3.1. Extension of velocity-transformations to 3-D non-equilibrium flows

Morkovin’s (Reference Morkovin1962) hypothesis has inspired various velocity transformations for statistically stationary 2-D flows, like channels and boundary layers, using mean property variations to collapse the compressible mean velocity onto an equivalent incompressible mean velocity in the near-wall region (Van Driest Reference Van Driest1951; Zhang et al. Reference Zhang, Bi, Hussain, Li and She2012; Trettel & Larsson Reference Trettel and Larsson2016; Griffin et al. Reference Griffin, Fu and Moin2021; Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023). However, extensions to statistically 3-D or temporally varying compressible flows have not yet been applied in the literature. Previous studies in transient 3-D non-equilibrium incompressible flows have shown that the initial $u_{\tau }$ and $\ell _{\nu }$ do not appropriately collapse the near-wall $\overline {u}$ (Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020), while rescaling the 3-D velocity magnitude, $\|\overline {\boldsymbol{u}}\|$ , with the local $u_{\tau }(t)$ and $\ell _{\nu }(t)$ can collapse the near-wall statistics to the canonical counterpart (Moin et al. Reference Moin, Shih, Driver and Mansour1990). This suggests that an appropriate compressible velocity transformation for the near-wall region must be local in time and account for the variation in the transport properties.

First, the mean stress balance is projected along the mean flow direction as

(3.1) \begin{equation} {\boldsymbol{e}_{s}\boldsymbol{\cdot }\int _{0}^{y}\frac {\partial \overline {\rho }\widetilde {\boldsymbol{u}}}{\partial t}{\rm d}y + \boldsymbol{e}_{s}\boldsymbol{\cdot }(\overline {\rho }\widetilde {v}\widetilde {\boldsymbol{u}} + \overline {\rho }\widetilde {v''\boldsymbol{u}''}) = \boldsymbol{e}_{s}\boldsymbol{\cdot }(\overline {{\tau }}\boldsymbol{e}_{y}- \boldsymbol{\tau }_{w}) + \boldsymbol{e}_{s}\boldsymbol{\cdot }\int _{0}^{y}\overline {\rho }\boldsymbol{g}{\rm d}y}. \end{equation}

Now this stress balance is considered in a region where $y\ll h$ . In this region, $\overline {\rho }\widetilde {v}\widetilde {\boldsymbol{u}} + \overline {\rho }\widetilde {v''\boldsymbol{u}''}$ is negligible because of the no slip condition. The no slip condition also causes $\int _{0}^{y}\overline {\rho }\boldsymbol{g}{\rm d}y \sim {O}(y)$ and $\int _{0}^{y}{\partial \overline {\rho }\widetilde {\boldsymbol{u}}}/ \partial {t}\,{\rm d}y \lesssim {O}(y)$ . Furthermore, $\overline {{\tau }}\boldsymbol{e}_{y}\approx \overline {\mu }\partial _{y}\widetilde {\boldsymbol{u}}$ such that the dominant stress balance is $\overline {\mu }\partial _{y}\|\widetilde {\boldsymbol{u}}\| = \boldsymbol{e}_{s}\boldsymbol{\cdot }\tau_{w}$ . This implies that the near-wall $\|\widetilde {\boldsymbol{u}}\|$ should be agnostic to the spanwise acceleration and determined only by its viscous scales, $u_{\tau }(t)$ and $\ell _{\nu }(t)$ , and the near-wall mean property variations. Since the time derivatives are negligible here, the transport properties and viscous scales can be assumed to be quasisteady. In figure 3(a,b) $\|\widetilde {\boldsymbol{u}}\|$ is plotted using the incompressible viscous scaling by normalizing with $u_{\tau }(t)$ and $\ell _{\nu }(t)$ . These wall-evaluated scales are not capable of ensuring near-wall self-similarity as even near $y/\ell _{\nu }(t) = 10$ , the $\textit{Ma} = 3.0$ profiles begin to depart from the incompressible law-of-the-wall. While outer-layer self-similarity is not ensured with these viscous scales, even for an incompressible flow (Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020), these wall-scaled flows exhibit a clear ${\textit{Ma}}$ dependence in the wakes seen most prominently in the $\varPi = 10$ cases in figure 3(a).

Figure 3.

Quasisteady friction scaling of $\|\widetilde {\boldsymbol{u}}\|$ for $\varPi = 10$ (a) and $\varPi = 40$ (b). Here $\widetilde {T}$ normalized by $T_{w}$ with $y$ normalized by $\ell _{\nu }(t)$ (c). The TL transformation of $\|\widetilde {\boldsymbol{u}}\|$ for $\varPi = 10$ (d) and $\varPi = 40$ (e) and friction scaling of $\widetilde {T}$ by normalizing with $T_{\tau }(t)$ (f). The colours and line styles follow from table 1. The colours from dark to light are offset by time increments of $\Delta t^{+} = 100$ . In (a), (b), (d) and (e), the green line plots the viscous sublayer of an incompressible flow as $u^{+} = y^{+}$ and the green dashed line plots the mean velocity of an ${\textit{Re}}_{\tau } = 550$ turbulent channel (Lee & Moser Reference Lee and Moser2015).

Based on the mean-stress balance and observations of near-wall self-similarity in the incompressible spanwise accelerated flow (Moin et al. Reference Moin, Shih, Driver and Mansour1990), a velocity transformation is pursued using the Trettel & Larsson (Reference Trettel and Larsson2016) (TL) velocity transformation. In Gomez (Reference Gomez2025), the Griffin et al. (Reference Griffin, Fu and Moin2021) (GFM) velocity transformation was presented. By construction, the GFM and TL transformations are equivalent near the wall where the self-similarity is expected to hold. With the TL transformation, the compressible mean stress balance is assumed to map to an equivalent incompressible balance as $\mu _{w}{\mathrm{d} \|\widetilde {\boldsymbol{u}}\|_{TL}}/{{\rm d}}{y^{*}} = \overline {\mu }\ell _{\nu }/u_{\tau }\,{\partial \|\widetilde {\boldsymbol{u}}\|}\partial {y} = 1$ where $y^{*} = \sqrt {\overline {\rho }/\rho _{w}}\mu _{w}/\overline {\mu } y/\ell _{\nu }(t)$ is the semilocal coordinate and $\|\widetilde {\boldsymbol{u}}\|_{TL}$ the transformed velocity. Following arguments from Trettel & Larsson (Reference Trettel and Larsson2016), the velocity transformation is then

(3.2) \begin{equation} {\|\widetilde {\boldsymbol{u}}\|_{TL} = \frac {1}{u_{\tau }}\int _{0}^{y}\sqrt {\frac {\overline {\rho }}{\rho _{w}}}\bigg (1 + \frac {1}{2}\frac {1}{\overline {\rho }}\frac {\partial \overline {\rho }}{\partial y} y' - \frac {1}{\overline {\mu }}\frac {\partial \overline {\mu }}{\partial y}y'\bigg )\frac {\partial \|\widetilde {\boldsymbol{u}}\|}{\partial y}{\rm d}y',}\, \end{equation}

where all the properties and wall quantities are evaluated locally in time. This transformation omits any viscous-scaling in $t$ since temporal gradients are not present in the mean stress balance close to the wall. The Zhang et al. (Reference Zhang, Bi, Hussain, Li and She2012) and Van Driest (Reference Van Driest1951) velocity transformations do not create an analogous transformation between the compressible and incompressible mean stress balance, thus they are not applicable here. In figure 3(d,e), the TL transformation is applied, illustrating an improved near-wall collapse to the wall scaling from figure 3(a,b). For the $\varPi = 10$ cases, the TL transformation mitigates the ${\textit{Ma}}$ dependence in the wakes collapsing them closer to the incompressible canonical flow in green. While the collapse in the near-wall region improves for the $\varPi =40$ case up to $y^{*}=10$ by reducing the wake and bringing the buffer layer closer to the wall for the supersonic cases, the rapid evolution in the wake limits the transformation from extending farther from the wall. As evidenced in (3.1), away from the wall the acceleration terms are dominant and not governed by viscous scales. Thus, velocity transformations relying on viscous scales cannot collapse the wake in strongly accelerated flows, even for incompressible flows. However, wake-deficit scaling with outer units was shown to collapse the wake in Gomez (Reference Gomez2025).

Following the success of the quasisteady scaling in the velocities and the use of the friction temperature in the literature (Kader Reference Kader1981; Kong et al. Reference Kong, Choi and Lee2000; Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2016), a quasisteady friction temperature, $T_{\tau }$ , is presented. The assumption here is that the near-wall temperature is governed by the total wall-shear stress such that $T_{\tau }(t) = q_{w}(t)/(c_{p}\rho _{w}(t)u_{\tau }(t))$ . This temperature scale is presented in figure 3(f) for $\widetilde {T}$ where the wall-normal coordinate is normalized by $\ell _{\nu }(t)$ . The near-wall collapse in $\widetilde {T}$ suggests that the near-wall temperature is also influenced by its friction scales and mean flow direction. The $\textit{Ma} = 1.5$ and $\textit{Ma} = 0.3$ cases also exhibit a near-wall $\widetilde {T}$ peak for $\varPi = 40$ . The emergence of this peak is casused by elevated viscous heating due to the significant increase in turbulent intensities attributed with the $\varPi = 40$ case after the initial decrease discussed in § 4.1.

3.2. A self-similar velocity transformation for the spanwise velocity

The $x$ - and $z$ -averaged spanwise momentum equation is

(3.3) \begin{equation} \frac {\partial }{\partial t}(\overline {\rho }\widetilde {w}) + \frac {\partial }{\partial y}(\overline {\rho }\widetilde {w''v''}) + \frac {\partial }{\partial y}(\overline {\rho }\widetilde {w}\widetilde {v}) - \overline {\rho }g_{z} = \frac {\partial \overline {\tau }_{yz}}{\partial y}= \frac {\partial }{\partial y}\bigg [\overline {\mu }\frac {\partial \overline {w}}{\partial y} + \overline {\mu ' \frac {\partial w'}{\partial y}} + \overline {\mu ' \frac {\partial v'}{\partial z}}\bigg ] . \end{equation}

Due to mass conservation and an unsteady $\overline {\rho }$ , $\widetilde {v}$ is non-zero for $t\gt 0$ . However, the magnitude of $\widetilde {v}$ remains negligible compared with $\widetilde {w}$ and $\widetilde {u}$ and its contribution is omitted for the rest of this section. For $t\leqslant 0$ , the spanwise Reynolds shear stress, $\overline {\rho }\widetilde {w''v''}$ , and spanwise turbulent viscous stresses, $\overline {\mu '\partial _{y}w'} + \overline {\mu '\partial _{z} v'}$ , are initially zero due to the lack of net transport in the spanwise homogeneous direction and spanwise homogeneity. As time advances, the development of $\widetilde {w}$ will lead to the generation of $\overline {\rho }\widetilde {v''w''}$ (Moin et al. Reference Moin, Shih, Driver and Mansour1990; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020) and turbulent viscous stresses via the net transport of $\mu '$ in $\boldsymbol{e}_{z}$ through $\overline {\mu '\partial _{y}w'}$ , though the latter are negligible (Bradshaw Reference Bradshaw1977).

