2.1. Modal Helmholtz decomposition

The Helmholtz decomposition allows any three-dimensional vector field to be expressed as a sum of solenoidal and dilatational components. Following the fundamental theorem of vector calculus (Murray Reference Murray1898), the perturbation velocity vector $\boldsymbol {u}'$ is cast as

(2.10a–c)\begin{equation} \boldsymbol{u}'=\boldsymbol{u}'^s+\boldsymbol{u}'^d,\quad \boldsymbol{u}'^s=\boldsymbol{\nabla} \times \boldsymbol{\varPsi},\quad \boldsymbol{u}'^d=\boldsymbol{\nabla} \phi. \end{equation}

Here, $\phi$ is the velocity potential, and $\boldsymbol {\varPsi }$ is the vector potential. The solenoidal velocity $\boldsymbol {u}'^s$ is divergence-free, and the dilatational part $\boldsymbol {u}'^d$ is irrotational by definition. The potentials $\phi$ and $\boldsymbol {\varPsi }$ are governed by the Poisson equations (Hirasaki & Hellums Reference Hirasaki and Hellums1970)

(2.11a,b)\begin{equation} \nabla^2\phi=\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}', \quad\nabla^2\boldsymbol{\varPsi}=-\boldsymbol{\nabla}\times\boldsymbol{u}'. \end{equation}

It is worth noting that the vector potential $\boldsymbol {\varPsi }$ is constrained to be solenoidal (Hirasaki & Hellums Reference Hirasaki and Hellums1970). The boundary conditions given by

(2.12a–d)\begin{equation} \left.\frac{\partial \phi}{\partial y}\right|_{0,l_y}=0,\quad \left.\varPsi_x\right|_{0,l_y}=0,\quad\left.\frac{\partial \varPsi_y}{\partial y}\right|_{0,l_y}=0,\quad \left.\varPsi_z\right|_{0,l_y}=0 \end{equation}

must be satisfied at the wall and the freestream boundary for a unique vector potential (Hirasaki & Hellums Reference Hirasaki and Hellums1970). Here, $\varPsi _x$, $\varPsi _y$ and $\varPsi _z$ are the streamwise, wall-normal and spanwise components of the vector potential, respectively.

As mentioned in the Introduction, we use Helmholtz decomposition in conjunction with linear stability analysis to gain further insight into the compressibility effects on compressible boundary layer stability. The velocity field of each eigenmode is subjected to Helmholtz decomposition. The potentials $\phi$ and $\boldsymbol {\varPsi }$ are expressed in the normal mode form as

(2.13a,b)\begin{equation} \phi=\hat{\phi}(y)\exp({\iota(\alpha x+\beta z-\omega t)}),\quad \varPsi_i=\hat{\varPsi}_i(y)\exp({\iota(\alpha x+\beta z-\omega t)}), \end{equation}

where $\hat {\phi }$ and $\hat {\varPsi }_i$ are the eigenfunctions of the velocity and vector potentials, respectively. Substituting the normal mode form of potentials (2.13a,b) in the Poisson equations (2.11a,b), the governing equations for the potential eigenfunctions are obtained:

(2.14)\begin{equation} \left.\begin{gathered} -\alpha^2\hat{\phi}+\frac{{\rm d}^2\hat{\phi}}{{{\rm d} y}^2}-\beta^2\hat{\phi}= \iota\alpha\hat{u}+\frac{{\rm d}\hat{v}}{{\rm d} y}+\iota\beta\hat{w}, \\ -\alpha^2\hat{\varPsi}_x+\frac{{\rm d}^2\hat{\varPsi}_x}{{{\rm d} y}^2}-\beta^2 \hat{\varPsi}_x=\iota\beta\hat{v}-\frac{{\rm d}\hat{w}}{{\rm d} y}, \\ -\alpha^2\hat{\varPsi}_y+\frac{{\rm d}^2\hat{\varPsi}_y}{{{\rm d} y}^2}-\beta^2\hat{\varPsi}_y= \iota\alpha\hat{w}-\iota\beta\hat{u}, \\ -\alpha^2\hat{\varPsi}_z+\frac{{\rm d}^2\hat{\varPsi}_z}{{{\rm d} y}^2}-\beta^2\hat{\varPsi}_z= \frac{{\rm d}\hat{u}}{{\rm d} y}-\iota\alpha\hat{v}. \end{gathered}\right\} \end{equation}

Here, $\hat {u}$, $\hat {v}$ and $\hat {w}$ are the eigenmode shapes of the streamwise, wall-normal and spanwise velocity components, respectively. The following boundary conditions must be satisfied for uniqueness of the potentials:

(2.15a–d)\begin{equation} \left.\frac{{\rm d} \hat{\phi}}{{\rm d} y}\right|_{0,l_y}=0,\quad \left.\hat{\varPsi}_x\right|_{0,l_y}=0,\quad\left.\frac{{\rm d} \hat{\varPsi}_y}{{\rm d} y}\right|_{0,l_y}=0,\quad \left.\hat{\varPsi}_z\right|_{0,l_y}=0. \end{equation}

A more detailed discussion on the boundary conditions is provided in Appendix A.

