Effects of wall transpiration on the supersonic boundary-layer oblique-type transition

The study of transpiration cooling is vital for the development of high-speed aircraft. In the current work, direct numerical simulation (DNS) is performed to investigate the impacts of wall transpiration on the boundary-layer oblique breakdown over a Mach 2 flat plate. The porous injection model is used to mimic the transpiration from the equally spaced circular pores. It has been observed from the numerical results that wall transpiration leads to the amplified growth rate of the imposed oblique mode waves, steady vortex waves, and other higher-harmonic waves. As a result, the occurrence of boundary-layer transition shifts upstream. Due to the presence of transpiration, the normal gradients of both streamwise velocity and temperature are decreased at the wall, which causes reduced skin friction and heat flux in the transpiration region. In addition, when upstream transpiration is present, reductions in skin friction and heat flux can also be observed within turbulent regions. This study provides insights into the DNS investigation on compressible boundary-layer natural transitions coupled with wall transpiration, and the results indicate that more systematic investigations addressing this problem are needed.


Introduction
The high-speed aircraft will experience severe aerodynamic heating on the surfaces.On the one hand, the presence of viscous effects and dissipation effects can result in large near-wall gradients of temperature.On the other hand, boundary-layer transition typically causes a sharp increase in aerodynamic drag and heat flux [1][2].To mitigate high heat loads experienced by the surface, transpiration cooling [3][4] has emerged as an active cooling technique widely applied in aerospace applications.In this case, transpiration cooling and laminar-to-turbulent flow transition are closely linked together.Thus, it is necessary to conduct the study of boundary-layer transition coupled with transpiration cooling.This is important for the development of the transpiration technique, as well as the accurate prediction for boundary-layer transition.
Recent research showed that transpiration cooling is an efficient and promising method for a longduration cruise flight [5].Thus, there are many studies devoted to studying the mechanism of wall transpiration in typical boundary-layer flows.By experiments, Meinert et al. [3] performed a study where the gas was injected into boundary layers of turbulent flows.Their findings demonstrated that the distributions of skin friction and heat transfer obtained by the experiments and the simple predicting equations show a good agreement.Bellettre et al. [6] used air, water vapor, and alcohol as coolants for transpiration cooling.Their experiment showed that alcohol can achieve the highest cooling efficiency with a fixed injection rate.In a high enthalpy shock tunnel, Camillo et al. [7] studied the hypersonic boundary-layer second-mode stability under the wall transpiration of nitrogen gases.The study revealed IOP Publishing doi:10.1088/1742-6596/2764/1/012029 2 that the boundary-layer transition can be delayed under some specific situations.Because of the challenges and expenses associated with designing transpiration cooling systems in experiments, numerical methods offer the convenience for preliminary study of wall transpiration.
The advancement of computer technology has enabled researchers to conduct direct numerical simulations (DNS) for studying transpiration cooling.It is very powerful for DNS to capture the flow details that would be difficult to measure and observe in experiments and other numerical methods.Avsarkisov et al. [8] used DNS to study the incompressible turbulent Poiseuille flow under the impact of wall transpiration and proposed a scaling law of mean velocity.Keller and Kloker [9] did the numerical work where the coolant was injected through slots under laminar and turbulent flow conditions.For air-into-air cooling, the cooling effectiveness is found to be decreased in turbulent flows compared with that in laminar flows.Christopher et al. [10] studied a Mach 0.3 incompressible flatplate turbulent boundary layer with transpiration by DNS, and the impacts of transpiration on the turbulent kinetic energy and heat flux were obtained.Recently, Cerminara [11] used DNS to study a Mach 5 turbulent boundary layer induced by bypass transition.They focused on the turbulent effects on transpiration cooling effectiveness with changing blowing ratio and pore diameter.These DNS works provide an in-depth understanding of transpiration cooling in turbulent flows.However, there is still a lack of DNS studies concerning the natural transition process interacting with wall transpiration.As the object of this paper, the DNS investigation on the boundary-layer natural transition under the wall transpiration will be very beneficial for the design of TPS.
In the low-disturbance flight environment, the natural transition is deemed as a usual path of boundary-layer transition [12].During the linear unstable stage of natural transition, the oblique wave of the first mode is deemed the most amplified in supersonic boundary layers (Ma<4) [13].While the well-known Mack-mode instability is often dominant in the hypersonic regime [14], following the linear stage, nonlinear wave-wave interactions lead to complicated flow mechanisms and various types of breakdowns [15].Among these scenarios, oblique breakdown is regarded as the most probable transition path for initiating the occurrences of boundary-layer transition in supersonic cases [16][17].The oblique breakdown process involves the nonlinear interactions of a pair of oblique waves with equal but opposite wave angles, which causes the formation of a wave triad.The significance of oblique breakdown has been emphasized over the past 30 years [18][19].
In this study, a Mach 2 boundary-layer oblique breakdown induced by first-mode waves over a flat plate is simulated by DNS.The aim is to obtain the impacts of transpiration on the disturbance evolutions, flow structures, and the distributions of skin friction and heat transfer during oblique breakdown.In the following, section II presents the description of simulation details.The DNS results about oblique breakdown coupled with transpiration cooling are given in Sec.III.Finally, Sec.IV gives the conclusions.