The time evolution of these terms is shown in figure 4(a–c) for $\varPi = 40$ and the three ${\textit{Ma}}$ . The acceleration term, $\partial _{t}(\overline {\rho }\widetilde {w})$ , is estimated from the saved snapshots using finite differencing. The acceleration term primarily balances $\overline {\rho }g_{z}$ in the outer region of the flow, while for short times, the viscous stress balances $\overline {\rho }g_{z}$ near the wall. As time advances, $\overline {\rho }\widetilde {v''w''}$ intensifies, balancing the viscous stresses. This can be seen more clearly in figure 4(d–f) where the magnitudes of each of the terms is integrated in $y$ to measure their effect across the channel. These plots show a period where the spanwise Reynolds stresses are negligible. An estimate for $\overline {\tau }_{yz}$ via $\overline {\mu }\partial _{y}\overline {w}$ is included, demonstrating that the turbulent viscous stresses are indeed negligible (Bradshaw Reference Bradshaw1977). Hence, for a short time interval, the effect of the Reynolds stresses and turbulent viscous stresses on the spanwise momentum equation may be neglected. For the supersonic cases, this interval extends much longer than the $\textit{Ma} = 0.3$ case due to the increase in near-wall viscosity that dampens the production of the spanwise Reynolds shear stress. The time interval, $t_{I}$ , is defined as the time where $\int _{0}^{2h}\left |\partial _{y}\overline {\tau }_{yz}\right |{\rm d}y = \int _{0}^{2h}\left |\partial_{y}(\overline{\rho}\widetilde{w''v''})\right |{\rm d}y$ , that is the time where the viscous stress is no longer dominant over the Reynolds stress.

Figure 4.

Spanwise mean momentum balance for $\varPi = 40$ and $\textit{Ma} = 0.3$ (a,d), $1.5$ (b,e), $3.0$ (e,f). In (a)–(c), the colours from dark to light denote different time instances, in increments of $\Delta t^{+} = 100$ . The quantity, $f$ , is labelled in the legend of (a).

To model the initial development of the spanwise mean flow for $t \ll {t_{I}}$ , several assumptions are made. Following the observations from figure 4, it is assumed that the turbulent viscous stresses can be neglected such that $\overline {\tau }_{yz} \approx \overline {\mu } \partial _{y}\widetilde {w}$ and that $\overline {\rho }\widetilde {v''w''}$ is negligible compared with $\overline {\tau }_{yz}$ . The third assumption is that the mean properties are quasisteady. This means that $\overline {\rho }$ and $\overline {\mu }$ are treated as constant in time with their values updated locally at each time and their temporal gradients are negligible. Finally, it is assumed that density fluctuations play a minor role such that $\overline {w} \approx \widetilde {w}$ . With these approximations, (3.3) can be simplified as

(3.4) \begin{equation} \frac {\partial \widetilde {w}}{\partial t} - g_{z} = \frac {1}{\overline {\rho }}\frac {\partial }{\partial y}\bigg [\overline {\mu }\frac {\partial \widetilde {w}}{\partial y}\bigg ] . \end{equation}

These assumptions essentially assume that a laminar prediction for $\widetilde {w}$ holds. This has been shown to hold only for short times in incompressible simulations similar to the cases studied here (Moin et al. Reference Moin, Shih, Driver and Mansour1990; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020) and for spanwise oscillating walls in drag reduction (Quadrio & Sibilla Reference Quadrio and Sibilla2000; Choi et al. Reference Choi, Xu and Sung2002). The extent of the laminar prediction is expected to hold for a much longer time for the supersonic cases because of the larger $t_{I}$ .

In Gomez (Reference Gomez2025), (3.4) was solved using separation of variables and a series solution with the eigenmodes of $1/\overline {\rho }\partial _{y}[\overline {\mu }\partial _{y} ]$ . The series solution is an exact solution to (3.4) under the assumptions described above. Here, a similarity solution is presented which provides a velocity transformation for $\widetilde {w}$ . First, it is assumed that

(3.5) \begin{equation} \widetilde {w} = g_{z} t f(\eta ) \end{equation}

where

(3.6) \begin{equation} \eta (y,t) = \sqrt {\frac {\overline {\rho }(y) \overline {\mu }(y)}{t}} \int _{0}^{y}\frac {{\rm d}\xi }{\overline {\mu }(\xi )} \end{equation}

is the similarity variable and the explicit time dependence of $\overline {\mu }$ and $\overline {\rho }$ is neglected. For an incompressible flow with constant transport properties, $\eta _{inc}(y,t) = y/\sqrt {\overline {\nu }t}$ , where $\nu = \mu /\rho$ is the kinematic viscosity, is the same similarity variable as that used in Stoke’s first problem (Schlichting & Gersten Reference Schlichting and Gersten2016). The similarity variable $\eta$ is different to the one introduced by the Lees–Dorodnitsyn transformation (Dorodnitsyn Reference Dorodnitsyn1942; Lees Reference Lees1956; Anderson Jr. Reference Anderson2006; Schlichting & Gersten Reference Schlichting and Gersten2016), even when converting $x$ to $u_{e}t$ , because the quantity in the square-root is wall-normally varying and the integrand is $\overline {\mu }^{-1}$ rather than $\overline {\rho }$ . It can then be shown that using (3.5), (3.4) becomes

(3.7) \begin{equation} f - \frac {\eta }{2}\frac {\mathrm{d} f}{\mathrm{d} \eta } -1 = \frac {{\rm{d}}^{2}{f} }{\mathrm{d} {\eta }^{2}}. \end{equation}

Note that (3.7) is the same for both incompressible and compressible flow. The difference in the solutions is the similarity variable, $\eta$ . Reducing (3.5) to the same incompressible similarity equation via $\eta$ demonstrates the applicability of Morkovin’s (Reference Morkovin1962) hypothesis where the same incompressible mechanisms govern the compressible spanwise response, provided that the mean property variations are accounted for.

For (3.5) to be a valid solution, it must satisfy the no-slip boundary conditions at $y=0,2h$ and the initial condition $\widetilde {w} = 0$ at $t = 0$ . Since (3.7) is a second-order differential equation, it can only satisfy two boundary conditions. To reconcile the boundary condition at $y = 2h$ , the solution proposed in (3.5) will only be valid for $y \in [0,h]$ and $\widetilde {w}$ will be assumed to be an even function in $y$ about $y = h$ so that its wall-normal derivative is continuous. The boundary condition at $y=0$ and the initial condition at $t = 0$ on $\widetilde {w}$ translate to the boundary conditions that $f = 0$ at $\eta = 0$ and $f=1$ at $\eta \rightarrow \infty$ , respectively. The latter condition ensures that $f$ is bounded as $t \rightarrow 0$ . Due to the additional assumption of symmetry about $y = h$ , there is an additional constraint that ${\mathrm{d} f}/{\rm d}{\eta } = 0$ at $\eta (h,t)$ . This presents an additional boundary condition that cannot be reconciled analytically. However, it will be assumed that $\eta (h,t) \gg 1$ so that this symmetry constraint becomes equivalent to ${\mathrm{d} f}/{\rm d}{\eta } = 0$ as $\eta \rightarrow \infty$ . This is equivalent to assuming that the viscous length scales in the laminar solution are much smaller than $h$ . An order of magnitude estimate for $\eta (h,t)$ can be found using viscous wall units and approximating $\eta$ with the incompressible counterpart as $\eta (h,t) \sim h/\sqrt {\overline {\nu }t} = {\textit{Re}}_{\tau }/\sqrt {t^{+}}$ . The smallest $\eta (h,t)$ can be is at $t^{+} \approx 550$ during the duration of the simulation, which for the cases studied here is $\eta \approx {\textit{Re}}_{\tau }/\sqrt {550} \approx 23$ . It will be shown that for this estimate, ${\mathrm{d} f}/{\rm d}{\eta } = 0$ can be satisfied to numerical round off error.

The solution to (3.7) that satisfies the boundary conditions at $\eta = 0$ and $\eta \rightarrow \infty$ is

(3.8) \begin{equation} f = 1 - \bigg (\frac {\eta ^2}{2} + 1\bigg )\textrm {erfc}\bigg (\frac {\eta }{2}\bigg ) + \frac {\eta }{\sqrt {\pi }}e^{-\frac {\eta ^2}{4}}, \end{equation}

where $\textrm {erfc}$ is the complimentary error function (Boas Reference Boas2006). The solution’s derivative, ${\mathrm{d} f}/{\rm d}{\eta } = 2/\sqrt {\pi }\exp (-\eta ^2/4) - \eta \textrm {erfc}(\eta /2)$ , using the incompressible estimate of $\eta \approx 23$ evaluates to ${O}{ (10^{-60} )}$ . The smallest value for all six compressible cases is $\eta (h,t) \approx 10$ at the end of the simulation, which evaluates ${\mathrm{d} f}d/{\eta } \approx 10^{-13}$ . Though not exactly zero, it is a sufficient approximation for symmetry about $y=h$ . The value of $\eta (h,t)$ is much larger during the interval $t \in [0,{t_{I}}]$ where the laminar assumptions are expected to hold which only improves the approximation regarding symmetry about $y=h$ .

Figure 5.

