2.1. Governing equations and system definition

This study investigates two typical wall-bounded turbulent systems: sheared ACRBC and spanwise RSPCF. In the RSPCF system, as shown in figure 1(a), the fluid is confined between two infinite parallel plates separated by a gap width $ L$ . The unit vectors of the $x$ -, $y$ - and $z$ -axes are denoted by $ \hat {\boldsymbol{x}}$ , $ \hat {\boldsymbol{y}}$ and $ \hat {\boldsymbol{z}}$ , respectively. The top and bottom walls move with equal and opposite velocities, creating a background shear with streamwise velocity difference $ [U]$ . The entire system rotates about the spanwise $z$ -axis with angular velocity $ \varOmega$ , and is subject to gravitational acceleration $ - g\hat {\boldsymbol{y}}$ acting in the negative $y$ -direction. A temperature difference $ [T]$ is maintained between the top and bottom walls. The velocity difference is adopted as the velocity reference scale. The non-dimensionalised governing equations are

(2.1a) \begin{align}\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} = 0, \\[-28pt] \nonumber \end{align}

(2.1b) \begin{align} \frac {\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{u} = -\boldsymbol{\nabla }p + \frac {1}{Re} {\nabla} ^2 \boldsymbol{u} + R_{\varOmega }\boldsymbol{u} \times \hat {\boldsymbol{z}} + \frac {Ra}{Re^2 \textit{Pr}} T\kern-0.5pt\hat {\boldsymbol{y}}, \\[-28pt] \nonumber \end{align}

(2.1c) \begin{align} \frac {\partial T}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }T = \frac {1}{\textit{Re} \textit{Pr}} {\nabla} ^2 T. \\[0pt] \nonumber \end{align}

Here, the boundary conditions are $ \boldsymbol{u}|_{y=\pm 0.5} = \pm 0.5 \hat {\boldsymbol{x}}$ and $ T|_{y=\pm 0.5} = \mp 0.5$ , with periodic boundary conditions applied in both the $x$ - and $z$ -directions.

For the ACRBC system, illustrated in figure 1(b), the unit vectors for $ r, \varphi , z$ are $ \hat {\boldsymbol{r}}, \hat {\boldsymbol{\varphi }}, \hat {\boldsymbol{z}}$ , respectively. An incompressible fluid is confined between two independently rotating coaxial cylinders with inner and outer radii $ r_i$ and $ r_o$ , respectively. The angular velocities of the inner and outer cylinders are $ \varOmega _i \hat {\boldsymbol{z}}$ and $ \varOmega _o \hat {\boldsymbol{z}}$ , respectively. The mean angular velocity is defined as $ \varOmega = (\varOmega _i + \varOmega _o)/2$ , and the radius ratio is $ \eta = r_i/r_o$ . The inner and outer cylinder walls are maintained at constant temperatures $ T_i$ and $ T_o$ , respectively, with temperature difference $[T] = T_o - T_i$ . In this system, the centrifugal force field generated by the mean rotation provides an equivalent radial gravity. The linear velocity difference between the inner and outer cylinders, $ [U]$ , serves as the velocity reference scale, and $ L = r_o - r_i$ is the gap width. As indicated by Zhang et al. (Reference Zhang, Xia and Chen2022), the buoyancy effect caused by the inertial term $\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}$ cannot be approximated by an explicit momentum source term with $O(\alpha _T [T])$ relative error. Therefore, for simplicity, a valid buoyancy correction is not included in this work. The non-dimensionalised momentum equation for the ACRBC system is

(2.2) \begin{align} \frac {\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol {\nabla }\boldsymbol{u} = -\boldsymbol {\nabla }p + \frac {1}{Re} {\nabla} ^2 \boldsymbol{u} + R_{\varOmega }\boldsymbol{u} \times \hat {\boldsymbol{z}} - T \frac {Ra}{Re^2 \textit{Pr}} \frac {2(1-\eta )}{1+\eta } \hat {\boldsymbol{r}}. \end{align}

Here, the velocity vector is $ \boldsymbol{u}=(u_r,u_\varphi ,u_z)$ . The no-slip and isothermal boundary conditions are given by:

(2.3a) \begin{align} & \text{for }\,r = r_i = \frac {\eta }{1-\eta }, \quad \boldsymbol{u} = \left (0, \frac {r_i}{r_o+r_i}, 0\right )\!, \quad T = -0.5; \\[-12pt] \nonumber \end{align}

(2.3b) \begin{align} & \text{for }\,r = r_o = \frac {1}{1-\eta }, \quad \boldsymbol{u} = \left (0, \frac {-r_o}{r_o+r_i}, 0\right )\!, \quad T = 0.5. \\[10pt] \nonumber \end{align}

Here, the domain is infinite in the axial direction.

