cooling wall. For the asymmetric subharmonic resonance case at an adiabatic wall temperature, the Cf begins to increase at x 30 mm, which means that this case has the earliest transition in all cases. However, at a cooling wall, the transition onset position in the asymmetric subharmonic resonance is almost the same as that in the subharmonic resonance. This also proves that the wall cooling has a stronger transition delaying effect in the asymmetric subharmonic resonance. For the detuned resonance, the transition onset occurs at x 37 mm at an adiabatic wall temperature and further moves backward to x 53 mm at a lower wall temperature. In addition, the Cf distribution matches well with the turbulent correlation in the downstream of the computational domain except for the fully laminar case FC, which indicates that the fully developed turbulence is achieved. For R e θ, its development is similar to that of Cf. For the shape factor, the DNS computations first keep at an almost constant value and then begin to rapidly decrease at where the transition occurs. After transition, H12 becomes a relatively low turbulent value. We can also observe that the cooling wall leads to a lower H12 in both the laminar and turbulent regions. In addition, because no transition is observed in case FC, its shape factor remains a high laminar value.

FIG. 7.

Time- and spanwise-averaged (a) skin friction coefficient, (b) momentum thickness Reynolds number, and (c) shape factor.

FIG. 7.

Time- and spanwise-averaged (a) skin friction coefficient, (b) momentum thickness Reynolds number, and (c) shape factor.

Close modal

The skin friction coefficient curves indicate overshoot phenomena in the four secondary-instability-induced transition processes. For further analysis, Figs. 8(a)–8(d) provide the time-averaged wall-normal gradient of the streamwise velocity ( L ref u ) / ( u y ) at the wall for cases at the adiabatic wall temperature. For the fundamental resonance case, the wall-normal velocity gradient begins to increase at x 45 mm and has two obvious streaks, corresponding with the two high-speed streaks in Fig. 3(a). A clear four-streak pattern of the wall-normal velocity gradient exists along the spanwise direction at the region where the skin friction overshoots ( x 50 mm). The same situation is observed in Fig. 8(b) for the subharmonic case S. For case A in Fig. 8(c), the wall-normal velocity gradient is not regular in the spanwise direction, but we still find obvious streaks in the overshoot region. For case D in Fig. 8(d), its peak value in Cf is less than the other three cases in the overshoot region. As a result, its wall-normal velocity gradient is obviously less than that in other cases. However, at least four streaks can still be found at x 50 mm.

FIG. 8.

Time-averaged wall-normal gradient of the streamwise velocity ( L ref u ) / ( u y ) at the wall for (a) case F, (b) case S, (c) case A, and (d) case D.

FIG. 8.

Time-averaged wall-normal gradient of the streamwise velocity ( L ref u ) / ( u y ) at the wall for (a) case F, (b) case S, (c) case A, and (d) case D.

Close modal

As discussed in the previous studies by Franko and Lele,20,21 overshoot during transition is due to transport in the wall-normal direction of the momentum caused by the streaks and streamwise vortexes. The Reynolds shear stress is analyzed to clearly identify this mechanism, and the results for cases F, S, A, and D are presented in Fig. 9. The superscript of double prime ″ represents the Favre averaged value. The Reynolds shear stress is positive in the laminar region but becomes negative in the transitional and turbulent regions. This negative Reynolds shear stress is responsible for the high (low)-momentum fluid moving toward (away from) the wall. The negative Reynolds shear stress has a pronounced peak for all four cases, and the peak position is in the overshoot region of the skin friction. The Reynolds shear stress then decreases obviously and maintains at a nearly constant value as the flow becomes fully turbulent.

FIG. 9.

Contours of the time- and spanwise-averaged Reynolds stress for (a) case F, (b) case S, (c) case A, and (d) case D.

FIG. 9.

Contours of the time- and spanwise-averaged Reynolds stress for (a) case F, (b) case S, (c) case A, and (d) case D.