The solenoidal and dilatational parts of the velocity field can also be expressed in the modal form

(2.16a,b)\begin{equation} u'^s_i=\hat{u}^s_i(y)\exp({\iota(\alpha x+\beta z-\omega t)}),\quad u'^d_i=\hat{u}^d_i(y)\exp({\iota(\alpha x+\beta z-\omega t)}). \end{equation}

The eigenfunctions $\hat {u}^s_i$ and $\hat {u}^d_i$ are then obtained using

(2.17)\begin{equation} \left.\begin{gathered} \hat{u}^d =\iota\alpha\hat{\phi},\quad\hat{v}^d=\frac{{\rm d}\hat{\phi}}{{\rm d}y}, \quad\hat{w}^d=\iota\beta\hat{\phi}, \\ \hat{u}^s =\frac{{\rm d}\hat{\varPsi}_z}{{\rm d} y}-\iota\beta\hat{\varPsi}_y, \quad \hat{v}^s=\iota\beta\hat{\varPsi}_x-\iota\alpha\hat{\varPsi}_z, \quad \hat{w}^s=\iota\alpha\hat{\varPsi}_y-\frac{{\rm d}\hat{\varPsi}_x}{{\rm d}y}. \end{gathered}\right\} \end{equation}

The fluctuating velocity eigenfunctions can be recovered from $\hat {u}_i^s$ and $\hat {u}_i^d$:

(2.18)\begin{equation} \hat{u}_i=\hat{u}_i^s+\hat{u}_i^d. \end{equation}

2.1.1. Momentum decomposition

In this work, we employ Helmholtz decomposition to partition the momentum perturbations ($m_i'$) as well. The momentum perturbations in the linear limit can be approximated as

(2.19)\begin{equation} m_i'=\rho u_i-\overline{\rho u}=\bar{\rho}u_i'+\rho'\bar{U}_i. \end{equation}

The momentum perturbations are then split into solenoidal and dilatational parts:

(2.20)\begin{equation} m_i'=m_i'^s+m_i'^d, \end{equation}

where $m_i'^s$ is the solenoidal component and $m_i'^d$ is the dilatational component of the momentum field. The eigenfunctions of $m_i'^s$ and $m_i'^d$ can also be obtained by following the procedure outlined in (2.14)–(2.17).

2.2. Pressure decomposition

The decomposition of pressure in general is not unique, and multiple formulations exist. Erlebacher et al. (Reference Erlebacher, Hussaini, Kreiss and Sarkar1990) define the solenoidal pressure as the field that satisfies the incompressible Poisson equation with a source term corresponding to the solenoidal velocity. Then the dilatational pressure is defined as the difference between the total pressure and the solenoidal pressure. More recently, Yu, Xu & Pirozzoli (Reference Yu, Xu and Pirozzoli2020) proposed a formal partition of pressure into the rapid, slow, viscous and mass flux related terms. The rapid and slow terms can be further split into solenoidal and dilatational parts using the Helmholtz decomposition of velocity. Building on the momentum potential theory approach (Doak Reference Doak1989), Unnikrishnan & Gaitonde (Reference Unnikrishnan and Gaitonde2020) also propose a split of pressure fluctuations into their hydrodynamic, acoustic and entropic components. In this subsection, a novel formulation to decompose the total pressure fluctuations into solenoidal ($p'^s$) and dilatational ($p'^d$) pressure in the linear limit is presented.

The divergence of the linearized perturbation momentum equation (2.6b) yields the following equation for pressure perturbations:

(2.21)\begin{align} -\frac{\partial^2 p'}{\partial x_i\,\partial x_i}+\frac{1}{\bar{\rho}}\,\frac{\partial \bar{\rho}}{\partial x_i}\,\frac{\partial p'}{\partial x_i}&= \left[\bar{\rho}\,\frac{\partial}{\partial t}+\bar{\rho}\bar{U}_k\,\frac{\partial}{\partial x_k}\right]\frac{\partial u_i'}{\partial x_i}+2\bar{\rho}\,\frac{\partial\bar{U}_k}{\partial x_i}\,\frac{\partial u_i'}{\partial x_k}\nonumber\\ &\quad -\frac{1}{Re}\left[\frac{\partial^2\tau_{ik}'}{\partial x_i\,\partial x_k}-\frac{1}{\bar{\rho}}\,\frac{\partial \bar{\rho}}{\partial x_i}\,\frac{\partial \tau_{ik}'}{\partial x_k}\right]. \end{align}

The Helmholtz decomposition of the velocity field $u_i'=u_i'^s+u_i'^d$ is substituted in (2.21). We assume that the evolution of $p'^s$ is determined completely by the solenoidal part of velocity field. The governing equation for the solenoidal pressure is

(2.22)\begin{equation} -\frac{\partial^2 p'^s}{\partial x_i\,\partial x_i}+\frac{1}{\bar{\rho}}\,\frac{\partial \bar{\rho}}{\partial x_i}\,\frac{\partial p'^s}{\partial x_i} = 2\bar{\rho}\,\frac{\partial\bar{U}_k}{\partial x_i}\,\frac{\partial u_i'^s}{\partial x_k} -\frac{1}{Re}\left[\frac{\partial^2\tau_{ik}'^s}{\partial x_i\,\partial x_k}-\frac{1}{\bar{\rho}}\,\frac{\partial \bar{\rho}}{\partial x_i}\,\frac{\partial \tau_{ik}'^s}{\partial x_k}\right]. \end{equation}

The components of the solenoidal viscous stress tensor $\tau _{ik}'^s$ are

(2.23)\begin{equation} \tau_{ik}'^s=\bar{\mu}\left(\frac{\partial u_i'^s}{\partial x_k}+\frac{\partial u_k'^s}{\partial x_i}\right). \end{equation}

The solenoidal pressure satisfies the following Neumann boundary condition at the wall and the freestream boundary:

(2.24)\begin{equation} \left.\frac{\partial p'^s}{\partial y}\right|_{(0,l_y)}=\frac{1}{Re}\,\frac{\partial \tau_{2k}'^s}{\partial x_k} . \end{equation}

It must be noted that in the absence of gradient of mean density and temperature, $p'^s$ is governed by the incompressible Poisson equation. Thus for a constant base density and temperature profile, the current decomposition of pressure reduces to the formulation presented in Erlebacher et al. (Reference Erlebacher, Hussaini, Kreiss and Sarkar1990).

