In the current research, the two-dimensional (2-D) DNS is employed to directly resolve the detailed BL flow fields among the metasurface microstructures, which are used to verify the effectiveness of the impedance boundary condition when employed in the LST and the stabilization performance of the designed broadband metasurface. The 2-D LST and *e*^{N} methods are mainly used to analyze the increment ratio of the unstable modes and determine the transition delay performance on the metasurface. It is worth noting that the three-dimensional (3-D) first mode is the most unstable in a supersonic BL on a flat plate [35]. In the Mach 4 BL flow, the wave angles [*Θ* ≡ arctan(*β/α*_{r}), *β* and *α*_{r} are respectively spanwise and streamwise wavenumbers] of these unstable 3-D waves calculated at *x*^{∗} = 0.1 m (Fig. 7a) fall in the range of 0 ~ 3.49, which is much smaller than \(\sqrt{M_{\infty}^2-1}\). That implicates that the first modes considered here belong to the inviscid regime [35]. According to the previous studies [36,37,38], the 2-D analyses of the metasurface effect on the first mode could apply equally to the 3-D cases. Additionally, the ideal metasurface with 2-D microslits can be regarded as an equivalent 3-D one with the same acoustic characteristics, provided that these have the same hydraulic diameter *d*_{h}, porosity *n*, and depth *H* [23].

### 2.1 Direct numerical simulation

The DNS governing equations are 2-D Navier–Stokes equations in dimensional conservation form:

$$\frac{\partial {\mathrm{Q}}^{\ast }}{\partial {t}^{\ast }}+\frac{\partial {\mathrm{E}}^{\ast }}{\partial {x}^{\ast }}+\frac{\partial {\mathrm{F}}^{\ast }}{\partial {y}^{\ast }}=0,$$

(1)

where *Q*^{*} is a vector of conservative variables and *E*^{*} and *F*^{*} are flux vectors in the *x* and *y* directions, respectively. These vectors are expressed as

$${\mathrm{Q}}^{\ast }=\left\{\begin{array}{c}{\rho}^{\ast}\\ {}{\rho}^{\ast }{u}^{\ast}\\ {}{\rho}^{\ast }{v}^{\ast}\\ {}{e}^{\ast}\end{array}\right\},\kern0.5em {\mathrm{E}}^{\ast }=\left\{\begin{array}{c}{\rho}^{\ast }{u}^{\ast}\\ {}{\rho}^{\ast }{u}^{\ast 2}+{p}^{\ast }-{\tau}_{xx}^{\ast}\\ {}{\rho}^{\ast }{u}^{\ast }{v}^{\ast }-{\tau}_{xy}^{\ast}\\ {}{u}^{\ast}\left({e}^{\ast }+{p}^{\ast}\right)\hbox{-} {u}^{\ast }{\tau}_{xx}^{\ast }+{v}^{\ast }{\tau}_{xy}^{\ast }-{q}_x^{\ast}\end{array}\right\},\kern0.5em {\mathrm{F}}^{\ast }=\left\{\begin{array}{c}{\rho}^{\ast }{v}^{\ast}\\ {}{\rho}^{\ast }{u}^{\ast }{v}^{\ast }-{\tau}_{xy}^{\ast}\\ {}{\rho}^{\ast }{v}^{\ast 2}+{p}^{\ast }-{\tau}_{yy}^{\ast}\\ {}{v}^{\ast}\left({e}^{\ast }+{p}^{\ast}\right)\hbox{-} {u}^{\ast }{\tau}_{xy}^{\ast }+{v}^{\ast }{\tau}_{yy}^{\ast }-{q}_y^{\ast}\end{array}\right\}.$$

(2)