Direct numerical simulation
The compressible non-dimensional Navier-Stokes equations in the Cartesian coordinate system are expressed as: Where ( ) Are conservative variables, inviscid flux vectors, and viscous flux vectors, respectively.Moreover, ij δ is the Kronecker delta, and the vector of velocity is denoted by u .The total energy e , the viscous q can be computed by , 3 Respectively.The thermal conductivity is represented by k .The present gas is air, and its pressure can be calculated by the state equation.In addition, Sutherland's equation is utilized to compute dynamic viscosity μ .The freestream parameters are used to normalize the flow variables in the DNS simulations, and the length variable is scaled by the reference length.The reference length in this paper is 1 m.
The present DNS code called OpenCFD has been tested to be reliable and accurate in many studies of boundary-layer transition and turbulence [20][21].The optimized sixth-order monotonicity-preserving scheme is utilized to discretize inviscid terms with Steger-Warming splitting.The sixth-order central difference scheme is used for the spatial discretization of viscous terms.Moreover, the third-order explicit Runge-Kutta method is applied for time marching.

Computational setup
The free-stream conditions in this study are selected to be the same as those of Sharma et al.'s study [22], and the detailed parameters are presented in Table 1.The Ma ∞ , unit Re , * T ∞ , * u ∞ , Pr, γ , and * p ∞ represent Mach number, unit Reynolds number, temperature, streamwise velocity, Prandtl number, specific heat ratio, and pressure, respectively.Hereinafter, subscripts " ∞ " and " w " represent flow parameters of free stream and wall, respectively.In addition, the superscript " * " denotes the dimensional variables, and the dimensionless variables are denoted with no superscript.In addition, equidistant grid spacing is employed in both the streamwise and spanwise directions.In the wall-normal direction, the grid of exponential stretching is utilized.The number of grid nodes in three directions is 2000 120 140.
The grid resolution in this study is max max 13.92 0.92 7.58 . Here w τ denotes the maximum wall shear stress.As studied by Coleman and Sandberg [23], the thresholds of grid resolution in DNS should be 15, 1, and 8 for Δ , x y + + Δ and Δz + , respectively.Correspondingly, it is evident that the present resolution of grid nodes meets the requirement of studying boundary-layer oblique breakdown using DNS.T T = is adopted for all cases in this study, and aw T corresponds to the adiabatic wall temperature in laminar boundary-layer flows.According to the studies of Sharma et al. [22], oblique breakdown can be triggered by wall blowing and suction strip expressed as: Where A is the forcing amplitude, which is set as 0.00034 in the present study.The disturbance strip extends from To model coolant injection through a porous surface in the region of wall transpiration, a streamwise and spanwise periodic function is employed.The specific modeling [11] is expressed as follows: Where F and D denote the injection rate and the pore's diameter, respectively.This injection strip extends from According to Cerminara [11], this model offers a simplified approach for simulating coolant injections over the porous surface.It circumvents the challenges and high computational cost associated with simulating flow passing through the entire porous layer using DNS, as the current object is to mimic the injection effects in the outer flows.In addition, the coolant gas is considered to be air, which is injected at wall temperature.The coolant is assumed to exit the pore with the same pressure as the outer gases at the wall.The density of blowing gas is calculated to adhere to the perfect gas law.These configurations for the coolant setup have been frequently adopted in previous DNS studies [10][11].For easy viewing, the simulation set-ups for present cases are summarized in Table 2.