The dilatational pressure can then be computed by subtracting $p'^s$ from the total pressure fluctuations $p'$:

(2.25)\begin{equation} p'^d=p'-p'^s. \end{equation}

The pressure decomposition proposed is also applied to each eigenmode. Thus, $p'^s$ and $p'^d$ are expressed in the modal form

(2.26a,b)\begin{equation} p'^s=\hat{p}^s(y)\exp({\iota(\alpha x+\beta z-\omega t)}),\quad p'^d=\hat{p}^d(y)\exp({\iota(\alpha x+\beta z-\omega t)}). \end{equation}

The modal form of pressure perturbations is substituted in (2.22) to obtain the following second-order ordinary differential equation (ODE) for $\hat {p}^s$:

(2.27)\begin{equation} \frac{{\rm d}^2\hat{p}^s}{{{\rm d}y}^2}-\left[\frac{1}{\bar{\rho}}\,\frac{{\rm d}\bar{\rho}}{{\rm d} y}\right]\frac{{\rm d}\hat{p}^s}{{\rm d}y}-(\alpha^2+\beta^2)\hat{p}^s=-\hat{\mathcal{S}}(\hat{u}_i^s;\bar{\rho},\bar{U},\bar{\mu}), \end{equation}

where $\hat {\mathcal {S}}(\hat {u}_i^s;\bar {\rho },\bar {U},\bar {\mu })$ is the source term dependent on the solenoidal velocity eigenfunction and the basic state

(2.28)\begin{equation} \hat{\mathcal{S}}=2\iota\alpha\bar{\rho}\,\frac{{\rm d}\bar{U}}{{\rm d} y}\,\hat{v}^s -\frac{1}{Re}\left[2\,\frac{{\rm d}^2\bar{\mu}}{{{\rm d} y}^2}-\frac{2}{\bar{\rho}}\,\frac{{\rm d}\bar{\rho}}{{\rm d} y}\,\frac{{\rm d}\bar{\mu}}{{\rm d} y}\right]\frac{{\rm d}\hat{v}^s}{{\rm d}y}-\frac{1}{Re}\left[2\,\frac{{\rm d}\bar{\mu}}{{\rm d} y}-\frac{\bar{\mu}}{\bar{\rho}}\,\frac{{\rm d}\bar{\rho}}{{\rm d} y}\right]\hat{\nabla}^2\hat{v}^s . \end{equation}

Here, $\hat {\nabla }^2\hat {v}^s=-(\alpha ^2+\beta ^2)\hat {v}^s+\textrm {d}^2\hat {v}^s/{\textrm {d} y}^2$ denotes the Laplacian of $\hat {v}^s$. The solenoidal component of pressure eigenmode is obtained by solving the ODE (2.22) alongside the boundary condition

(2.29)\begin{equation} \frac{{\rm d}\hat{p}^s}{{\rm d} y}=\frac{1}{Re}\left[\bar{\mu}\,\hat{\nabla}^2\hat{v}^s+\frac{{\rm d}\bar{\mu}}{{\rm d} y}\,\frac{{\rm d}\hat{v}^s}{{\rm d} y}\right]. \end{equation}

In high-speed flows, the velocity field develops a dilatational component allowing for pressure to perform work on the velocity field. The kinetic energy can therefore be diverted to internal energy via the pressure-dilatation mechanism (Sarkar Reference Sarkar1992; Mittal & Girimaji Reference Mittal and Girimaji2019). Thus, in dealing with compressible flows, it is important to consider flow–thermodynamic interactions and account for both kinetic and internal energy. In the linear limit, the perturbation kinetic energy contained in the velocity fluctuations and the internal energy in pressure fluctuations (Mittal & Girimaji Reference Mittal and Girimaji2019) are defined as

(2.30a,b)\begin{equation} k=\frac{1}{2}\,\bar{\rho}u_i'u_i',\quad e=\frac{p'p'}{2\gamma\bar{P}}. \end{equation}

The fluctuating kinetic energy can be partitioned into three components using the Helmholtz decomposition of the velocity field:

(2.31)\begin{equation} k=\underbrace{\tfrac{1}{2}\bar{\rho}u_i'^su_i'^s}_{k_s}+\underbrace{\tfrac{1}{2}\bar{\rho}u_i'^du_i'^d}_{k_d}+\underbrace{ \bar{\rho}u_i'^su_i'^d}_{k_{sd}}, \end{equation}

where $k_s$, $k_d$ and $k_{sd}$ are the solenoidal, dilatational and covariance components of kinetic energy, respectively. Here, $k_s$ and $k_d$ quantify the energy in the solenoidal and dilatational velocity field, and are always positive; $k_{sd}$ represents the correlation between the solenoidal and dilatational velocity field, and is not positive definite. Similarly, the internal energy in pressure fluctuations can also be decomposed into solenoidal ($e_s$), dilatational ($e_d$) and covariance ($e_{sd}$) components:

(2.32)\begin{equation} e=\underbrace{\frac{p'^sp'^s}{2\gamma\bar{P}}}_{e_s}+\underbrace{\frac{p'^dp'^d}{2\gamma\bar{P}}}_{e_d}+\underbrace{\frac{p'^sp'^d}{\gamma\bar{P}}}_{e_{sd}}. \end{equation}

2.3. Numerical methodology

The eigenmodes are computed by solving the global eigenvalue problem (2.9) on $N=199$ collocation points using Chebyshev polynomials (Malik Reference Malik1990). The Chebyshev polynomials are defined on the following Gauss–Labato points ($\xi _i$) in the interval $[-1,1]$:

(2.33)\begin{equation} \xi_i=\cos{\frac{{\rm \pi} i}{N}},\quad i=0,1,\ldots, N. \end{equation}

The physical domain ($y \in [0,l_{y}]$) is mapped to the computational domain using an algebraic stretching function (Malik Reference Malik1990)

(2.34)\begin{equation} y=a\,\frac{1+\xi}{b-\xi},\quad\mbox{where } b=1+\frac{2a}{l_{y}}\mbox{ and } a=\frac{y_ll_{y}}{l_{y}-2y_l}. \end{equation}

Here, $l_{y}$ is the length of the physical domain, and the parameter $y_l$ is set equal to half of the 99 % boundary layer thickness ($y_{99}$) in all stability calculations.