Here, *ρ*^{*} is the density; *u*^{*}, *v*^{*} are velocity components in Cartesian coordinates; *e*^{*} = *p*^{*}/(*γ*-1) + *ρ*^{*}(*u*^{*2} + *v*^{*2})/2 is the total energy; and *p*^{*} is the pressure. The superscript ^{*} denotes dimensional variables. *τ*^{∗} and *q*^{*} are the stress tensor and heat flux with the components

$${\tau}_{xx}^{\ast }=2{\mu}^{\ast}\frac{\partial {u}^{\ast }}{\partial {x}^{\ast }}-\frac{2}{3}{\mu}^{\ast}\left(\frac{\partial {u}^{\ast }}{\partial {x}^{\ast }}+\frac{\partial {v}^{\ast }}{\partial {y}^{\ast }}\right),\kern1em {\tau}_{xy}^{\ast }={\mu}^{\ast}\left(\frac{\partial {u}^{\ast }}{\partial {y}^{\ast }}+\frac{\partial {v}^{\ast }}{\partial {x}^{\ast }}\right),\kern1em {\tau}_{\mathrm{yy}}^{\ast }=2{\mu}^{\ast}\frac{\partial {v}^{\ast }}{\partial {y}^{\ast }}-\frac{2}{3}{\mu}^{\ast}\left(\frac{\partial {u}^{\ast }}{\partial {x}^{\ast }}+\frac{\partial {v}^{\ast }}{\partial {y}^{\ast }}\right),$$

(3)

$${\displaystyle \begin{array}{cc}{q}_x^{\ast }=-\frac{\mu^{\ast}\gamma R}{{Pr} \left(\gamma \hbox{-} 1\right)}\frac{\partial {T}^{\ast }}{\partial {x}^{\ast }},& {q}_y^{\ast }=-\frac{\mu^{\ast}\gamma R}{{Pr} \left(\gamma \hbox{-} 1\right)}\frac{\partial {T}^{\ast }}{\partial {y}^{\ast }}\end{array}},$$

(4)

where *T*^{*} denotes the temperature. The dynamic viscosity *μ*^{*} is calculated using Sutherlands law. The specific heat ratio and Prandtl number of perfect gas flows are assumed to be *γ* = 1.4 and *Pr* = 0.72, respectively. *R* is a constant with *R* = *R*_{0}*/M*, where gas constant *R*_{0} = 8.314 J/(mol·K); molar mass *M* = 0.029 kg/mol.

### 2.2 Linear stability theory

In this study, 2-D instabilities are considered, and the small nondimensionalized perturbation is expressed in harmonic wave form:

$${\left[{u}^{\prime },{v}^{\prime },{p}^{\prime },{T}^{\prime}\right]}^T=\Psi (y){e}^{i\left(\alpha x-\omega t\right)}.$$

(5)

The dimensionless coordinates *x* and *y* are normalized by the BL thickness scale \({l}^{\ast }=\sqrt{\mu_e^{\ast }{x}^{\ast }/{\rho}_e^{\ast }{u}_e^{\ast }}\). The flow velocity components *u*^{*}, *v*^{*}*,* temperature *T*^{*} and pressure *p*^{*} are normalized by *u*_{e}^{*}, *T*_{e}^{*} and *ρ*_{e}^{*}*u*_{e}^{*}^{2}, respectively. The subscript *e* denotes the mean flow variables at the BL edge. In Eq. (5), \(\Psi ={\left[\hat{u},\hat{v},\hat{p},\hat{T}\right]}^T\), *α* is the dimensionless streamwise wavenumber normalized by 1/*l*^{*}, and *ω* is the angular frequency normalized by *u*_{e}^{*}/*l*^{*}. With an assumption of a local-parallel boundary layer, the small disturbances governing equations is derived by substituting disturbance quantities into compressible Navier–Stokes equations and equations of state and then subtracting the mean flow parts. The details can be found in Malik [39]. Substituting Eq. (5) into the disturbances governing equations and neglecting the nonlinear terms yields the linearized dispersion relation

$$L\left(\alpha, \omega \right)\Psi =0,$$

(6)

where nonzero elements of the matrix *L* are evaluated by the local mean flow velocity and temperature, which can be obtained from a compressible self-similar solution [39]. The boundary conditions are

$${\displaystyle \begin{array}{c}\kern0.75em \hat{u}=\hat{T}=0,\hat{v}=A\hat{p},\kern1.75em y=0\\ {}\kern1.25em \hat{u}=\hat{v}=\hat{T}=0,\kern3.25em y\to \infty \end{array}},$$