Flow-field characteristics
The instantaneous flow field of the streamwise velocity at in the x-z plane for cases with and without transpiration is presented in Figure 2. It is found that these low-speed streaks are breaking down in both cases, which is consistent with the previous finding that turbulent flows are induced due to the bursting of low-speed streaks that exist in close proximity to the wall [15].The spanwise spacing of the steady streaks is approximately half the width of the spanwise computational domain.This spacing corresponds to the spanwise spacing of the initial introduced oblique waves, which suggests that the dominated mode is mode (0,2).This observation agrees with the observation made by Chang and Malik [16] and Guo et al. [15].Additionally, it is also found that the local low-speed region is generated by the wall transpiration.Meanwhile, the breakdown of low-speed streaks shifts noticeably upstream due to the transpiration.In other words, wall transpiration leads to an earlier occurrence of transition.for both cases are presented in Figure 3.The presence of rope-like structures is evident in the x-y plane.Initially, these structures emerge from the wall and gradually lift away.As they move downstream, the tips of these structures begin to break up into small-scale structures.Within the high-shear region, the spanwise vorticity is prominently higher compared to the surrounding regions.As these high-shear structures move downstream and become unstable, the instability of the high-shear layer gets further induced within the transition region.Thus, a breakdown of the laminar flow occurs.It is evident that the spanwise vorticity in the high-shear region increases under wall transpiration, enhancing the instability within this region.Correspondingly, the appearance of breakdown is shifted upstream.The streamwise distributions of boundary-layer thickness 99 δ and shape factor for both cases are presented in Figure 4.The shape factor is typically denoted as the ratio of the displacement thickness to the momentum thickness.The figure shows that the growth rate of boundary-layer thickness increased within the region of wall transpiration.As the flow progresses downstream, the streamwise position of the second increase in the boundary-layer thickness is shifted upstream for the baseline case without transpiration.This shift is due to an earlier occurrence of the transition with the presence of transpiration.Meanwhile, within the turbulent region, the boundary layer experiences thickening because of the upstream wall transpiration.Similarly, the wall transpiration increases the boundary-layer shape factor initially.However, the shape factor decreases sharply to a very low value in advance due to the earlier onset of transition.Furthermore, in turbulent regions, the shape factor is nearly not affected by upstream transpiration.Figure 5 shows the wall-normal profiles of streamwise velocity and temperature at a transpiration location in both cases with and without transpiration.This figure clearly illustrates that the thickness of boundary layer is increased within the wall transpiration region, which is consistent with the finding in Figure 4.In addition, both the gradient of streamwise velocity u y ∂ ∂ and the gradient of temperature T y ∂ ∂ are also increased at the wall due to the wall transpiration.Given that skin friction is primarily influenced by the gradient of streamwise velocity at the wall, and heat flux is predominantly determined by the temperature gradient at the wall, it can be inferred that both skin friction and heat flux will decrease due to wall transpiration.

Disturbance evolutions
The fast Fourier transformations (FFT) are performed in both spanwise and time directions for each case to quantify the evolution of various disturbances.Figure 6 shows the streamwise evolution of different disturbances in both cases with and without transpiration.Figure 6(a) shows that the growth rate of the oblique mode waves (1,1) increases due to wall transpiration.In other words, mode (1,1) is destabilized by the wall transpiration.In Figure 6(b), the growth rate of steady vortex mode (0,2) also experiences an increase in the second half region of wall transpiration.However, in the first half region of transpiration, the amplification rate of mode (0,2) remains unaffected by the presence of transpiration.This suggests that transpiration does not directly affect the steady modes, and the amplified growth rates of steady mode (0,2) in the region of transpiration are mainly caused by the amplified mode (1,1).This is because the fundamental mode (1,1) functions as introducing disturbances into modes (0,2) [16].As shown in Figure 6(c) and Figure 6(d), the growth rates of modes (1,3) and (1,5) are also increased in the case where wall transpiration is imposed.This phenomenon is attributed to the strengthening wavevortex triad, which consists of oblique modes and steady modes, as modes (1,3) and (1,5) are triggered by nonlinear interactions of the wave-vortex triad [16].Eventually, the amplitude of these disturbances has grown to the same level in advance, and oblique breakdown occurs earlier in the case of transpiration.In a word, the enhanced destabilization of the fundamental first mode by wall transpiration leads to the earlier onset of transition.Thus, the control strategy of suppressing mode (1,1) might be most efficient The relevant study might be helpful for the development of TPS design in high-speed aircraft.