Similarity scaling of the spanwise flow for (a) $\varPi = 10$ and (b) $\varPi = 40$ . The solid and dashed lines colour coded with table 1 are DNS data, with the dashed lines plotted for $t \gt {t_{I}}$ , and colours from dark to light indicate increasing time, plotted from $t^+ = 60$ in increments of $\Delta t^{+} = 60$ . The green dashed line is the compressible similarity solution, $f(\eta (y,t))$ . The magenta dotted line is the incompressible similarity solution, $f(\eta _{inc}(y,t))$ , against $\eta (y,t)$ at $t^{+} = 60$ . The $\textit{Ma} = 1.5$ and $\textit{Ma} = 3.0$ cases are offset horizontally for visibility.

The instantaneous $\widetilde {w}$ are plotted in figure 5(a,b) for all six cases, normalized by $g_{z} t$ , against $\eta (y,t)$ . The similarity solution, $f$ , is also included where $\eta$ is evaluated using (3.6) for the compressible solution and $\eta _{inc}$ to illustrate an incompressible approximation. For short time intervals, $f(\eta )$ and $\widetilde {w}/g_{z}t$ agree because of the absence of spanwise turbulent stresses for all six cases. As time advances, the laminar approximation breaks creating departures from the compressible-laminar similarity solution. This departure is most severe in the $\textit{Ma} = 0.3$ cases where $t_{I}$ , the time it takes $\overline {\rho }\widetilde {v''w''}$ to dominate over $\overline {\tau }_{yz}$ , is smallest. For the $\textit{Ma} = 3$ cases, $\widetilde {w}/g_{z}t$ and $f(\eta )$ agree during the full simulation time as $\overline {\rho }\widetilde {v''w''}$ remains negligible compared with $\overline {\tau }_{yz}$ . Although figure 2(a) demonstrates significant temporal variation in $\overline {\rho }$ for case 6, the agreement between the data and the similarity solution demonstrates that the quasisteady approximation is valid and any discrepancies in the smaller ${\textit{Ma}}$ flows are indeed from the turbulent stresses. Finally, the incompressible solution, $f(\eta _{inc})$ , agrees well with the $\textit{Ma} = 0.3$ cases, but exhibits significant departures from $f(\eta )$ and $\widetilde {w}/g_{z}t$ in the near-wall region as the ${\textit{Ma}}$ increases. Properly accounting for the mean property variations in $\eta$ allows the similarity solution, $f$ , to accurately predict $\widetilde {w}$ for small times.

3.3. Mean shear and heat transfer

The evolution of $\tau _{w,x}$ is shown in figure 6(a) for all six cases. The temporal evolution of $\tau _{w,x}$ depends both on ${\textit{Ma}}$ and $\varPi$ and is a direct result of the evolution of $u_{b}$ , which is not here constrained as the streamwise flow is driven with a constant $g_{x}$ . In incompressible studies of sudden spanwise acceleration (Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020) and in the drag reduction observed in spanwise oscillations with constant ${\rm d}p/{\rm d}x$ (Ge & Jin Reference Ge and Jin2017; Ricco et al. Reference Ricco, Skote and Leschziner2021), $\tau _{w,x}$ is also affected by the spanwise acceleration.

Figure 6.

Temporal variation of (a) $\tau _{w,x}$ , (b) $\tau _{w,z}$ , (c) $q_{w}$ and (d) $C_{f}/(S_{t}Pr )$ . In (b), the symbols denote the laminar prediction, $2g_{z}\sqrt {\rho _{w}\mu _{w}t/\pi }$ , with the circles and triangles denoting $\varPi = 40$ and $\varPi = 10$ , respectively. The colours and line styles are defined in table 1.

Owing to the success of the laminar prediction of $\widetilde {w}$ for initial times, a laminar prediction of $\tau _{w,z}$ is also included via $\tau _{w,z} = \mu _{w}g_{z}\partial _{y}f(\eta )\vert _{y=0} = 2g_{z}\sqrt {\rho _{w}\mu _{w}t/\pi }$ in figure 3(b). This prediction agrees well for $\varPi = 10$ and for $\varPi = 40$ begins to deviate substantially for $t \lt {t_{I}}$ despite the success of the similarity solutions. The departure is present even for $\textit{Ma} = 3$ , indicating discrepancies in the near-wall prediction of $\widetilde {w}$ from the similarity solution because of the production of turbulent stresses. Due to the net acceleration in the flow, $\tau _{w,z}$ increases monotonically during the transient. The increase in kinetic energy then results in additional viscous heating (Lele Reference Lele1994), raising both $\overline {T}$ and $q_{w}$ , as shown in figure 6(c). The temporal evolution of $\tau _{w,z}$ and $q_{w}$ show more similarity for matched value of $\varPi$ rather than ${\textit{Ma}}$ .

A common predictive tool in turbulent boundary layers is the Reynolds analogy factor, relating the friction coefficient, $C_{f}$ and the Stanton number, $S_{t}$ (Bradshaw Reference Bradshaw1977). While Fanning flow uses a friction coefficient based on the bulk velocity, here $C_{f}$ is related to the centreline quantities as an analogy for the free stream conditions. Here, $C_{f} = 2\|\boldsymbol{\tau }_{w}\|/ (\rho _{c}\|\widetilde {\boldsymbol{u}}_{c}\|^{2} )$ and $S_{t} = q_{w}/ (\rho _{c}c_{p}\|\widetilde {\boldsymbol{u}}_{c}\|(T_{c} - T_{w}) )$ . Other definitions for these quantities have been used in the literature, primarily in the context of boundary layers. The ratio, $C_{f}/S_{t}Pr$ , begins at $1$ and promptly evolves as $g_{z}$ is applied in figure 3(d). The Reynolds analogy factor, $f_{RA} = 2S_{t}/C_{f}$ , (Bradshaw Reference Bradshaw1977) is observed to be approximately $2Pr$ for the canonical channels simulated herein at $t = 0$ and during the simulation time of the $\varPi = 10$ cases whereas $f_{RA}$ varies significantly for $\varPi = 40$ . The choice of $C_{f}$ and $S_{t}$ is non-unique. It is possible that different choices could lead to a constant ratio throughout the simulation time, though this is outside of the scope of this work.

4.1. Drop in TKE and thermal fluctuations

Spanwise acceleration has been shown to reduce the TKE via flow control strategies with active walls (Quadrio & Sibilla Reference Quadrio and Sibilla2000; Choi et al. Reference Choi, Xu and Sung2002; Ge & Jin Reference Ge and Jin2017; Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021; Ricco et al. Reference Ricco, Skote and Leschziner2021; Chandran et al. Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) or the initial development of a spanwise flow (Bradshaw & Pontikos Reference Bradshaw and Pontikos1985; Moin et al. Reference Moin, Shih, Driver and Mansour1990; Coleman et al. Reference Coleman, Kim and Moser1995; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020). From the sudden spanwise acceleration, incompressible studies (Moin et al. Reference Moin, Shih, Driver and Mansour1990; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020) have concluded that due to the spanwise flow, the pressure-strain term, $\overline {p'\partial _{z}v'}$ , drops, leading to a reduction in $\overline {v'v'}$ . This leads to a reduction of the streamwise Reynolds shear stress production, $\overline {v'v'}\partial _{y}\overline {u}$ , which reduces the magnitude of $\overline {u'v'}$ . Finally, this causes a reduction in the production of the TKE, $\overline {u'v'}\partial _{y}\overline {u}$ , leading to the reduction in the TKE. Once the spanwise flow is sufficiently developed, the additional spanwise Reynolds shear stresses lead to additional production in the TKE, ultimately increasing it. Mechanistically, the reduction in the Reynolds shear stresses occurs because of a misalignment between the near-wall structures and those farther away from the wall leading to less efficient Reynold shear stress production (Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020). The misalignment in the flow structures is described in more detail in § 4.4. In this section, the mechanisms described will be shown to be similar for the compressible Reynolds stresses and concludes with a description of the thermal turbulent transport.

The TKE, $\overline {\rho }\widetilde {k} = \overline {\rho } (\widetilde {u''u''} + \widetilde {v''v''} + \widetilde {w''w''} )/2$ and some representative Reynolds stresses are plotted in figure 8(a–f) for $\textit{Ma} = 1.5$ and $\varPi = 40$ . The evolution of the Reynolds stresses can be divided into two stages. The first stage is characterized by a reduction in $\overline {\rho }\widetilde {u''u''}$ due to decreased production from the drop in $\overline {\rho }\widetilde {u''v''}$ . This coincides with a reduction in $\overline {\rho }\widetilde {v''v''}$ due to the decrease in the production of $\overline {\rho }\widetilde {u''v''}$ (Moin et al. Reference Moin, Shih, Driver and Mansour1990; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020). While this is occurring, there is an increase in $\overline {\rho }\widetilde {w''w''}$ due to the net transport in the spanwise direction’s role in generating $\overline {\rho }\widetilde {v''w''}$ , leading to production of spanwise turbulent fluctuations. Despite the increase in the mean kinetic energy and spanwise turbulent fluctuations, there is a reduction in the TKE driven by the reduction in production from $\overline {\rho }\widetilde {u''v''}$ in agreement with various incompressible studies with imposed spanwise flows (Bradshaw Reference Bradshaw1977; Moin et al. Reference Moin, Shih, Driver and Mansour1990; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020). During the second stage, the $\overline {\rho }\widetilde {u''v''}$ intensifies, leading to an increase in $\overline {\rho }\widetilde {u''u''}$ while $\overline {\rho }\widetilde {w''w''}$ continues to rise. As a result, the TKE increases at this point as well. In Gomez (Reference Gomez2025), the $\varPi = 10$ and $\textit{Ma} = 1.5$ case was shown to follow the same trends as the $\varPi = 40$ case, although the decrease in the magnitude of $\overline {\rho }\widetilde {u''v''}$ was smaller, leading to a smaller reduction in the TKE. Furthermore, the TKE and Reynolds stresses of the other cases were shown to follow similar trends, with cases 1 and 2 demonstrating almost identical temporal evolution.

Figure 8.

Temporal variation of (a–e) the Reynolds stresses and (f) TKE for $\textit{Ma} = 1.5$ and $\varPi = 40$ , normalized by the initial $\tau _{w,x}$ . The lines are coloured with the colourbar at every $80\ell _{\nu }(0)/u_{\tau }(0)$ in (a) and the arrows illustrate the direction of time in the statistics.

Figure 9.