The two systems share some common definitions of non-dimensional parameters, involving the kinematic viscosity $\nu$ , thermal diffusivity $\kappa$ , and thermal expansion coefficient $ \alpha _T$ of the fluid. The primary independent control parameters are defined as follows: Reynolds number $ \textit{Re}=[U]L/\nu$ , rotation number $R_\varOmega = 2 \varOmega L/[U]$ , Rayleigh number $Ra=g\alpha _T [T] L^3/(\nu \kappa )$ , and Prandtl number $ \textit{Pr}=\nu /\kappa$ . Correspondingly for RSPCF, the thermal Nusselt number and the rotating-frame momentum Nusselt number are defined as $ \textit{Nu}_T = - {\rm d}\langle T \rangle / {\rm d}y \big |_{\pm 0.5}$ and $ \textit{Nu}_U = {\rm d}\langle U \rangle / {\rm d}y \big |_{\pm 0.5}$ , respectively, where $\langle {\cdot }\rangle$ denotes averaging over time and the horizontal plane.

2.2. Twin-force analogy and inertial-frame Reynolds analogy

To unify the analysis and quantitative description of the coupling effects of rotation (Coriolis force) and stratification (buoyancy) in wall-bounded turbulence, we draw upon and extend the theoretical framework proposed by Zhang & Sun (Reference Zhang and Sun2024). The core of this framework lies in revealing the profound similarity between Coriolis and buoyancy effects, and systematically describing their coupling through two central concepts: the twin-force analogy and the inertial-frame Reynolds analogy. This framework introduces a variable transformation $\theta =(R_{\varOmega } y - u)/(1-R_{\varOmega })$ , and defines the inertial-frame momentum Nusselt number as $ \textit{Nu}_\theta = - {\rm d}\langle \theta \rangle / {\rm d}y \big |_{\pm 0.5}$ , thereby absorbing the influence of rotation. Hereafter, the classical Reynolds analogy refers to the similarity between $ \textit{Nu}_U$ and $ \textit{Nu}_T$ , whereas the inertial-frame Reynolds analogy refers to that between $ \textit{Nu}_\theta$ and $ \textit{Nu}_T$ . Consequently, the streamwise momentum equation can be rewritten in a form similar to the temperature transport equation; this transformation reveals that the action of the Coriolis force on the wall-normal velocity is analogous to a buoyancy effect, thus mathematically establishing the connection between these two forces. On this basis, to unify the quantification of rotation and stratification effects, the following non-dimensional parameters are introduced: the Coriolis Richardson number $Ri_\theta =R_{\varOmega }(1-R_{\varOmega })$ , and the Coriolis Rayleigh number $Ra_\theta =Re^2 Ri_\theta$ , where, under the present sign convention, positive $Ra_\theta$ corresponds to destabilising rotation, and negative $Ra_\theta$ corresponds to stabilising rotation. The relationship between the rotation number $R_{\varOmega }$ and the Coriolis Rayleigh number is given by

(2.4) \begin{align} R_{\varOmega } = \frac {1}{2} \pm \sqrt {\frac {1}{4} - \frac {\textit{Ra}_{\theta }}{\textit{Re}^2}}, \end{align}

where the solutions correspond to the strong and weak rotation regimes, respectively. The buoyancy Richardson number is defined as $Ri_T=Ra/Re^2$ for $ \textit{Pr} \leqslant 1$ , and $Ri_T=Ra\, Pr/Re^2$ for $ \textit{Pr} \gt 1$ , and the global Richardson number is $ Ri_\chi = Ri_\theta + Ri_T$ . In the Rayleigh number formulation used for the LSA, the global Rayleigh number is defined as $ \textit{Ra}_\chi = Ra_\theta + Ra$ without $ \textit{Pr}$ correction. Two relative strength parameters, $ \gamma = (Ri_T - Ri_\theta )/(Ri_T + Ri_\theta )$ and $ \lambda = (Ra - Ra_\theta )/(Ra + Ra_\theta )$ , are used to describe the relative contributions of buoyancy and the Coriolis force in Richardson number and Rayleigh number forms, respectively. Here, $ \gamma = -1$ and $ \lambda = -1$ represent pure rotation, whereas $\gamma =1$ and $\lambda =1$ represent pure stratification. Correspondingly, a global Nusselt number $ \textit{Nu}_\chi$ is defined as the weighted average of the heat and inertial-frame momentum transport coefficients:

(2.5) \begin{align} \textit{Nu}_\chi = \frac {1+\lambda }{2}\textit{Nu}_T + \frac {1-\lambda }{2}\textit{Nu}_\theta, \end{align}

which is consistent with the calculated data in Zhang & Sun (Reference Zhang and Sun2024). Based on this, the core idea of the twin-force analogy is proposed: buoyancy and the Coriolis force act as ‘twin forces’, whose effects on the flow field can be linearly superimposed or counteracted. Therefore, the overall transport efficiency of the system, $ \textit{Nu}_\chi$ , is primarily determined by the Reynolds number $ Re$ and the global Richardson number $ Ri_\chi$ , showing weak dependence on the relative strength of the two effects. In previous studies at $ \textit{Pr}=1$ , the similarity between TCF and RBC in two dimensions was strictly proven through reasonable mathematical transformations (Tanaka et al. Reference Tanaka, Kida, Yanase and Kawahara2000; Zhang et al. Reference Zhang, Xia, Shi and Chen2019). Although explicit correspondence based on the original variables is incomplete in three-dimensional flows (Jeganathan et al. Reference Jeganathan, Alba and Ostilla-Mónico2023), the inertial-frame Reynolds analogy based on the $\theta$ transformation has been verified for specific $\gamma$ at $ \textit{Pr}=1$ . However, the combined effects of buoyancy and the Coriolis force on the flow still require verification for cases with $ \textit{Pr}=1$ (specifically $ Ri_\chi =0$ ) and $ \textit{Pr} \neq 1$ . Therefore, this study will systematically investigate these key parameter ranges via DNS to reveal the core laws that characterise the coupling of multiphysics involving rotation, stratification and shear.

2.3. The LSA

We employ LSA to quantitatively establish the theoretical connection between ACRBC and RSPCF at the critical point of instability. The relevant physical quantity $ \tilde {\psi }$ is decomposed into the sum of a steady base flow $ \varPsi _0$ and an infinitesimal perturbation $ \hat {\psi }$ . For the ACRBC system, we adopt the LSA framework detailed in Zhong et al. (Reference Zhong, Wang and Sun2023). For the RSPCF system, perturbations are imposed on the linear base flow $ U_0(y) = y$ and $ T_0(y) = -y$ . Its governing equations can be simplified by introducing the wall-normal vorticity $ \varpi = \partial u/ \partial z - \partial w/ \partial x$ , yielding the following system of equations (Wallin et al. Reference Wallin, Grundestam and Johansson2013):

(2.6a) \begin{align} s\varpi = - {\rm i}\alpha U_0 \varpi + \frac {1}{\textit{Re}} \big({\rm D}^2 - k^2 \big)\varpi - {\rm i}\beta \big(U'_0 - R_{\varOmega } \big)v, \\[-28pt] \nonumber \end{align}

(2.6b) \begin{align} s \big({\rm D}^2 - k^2 \big)v = - {\rm i}\alpha U_0 \big({\rm D}^2 - k^2 \big)v + \frac {1}{Re} \big({\rm D}^2 - k^2 \big)^2 v - {\rm i}\beta R_{\varOmega }\varpi - \frac {Ra}{Re^2 \textit{Pr}}k^2 T, \\[-28pt] \nonumber \end{align}

(2.6c) \begin{align} sT = \left (-{\rm i}\alpha U_0 + \frac {{\rm D}^2-k^2}{\textit{Re}\, \textit{Pr}}\right )T - v{\rm D}T_0. \\[0pt] \nonumber \end{align}

The boundary conditions are $ y=\pm 0.5$ , $ v = {\partial v}/{\partial y} = \varpi = 0$ . Here, $ {\rm D} = {\rm d}/{\rm d}y$ , and $ k^2 = \alpha ^2 + \beta ^2$ , with $ \hat {\psi } = \psi (y)\,{\rm e}^{st+{\rm i}(\alpha x+\beta z)}$ , where $ \alpha$ and $ \beta$ are the streamwise and spanwise wavenumbers, respectively, and $ s$ is the complex growth rate. The governing equations of both systems constitute a generalised eigenvalue problem. We solve them numerically using the Chebyshev spectral collocation method.