(7)

where *A* is admittance of acoustic metasurface. Notably, for a smooth wall, *A* is equal to zero.

The spatial theory is able to demonstrate the changes in amplitudes of eigenmodes with distance; therefore, the dispersion problem is converted into a spatial problem, in which *ω* is given and *α* is unknown. *α* is obtained by employing the single-domain spectral collocation method [39]. The growth rate of a wave perturbance is *σ =* − *α*_{i}, where *α*_{i} is the imaginary part of *α*. The disturbance wave is unstable with a negative *α*_{i}. A regular acoustic metasurface structure consisting of microslits is shown in Fig. 1, with width 2*b*, spacing *s*, and depth *H* nondimensionalized by *l*^{*}*.*

The acoustic metasurface admittance for normal incident waves was derived in Ref. [18, 40]

$$A=\frac{1}{Z}={nM}_e\sqrt{T_w\left(1-\frac{\tan \left({k}_vb\right)}{k_vb}\right)\left(1+\frac{\left(\gamma -1\right)\tan \left({k}_tb\right)}{k_tb}\right)}\tanh \left( i\omega {HM}_e\sqrt{\frac{1+\frac{\left(\gamma -1\right)\tan \left({k}_tb\right)}{k_tb}}{\tau_w\left(1-\frac{\tan \left({k}_vb\right)}{k_vb}\right)}}\right),$$

(8)

where *A* is the metasurface admittance, *Z* is the metasurface impedance, *n* (≡2*b/s*) is the porosity of the acoustic metasurface, and the dimensionless viscous wavenumber *k*_{v} and thermal wavenumber *k*_{t} are given by \({k}_v\equiv \sqrt{i\omega R/\left({T}_w{\mu}_w\right)}\) and \({k}_t\equiv \sqrt{Pr}{k}_v\), respectively [41]. The admittance of the acoustic metasurface is a complex number and can be expressed as *A* = |*A*|*e*^{iθ}, where |*A*| denotes the admittance magnitude and *θ* denotes the admittance phase. Accordingly, the disturbance energy flux contributed by the acoustic metasurface per unit area is

$${E}_w=\frac{1}{4}\left({p}_w^{\prime }+{c.c.}\right)\left({v}_w^{\prime }+{c.c.}\right)=\mid A\mid {\left|{p}^{\prime}\right|}^2\cos \left(\theta \right),$$

(9)

where *c.c.* denotes the corresponding complex conjugate. To suppress the disturbances, i.e., *E*_{w} *<* 0*,* the admittance phase angle should be in the range of *θ*∈[0.5π,1.5π].

The *e*^{N} method is a semiempirical method based on LST, which predicts the transition location by accumulating the growth rate of unstable waves [42]. This method is applied to evaluate the transition delay performance by the designed acoustic metasurface. In the *e*^{N} method, the logarithmic amplification ratio of the amplitude versus its initial value for each wave is:

$$N\left(\omega, x\right)=\ln \left(A/{A}_0\right)={\int}_{x_0}^x\sigma \left(\omega, x\right)\mathrm{d}x,$$

(10)

where *x*_{0} is the location of onset of instability, and *A*_{0} is the disturbance amplitude there. The envelope of these ratios, i.e., the *N* factor that represents the amplitude evolution of the most amplified disturbance, is:

$$N(x)\equiv {\max}_{\omega }N\left(\omega, x\right).$$

(11)