Temporal variation of (a–c,e–g) the velocity–temperature covariances normalized by $u_{\tau}(0)$ and $T_{\tau }(0)$ and (d,h) the thermal fluctuations normalized by $T_{\tau }(0)^{2}$ for $\textit{Ma} = 1.5$ (a–d) $\varPi = 10$ and (e–h) $\varPi = 40$ . The lines are coloured with the colourbar in figure 8(a) and the arrows illustrate the direction of time in the statistics.

While the Reynolds stresses follow similar behaviour between the $\varPi = 10$ and $\varPi = 40$ cases, the velocity–temperature covariances, $\widetilde {\boldsymbol{u}''T''}$ , have qualitatively different responses to the spanwise acceleration. Focusing first on the $\varPi = 10$ and $\textit{Ma} = 1.5$ case in figure 9(a–d), the covariances illustrate a slight reduction in $\widetilde {u''T''}$ and $\widetilde {v''T''}$ and an increase in $\widetilde {w''T''}$ . The latter reflects the increased transport in the spanwise direction. The slight changes in $\widetilde {u''T''}$ reflect that the $u$ fluctuations remain correlated with the $T$ fluctuations while the anticorrelation between $v''$ and $T''$ reflect the importance of sweep and ejection events even in the presence of spanwise acceleration. Despite the change in the magnitudes, $\widetilde {u''T''}$ and $\widetilde {v''T''}$ behave similar to canonical compressible flows (Coleman et al. Reference Coleman, Kim and Moser1995). Finally, $\widetilde {T''T''}$ decreases despite the net increase in $\widetilde {T}$ , similar to the reduction in TKE.

The $\varPi = 40$ and $\textit{Ma} = 1.5$ case’s velocity–temperature covariances and thermal fluctuations in figure 9(e–h), illustrate a significantly different response. For initial times, $\widetilde {u''T''}$ and $\widetilde {v''T''}$ decrease in magnitude while $\widetilde {w''T''}$ increases. Eventually, the velocity–temperature covariances change signs along $y$ at $t^{+} \approx 280$ . This behaviour can be observed in the instantaneous visualizations of $u'$ and $T'$ in figure 7(b,d,f,h) and highlights a difference in the turbulent thermal transport absent in the $\varPi = 10$ case. These differences can be explained by comparing $\widetilde {T}$ in figure 3. For $\varPi = 10$ , $\widetilde {T}$ remains monotonic in the wall-normal direction, whereas for $\varPi = 40$ , a near-wall $\widetilde {T}$ peak emerges for $t^{+} \approx 200$ . Thus, for $t\gtrapprox 200$ , the lift-up mechanisms change the transport of $T'$ above this near-wall $\widetilde {T}$ peak. Below this peak, $\widetilde {u}$ , $\widetilde {w}$ and $\widetilde {T}$ are all monotonically increasing like in the $\varPi = 10$ case, maintaining the same behaviour in the velocity–temperature covariances. Away from the peak, $\widetilde {T}$ is monotonically decreasing whereas $\widetilde {u}$ and $\widetilde {w}$ increase in $y$ . This means that the lift-up mechanism advects low-speed, high temperature fluid up away from the temperature peak and high-speed, low temperature fluid down towards the peak. Thus, the change in sign in $\widetilde {u_{i}^{\prime\prime}T''}$ above the near-wall $\widetilde {T}$ peaks stems from changes in the lift-up mechanism. Similar observations have been made in turbulent boundary layers (Duan et al. Reference Duan, Beekman and Martin2010; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011). In § 4.2, the lift-up mechanisms are discussed in more detail. Finally, $\widetilde {T''T''}$ also decreases for initial times as shown in figure 9(h), similar to what happened in the $\varPi = 10$ case. For later times, $\widetilde {T''T''}$ increases and produces two peaks likely stemming from decreased production near the zero-crossing of $\widetilde {v''T''}$ and $\partial _{y}\widetilde {T}$ .

The reduction in the TKE and $\overline {\rho }\widetilde {T''T''}$ can be explained by considering their budgets. First, the Reynolds stress budget is

(4.1) \begin{equation} {\partial _{t}\left(\overline {\rho }\widetilde {u_{i}^{\prime\prime}u_{\!j}^{\prime\prime}}\right) = \mathcal{P}_{\textit{ij}} + \mathcal{T}_{\textit{ij}} + \mathcal{S}_{\textit{ij}} + \mathcal{E}_{\textit{ij}} + \mathcal{A}_{\textit{ij}} + \mathcal{C}_{\textit{ij}}} \end{equation}

where $\mathcal{P}_{\textit{ij}} = -\overline {\rho }(\widetilde {u_{i}^{\prime\prime}u_{k}''}\partial _{x_{k}}\widetilde {u}_{\!j} + \widetilde {u_{\!j}^{\prime\prime}u_{k}''}\partial _{x_{k}}\widetilde {u}_{i})$ , is the production, $\mathcal{T}_{\textit{ij}} = -\partial _{x_{k}} \widetilde {\rho u_{i}^{\prime\prime}u_{\!j}^{\prime\prime}u_{k}''}$ is the turbulent transport, $\mathcal{S}_{\textit{ij}} = -( \overline {u_{\!j}^{\prime\prime}\partial _{x_{i}}p'} + \overline {u_{i}^{\prime\prime}\partial _{x_{\!j}}p'} )$ is the pressure-strain, $\mathcal{E}_{\textit{ij}} = -( \overline {u_{i}^{\prime\prime}\partial _{x_{k}}\tau _{jk}'} + \overline {u_{\!j}^{\prime\prime}\partial _{x_{k}}\tau _{ik}'} )$ is the dissipation, $\mathcal{A}_{\textit{ij}} = -\widetilde {v}\partial _{y}(\overline {\rho }\widetilde {u_{i}^{\prime\prime}u_{\!j}^{\prime\prime}})$ is the wall-normal advection and $\mathcal{C}_{\textit{ij}} = \overline {u_{i}^{\prime\prime}}\partial _{x_{k}}(\overline {\tau }_{jk} - \overline {p}\delta _{jk}) +\overline {u_{\!j}^{\prime\prime}}\partial _{x_{k}}(\overline {\tau }_{ik} - \overline {p}\delta _{ik})$ are the compressibility terms. The thermal budget is

(4.2) \begin{equation} {\partial _{t}\left(\overline {\rho }\widetilde {T''T''}\right) = \mathcal{P}_{\textit{TT}} + \mathcal{T}_{\textit{TT}} + \mathcal{S}_{\textit{TT}} + \mathcal{E}_{\textit{TT}} + \mathcal{A}_{\textit{TT}} + \mathcal{C}_{\textit{TT}}}, \end{equation}

where $\mathcal{S}_{\textit{TT}} = 2c_{v}^{-1}( \overline { T''S_{\textit{ij}}''\tau '_{\textit{ij}} } - \overline { T'' \partial _{x_{i}}u''_{i}p' } )$ is the thermal pressure-stress strain, $\mathcal{P}_{\textit{TT}} = 2(-\overline {\rho }\widetilde {v''T''}\partial _{y}\widetilde {T} -(\gamma -1)\overline {\rho }\widetilde {T''T''}\partial _{y}\widetilde {v} + c_{v}^{-1}\overline {T''\tau _{yi}}\partial _{y}\widetilde {u}_{i} + c_{v}^{-1} \overline {S_{\textit{ij}}''T''}(\overline {\tau }_{\textit{ij}} - \overline {p}\delta _{\textit{ij}}))$ is the thermal production, $\mathcal{E}_{\textit{TT}} = 2c_{v}^{-1}\overline {\partial _{x_{i}}T''q_{i} }$ is the heat dissipation, $\mathcal{T}_{\textit{TT}} = -\partial _{y}(\overline {\rho }\overline {T''T''v''} ) -2 c_{v}^{-1}\partial _{y}\overline {T''q_{y}}$ is the turbulent and heat transport, $C_{\textit{TT}} = 2 c_{v}^{-1} \overline {T''}\overline {\tau }_{yi}\partial _{y}\overline {u}_{i}$ is the compressibility term and $\mathcal{A}_{\textit{TT}} = \partial _{y}(\overline {\rho }\widetilde {v}\widetilde {T''T''})$ the wall-normal convection. Here, $S_{\textit{ij}} = (\partial _{x_{\!j}}u_{i} + \partial _{x_{i}}u_{\!j} )/2$ is the strain rate tensor and $\delta _{\textit{ij}}$ is the Kronecker delta. Each individual term in (4.1)–(4.2) is integrated as $\phi _{I}(t) = \int _{5\ell _{\nu }(0)}^{75\ell _{\nu }(0)}\phi (y,t){\rm d}y$ with the gains computed as $(\phi _{I}(t)-\phi _{I}(0))$ as in Lozano-Durán et al. (Reference Lozano-Durán, Giometto, Park and Moin2020).

Figure 10.

The gains of the $\overline {\rho }\widetilde {v''v''}$ (a), $\overline {\rho }\widetilde {u''v''}$ (b), $\overline {\rho }\widetilde {u''u''}$ (c) and $\overline {\rho }\widetilde {T''T''}$ (d) budgets integrated from $y/\ell _{\nu }(0)\in [5,75]$ for $\textit{Ma} = 1.5$ and $\varPi = 40$ . The gains in (a) are normalized by the initial pressure-strain whereas (b) and (d) are normalized by their initial production terms. In (d), the red-dashed line is $-2\overline {\rho }\widetilde {v''T''}\partial _{y}\widetilde {T}$ .