2.4. Numerical methodology

To quantitatively validate our hypothesis under more complex conditions, DNS were performed using the open-source spectral element code NEK5000. The computational configuration largely follows Zhang & Sun (Reference Zhang and Sun2024), with the parameters selected as $ Re \in \{5000, 10\,000\}$ , $ \textit{Pr} \in \{1, 4,0.25\}$ , $ Ri_{\chi } \in \{0, 0.16\}$ and $ \gamma \in \{0, \pm 1, \pm 2\}$ . Notably, we have supplemented the study with a series of specific cases at $Ri_\chi = 0$ , where the Coriolis and buoyancy forces exhibit identical magnitudes but act with opposing physical effects. These cases provide a crucial baseline for a detailed analysis of the superposition and cancellation mechanisms between the two forces. The computational domain has dimensions $L_x \times L_y \times L_z = 9\pi \times 1 \times 4\pi$ , and the spectral element mesh consists of $N_x \times N_y \times N_z = 81 \times 10 \times 72$ elements. The polynomial order of the spectral elements was $N=7$ , corresponding to $N+1=8$ Gauss–Lobatto–Legendre points in each spatial direction within each element.

Consistent with the LSA, the DNS employ no-slip isothermal walls and periodic boundary conditions in the horizontal directions. The grid resolution strictly follows the standards validated in Zhang & Sun (Reference Zhang and Sun2024), ensuring sufficient accuracy for the current parameter space.

4.1. Validation and breakdown of analogies at $ \textit{Pr}=1$

We first consider the most ideal case, $ \textit{Pr}=1$ . According to the prediction of the twin-force analogy, the overall transport efficiency of the system is primarily determined by the global Richardson number $ Ri_\chi$ , and is independent of the specific ratio of rotation to stratification (Zhang & Sun Reference Zhang and Sun2024). A direct corollary is that when $ Ri_\chi =0$ , regardless of the specific combination of the Coriolis force and buoyancy, the flow state should degenerate into ordinary plane Couette flow driven by shear. However, our DNS results reveal significant deviations from this prediction. As shown in figure 4, with fixed $ Re=10\,000$ , $ \textit{Pr}=1$ , $ Ri_\chi =0$ and $Ra/10^6 = 0, 1, -1, -10, 10, -20, 20$ , different combinations of rotation and stratification lead to distinct flow structures. When rotation acts to destabilise the flow (figure 4 a), the flow evolves into ordered, nearly streamwise-homogeneous longitudinal vortices dominated by spanwise wavenumbers. It should be emphasised that the spanwise vortices identified in the preceding LSA and the streamwise-aligned longitudinal vortices observed here in the DNS correspond to different parameter regimes and are governed by different dominant mechanisms. Conversely, when buoyancy acts as the destabilising agent (figure 4 b), the vortex structures are more chaotic, exhibiting small-scale features in both streamwise and spanwise directions.The differences in transport characteristics are equally significant; as shown in figure 4(c), when buoyancy effects are weak ( $ \textit{Ra} \leqslant 10^6$ ), $ \textit{Nu}_T$ and $ \textit{Nu}_U$ are nearly identical, consistent with the classical Reynolds analogy, but differ slightly more from $ \textit{Nu}_\theta$ . However, in the strong destabilising buoyancy regime ( $ \textit{Ra} \gt 10^6$ ), the thermal Nusselt number increases significantly with $ \textit{Ra}$ , whereas the two momentum Nusselt numbers (influenced by the stabilising Coriolis force) show smaller variations, leading to the complete breakdown of both the classical Reynolds analogy and the inertial-frame Reynolds analogy.

Figure 4.

Results for $ \textit{Re}=10\,000$ , $ \textit{Pr}=1$ under different rotation and stratification intensities with $Ra_\chi =0$ . (a,b) Temperature contours in the $x{-}z$ plane at $y=0$ for $Ra/10^6 = -1$ and $20$ , respectively. (c) Corresponding Nusselt number results. (d) Energy fraction of the modes with $|\beta /\alpha | \gt 8$ for $Ra_\chi =0$ , $ \textit{Re}=5000$ and $10\,000$ under different rotation and stratification intensities. The red dashed line marks the $25\,\%$ threshold.

Figure 5.

Spectral characteristics at $y=0$ after virtually introducing buoyancy and rotation contributions to vorticity of plane Couette flow at $ \textit{Re}=10\,000$ : (a) rotation contribution; (b) buoyancy contribution.