Close modal
The mean flow would gradually develop from a laminar profile to a turbulent profile. Figure 10 plots the time- and spanwise-averaged van Driest velocity profile at x = 85 mm in cases at the adiabatic wall temperature and the cooling wall temperature. The DNS results are compared with correlations in the viscous sublayer ( U V D + = y +) and the log law ( U V D + = 2.5 ln y + + 5.5). The van Driest transform is defined as47 
y + = ρ w u τ y μ w , U V D + = U V D u τ , u τ = τ w ρ w , U V D = 0 u ρ ρ w d u .
(11)
FIG. 10.

Time- and spanwise-averaged van Driest velocity profiles at x 85 mm: (a) adiabatic wall temperature cases and (b) wall cooling cases.

FIG. 10.

Time- and spanwise-averaged van Driest velocity profiles at x 85 mm: (a) adiabatic wall temperature cases and (b) wall cooling cases.

Close modal

At Tw = Taw, the velocity profiles match with the excepted turbulent profile very well, which indicates that the flow has developed to full turbulence in all cases. This phenomenon corresponds well with the Cf distributions in Fig. 7(a). At a cooling wall of T w = 0.84 T a w, the velocity profile in the fundamental resonance case keeps the linear behavior because no transition was observed in this case, and the flow is fully laminar [see Figs. 5(a) and 7(a)].

This section shows the transition control scenario with the wall cooling effect. The results of fundamental resonance cases are presented first. Figure 11 plots the distributions of the streamwise velocity in the controlling cases and the reference case F. In the first case FC1, which has the earliest cooling strip, the flow keeps laminar until two obvious high-speed streaks appeared at x 66 mm. After that, the breakdown happens, and the flow develops to full turbulence at x 76 mm. In the second case FC2, the cooling strip moves backward, and the two high-speed streaks appear at x 63 mm, which is a little earlier than that in case FC2. In the last case FC3, the cooling strip further moves backward, and the breakdown significantly moves forward to x 45 mm. In addition, we can find that the streamwise streaks in case FC3 become weak, and the Λ-type vortex pattern becomes obvious.

FIG. 11.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case F, (b) case FC1, (c) case FC2, and (d) case FC3.

FIG. 11.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case F, (b) case FC1, (c) case FC2, and (d) case FC3.

Close modal

The streamwise evolution of selected modes in fundamental resonance cases is plotted in Fig. 12. The results for the adiabatic wall temperature case F are also shown for comparison. The two vertical dashed lines represent the start and end positions of the cooling strip. In case FC1, the primary mode (1, 0) is stabilized by the cooling strip, and its amplitude decreases quickly until it begins to increase at x 28 mm. At the same time, the secondary mode (1, 1) is also suppressed, whose amplitude experiences an increase–decrease process in the cooling region. After that, mode (1, 1) begins to increase at x 28 mm. However, the secondary growth rate in case FC1 is much lower than that in case F, which makes a much slower growth in the disturbance energy. At x 66 mm, a sudden increase in both primary and secondary modes was observed, which indicates the final breakdown to turbulence. In addition, this position corresponds well with the 3D wave pattern in Fig. 11(b). For the second control case FC2, the wall cooling effect also significantly suppresses the primary mode (1, 0). As a result, the growth rate of the secondary mode (1, 1) also decreases until x 36 mm. After that, mode (1, 0) begins to increase, and the growth rate of mode (1, 1) also increases. Finally, breakdown occurs at x 65 mm. For the last case FC3, because the cooling strip is very close to the breakdown position, the transition delaying effect is very weak. Although the wall cooling also significantly suppresses the primary and secondary modes, the amplitude of disturbance has already increased to a relatively high level, and the transition appears at x 45 mm.

FIG. 12.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases F and FC1, (b) cases F and FC2, and (c) cases F and FC3.

FIG. 12.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases F and FC1, (b) cases F and FC2, and (c) cases F and FC3.

Close modal
FIG. 13.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case S, (b) case SC1, (c) case SC2, and (d) case SC3.