The gains for $\overline {\rho }\widetilde {v''v''}$ , $\overline {\rho }\widetilde {u''v''}$ , $\overline {\rho }\widetilde {u''u''}$ and $\overline {\rho }\widetilde {T''T''}$ budgets are plotted in figure 10(a–d) for $\textit{Ma} = 1.5$ and $\varPi = 40$ . The temporal evolution of the Reynolds stress budgets follows the incompressible pattern (Moin et al. Reference Moin, Shih, Driver and Mansour1990; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020) in that a drop in $\mathcal{S}_{22}$ causes a reduction in $\overline {\rho }\widetilde {v''v''}$ . The decrease in $\overline {\rho }\widetilde {v''v''}$ then causes a drop in $\mathcal{P}_{12}$ via $\overline {\rho }\widetilde {v''v''}\partial _{y}\widetilde {u}$ , which subsequently reduces $\overline {\rho }\widetilde {u''v''}$ . This reduction further causes a drop in $\mathcal{P}_{11}$ , leading to a decrease in both $\overline {\rho }\widetilde {u''u''}$ and the TKE. The temporal evolution of the gains are similar to those reported by Lozano-Durán et al. (Reference Lozano-Durán, Giometto, Park and Moin2020), including even the initial drop in $\mathcal{S}_{12}$ before it rises. The decrease in $\overline {\rho }\widetilde {T''T''}$ comes from a reduction in $\mathcal{P}_{\textit{TT}}$ despite the increase in $\mathcal{E}_{\textit{TT}}$ , $\mathcal{T}_{\textit{TT}}$ and $\mathcal{S}_{\textit{TT}}$ similar to what is observed in $\overline {\rho }\widetilde {u''u''}$ . The term most responsible for the decrease in $\mathcal{P}_{\textit{TT}}$ is the $-2\overline {\rho }\widetilde {v''T''}\partial _{y}\widetilde {T}$ term, which is analogous to the $\overline {\rho }\widetilde {v''u''}\partial _{y}\widetilde {u}$ production term in the TKE and $\overline {\rho }\widetilde {u''u''}$ budget.

4.2. Organization of turbulent and thermal transport

In § 4.1, the presence of the near-wall peak in $\widetilde {T}$ leads to a change in sign in the velocity–temperature covariances that is only observed in the $\varPi = 40$ case. Additionally, the development of $\widetilde {w}$ leads to the production of $\overline {\rho }\widetilde {w''w''}$ via the generation of $\overline {\rho }\widetilde {v''w''}$ . In this section, quadrant decompositions (Wallace Reference Wallace2016) will be used to study the organization of the turbulent and thermal transport. While the quadrant decomposition is most commonly used to quantify contributions of $u$ and $v$ based on their signs (Wallace et al. Reference Wallace, Eckelmann and Brodkey1972), the quadrant decomposition has also been used to bin $T$ and $v$ fluctuations (Perry & Hoffmann Reference Perry and Hoffmann1976; Nagano & Tagawa Reference Nagano and Tagawa1988; Kong et al. Reference Kong, Choi and Lee2000). For a variable $a$ and wall-normal velocity $v$ , the quadrants are organized into four quadrants, $Q_{i}$ , where $Q_{1} = \{(a,v) : a''\gt 0\textrm { and } v''\gt 0\}$ , $Q_{2} = \{(a,v) : a''\lt 0\textrm { and } v''\gt 0\}$ , $Q_{3} = \{(a,v) : a''\lt 0\textrm { and } v''\lt 0\}$ and $Q_{4} = \{(a,v) : a''\gt 0\textrm { and } v''\lt 0\}$ . For $u$ , the $Q_{4}$ and $Q_{2}$ events are commonly denoted as sweeps and ejections, respectively. To avoid additional nomenclature, the $Q_{4}$ and $Q_{2}$ events for $w$ and $T$ will also be denoted as sweeps and ejections.

The time evolution of the probability distributions of the $Q_{i}$ events was presented in Gomez (Reference Gomez2025). Here, the contributions of the $Q_{i}$ events to $\overline {\rho }\widetilde {w''v''}$ and $\overline {\rho }\widetilde {T''v''}$ are measured through

(4.3) \begin{equation} \langle \overline {\rho }a''{v}''|Q_{i}\rangle = \frac {1}{h}\int _{0}^{h}\overline {\rho }(\widetilde {a''{v}''}:(a,{v})\in Q_{i}){\rm d}y, \end{equation}

for $a = w,T$ where $\sum _{i=1}^{4}\langle \overline {\rho }a''{v}''|Q_{i}\rangle = \langle \overline {\rho }a''{v}''\rangle$ and are normalized by $W_{av}$ where

(4.4) \begin{equation} W_{av}(t)^{2} = \frac {1}{h^{2}}\int _{0}^{h}\overline {\rho }\widetilde {a''a''}{\rm d}y\int _{0}^{h}\overline {\rho }\widetilde {v''v''}{\rm d}y = \langle \overline {\rho }a''a''\rangle \langle \overline {\rho }v''v''\rangle \gt 0, \end{equation}

for $a = w,T$ . Normalizing with $\langle \overline {\rho }\widetilde {a''v''}\rangle$ can be problematic since at $t = 0$ , $\langle \overline {\rho }\widetilde {w''v''}\rangle = 0$ and for the $\varPi = 40$ cases, due to the change in sign in $\widetilde {v''T''}$ , $\langle \overline {\rho }\widetilde {T''v''}\rangle$ could be zero. Additionally, $\langle \overline {\rho }a''{v}''|Q_{i}\rangle /W_{av}(t) \in [-1,1]$ via the Cauchy–Schwarz inequality.

The $Q_{i}$ contributions to $\langle \overline {\rho }\widetilde {w''v''}\rangle$ are shown in figure 11. At $t = 0$ , they all contribute roughly equal amounts since $\langle \overline {\rho }\widetilde {w''v''}\rangle = 0$ initially due to the lack of net transport in the spanwise direction. As the flow develops a net spanwise shear, $\widetilde {w}_{y}$ , the sweeps and ejection events are expected to play a role via lift-up effects. Indeed, as time advances, $Q_{2}$ and $Q_{4}$ events play a larger role while the $Q_{1}$ and $Q_{3}$ become less relevant in contribution to $\langle \overline {\rho }\widetilde {w''v''}\rangle$ . There is a slight preference towards ejections rather than sweeps by comparing figure 11(a,d), especially as the ${\textit{Ma}}$ increases.

Figure 11.

Temporal evolution of the (a) $Q_{2}$ , (b) $Q_{1}$ , (c) $Q_{3}$ and (d) $Q_{4}$ contributions to $\langle \overline {\rho }\widetilde {w''v''}\rangle$ , normalized by $W_{wv}(t)$ . The colours and line styles are defined in table 1.

Figure 12.

Temporal evolution of the (a) $Q_{2}$ , (b) $Q_{1}$ , (c) $Q_{3}$ and (d) $Q_{4}$ contributions to $\langle \overline {\rho }\widetilde {T''v''}\rangle$ , normalized by $W_{Tv}(t)$ . The colours and line styles are defined in table 1.

The contributions to $\langle \overline {\rho }\widetilde {T''v''}\rangle$ from the $Q_{i}$ events are presented in figure 12. At $t = 0$ , the $Q_{2}$ and $Q_{4}$ contributions to $\langle \overline {\rho }\widetilde {T''v''}\rangle$ are the most dominant, in agreement with the dominance of sweep and ejection events in the thermal transport of canonical shear flows (Perry & Hoffmann Reference Perry and Hoffmann1976; Nagano & Tagawa Reference Nagano and Tagawa1988; Kong et al. Reference Kong, Choi and Lee2000). As expected from the change in sign in the velocity–temperature covariances in figure 9, the $Q_{i}$ contributions reveal distinct behaviour for the $\varPi = 10$ and $\varPi = 40$ cases once the spanwise flow is sufficiently developed. For $\varPi = 10$ , the $Q_{2}$ and $Q_{4}$ event contributions are dominant throughout the duration of the simulation, with only slight reductions in their magnitudes towards the end. The $\varPi = 40$ cases behave similar to the $\varPi = 10$ cases initially. For $t^{+} \approx 200$ , coinciding with the time that the near-wall $\widetilde {T}$ peak emerges, the $Q_{2}$ and $Q_{4}$ events begin to lose their dominance while the $Q_{1}$ and $Q_{3}$ contributions dominate. The change from $Q_{2}$ and $Q_{4}$ events to $Q_{1}$ and $Q_{3}$ events in $T$ support the explanation for the change in sign in the velocity–temperature covariances. That is, between the wall and the near-wall peak, the lift-up mechanism is dominated by sweeps and ejections such that $\mp v''$ coincide with $\pm T''$ . Between the near-wall peak and $h$ , the sweeps and ejections are no longer dominant as $\pm v''$ coincide with $\pm T''$ which give rise to $Q_{1}$ and $Q_{3}$ events. Since the near-wall peaks occur for $y/\ell _{\nu }(0) \lt 100$ , the majority of the channel is no longer governed by sweeps and ejections of $T$ . Because of the dominance in the $Q_{1}$ and $Q_{3}$ events, $\langle \overline {\rho }\widetilde {v''T''}\rangle$ changes sign in the $\varPi = 40$ case. The trends in the $Q_{i}$ events for $T$ align with the qualitatively different behaviour in the velocity–temperature covariances from figure 9 for the two $\varPi$ . They further suggest that the ${\textit{Ma}}$ is not responsible in changing the overall organization of the turbulent-thermal transport. The observations in figure 12 agree with the results from Gomez (Reference Gomez2025) where the contributions of the $Q_{i}$ events were measured through the time evolution of the probabilities of the $Q_{i}$ across the entire channel.

4.3. Temporal evolution of the wavenumber spectra

The previous sections focused on one-point statistics highlighting differences between the $\varPi = 10$ and $\varPi = 40$ cases. Here, the effects of $\varPi$ on the turbulent structure will be further studied by comparing the temporal evolution of the spectra. While this section considers Fourier transforms along the streamwise and spanwise directions, it could be possible to define the the Fourier transforms along $\boldsymbol{e}_{s}$ and its perpendicular, $\boldsymbol{e}_{s}\wedge \boldsymbol{e}_{y}$ , or other directions defined by the flow statistics. However, the alignment of the identified structures in the next section is shown to lie between $\boldsymbol{e}_{s}$ and the Reynolds shear stress direction, suggesting that axes based on the turbulent statistics do not capture the alignment of the structures. Thus, for ease of interpretation and avoiding wall-normally varying axes, the Fourier transforms here are taken along the fixed streamwise and spanwise directions, $\boldsymbol{e}_{x}$ and $\boldsymbol{e}_{z}$ , respectively.