For each eigenmode, the boundary value problem (2.14)–(2.15a–d) is solved to compute the potential eigenfunctions $\hat {\phi }$ and $\hat {\varPsi }_i$. The second-order ODE (2.14) is discretized using Chebyshev polynomials in the wall-normal direction. The mode shapes for the solenoidal and dilatational parts of velocity are then obtained from (2.17). Subsequently, the ODE (2.27) is also solved using Chebyshev polynomials to compute $\hat {p}^s$. The mode shape of the dilatational pressure $\hat {p}^d$ is then obtained from the residual of $\hat {p}$ and $\hat {p}^s$.

To verify the accuracy of the modal formulation of Helmholtz decomposition, the velocity eigenfunctions obtained from (2.18) are compared against the standard velocity eigenfunctions obtained from (2.9). The mode shapes for the streamwise and wall-normal velocity at $M=6$ and $Re=4000$ obtained from the two approaches are shown in figure 1. The mode shapes correspond to the most unstable eigenmode at $\alpha =0.175$ and $\beta =0$. The sum total of the solenoidal and dilatational velocity recovers the velocity mode shapes obtained from the standard eigenmode analysis. The profiles of the velocity divergence ($\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u}'$) and vorticity magnitude ($|\boldsymbol {\varOmega }|=|\boldsymbol {\nabla }\times \boldsymbol {u}'|$) are presented in figure 2. It is evident from the figures that the solenoidal part of the velocity field is divergence-free as the dilatational component contributes to all of the dilatation in the flow. Similarly, $\boldsymbol {u}'^d$ is irrotational, and vorticity is entirely contained in the solenoidal mode. It is worth noting that the solenoidal field does not include acoustic waves and is entirely vortical in nature. The dilatational field, on the other hand, is not restricted to acoustic phenomena and incorporates both acoustic and entropy effects (Sagaut & Cambon Reference Sagaut and Cambon2008).

Figure 1. Mode shapes of (a) streamwise velocity ($\hat {u}$) and (b) wall-normal velocity ($\hat {v}$), at $M=6$, $Re=4000$, $\alpha =0.175$ and $\beta =0$. Black solid lines correspond to the mode shapes obtained from the standard eigenmode formulation (2.9). Dashed red lines denote the sum total of the dilatational and solenoidal parts of the velocity.

Figure 2. Mode shapes of the (a) velocity divergence ($\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u}'$) and (b) vorticity magnitude ($|\boldsymbol {\varOmega }|=|\boldsymbol {\nabla }\times \boldsymbol {u}|$), at $M=6$, $Re=4000$, $\alpha =0.175$ and $\beta =0$. Red dashed lines correspond to the (a) divergence and (b) vorticity of the solenoidal velocity $\boldsymbol {u}'^s$. Blue dash-dotted lines represent the (a) divergence and (b) vorticity of the dilatational velocity $\boldsymbol {u}'^d$.

3.1. Most unstable first and second modes

The solenoidal and dilatational contributions to the most unstable first and second modes are considered now. The growth rates of the most unstable first and second modes at different $M$ are shown in figure 3. The first mode is the dominant instability for $M<4$, while the second mode dominates at higher $M$. The most unstable first mode is oblique for $M\geqslant 1$, while the most unstable second mode is always streamwise. Figure 4 plots the solenoidal, dilatational and covariance energy fractions at different $M$ for both first and second modes. The first mode is purely solenoidal as the dilatational and covariance components have negligible energy. This is not surprising, as the first mode instability is a TS wave, which is a vortical instability. The second mode, on the other hand, is dominantly dilatational and has non-negligible solenoidal energy. The solenoidal energy fraction increases with Mach number. The dilatational and solenoidal modes are negatively correlated for the second mode. The covariance value is less than $-5\,\%$ for $M<6$, and gradually increases in magnitude up to $-15\,\%$ at $M=8$. The dominantly dilatational nature of the second mode indicates that flow–thermodynamic interactions are significant for the second mode. For the first mode, however, flow–thermodynamic interactions are not important as the dilatational velocity is negligible.

Figure 3. Growth rates ($\omega _i$) of the most unstable first and second modes at different $M$.

Figure 4. Solenoidal ($k_s^g$), dilatational ($k_d^g$) and covariance ($k_{sd}^g$) components of the average kinetic energy for the most unstable (a) first mode and (b) second mode, at different $M$.