Quantitative analysis of energy spectra (figure 4 d) further confirms that the validity of the classical Reynolds analogy is closely related to the ‘two-dimensionality’ of the flow. We also verified six cases with $ Re=5000,\ Ra/10^6=1, -1, 2, -2, 6, -6$ , and found that $ \textit{Nu}_T$ and $ \textit{Nu}_U$ exhibit similarity only when the energy fraction of structures characterised by a wavenumber ratio $ |\beta /\alpha | \gt 8$ (explicitly including the $ \alpha =0$ mode) is relatively high – specifically, exceeding approximately 25 % (the red dashed line in figure 4 d) in the present parameter space. The physical mechanism is that the flow field exhibits significant quasi-two-dimensional characteristics. This streamwise homogeneity suppresses the three-dimensional effects that cause the discrepancy between momentum and scalar transport, thereby aligning their transport mechanisms. To analyse the effects of body forces on different modes, we define the buoyancy characteristic spectrum $ \chi _b$ and the Coriolis force characteristic spectrum $ \chi _c$ as follows:

(4.1a,b) \begin{align} \chi _b = (\alpha ^2 + \beta ^2) E_T, \quad \chi _c = \beta ^2 E_k, \end{align}

where $ E_T$ and $ E_k$ are the squared moduli of the spectral coefficients for temperature and velocity, respectively. These definitions are based on the terms on the right-hand side of the vorticity equation derived by Zhang & Sun (Reference Zhang and Sun2024). Figures 5(a,b) present the spectral signatures of vorticity contributions resulting from the virtual imposition of rotation and buoyancy effects upon a pure shear flow. The rotation-dominated case (refer to figure 4 a) manifests strong anisotropy, with energy confined to a narrow band near $ \alpha \approx 0$ , and elongated along $ \beta$ , consistent with the potential spectral characteristics of the Coriolis force (figure 5 a). In contrast, the flow field in the buoyancy-dominated regime (figure 4 b) mirrors the buoyancy spectrum in figure 5(b), characterised by energy diffusion around the origin, and a broad distribution in both $ \alpha$ and $ \beta$ directions. This suggests that thermal plume instability triggers complex three-dimensional, multi-scale motions. Consequently, unstable buoyancy promotes structural complexity, whereas the Coriolis force acts to organise the flow into coherent, anisotropic structures.

To fundamentally explain the impact of different driving forces on flow structure, we examined the budgets of the individual velocity-component contributions to turbulent kinetic energy (TKE) and enstrophy. The equation for the contribution of the $i$ th velocity component to TKE is

(4.2) \begin{align} \frac {\partial \langle k_{u_i} \rangle }{\partial t} + \langle u_{\kern-1pt j} \rangle \frac {\partial \langle k_{u_i} \rangle }{\partial x_{\kern-1pt j}} & = - \langle u'_i u'_{\kern-1pt j} \rangle \frac {\partial \langle u_i \rangle }{\partial x_{\kern-1pt j}} - \frac {1}{Re} \left \langle \frac {\partial u'_i}{\partial x_{\kern-1pt j}} \frac {\partial u'_i}{\partial x_{\kern-1pt j}} \right \rangle \nonumber \\& \quad + \frac {\partial }{\partial x_{\kern-1pt j}} \left ( \langle p' u'_i \rangle \delta _{\textit{ij}} + \langle k_{u_i} u'_{\kern-1pt j} \rangle + \frac {1}{Re} \frac {\partial \langle k_{u_i} \rangle }{\partial x_{\kern-1pt j}} \right ) + W_{C, u_i} + W_{B, u_i}, \end{align}

without summation over $i$ , where $\phi '=\phi -\langle \phi \rangle$ denotes the fluctuating component of the variable $\phi$ . Here, $k_{u_i} = {u'_i}{u'_i}/2$ (no summation) represents the instantaneous contribution of the $i$ th velocity component to the TKE; $\langle u_i'u_i'\rangle$ denotes the $i$ th normal Reynolds stress component. The Coriolis force contribution is $ W_{C, u_i}$ , and the buoyancy contribution is $ W_{B, u_i}$ . The only non-zero terms of $ W_{B, u_i}$ and $ W_{C, u_i}$ are $W_{B, v} = (Ra/(Re^2 Pr)) \langle v' T' \rangle$ and $ W_{C, v} = -W_{C, u} = -R_\varOmega \langle v' u' \rangle = Ri_\theta \langle v'\theta '\rangle$ .