FIG. 13.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case S, (b) case SC1, (c) case SC2, and (d) case SC3.

Close modal

The control cases of fundamental resonance are discussed below. Figure 13 first plots the contours of the instantaneous streamwise velocity for the subharmonic resonance cases. For the first case SC1, the X-type wave pattern appears at x 52 mm. After that, a streamwise streak pattern can be observed, and the breakdown happens. For the second case SC2, the X-type wave pattern becomes weak, and the streamwise streak becomes more obvious. However, because the cooling strip moves downstream, the transition suppression effect weakens, and the breakdown position slightly moves forward to x 50 mm. For the last case SC3, the X-type wave pattern further weakens, and the two streamwise streaks further enhance. In this case, the transition delaying effect is very weak, and the transition appears at x 45 mm.

The streamwise evolution of different modes is plotted in Fig. 14. Similar to the fundamental resonance cases, the wall cooling strip significantly suppresses the primary mode (1, 0) in the subharmonic resonance cases. As a result, the secondary growth rate of the secondary mode (1/2, ±1) decreases. At the same time, the high-spanwise wavenumber modes (1, 2) and (0, 2) have a lower growth rate, but the two modes become much more dominant in front of the breakdown position. Therefore, the two streamwise streaks become more obvious in the three control cases.

FIG. 14.

Development of the maximum streamwise velocity disturbances for different modes: (a) case S, (b) case SC1, (c) case SC2, and (d) case SC3.

FIG. 14.

Development of the maximum streamwise velocity disturbances for different modes: (a) case S, (b) case SC1, (c) case SC2, and (d) case SC3.

Close modal

The local wall cooling effect on the asymmetric subharmonic resonance is discussed later. The counters of streamwise velocity are shown in Fig. 15. Obviously, with a wall cooling strip at 17.5 mm < x < 23.5 mm, the transition is significantly delayed to x 80 mm. When the wall cooling strip moves backward to 23.5 mm < x < 29.5 mm, the transition delaying effect is weakened, and the breakdown position moves upstream to x 70 mm. As the wall cooling strip further moves upstream to 29.5 mm < x < 35.5 mm, the transition just occurs behind the strip, and the breakdown position is almost the same as that in the baseline case A. The aforementioned phenomena indicate that the local wall cooling has a stronger transition suppression effect in the asymmetric subharmonic resonance than those in the fundamental and subharmonic resonances. However, a similar conclusion is that an earlier wall cooling always has a stronger suppression effect on transition.

FIG. 15.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases A and AC1, (b) cases A and AC2, and (c) cases A and AC3.

FIG. 15.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases A and AC1, (b) cases A and AC2, and (c) cases A and AC3.

Close modal

The streamwise evolution of selected modes is plotted in Fig. 16. It can be seen that the local wall cooling strip has a stronger suppression effect on the oblique primary mode (1, 1) than that on the 2D primary mode (1, 0) in the fundamental and subharmonic resonances, and the amplitude of primary mode has a much greater decrease. As a result, the growth of secondary modes (1/2, 2) and (1/2, −1) is significantly suppressed, and the transition is obviously delayed. In case AC2, because the cooling strip moves backward, the suppression on primary mode (1, 1) becomes weaker, and it has a higher amplitude. Therefore, modes (1/2, 2) and (1/2, −1) have greater secondary growth rates, and transition occurs earlier. In the third case, the amplitude of the oblique mode has already grown to a relatively high level, thus although mode (1, 1) is also suppressed by the wall cooling strip, the secondary growth rate of modes (1/2, 2) and (1/2, −1) decreases slightly. As a result, the transition happens quickly.

FIG. 16.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases A and AC1, (b) cases A and AC2, and (c) cases A and AC3.

FIG. 16.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases A and AC1, (b) cases A and AC2, and (c) cases A and AC3.