The solenoidal and dilatational parts of the velocity eigenfunctions for the most unstable mode at $M=0.5$ are shown in figure 5. The most unstable mode at $M=0.5$ is a streamwise first mode. The eigenfunctions are normalized by the magnitude of pressure perturbation at the wall. The dilatational velocity is negligible in both the streamwise and wall-normal directions as the first mode is a vortical instability. The streamwise solenoidal velocity has a strong peak near the critical layer ($y_c=0.83$), while the wall-normal solenoidal velocity peaks near the boundary layer edge ($y_{99}=4.95$). Figure 6 plots the eigenfunctions of the dilatational and solenoidal parts of velocity for the most unstable mode at $M=3$. The most unstable mode at $M=3$ is an oblique first mode. Much like the first mode at $M=0.5$, the dilatational contribution to the velocity field is negligible. The solenoidal part of the velocity dominates and has a strong peak near the critical layer ($y_c=3.85$) in the streamwise and spanwise directions. The solenoidal and dilatational parts of the most unstable first mode at $M=6$ are presented in figure 7. The most unstable first mode at $M=6$ is also oblique. Similar to the $M=0.5$ and $M=3$ cases, the velocity field is also dominantly solenoidal for the first mode at $M=6$. The solenoidal and dilatational contributions to velocity for the second mode at $M=6$ are shown in figure 8. Unlike the first mode, the dilatational part dominates for the second mode. Both solenoidal and dilatational components of the streamwise velocity peak at the wall but are of opposite sign. It must be noted that $\hat {u}^s$ and $\hat {u}^d$ do not satisfy the no-slip condition independently. However, the sum of the components is zero at the wall. Physically, it is not necessary for each component to satisfy the no-slip boundary condition independently as long as the total field satisfies the no-slip condition (Jenvey Reference Jenvey1989). Moreover, enforcing the condition separately on each component can misrepresent the coupling and phase relationship between the different components at the boundary. The dilatational contribution to streamwise velocity decays gradually from its peak value at the wall. On the other hand, $|\hat {u}^s|$ decreases sharply near the wall before increasing beyond $y=3$. The minimum of $|\hat {u}^s|$ corresponds to the location of peak streamwise velocity ($\hat {u}$). The solenoidal component of streamwise velocity has a local maximum near the sonic line ($y_s=7$) and is comparable to $\hat {u}_d$ beyond the sonic line. The dilatational part of the wall-normal velocity peaks near the sonic line and is the dominant component inside the boundary layer. The solenoidal part of $\hat {v}$ peaks near the boundary layer edge ($y_{99}=15.3$) and is comparable to $\hat {v}^d$ outside the boundary layer. Overall, the dilatational effects and hence flow–thermodynamic interactions are most prominent below the sonic line.

Figure 5. Mode shapes of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$) and (b) wall-normal velocity ($\hat {v}$), for the most unstable first mode at $M=0.5$.

Figure 6. Mode shapes of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$), (b) wall-normal velocity ($\hat {v}$) and (c) spanwise velocity velocity ($\hat {w}$), for the most unstable first mode at $M=3.0$.

Figure 7. Modes shapes of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$), (b) wall-normal velocity ($\hat {v}$) and (c) spanwise velocity ($\hat {w}$), for the most unstable first mode at $M=6.0$.

Figure 8. Modes shapes of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$) and (b) wall-normal velocity ($\hat {v}$), for the most unstable second mode at $M=6.0$.

As mentioned earlier, the potential energy contained in pressure fluctuations plays an important role in high-speed flows. The energy partition between the kinetic and internal modes is quantified by the parameters $f_{k}$ and $f_{d}$ defined by

(3.3a,b)\begin{equation} f_k=\frac{e^g}{e^g+k^g},\quad f_d=\frac{e^g}{k_d^g+e^g}. \end{equation}

Equipartition between the dilatational kinetic and internal energy has been observed previously in decaying compressible (Sarkar et al. Reference Sarkar, Erlebacher, Hussaini and Kreiss1991; Lee & Girimaji Reference Lee and Girimaji2013) and forced compressible (Jagannathan & Donzis Reference Jagannathan and Donzis2016) turbulence at high $M_t$. These flows are dominated by broad spectra and nonlinear interactions. Here, we will examine the energy partition in the linear limit. Figure 9 plots the fraction with respect to total kinetic ($\,f_k$) and dilatational kinetic ($\,f_d$) energy for the most unstable first and second modes at different $M$. For the first mode, the internal energy content is negligible compared to the total kinetic energy. However, the internal energy content is significantly larger than the dilatational kinetic energy. In contrast to the first mode, the second mode has substantial internal energy content (over 25 %). The dilatational kinetic energy of the second mode is at least 70 % higher than $e^g$ at all $M$. Both the energy fractions decrease slightly with $M$ for the second mode.

Figure 9. Plots of (a) $f_k$ and (b) $f_d$ energy fractions for the most unstable first and second modes at different Mach numbers.

Generally, the perturbation field draws energy from the mean flow by the instability-enabled production mechanism. To gain further insight on the instability mechanisms, the production of total kinetic energy ($P_k$) is decomposed into solenoidal ($P_{ss}$), dilatational ($P_{dd}$) and cross ($P_{sd}$) components as follows:

(3.4)\begin{equation} P_k=-\bar{\rho}u_i'u_j'\,\frac{\partial\bar{U}_i}{\partial x_j}=\underbrace{-\bar{\rho}u_i'^su_j'^s\,\frac{\partial\bar{U}_i}{\partial x_j}}_{P_{ss}}\underbrace{{}-\bar{\rho}u_i'^du_j'^d\,\frac{\partial\bar{U}_i}{\partial x_j}}_{P_{dd}}\underbrace{{}-\bar{\rho}(u_i'^su_j'^d+u_i'^du_j'^s)\,\frac{\partial\bar{U}_i}{\partial x_j}}_{P_{sd}}. \end{equation}