First, the streamwise Fourier modes of a variable $f$ are defined as

(4.5) \begin{equation} \widehat {f}(y,t;k_{x}) = \frac {1}{L_{z}L_{x}}\int _{0}^{L_{z}}\int _{0}^{L_{x}}f(x,y,z,t)e^{ik_{x}x}{\rm d}x{\rm d}z, \end{equation}

averaged over the eight ensembles and over the channel half-height, where $k_{x} = 2\pi /\lambda _{x}$ is the streamwise wavenumber and $\lambda _{x}$ is the wavelength. The streamwise premultiplied power spectrum of the variable $f$ is then $E_{ff}(y,k_{x},t) = k_{x}\left |\widehat {f}(y,t;k_{x})\right |^{2}$ for $k_{x}\gt 0$ . The spanwise premultiplied power spectrum, $E_{ff}(y,k_{z},t)$ , is defined similarly by swapping $x$ and $z$ using the spanwise wavenumber, $k_{z}$ , and spanwise wavelength, $\lambda _{z}$ .

The premultiplied kinetic energy spectra, $E_{kk} = (E_{uu} + E_{vv} + E_{ww})/2$ , and premultiplied temperature spectra, $E_{\textit{TT}}$ , are compared for the $\textit{Ma} = 1.5$ and $\varPi = 40$ case in figure 13 for four representative times and the initial canonical configuration. At $t^{+} = 0$ , $E_{kk}(y,k_{x},t)$ and $E_{\textit{TT}}(y,k_{x},t)$ peak near $(y/\ell _{\nu }(0),\lambda _{x}/\ell _{\nu }(0)) = (20,1000)$ and $(10,1000)$ , respectively, as expected for the near-wall cycle in wall-bounded turbulent flows (Lee & Moser Reference Lee and Moser2015). Figure 13(a,e) reveal that the most significant initial temporal evolution in the spectra is seen in the near-wall small-scales while the spectra in the large-scale outer region mostly coincide with the canonical spectra. As time advances, the near-wall peaks move to smaller values of $\lambda _{x}/\ell _{\nu }(0)$ for both $E_{kk}$ and $E_{\textit{TT}}$ . For $t^{+} \leqslant 200$ , the contours for large $\lambda _{x}$ away from the wall still coincide with the contours from $t^{+} = 0$ , suggesting that the effect of $g_{z}$ on the large-scales lags behind its effect on the small-scales. Similar observations were made with structural observations of wall-attached Reynolds shear-stress carrying eddies in incompressible flow (Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020) and are expanded in § 4.4. Eventually, the large-scale structures become less energetic as they too orient away from $\boldsymbol{e}_{x}$ .

The movement of the near-wall peak to smaller $\lambda _{x}$ and $y$ is related to two mechanisms. The first is as $\tau _{w}$ increases, the viscous length scale decreases causing the characteristic size of the near-wall structures to subsequently shrink. The second is that the extent of the near-wall structures along $\boldsymbol{e}_{x}$ decreases as these structures turn in the direction of the net acceleration. As a result of these two mechanisms and assuming these near-wall structures have an orientation of $\theta$ , the characteristic streamwise wavelength, $l$ , and wall-normal location, $y_{l}$ , of the structures in the near-wall peak can be approximated as

(4.6) \begin{equation} (l(t),y_{l}(t)) \approx (1000 \cos (\theta (t)),y_{0})\ell _{\nu }(t), \end{equation}

where $y_{0} = 20\ell _{\nu }(0)$ and $10 \ell _{\nu }(0)$ for $E_{kk}(y,k_{x},t)$ and $E_{\textit{TT}}(y,k_{x},t)$ , respectively. These are shown to approximately fall near the location of the near-wall peaks, albeit in log-space.

Figure 13.

Instantaneous contours of the streamwise premultiplied spectra for $\textit{Ma} = 1.5$ and $\varPi = 40$ with ${h}E_{kk}(y,k_{x},t)/u_{\tau }(0)^{2}$ (a–d) and ${h}E_{\textit{TT}}(y,k_{x},t)/T_{\tau }(0)^{2}$ (e–h). The solid lines are isocontours of $5\,\%$ , $10\,\%$ , $20\,\%$ , $50\,\%$ and $95\,\%$ of the maximum instantaneous premultiplied spectra. The black solid lines and coloured contours are at the time listed while the red solid lines are taken at $t^{+} = 0$ as a comparison. The solid orange crosses denote the predicted location of the near-wall peak via (4.6).

Figure 14.

Similar to figure 13, except $E_{kk}(y,k_{z},t)$ (a–d) and $E_{\textit{TT}}(y,k_{z},t)$ (e–h) are plotted for $\textit{Ma} = 1.5$ , $\varPi = 40$ in (a), (b), (e) and (f) and $\textit{Ma} = 1.5$ , $\varPi = 10$ in (c), (d), (g) and (h).

The premultiplied spanwise spectra, $E_{kk}(y,k_{z},t)$ and $E_{\textit{TT}}(y,k_{z},t)$ , at $\textit{Ma} = 1.5$ are compared between the $\varPi = 40$ case in figure 14(a,b,e,f) and the $\varPi = 10$ case in figure 14(c,d,g,h). At $t=0$ , both $E_{kk}$ and $E_{\textit{TT}}$ peak around $y/\ell _{\nu } \approx 15$ and $\lambda _{z}/\ell _{\nu }(0) \approx 200$ , which are characteristic of the near-wall cycle in both incompressible and compressible flows (Lee & Moser Reference Lee and Moser2015; Cogo et al. Reference Cogo, Salvadore, Picano and Bernardini2022). Unlike the streamwise spectra, $E_{kk}(y,k_{z},t)$ , reacts slower to the spanwise acceleration such that the isocontours for up to $t^{+} = 199$ approximately overlap with those at $t^{+} = 0$ for $\varPi = 40$ . As time advances, the peak $E_{kk}$ move to larger $\lambda _{z}$ due to the increased alignment of the kinetic energy structures towards $\boldsymbol{e}_{z}$ . The isocontours of $E_{\textit{TT}}$ for the $\varPi = 40$ case depart from the $t^{+} = 0$ isocontours faster than $E_{kk}$ . The near-wall peaks of $E_{\textit{TT}}$ move significantly closer to the wall while also moving towards larger $\lambda _{z}$ . Both figures 13(h) and 14(f), reveal a secondary peak in $E_{\textit{TT}}$ of small-scales in the buffer layer away from the primary near-wall peak. This secondary peak is responsible for the secondary peak in the plots of $\widetilde {T''T''}$ from figure 9(h). On the other hand, the temporal evolution in the $\varPi = 10$ case does not substantially affect the spanwise spectra. This suggests a small gradual change in the spanwise organization of the flow due to the weaker acceleration, despite the changes to the mean flow during the simulation time.

The structure of the outer-layer turbulent eddies is now studied by extending the characteristic length scale, $\ell _{1,2}(y)\sim (u_{\tau } h)^{1/2}(\partial _{y}\overline {u})^{-1/2}$ , introduced in Pirozzoli (Reference Pirozzoli2012) for canonical wall-bounded flows to the compressible transient 3-D flow. The success of $\ell _{1,2}$ is in agreement with the attached eddy model (Townsend Reference Townsend1976; Perry & Marusic Reference Perry and Marusic1995) which argues that the length scales of the turbulent eddies grow with wall-normal height. By consideration of the ${\textit{Ma}}$ -independence under an appropriate velocity transformation, $\ell _{1,2}$ has been extended to compressible flows by replacing $\partial _{y}\overline {u}$ in $\ell _{1,2}$ with the appropriately transformed mean shear. Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016) used the Van Driest (Reference Van Driest1951) transformation whereas Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022) noted an improvement when using the GFM transformation due to a better agreement with the incompressible flow in the mean flow field. Using the velocity transformation in (3.2), a time-dependent, compressible $\ell _{1,2}$ is introduced as

(4.7) \begin{equation} \ell ^{*}_{1,2}(y,t) = (u_{\tau }(t) h)^{1/2}\bigg (\frac {u_{\tau }(t)}{\ell _{\nu }(t)}\frac {\partial \|\widetilde {\boldsymbol{u}}\|_{TL}}{\partial y^{*}}\bigg )^{-1/2} = (h \ell _{\nu }(t))^{1/2}\bigg (\frac {\partial \|\widetilde {\boldsymbol{u}}\|_{TL}}{\partial y^{*}}\bigg )^{-1/2}. \end{equation}

Since $\ell _{1,2}$ relied on an assumption that the outer scale is characterized by $u_{\tau }$ , $h$ and the local shear, it is expected then that $\ell _{1,2}^{*}$ should only hold in regimes where $u_{\tau }$ , $h$ and the velocity transformation define a ${\textit{Ma}}$ - and time-independent outer region. From figure 3(d,e), this is only satisfied for the $\varPi = 10$ cases. The normalized premultiplied spanwise spectra, $E_{kk}(y,kz,t)/\widetilde {k}(y,t)$ , are plotted as a function of $\lambda _{z}/h$ in figure 15(a–d) and a function of $\lambda _{z}/\ell ^{*}_{1,2}(y,t)$ in figure 15(e–h). For the $\varPi = 10$ cases, $\ell _{1,2}^{*}(y,t)$ collapses the spectra at the different wall-normal heights by accounting for the temporal and wall-normal variation in the transport properties, further demonstrating that ideas presented in Pirozzoli (Reference Pirozzoli2012), Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016) and Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022) extend to this 3-D transient compressible flow. Similar to Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022), $E_{\textit{TT}}(y,t;k_{z})$ also exhibits the same outer-layer self-similarity as a function of $\lambda _{z}/\ell _{1,2}^{*}$ for $\varPi = 10$ , though this result is omitted for brevity. In line with the theory, the $\textit{Ma} = 1.5$ , $\varPi = 40$ case is not able to appropriately collapse the spectra due to the time-dependence in the wake using the TL transformation. Though not shown, the other $\varPi = 40$ cases do not admit this $\ell _{1,2}^{*}$ scaling. The lack of collapse for $\varPi = 40$ is a consequence of the strong acceleration creating non-equilibrium effects in $\widetilde {\boldsymbol{u}}$ rather than compressibility effects. The success of $\ell _{1,2}^{*}$ in characterizing the spanwise scales may suggest that the eddies in the $\varPi = 10$ cases evolve in quasiequilibrium whereas the eddies of the $\varPi = 40$ cases are in non-equilibrium, using the definition of equilibrium from Lozano-Durán et al. (Reference Lozano-Durán, Giometto, Park and Moin2020). Additionally, it may be a consequence of a strong deviation from the log-law. To remedy this, $\ell_{1,2}^{*}(y,t)$ is replaced with $\widetilde {\ell }^{*}_{1,2}(y,t) = \sqrt {h \ell _{\nu } y^{*}}$ in line with the attached-eddy theory. Using this scaling in figure 15(e–h) provides an improvement for the scaling, even the $\varPi = 40$ case. Thus, despite the non-equilibrium conditions, the typical eddy size grows with the vertical distance using the semilocal coordinate as $\sqrt {\ell _{\nu }y^{*}}$ . The success of the $\widetilde {\ell }^{*}_{1,2}$ scaling highlights the delayed response in the turbulent organization in the outer region to the spanwise acceleration relative to the near-wall, as discussed in § 4.4.