Close modal

The results of detuned resonances with controlled wall cooling strips are shown later. Figure 17 first shows the counters of the streamwise velocity. In the first controlled case DC1, the wall cooling strip is located at 16 mm < x < 22 mm, and the transition is delayed to x 58 mm. In addition, the vortex pattern in front of the breakdown position becomes much more messy. As the wall cooling strip moves downstream to 22 mm < x < 28 mm, the transition delaying effect becomes weaker, and the breakdown position moves upstream to x 53 mm. The transition delaying effect further weakens when the wall cooling strip is located at 28 mm < x < 34 mm. Generally, the transition delaying effect is not very obvious in the detuned resonance cases, which is similar to the phenomena in the subharmonic resonance cases.

FIG. 17.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case D, (b) case DC1, (c) case DC2, and (d) case DC3.

FIG. 17.

Contours of the instantaneous streamwise velocity at y / δ in = 0.5: (a) case D, (b) case DC1, (c) case DC2, and (d) case DC3.

Close modal

The streamwise evolution of selected modes in detuned resonance cases is plotted in Fig. 18. In the first control case DC1, the wall cooling makes the amplitude of the primary mode (1, 0) sharply decrease in the region 16 mm < x < 24 mm. At the same time, the secondary mode (1/4, 1) has a lower growth rate. As for mode (3/4, −1), it keeps damping until the primary mode (1, 0) recovers to grow at x 24 mm. Behind x 34 mm, modes (1/4, 1) and (3/4, −1) grow at about the same growth rate until the flow breaks down. In the second case DC2, because of the later wall cooling, the primary mode (1, 0) has increased to a higher amplitude; therefore, the suppression on the secondary modes becomes weaker, and the transition occurs earlier. A similar situation is observed in the third case DC3, in which the wall cooling further moves backward to 28 mm < x < 34 mm. In the wall cooling region, the primary mode (1, 0) damps, and the growth rates of other modes decrease. However, because both modes have increased to a high level, the breakdown quickly occurs just downstream of the cooling strip.

FIG. 18.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases D and DC1, (b) cases D and DC2, and (c) cases D and DC3.

FIG. 18.

Development of the maximum streamwise velocity disturbances for different modes: (a) cases D and DC1, (b) cases D and DC2, and (c) cases D and DC3.

Close modal
FIG. 19.

Time- and spanwise-averaged skin friction coefficient for (a) fundamental, (b) subharmonic, (c) asymmetric subharmonic, and (d) detuned resonances.

FIG. 19.

Time- and spanwise-averaged skin friction coefficient for (a) fundamental, (b) subharmonic, (c) asymmetric subharmonic, and (d) detuned resonances.

Close modal

Finally, we plot the mean Cf results for the control cases in the four secondary instability cases in Fig. 19. In the distributions of Cf, we can see that the local wall cooling strip first makes a sharp increase and decrease at the beginning and the end of the cooling region. After that, Cf recovers to the typical laminar profile. In the fundamental cases, case FC1 has the latest transition position ( x 68 mm), and that case FC2 is a litter earlier ( x 63 mm). However, in case FC3, the transition position sharply moves forward to x 45 mm. In the subharmonic resonance cases, the transition delaying effect is not as efficient as that in fundamental resonance cases. Case SC1 has the latest transition onset position. As the cooling strip moves backward, the transition onset position gradually moves upstream. Different from the sudden changes of transition onset positions in cases FC2 and FC3, the transition onset positions in subharmonic resonances change nearly linearly as the location of the wall cooling strip moves. In the asymmetric subharmonic cases, the transition delaying effect is the strongest. Case AC1 has the latest transition position ( x 78 mm), and that case AC2 is earlier ( x 70 mm). In case AC3, the transition onset position moves forward significantly. In the detuned resonance cases, the situation is similar to those in the subharmonic cases. The first case DC1 has the strongest transition delaying effect, and the transition onset position moves forward approximately linear as the wall cooling strip moves backward.

Direct numerical simulation is used to investigate secondary instabilities in supersonic flat-plate boundary layers. The entire process of transition to turbulence is simulated, and four different nonlinear transition mechanisms are studied. The first two are the classical fundamental and subharmonic resonances, the third is termed asymmetric subharmonic resonance, and the last is the detuned resonance.