The global averages of the production components $P_{ss}^g$, $P_{dd}^g$ and $P_{sd}^g$ are shown in figure 10. The solenoidal production is the dominant component of $P_k^g$ for the first mode. The cross production is negative for the first mode and increases in magnitude with $M$, whereas the dilatational part is negligible. The magnitude increase of the cross production is partially responsible for the reduced growth rate of the first mode at high Mach numbers. The dilatational component of production contributes significantly to the total production of the second mode. The cross production is positive and is the dominant contributor to total production for the second mode. The positive nature of $P_{sd}^g$ leads to much higher growth rate of the second mode compared to the first mode. The profiles of $P_{ss}$, $P_{dd}$ and $P_{sd}$ at $M=6$ are shown for both first and second mode in figure 11. Most of the kinetic energy production for the first mode occurs in a region around the critical layer ($y_c=11.94$). The production components in the near‐wall region are negligible for the first mode. The solenoidal component of production peaks at the critical layer and drives the first mode instability. The cross component of production is always negative and inhibits the instability growth. On the other hand, for the second mode, $P_{sd}$ is positive in the majority of the boundary layer and is the dominant production mechanism beyond the sonic line. The cross production peaks near the generalized inflection point ($y_I=13.11$) for the second mode. The dilatational component of production is significant only in the near‐wall region below the sonic line. To better assess the role of second mode production physics in different regions of the boundary layer, the profiles of solenoidal and dilatational kinetic energy are shown in figure 12(a). The second-mode acoustic impedance (Kinsler et al. Reference Kinsler, Frey, Coppens and Sanders2000; Kuehl Reference Kuehl2018) based on linear stability $Z=p'/v'$ is also presented in figure 12(b). An acoustic impedance well is formed between the wall (infinite impedance) and the secondary peak in impedance near the generalized inflection line. In a recent study, Kuehl (Reference Kuehl2018) has demonstrated that acoustic energy in the second mode is trapped between the increasing impedances on the sides of the well. Consistent with the thermoacoustic interpretation of Kuehl (Reference Kuehl2018), we observe that dilatational kinetic energy dominates in the well region. In the region below the sonic line, kinetic energy is primarily dilatational and the ensuing wave motion is sustained by the dilatational production. Beyond the sonic line, as dilatational effects gradually weaken, the instability derives energy via the cross production mechanism.

Figure 10. Solenoidal ($P_{ss}^g$), dilatational ($P_{dd}^g$) and cross ($P_{sd}^g$) components of the averaged production for the most unstable (a) first mode and (b) second mode, at different $M$.

Figure 11. Profiles of the production components for the most unstable (a) first mode and (b) second mode, at $M=6$. The sonic and generalized inflection lines for the second mode are marked using dashed lines.

Figure 12. (a) Solenoidal and dilatational kinetic energy profiles $k_s, k_d$ for the second mode at $M=6$. (b) The acoustic impedance $Z$ based on linear stability theory for the second mode at $M=6$. The sonic and generalized inflection lines are marked using dashed lines.

The effect of obliqueness ($\psi =\tan ^{-1}(\beta /\alpha )$) on the kinetic energy components $k_{s}^g$, $k_{d}^g$ and $k_{sd}^g$ is considered next. The energy components as a function of $\psi$ for both first and second modes at $M=6$ are presented in figure 13. The solenoidal kinetic energy is the dominant component for first mode at all $\psi$. The dilatational kinetic energy is non-negligible for the streamwise first mode, and gradually decreases with increasing obliqueness. The covariance component for the first mode is negative and also decreases in magnitude as $\psi$ increases. The dilatational energy content decreases and the solenoidal contribution increases significantly for oblique second modes. Overall, the dilatational kinetic energy decreases with increasing obliqueness for both first and second modes. This finding is consistent with the behaviour of oblique perturbations reported in compressible homogeneous shear flows (Kumar et al. Reference Kumar, Bertsch and Girimaji2014).

Figure 13. Effect of obliqueness on the solenoidal, dilatational and covariance components of kinetic energy for (a) first mode and (b) second mode, at $M=6$.

3.2. Fast and slow modes

We now examine the solenoidal and dilatational contributions to the fast and slow modes. At the long-wave limit ($\alpha \to 0$), two discrete modes (fast and slow) are synchronized with the fast/slow acoustic wave ($c_{a\pm }=1\pm 1/M$) speeds (Fedorov & Tumin Reference Fedorov and Tumin2011). The phase speed of the fast mode decreases with increasing $\alpha$, and it synchronizes with the slow mode. The synchronization leads to the branching of the discrete spectrum and causes the second mode instability (Fedorov & Tumin Reference Fedorov and Tumin2011). The fast and slow modes exhibit a peak/trough in growth rates beyond the synchronization point. The growth rates of the fast and slow modes at $M=4$ and $M=6$ are shown in figure 14. The slow mode is unstable at low wavenumbers for both $M$ values. The synchronization between the discrete modes occurs at $\alpha \approx 0.3$ for $M=4$, and $\alpha \approx 0.15$ for $M=6$. The branching pattern of the discrete spectrum is different at $M=4$ and $M=6$. The fast mode after synchronization with the slow mode is the dominant instability at $M=4$. On the other hand, the slow mode becomes the dominant instability at $M=6$ after synchronization. Therefore, the second mode is the fast mode at $M=4$, whereas the slow mode is associated with the second mode at $M=6$.

Figure 14. Growth rates of the fast and slow modes at (a) $M=4$ and (b) $M=6$.