Figure 15.

Normalized premultiplied spanwise spectra as a function of $\lambda _{z}/h$ (a,b,c,d), $\lambda _{z}/\ell _{1,2}^{*}(y,t)$ and $\lambda _{z}/\widetilde {\ell }_{1,2}^{*}(y,t)$ (off-set vertically by $0.3$ for visibility) (e,f,g,h) for case 1 (a,e), case 2(b,f), case 3(c,g) and case 5(d,h). The different colours denote different $y/h$ and dark to light indicate increasing time, in increments of $160\ell _{\nu }(0)/u_{\tau }(0)$ .

4.4. Transport of turbulent and thermal coherent structures

Although the spectra provide insight into the structural organization of the flow, it is limited to scales defined along $\boldsymbol{e}_{x}$ and $\boldsymbol{e}_{z}$ despite the 3-D statistics. To study the organization of the turbulent eddies in an axis-free manner, the structure identification proposed by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) is used. Based on the spectra in § 4.3 and previous work on the structural characteristics of 3-D boundary layers, the near-wall structures respond to the spanwise acceleration before the large-scale structures creating misalignment. This misalignment is the mechanism responsible for inhibiting the Reynolds stress production (Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020). This conceptual picture is illustrated in figure 16(a,b) for the canonical channel at $t=0$ and sometime after the spanwise body force has been imposed. Here, the organization of the kinetic energy- and temperature-carrying eddies will be studied to quantify the misalignment between the near-wall and large-scale structures.

Figure 16.

Cartoon of the near-wall small scale structures and large-scale structures at $t = 0$ (a) and $t\gt 0$ (b) demonstrating the faster alignment of the near-wall structures than the larger structures farther from the wall. Representative near-wall kinetic energy structures identified at $t^{+} = 0$ and $t^{+} = 145$ with their spines in red (c). Instantaneous realizations of identified kinetic energy structures (d,f) and their spines (e,g) at $t^{+} = 0$ (d,e) and $t^{+} = 145$ (f,g) for $\textit{Ma} = 1.5$ and $\varPi = 40$ . The shading of the isosurfaces reflects the wall-normal distance, darker colours are closer to the wall, and the blue and red denotes positive and negative fluctuations, respectively. The spines in (e) and (g) are coloured based on their angle with respect to $\boldsymbol{e}_{x}$ as shown in the colourbar. Note that the figures plot a subset of the full computational domain to highlight some representative structures and their spines.

To define the kinetic energy structures, the instantaneous kinetic energy is first defined as $K = (u^2+v^2+w^2)/2$ and the fluctuating kinetic energy is then $K' = K - \overline {K}$ . The kinetic energy structures and temperature structures are defined as connected regions, $\varOmega$ , where

(4.8) \begin{equation} \left |K'\right | \geqslant 1.75\sqrt {\overline {K'K'}} \end{equation}

and

(4.9) \begin{equation} \left |T'\right | \geqslant 1.75\sqrt {\overline {T'T'}}, \end{equation}

respectively. Note that because $\overline {K} = (\overline {u_{i}'u_{i}'} + \overline {u}_{i}\overline {u}_{i})/2$ , $\overline {K'K'}$ is different from the TKE – it is the variance of the instantaneously fluctuating kinetic energy. The coefficient $1.75$ determines the threshold size and this structure identification has been shown to be robust in response to changes in its value (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). Connected regions whose volume is smaller than $(30\ell _{\nu }(0))^{3}$ are rejected to avoid the accumulation of small disconnected structures. Each of the structure’s sign can be computed based on the sign of $K'$ or $T'$ within each of the kinetic energy or temperature structures. To compute the angle of each structure, the centre of mass of each structure, $\boldsymbol{x}_{m} = x_{m}\boldsymbol{e}_{x} + y_{m}\boldsymbol{e}_{y} + z_{m}\boldsymbol{e}_{z}$ , is first computed as

(4.10) \begin{equation} \boldsymbol{x}_{m}V = \int _{\varOmega }\boldsymbol{x}{\rm d}x{\rm d}y{\rm d}z, \end{equation}

where $V$ is the volume of each $\varOmega$ . Then, for each structure, the matrix, $\boldsymbol{X}$ is computed as

(4.11) \begin{equation} X_{jk}V = \int _{\varOmega } (x_{\!j} - x_{j,m})(x_{k} - x_{k,m}){\rm d}x{\rm d}y{\rm d}z. \end{equation}

Since, $\boldsymbol{X}$ is symmetric, it can be written in its spectral representation as $\boldsymbol{X} = \sum _{\!j}^{3}\alpha _{\!j}\boldsymbol{p}_{\!j}\otimes \boldsymbol{p}_{\!j}$ where $\alpha _{1} \geqslant \alpha _{2} \geqslant \alpha _{3}$ and $\boldsymbol{p}_{\!j}$ are orthonormal eigenvectors often denoted as the principal axes. The principal direction of each structure is then $\boldsymbol{p}_{1}$ . The direction along the wall-parallel plane is $\boldsymbol{e}_{p}$ , the unit vector parallel to $\boldsymbol{p}_{1} - (\boldsymbol{p}_{1}\boldsymbol{\cdot }\boldsymbol{e}_{y})\boldsymbol{e}_{y}$ . Finally, the length of each structure is defined as $r_{1} = \textrm {max}(d) - \textrm {min}(d)$ , where

(4.12) \begin{equation} d = [(x-x_{m})\boldsymbol{e}_{x} + (z-z_{m})\boldsymbol{e}_{z}]\boldsymbol{\cdot }\boldsymbol{e}_{p} : (x,y,z) \in \varOmega \end{equation}

is the wall-parallel distance of $\varOmega$ along $\boldsymbol{e}_{p}$ . The angle of the structure is defined as $\theta$ , the angle between $\boldsymbol{e}_{p}$ and $\boldsymbol{e}_{x}$ , as $\cos (\theta ) = \boldsymbol{e}_{p}\boldsymbol{\cdot }\boldsymbol{e}_{x}$ . Finally, the spines are defined as the line segment that intersects $\boldsymbol{x}_{m}$ parallel to $\boldsymbol{p}_{1}$ with a wall-parallel length of $r_{1}$ .

In figure 16(c), the structure identification is illustrated for two representative near-wall kinetic energy structures at $t^{+} = 0$ and $t^{+} = 145$ for the $\textit{Ma} = 1.5$ and $\varPi = 40$ case. Their spines are included as well for visualization, illustrating that they can recover the statistical length of the structures and their change in orientation as the flow is sufficiently accelerated. Two snapshots at $t = 0$ and $t^{+} = 145$ are presented in figure 16(d,f) with the surface contours of the identified structures within a subset of the channel domain. At $t^{+} = 0$ , one can qualitatively note that the small-scale near-wall structures and large-scale structures are aligned along the streamwise direction, like in the cartoon presented in figure 16(a). At $t^{+} = 145$ , the data reflects figure 16(b) as the near-wall small-scale structures are aligning towards the spanwise direction while the large-scales are mostly streamwise aligned. Figure 16(e,g) plot the spines of each of the identified structures in figure 16(d,f), respectively, colour-coded by their angle $\theta$ , the angle between the spine and $\boldsymbol{e}_{x}$ . At $t^+ = 0$ , the spines all primarily have $\theta \approx 0^{\circ }$ . There are some spines with large $\theta$ corresponding to smaller-scale structures that are longer in their spanwise extent than their streamwise extent. Thus, the spine detection algorithm treats these structures as spanwise aligned. At $t^{+} = 145$ , the near-wall small-scale spines have $\theta \gtrsim 45^{\circ }$ while the longer spines away from the wall’s $\theta$ reflect their streamwise alignment with smaller $\theta$ . After a long enough time, the large-scale structures will eventually align with and equilibrate with the new direction of the flow. However, the focus of this section is on the initial period of the sudden spanwise acceleration.

To further quantify the orientation of the spines, instantaneous weighted histograms of $y_{m}$ and $\theta$ , $H(y_m,\theta )$ , are computed for each time instance across all $8$ ensembles for each case and are weighted by the length $r_{1}(y_{m},\theta )$ . The weighted histograms are computed as

(4.13) \begin{equation} R_{1}H(\theta ,y_{m}) = \sum _{\breve {\theta } -\theta = \Delta \theta _{1}}^{\Delta \theta _{2}}\sum _{\breve {y}_{m} -y_{m} = \Delta y_{m,1}}^{\Delta y_{m,2}}r_{1}(\breve {y}_{m},\breve {\theta }), \end{equation}

where $R_{1} = \sum _{\theta }\sum _{y_{m}}r_{1}(y_{m},\theta )$ is the total length of all the structures and $\Delta \theta _{1},$ $\Delta \theta _{2}$ , $\Delta y_{m,1}$ and $\Delta y_{m,2}$ denote the edges of the bins. The histograms use $22$ linearly spaced bins in $\theta \in (-20^\circ ,70^\circ )$ and $16$ logarithmically spaced bins in $y_{m}/\ell _{\nu }(0) \in (2,450)$ . The structures above the channel half-height are mapped to $y_{m} \rightarrow 2h - y_{m}$ in (4.13) to take advantage of the statistical symmetry across the channel half-height. Weighing the histograms with $r_{1}$ removes the influence of small-scale structures away from the wall whose alignment is poorly defined, like those pictured in figure 16(e,g). The orientation of the identified structures are also compared with the angle between two statistical quantities, $\widetilde {a}$ and $\widetilde {b}$ , as $\theta _{\widetilde {a},\widetilde {b}} = \tan ^{-1}(\widetilde {a}/\widetilde {b})$ , where $\widetilde {a}$ and $\widetilde {b}$ represent a streamwise and spanwise flow statistic, respectively. These angles are used to measure the orientation of the wall-parallel flow field, $\theta _{\widetilde {u},\widetilde {w}}$ , the Reynolds shear stresses, $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ , and velocity–temperature covariances, $\theta _{\widetilde {u''T''},\widetilde {w''T''}}$ , with respect to $\boldsymbol{e}_{x}$ . Both $\theta _{\widetilde {u},\widetilde {w}}$ and $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ have been used in incompressible studies, demonstrating a lag between the mean flow field and Reynolds shear stresses (Bradshaw & Pontikos Reference Bradshaw and Pontikos1985; Moin et al. Reference Moin, Shih, Driver and Mansour1990; Lozano-Durán et al. Reference Lozano-Durán, Giometto, Park and Moin2020). The addition of $\theta _{\widetilde {u''T''},\widetilde {w''T''}}$ serves to define a direction based on the velocity–temperature covariances.