Skin friction and heat transfer
The oblique breakdown naturally leads to a rapid increase in skin friction and heat transfer, which is very important for the TPS design of modern aircraft.The skin-friction coefficient f C and the heat- transfer coefficient h C , i.e., Stanton number, are typically computed as [19] 2 1 2 And ( ) Respectively, where is wall shear stress and w q is wall heat flux.In addition, recovery temperature is calculated by . The f C and h C calculated by the above equations for both cases are shown in Figure 7.This figure also includes a comparison with the values obtained from the empirical correlations.The specific empirical correlations for f C of laminar and turbulence can refer to Franko et al. [19], and the empirical equation of h C in the turbulent region is computed based on Reynolds analogy expressed as: 2 / .
The Reynold analogy factor s is given by 2 3 Pr − .Figure 7 shows that the values of both f C and h C exhibit good agreement with the empirical correlation in the laminar and turbulent region for the baseline case without wall transpiration, indicating the reliability of present simulations.Moreover, oblique breakdown in both cases results in a rapid increase in f C and h C , and the transition onset is clearly shifted upstream in the transpiration case.
Meanwhile, the overshoot of f C and h C are observed in both cases.By comparing the distribution of f C and h C between both cases, it is known that wall transpiration reduces the local skin friction and heat flux in the transpiration region.This is because the wall gradients of temperature and streamwise velocity are decreased, which has been previously mentioned in Figure 5. Additionally, in the turbulent regime, both f C and h C are also decreased as the result of wall transpiration in the upstream.It is worth mentioning that due to the wall transpiration, the maximum skin friction in the overshoot region is slightly increased, whereas the maximum heat flux is slightly decreased.These observed phenomena highlight the impacts of wall transpiration on skin friction and heat flux in the natural transitions.These findings are valuable in the design and implementation of transpiration techniques, as they provide insights into the potential benefits of utilizing wall transportation for controlling and manipulating flow characteristics.However, the more in-depth reasons and mechanisms of these phenomena remain unknown, more systematic relevant studies on this aspect should be addressed in the future.The current DNS results reveal that low-speed streaks are dominated during the oblique breakdown, and the imposed wall transpiration causes a noticeable shift in the breakdown of these low-speed streaks upstream.In other words, wall transpiration promotes the boundary-layer transition.Subsequently, the fast Fourier transformation is used to conduct the modal analysis.The results of modal analysis reveal that the instability of oblique first-mode waves (1, 1) is strengthened due to wall transpiration.In turn, the growth rate of steady mode (0, 2) and other harmonic modes are also increased because of the induced first mode waves.Thus, the amplitude of these modes grows to the same level in advance.This can explain the earlier occurrence of transition in the case of wall transpiration.In addition, the thickness of the boundary layer is increased within the local transpiration region, and this increase gets more significant in the turbulent regime.However, the shape factors of the boundary layer are nearly not affected within the turbulent regions.Meanwhile, the skin friction and heat flux are indeed decreased in the region of transpiration.This reduction can be attributed to the decrease in the wall gradients of streamwise velocity and temperature.Furthermore, decreased skin frictions and heat fluxes can also be observed in the turbulent region due to wall transpiration.It is also very interesting to have found that transpiration leads to a decrease in maximum heat flux while simultaneously increasing the maximum skin friction.These findings are very helpful for the design of TPS.

Figure 1 .
Figure 1.The schematic diagram of numerical simulation settings.The computational domain in this study is shown Figure.1.The inlet of computational domain is at pair (h, k) typically denotes a disturbance mode with frequency 0 hω and spanwise wavenumber 0 k β .In this study, (h, k) represents the sum of (h, +k) and (h, -k), since the disturbance strip simultaneously generates the modes (h, +k) and (h, -k) with the same amplitude.It should be noted that only modes (1, ±1), which denote the pair of initial oblique waves, are introduced through this disturbance strip.
entire width in the spanwise direction.To study the oblique breakdown interacting with wall transpiration, F is set as 0.002, while D is uniformly set as 0.0002 in this paper.It should be noted that the surface porosity of this model is

Figure 2 .
Figure 2. The instantaneous contours of streamwise velocity at

Figure 3 .
Figure 3.The Instantaneous flow field of spanwise vorticities z ω at / 2 z z L = : (a) Cref and (b) C1.The region of wall transpiration is denoted by vertical dashed lines.

Figure 4 .
Figure 4.The streamwise distributions of (a) boundary-layer thickness and (b) shape factor for the cases with and without transpiration.The region of transpiration cooling is denoted by vertical dashed lines.Figure5shows the wall-normal profiles of streamwise velocity and temperature at a transpiration location in both cases with and without transpiration.This figure clearly illustrates that the thickness of boundary layer is increased within the wall transpiration region, which is consistent with the finding in Figure4.In addition, both the gradient of streamwise velocity u y ∂ ∂ and the gradient of temperature

Figure 7 .
Figure 7.The streamwise distribution of Reynolds averaged (a) f C and (b) h C for both cases.
This work mainly aims to obtain the impacts of wall transpiration on supersonic boundary-layer oblique breakdown over the flat plate.The DNS has been performed to obtain the high-fidelity flow fields in the complete oblique breakdown process.The 3D first-mode waves are triggered by periodic disturbances of blowing and suction, and the model incorporating injection through the porous surface is utilized to simulate the wall transpiration.The use of this injection model allows for the investigation of the impacts of transpiration cooling on the outer flow field and associated phenomena.

Table 1 .
Free-stream conditions.The setup of boundary conditions is illustrated in Figure.1.At the boundary of the inlet, the flow profiles are specified by the laminar compressible Blasius solutions with the shooting method.The laminar profile at the inlet has been validated by comparing it with the numerical results obtained from solving 2D N-S equations with high-order finite difference schemes.In addition, no-reflecting boundary conditions have been utilized at both boundaries of the upper and outlet.Periodic conditions are specified at two lateral boundaries.An isothermal no-slip wall with

Table 2 .
The computational setups of all the cases.
to remove the negative impacts brought by the transpiration technique, i.e., the promotion of transition.