The kinetic energy components $k_s^g$, $k_d^g$ and $k_{sd}^g$ for the fast and slow modes at $M=4$ are shown in figure 15. The fast mode is dominantly dilatational at all wavenumbers. Initially, the solenoidal energy content of the slow mode increases considerably with $\alpha$ as it becomes unstable. As the slow mode stabilizes ($0.1<\alpha <0.3$), its solenoidal energy component decreases and the dilatational contribution increases. The solenoidal energy content of the slow mode increases again rapidly once it synchronizes with the fast mode. The slow mode is therefore dominantly solenoidal at high wavenumbers ($\alpha \geqslant 0.32$) where the second mode is unstable. For the slow mode at low $\alpha$, the covariance component of kinetic energy is positive and decreases gradually, becoming negative for $\alpha >0.05$. The covariance energy component is always negative for the fast mode. The solenoidal and dilatational parts of both fast and slow modes are uncorrelated beyond the synchronization point. Figure 16 plots the kinetic energy components for the discrete modes at $M=6$. Similar to the $M=4$ case, the fast mode is dominantly dilatational at all $\alpha$ for $M=6$ as well. At low wavenumbers ($\alpha <0.06$), the solenoidal contribution to kinetic energy increases with $\alpha$ for the slow mode as it becomes unstable. The solenoidal energy content of the slow mode decreases for $\alpha \in (0.06,0.15)$, and it becomes dilatationally dominant at the synchronization point ($\alpha \approx 0.15$). Contrary to the $M=4$ case, beyond the synchronization point, the solenoidal contribution to the kinetic energy of the slow mode decreases further at $M=6$. Consequently, in the region of second-mode instability ($\alpha \in [0.15,0.19]$), the slow mode is also dominantly dilatational at $M=6$. The dilatational and solenoidal parts of both modes are uncorrelated beyond the synchronization point.

Figure 15. (a) Solenoidal ($k_s^g$), (b) dilatational ($k_d^g$) and (c) covariance ($k_{sd}^g$) components of kinetic energy for the fast and slow modes at $M=4$. Black and red lines with symbols correspond to the fast and slow modes, respectively.

Figure 16. (a) Solenoidal ($k_s^g$), (b) dilatational ($k_d^g$) and (c) covariance ($k_{sd}^g$) components of kinetic energy for the fast and slow modes at $M=6$. Black and red lines with symbols correspond to the fast and slow modes, respectively.

We now examine the mode shapes of the solenoidal and dilatational parts of the velocity for the fast and slow modes at different $\alpha$. The mode shapes at $M=6$ and $\alpha =0.05$ are presented in figure 17. For the fast mode, the solenoidal part of the streamwise velocity peaks near the generalized inflection point ($y_I=13.3$), while the dilatational contribution to streamwise velocity is high in the near‐wall region. Generally, the dilatational component of the fast mode is more prominent than the solenoidal part. On the contrary, for the slow mode, the solenoidal part $u'^s$ is greater than the dilatational part in the majority of the boundary layer. The dilatational contribution to streamwise velocity is significant only in the near‐wall region ($y\leq 5$), and decays away from the wall. The dilatational part of the wall-normal velocity is greater than $\hat {v}^s$ throughout the boundary layer for the fast mode. For the slow mode, the dilatational contribution to wall-normal velocity dominates near the wall ($y\leq 5$). The solenoidal contribution is stronger away from the wall, and peaks near the boundary layer edge.

Figure 17. Profiles of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$) and (b) wall-normal velocity ($\hat {v}$), for the fast and slow modes at $M=6$, $\alpha =0.05$ and $\beta =0$. Black lines denote the components of the fast mode, and red lines correspond to the slow mode components. Solid lines and dash-dotted lines with symbols represent the solenoidal and dilatational parts of velocity, respectively.

Figure 18 plots the mode shapes just before the discrete spectrum branching at $M=6$ and $\alpha =0.15$. It is evident from the figure that the profiles for both the fast and slow mode are similar. The dilatational component of streamwise velocity is strongest near the wall and dominates below the sonic line ($y_s=7.64$). The solenoidal contribution to streamwise velocity exhibits a phase change at $y=3$. The dilatational component of wall-normal velocity peaks near the sonic line. The solenoidal part $\hat {v}^s$ has a phase change near the sonic line and increases beyond the sonic line peaking at the boundary layer edge.

Figure 18. Profiles of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$) and (b) wall-normal velocity ($\hat {v}$), for the fast and slow modes before the branch point at $M=6$, $\alpha =0.15$ and $\beta =0$. Black lines denote the components of the fast mode, and red lines correspond to the slow mode components. Solid lines and dash-dotted lines with symbols represent the solenoidal and dilatational parts of velocity, respectively.

The mode shapes beyond the synchronization point and near the peak/trough of growth rates at $M=6$ and $\alpha =0.175$ are shown in figure 19. The mode shapes of the slow mode at peak growth rate are similar to the slow mode profiles just before the branch point. Both solenoidal and dilatational contributions to the fast mode also exhibit similar profiles before and after the synchronization point besides the critical point ($y_c\approx 13.5$). The solenoidal part of the fast mode after synchronization exhibits a local maximum near the critical layer ($y_c=13.5$). Finally, the mode shapes for the solenoidal and dilatational part of the fast and slow mode near peak growth rate at $M=4$ are presented in figure 20. The fast mode at $M=4$ is unstable at high wavenumbers and, much like the $M=6$ case, is dilatationally dominant with strong dilatational motions in the near wall region. The solenoidal part of the slow mode, however, exhibits a strong peak near the critical layer, leading to the dominantly solenoidal nature of the slow mode beyond the synchronization point.

Figure 19. Profiles of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$) and (b) wall-normal velocity ($\hat {v}$) for the fast and slow modes near peak growth rate at $M=6$, $\alpha =0.175$ and $\beta =0$. Black lines denote the components of the fast mode, and red lines correspond to the slow mode components. Solid lines and dash-dotted lines with symbols represent the solenoidal and dilatational parts of velocity, respectively.