Beginning with $\textit{Ma} = 1.5$ and $\varPi = 40$ , figure 17(a–d) present $H(\theta ,y_{m})$ for the kinetic energy structures. As expected, the structures begin statistically likely to be streamwise aligned at $t=0$ regardless of their wall-normal centroid, $y_{m}$ . As evidenced by $H(\theta ,y_{m})$ , the kinetic energy structures closest to the wall are the first to respond to the spanwise acceleration while the structures farther from the wall are more likely to be aligned closer to $\boldsymbol{e}_{x}$ . The $H(\theta ,y_{m})$ of the temperature structures in figure 17(e–h) follow similar trends as the kinetic energy structures with near-wall structures more statistically likely to align in the spanwise direction. The kinetic energy and temperature structures’ $H(\theta ,y_{m})$ suggest that the temperature structures are more likely to be closer to the wall than the kinetic energy structures once $g_{z}$ has been applied. Similar to incompressible channels, for $t\gt 0$ , $\theta _{\widetilde {u},\widetilde {w}}$ is greater than $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ such that the direction of the Reynolds shear stresses lag compared with the direction of the mean flow field. For initial times ( $t^{+} \lessapprox 54$ ), both $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ and $\theta _{\widetilde {u''T''},\widetilde {w''T''}}$ are similar, albeit with a slight lag in the orientation of the velocity–temperature covariances’ orientation with respect to the Reynolds stresses. For later times, near the wall, the $\theta _{\widetilde {u''v''},\widetilde {w''v''}} \lt \theta _{\widetilde {u''T''},\widetilde {w''T''}}$ , coinciding the the increased presence of temperature structures closer to the wall. At the same time, $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ and $\theta _{\widetilde {u''T''},\widetilde {w''T''}}$ differ away from the wall as the velocity–temperature covariances change sign, reflecting differences in the thermal fluctuation transport not reflected in the Reynolds stresses. For the times plotted here, the peaks of $H(\theta ,y_{m})$ occur between $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ and $\theta _{\widetilde {u},\widetilde {w}}$ .

Figure 17.

Weighted histogram of identified (a–d) kinetic energy structures and (e–h) temperature structures as a function of $y_{m}$ and $\theta$ for Ma = 1.5 and $\varPi = 40$ . The solid black, dashed black and solid red lines denote $\theta _{\widetilde {u},\widetilde {w}}$ , $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ and $\theta _{\widetilde {u''T''},\widetilde {w''T''}}$ , respectively.

Figure 18.

Weighted histogram of identified (a,b) kinetic energy structures and (c,d) temperature structures as a function of $y_{m}$ and $\theta$ for Ma = 1.5 and $\varPi = {1}0$ . The solid black, dashed black and solid red lines denote $\theta _{\widetilde {u},\widetilde {w}}$ , $\theta _{\widetilde {u''v''},\widetilde {w''v''}}$ and $\theta _{\widetilde {u''T''},\widetilde {w''T''}}$ , respectively.

Figures 18(a,b) and 18(c,d) plot the $H(\theta ,y_{m})$ of the kinetic energy and temperature structures, respectively, for $\textit{Ma} = 1.5$ and $\varPi = 10$ for two representative times. Similar to the $\varPi = 40$ case, the near-wall structures align away from $\boldsymbol{e}_{x}$ before the outer layer structures and the temperature structures are closer to the wall than the kinetic energy structures. However, the orientation angles of $\varPi = 10$ are much smaller while the misalignment between the near-wall and large-scale structures is less than $10^\circ$ for $\varPi = 10$ and up to $30^{\circ }$ in $\varPi = 40$ . Similar to the observations of Lozano-Durán et al. (Reference Lozano-Durán, Giometto, Park and Moin2020), the more substantial misalignment in the strongly accelerated flow inhibits the production of $\overline {\rho }\widetilde {u''v''}$ and $\overline {\rho }\widetilde {T''v''}$ by pushing the turbulent structures out of equilibrium and creating a less efficient state for production. The success of the $\ell _{1,2}^{*}$ scaling of the spectra in figure 15(e–g) for $\varPi = 10$ suggested a quasiequilibrium state in the turbulent flow. The organization of this quasiequilibrium state reveals that the turbulent eddies are closely aligned such that the underlying turbulent structure is unchanged, consistent with the definition in Lozano-Durán et al. (Reference Lozano-Durán, Giometto, Park and Moin2020).

5.1. Discussion

The two choices of $\varPi$ create two qualitatively distinct responses to the spanwise acceleration despite the ${\textit{Ma}}$ . These effects are primarily observed in the temperature field where a secondary peak in $\widetilde {T}$ emerges for $\varPi = 40$ that is not observed in the $\varPi = 10$ cases. This is indicative of increased viscous heating from increased turbulent intensity. The near-wall $\widetilde {T}$ peak is also associated with an increased $q_{w}$ for the $\varPi = 40$ cases compared with the $\varPi = 10$ cases. Furthermore, the presence of the near-wall $\widetilde {T}$ peak changes the turbulent transport of temperature fluctuations by creating a change in sign across the peak in the velocity–temperature variances and an increased contribution of $Q_{1}$ and $Q_{3}$ in $\widetilde {T''v''}$ . This change in sign is not reflected in the Reynolds shear stresses, making $\widetilde {\boldsymbol{u}''v''}$ and $\widetilde {T''v''}$ uncorrelated, which ultimately limits the predictive capabilities of the GRA. Despite their differences, both the $\varPi = 10$ and $\varPi = 40$ cases observed a decrease in $\widetilde {T''T''}$ across the channel for early times despite the net increase in $\widetilde {T}$ .

Figure 21.

Measures of the sweep angle via $\beta _{c}$ , the angle of the centreline velocity relative to the angle of the wall shear stress (a) and $\theta _{c}$ , the angle of the centreline velocity (b) against $t^{+}$ in degrees. The colours and line symbols are denoted in table 1. Note that in (b), the lines corresponding to $\textit{Ma} = 0.3$ are just underneath the lines for $\textit{Ma} = 1.5$ .

If the angle of $\boldsymbol{u}_{c}$ relative to $\boldsymbol{\tau }_{w}$ ( $\beta$ ) is taken as a measure for the sweep angle, the sweep angles are at their peak, above $20^\circ$ for $\varPi = 40$ . Taking instead the relative velocity of $\boldsymbol{u}_{c}$ to $\boldsymbol{e}_{x}$ as the sweep angle gives, for the largest $\varPi$ , sweep angles around $60^{\circ }$ at the end of the simulation time. These sweep angles, plotted in figure 21, are similar to the sweep angles of supersonic aircraft, such as $\varLambda = 55^\circ$ for Concorde. However, the acceleration is much different in the case of the aircraft as opposed to the ones shown herein. Using Concorde as an example once again, its spanwise acceleration can be estimated as $(U_{\infty }\sin (\varLambda ))^2/L_{c}$ (Vos & Farokhi Reference Vos and Farokhi2015), where the cruising velocity is $U_{\infty } \approx 600\, { \rm m\,s^{- 1}}$ and the chord is $L_{c}\approx 20 \textrm { m}$ . This will be normalized with $a_{w}^{2}/\delta$ , where the boundary layer thickness, $\delta$ , is estimated as $\delta \approx 0.1\textrm { m}$ using a $1/5$ power-law estimate (Schlichting & Gersten Reference Schlichting and Gersten2016). Concorde’s normalized acceleration is then $A_{a} = U^{2}_{\infty }\sin (\varLambda )^{2}\delta /L_{c}a_{w}^{2} \approx 1.2\times 10^{-2}$ . The normalized acceleration can also be calculated for the channel as $A_{c} = g_{z}h/a_{w}^{2}$ . For the $\varPi = 10$ cases used herein, the normalized acceleration is $A_{c} \approx 2.5\times 10^{-3}$ , $6.9\times 10^{-2}$ and $3.3\times 10^{-1}$ for $\textit{Ma} = 0.3$ , $1.5$ and $3.0$ , respectively. Thus, for similar ${\textit{Ma}}$ as the Concord, case 2 would be at a similar normalized spanwise acceleration as those expected from real operations. Both $T_{m}$ and $\widetilde {T''T''}_{m}$ showed the best predictive capability for $\varPi = 10$ , suggesting that these approaches may have applicability in realistic conditions.

The success of $\widetilde {T}$ can help the development of wall models in non-equilibrium conditions. In Griffin et al. (Reference Griffin, Fu and Moin2023), the GRA of Zhang et al. (Reference Zhang, Bi, Hussain and She2014) and GFM velocity transformation were combined to develop a wall-model for supersonic flows. The extensions to the TL velocity transformation in § 3.1 and the GRA may also show promise in predicting $\widetilde {\boldsymbol{u}}$ and $\widetilde {T}$ in 3-D non-equilibrium conditions. A challenge for a wall-model of this type is that the self-similarity only extends to the viscous sublayer for large spanwise acceleration which can limit grid-size requirements. As a possible remedy for this, the Lagrangian relaxation towards equilibrium plus laminar non-equilibrium wall model (Fowler, Zaki & Meneveau Reference Fowler, Zaki and Meneveau2022) can show some promise by treating the equilibrium and non-equilibrium effects separately. Future work will use the DNS data along with extensions to these models in pursuit of wall-models for non-equilibrium compressible flow.