Direct numerical simulations indicate that all four types of secondary instabilities can induce transition to turbulence, and the asymmetric subharmonic resonance induces transitions much earlier, while the fundamental resonances are the latest. The fundamental resonance presents a regularly aligned lambda-type pattern before the final breakdown, while the subharmonic resonance exhibits a symmetric staggered pattern. The flow pattern is messy and not symmetric in the spanwise direction for asymmetric subharmonic and detuned resonance. Modal composition shows that nonlinear growth is caused by the wave triad composed of the primary and secondary modes. The wall cooling effects are studied at a lower wall temperature. In this situation, the fundamental resonance is fully suppressed, and no transition is observed. The other three resonances are partially suppressed, and transition moves downstream.

Transition controlled by local wall cooling strip is studied. The local wall cooling strip is placed at three different positions with half of the adiabatic wall temperature. The comparison with the baseline cases indicates that the local wall cooling can significantly stabilize the growth of the primary mode and make it to have a lower amplitude. As a result, the growth rate of secondary modes is also suppressed, and the breakdown is obviously delayed. An upstream wall cooling strip generally shows a stronger transition delaying effect than a downstream one because the former leads to a lower amplitude in the primary mode. In addition, the local wall cooling has a much stronger transition delaying effect in the fundamental and asymmetric subharmonic resonances than those in the subharmonic and detuned resonances.

This work was funded by the National Natural Science Foundation of China (Grant Nos. 12272177, U20A2070, and 12172175), the National Science and Technology Major Project (No. J2019-II-0014-0035), Young Talent lift project (2021-JCJQ-QT-064), and the priority academic program development of Jiangsu higher education institutions.

The authors would like to acknowledge Professor Li Xinliang from the Institute of Mechanics, Chinese Academy of Sciences, for the finite-difference code OpenCFD. The computations are supported by High Performance Computing Platform of Nanjing University of Aeronautics and Astronautics.

The authors have no conflicts to disclose.

Zaijie Liu: Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead). He-Xia Huang: Resources (lead); Writing – original draft (supporting); Writing – review & editing (lead). Mengying Liu: Validation (supporting); Visualization (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This section presents the results of the grid independence study for the high-amplitude forced fundamental resonance case F5. A baseline grid with n x × n y × n z = 3163 × 109 × 269 and a fine grid with n x × n y × n z = 3934 × 122 × 324 are used for the computations. The corresponding results are marked by F and Ffine, respectively. Figure 20(a) compares the skin friction coefficient distributions. There are almost no differences in the transition region between the different grids, only slight differences are observed in the fully developed turbulent region. Figure 20(b) compares the streamwise developments of the selected modes computed from the different grids, showing excellent agreement. These results demonstrate that the present grid can sufficiently simulate the secondary-instability-induced transition.

FIG. 20.

Grid independence study of case F: (a) time- and spanwise-averaged skin friction coefficient and (b) streamwise development of maximum streamwise velocity disturbance for different modes.

FIG. 20.

Grid independence study of case F: (a) time- and spanwise-averaged skin friction coefficient and (b) streamwise development of maximum streamwise velocity disturbance for different modes.

Close modal

The DNS solver and numerical methods are validated by an oblique breakdown case on an adiabatic flat plate. The flow conditions are M = 2.0 , T = 160 K, and R e = 2.407 × 10 7/m. The computational setup is the same as the previous DNS studies.48, Figure 21(a) shows the streamwise evolution of select modes in the oblique breakdown process, with the comparison between the DNS result of Kneer et al.48 In general, the overall agreement is good, demonstrating the reliability and numerical accuracy of the present code.

FIG. 21.

Streamwise development of maximum streamwise velocity disturbance for different modes in the oblique breakdown validation case.

FIG. 21.

Streamwise development of maximum streamwise velocity disturbance for different modes in the oblique breakdown validation case.

Close modal
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