Figure 20. Profiles of the solenoidal and dilatational parts of (a) streamwise velocity ($\hat {u}$) and (b) wall-normal velocity ($\hat {v}$) for the fast and slow modes near peak growth rate at $M=4$, $\alpha =0.35$ and $\beta =0$. Black lines denote the components of the fast mode, and red lines correspond to the slow mode components. Solid lines and dash-dotted lines with symbols represent the solenoidal and dilatational parts of velocity, respectively.

In summary, dilatational contributions to the fast mode dominate at all $\alpha$ for both $M=4$ and $M=6$. At low $\alpha$, in the region of first-mode instability, the solenoidal contribution to slow mode dominates for both $M=4$ and $M=6$. Beyond the synchronization point, the slow mode exhibits contrasting characteristics at $M=4$ and $M=6$. The slow mode is dominantly solenoidal at $M=4$ and dilatational at $M=6$.

3.3. Momentum decomposition

In this subsection, we examine the character of the perturbation momentum field ($m_i'$). The profiles of the solenoidal and dilatational components of $m_i'$ for the most unstable first mode at $M=6$ are shown in figure 21. The streamwise momentum perturbation ($m_x'$) is much larger than the wall-normal and spanwise components. The streamwise perturbation $m_x'$ is dominantly solenoidal, and the dilatational contribution is negligible. The solenoidal part $m_x'^s$ peaks near the critical layer ($y_c=11.94$). The solenoidal and dilatational parts of the wall-normal and spanwise momentum perturbations are similar in magnitude. Figure 22 plots the profiles of the solenoidal and dilatational components of $m_i'$ for the second mode at $M=6$. We observe that even for the second mode, the solenoidal contribution to $m_x'$ is much larger than the dilatational part. The observation is consistent with the findings of Unnikrishnan & Gaitonde (Reference Unnikrishnan and Gaitonde2019), wherein vortical content of momentum perturbations is shown to dominate for both first and second modes. Although the solenoidal contribution to $m_x'$ dominates for both modes, the dilatational part is more energetic for the second mode compared to the first mode. As in the case of the first mode, the dilatational and solenoidal parts of wall-normal momentum perturbations are equally energetic for the second mode.

Figure 21. Profiles of the solenoidal and dilatational parts of (a) streamwise ($m_x'$), (b) wall-normal ($m_y'$) and (c) spanwise momentum perturbations ($m_z'$), for the most unstable first mode at $M=6$.

Figure 22. Profiles of the solenoidal and dilatational parts of (a) streamwise ($m_x'$) and (b) wall-normal ($m_y'$) momentum perturbations for the most unstable second mode at $M=6$.

Overall, both momentum density and velocity perturbations are dominantly solenoidal for the first mode. However, the momentum perturbation characteristics of the second mode are different from the velocity fluctuations. The velocity field is dominantly dilatational for the second mode, whereas the solenoidal contribution to momentum is larger.

5.1. First mode

The effect, or lack thereof, of the flow-thermodynamics interactions on the first mode can be summarized as follows.

1. (i) As anticipated, the first mode is nearly exclusively solenoidal, and in general the obliqueness angle of the most unstable perturbation increases with Mach number. The velocity eigenfunctions of the first mode are almost entirely vortical.

2. (ii) The dilatational field is generally small and decreases with obliqueness angle of the perturbation.

3. (iii) At high speeds, the small dilatational field has a marked effect on the instability manifesting via the cross production – which arises from dilatational–solenoidal velocity covariance interacting with the mean flow. The cross production is negative and inhibits the first-mode growth.

4. (iv) The internal energy associated with the first mode is generally very small. The solenoidal pressure makes the highest contribution toward this internal energy.

5. (v) The pressure-dilatation is generally very small compared to production.

5.2. Second mode

The second or Mack mode arises due to flow–thermodynamics interactions, and the details are as follows.

1. (i) The second mode is dominantly dilatational, but the solenoidal part is not insignificant. The dilatational effects are strongest in the near‐wall region below the sonic line. Beyond the sonic line, both solenoidal and dilatational contributions to the velocity field are comparable for the second mode.

2. (ii) The most unstable second mode is always oriented in the streamwise direction. Once again, the intensity of the dilatational field diminishes with obliqueness angle.

3. (iii) The flow–thermodynamics effect manifests through the dilatational and cross production term. Beyond the sonic line where solenoidal and dilatational effects are comparable, the cross production is the dominant production mechanism of the second mode. The acoustic waves trapped below the sonic line (Kuehl Reference Kuehl2018) are sustained by the dilatational production.

4. (iv) The internal energy of the second mode is substantial and can be as much as 40 % of the kinetic energy. Thus a significant fraction of kinetic energy extracted from the mean flow is diverted to the internal energy of perturbations. For the second mode, the pressure near the wall up to the sonic line is dominantly dilatational. Beyond the sonic line, the solenoidal and dilatational contributions are comparable.

5. (v) The dilatational contribution to pressure-dilatation is significant in the near‐wall region and transfers energy from kinetic to internal form. The transfer facilitated by this mechanism provides an energy source for near‐wall heating.

5.3. Fast and slow modes

The compressibility effects on the discrete modes (fast and slow) are dependent on $M$. The fast mode is dilatationally dominant for both $M=4$ and $M=6$. In the region of first-mode instability, the solenoidal contribution to the slow mode dominates. Beyond the synchronization point, the slow mode has opposing characteristics at $M=4$ and $M=6$. The slow mode is dominantly solenoidal at $M=4$, whereas the dilatational contribution dominates at $M=6$.

The findings of this study highlight clearly the fundamental differences between the flow physics of low-speed and high-speed instability, and the transition process. It is evident that transition prediction tools must account for the differences.