The work done by the Coriolis force appears only in the wall-normal and streamwise TKE equations, and since the Coriolis force does no net work, it serves only to transfer energy. As neither the Coriolis force nor buoyancy acts directly on the velocity field – requiring pressure projection to eliminate the longitudinal component (Zhang et al. Reference Zhang, Xia, Zhou and Chen2020) – their forms in the vorticity equation are more fundamental. Therefore, one of the crucial tools for analysing the twin-force analogy is the enstrophy equation:

(4.3) \begin{align} \frac {\partial \langle E'_\omega \rangle }{\partial t} + \langle u_{\kern-1pt j} \rangle \frac {\partial \langle E'_\omega \rangle }{\partial x_{\kern-1pt j}} + \frac {\partial \langle u'_{\kern-1pt j} E'_\omega \rangle }{\partial x_{\kern-1pt j}} & = - \big \langle u'_{\kern-1pt j} \omega '_i \big \rangle \frac {\partial \langle \omega _i \rangle }{\partial x_{\kern-1pt j}} + \langle \omega _{\kern-1pt j} \rangle \left \langle \omega '_i \frac {\partial u'_i}{\partial x_{\kern-1pt j}} \right \rangle + \big\langle \omega '_i \omega '_{\kern-1pt j} \big\rangle \frac {\partial \langle u_i \rangle }{\partial x_{\kern-1pt j}} \nonumber\\& \quad + \left \langle \omega '_i \omega '_{\kern-1pt j} \frac {\partial u'_i}{\partial x_{\kern-1pt j}} \right \rangle + \frac {1}{Re} \frac {\partial ^2 \langle E'_\omega \rangle }{\partial x_{\kern-1pt j}^2} - \frac {1}{Re} \left \langle \frac {\partial \omega '_i}{\partial x_{\kern-1pt j}} \frac {\partial \omega '_i}{\partial x_{\kern-1pt j}} \right \rangle \nonumber\\& \quad + E_B + E_C, \end{align}

where $\boldsymbol{\omega }$ represents the vorticity, $ E'_\omega = \omega '_i \omega '_i /2$ is enstrophy, $E_B = -(Ra/(Re^2 Pr)) \langle \omega '_x\, \partial T'/ \partial z \rangle + (Ra/(Re^2 Pr)) \langle \omega '_z\, \partial T'/ \partial x \rangle$ is the buoyancy contribution, and $ E_C = R_{\varOmega } \langle \omega '_x\, \partial u'/ \partial z \rangle + R_{\varOmega } \langle \omega '_y\, \partial v'/ \partial z \rangle + R_{\varOmega } \langle \omega '_z\, \partial w'/ \partial z \rangle$ is the Coriolis force contribution. Vorticity is the curl of the velocity field; when a material line element aligned with vorticity is stretched by velocity gradients, the magnitude of vorticity increases correspondingly due to the conservation of angular momentum, a process known as vortex stretching. The distinction between three-dimensional and two-dimensional flows is that the vortex stretching term is zero in two-dimensional flows. Notably, the formal similarity between the Coriolis force term and the vortex stretching term reveals the core mechanism by which system rotation affects vorticity dynamics.

To fundamentally reveal the physical mechanism underlying the breakdown of this analogy, we analysed the budgets of wall-normal TKE and enstrophy. Here, the enstrophy budget is used not as a direct measure of $ \textit{Nu}_U$ , $ \textit{Nu}_\theta$ or $ \textit{Nu}_T$ , but as a diagnostic of how buoyancy and Coriolis forces generate or maintain vortical structures that subsequently influence the associated turbulent fluxes. We first examine the case $ \textit{Pr}=1$ , systematically varying the relative strength of rotation and stratification under the condition that $Ri_\chi$ is non-zero. The energy budget analysis ultimately clarifies the root cause of the analogy’s breakdown. In the wall-normal TKE equation, the two forces exhibit a misleading similarity: as shown by the solid lines in figure 6(a), when the contributions of buoyancy and the Coriolis force are consistent ( $ \gamma =0$ ), their contribution profiles almost completely overlap. This overlap arises because at $ \textit{Pr}=1$ and $\gamma =0$ , $Ri_\theta =Ri_T$ and the profiles of $\langle v'\theta '\rangle$ and $\langle v'T'\rangle$ are nearly identical. This explains why the two forces are analogous at the level of overall energy injection, leading to similarities in global transport coefficients. However, fundamental differences exist in the physical mechanisms of the two forces, which are vividly reflected in the enstrophy equation. In an ideal two-dimensional streamwise-homogeneous flow, the contribution terms of the Coriolis force and buoyancy to enstrophy are strictly symmetric in mathematical form, which would lead to a perfect inertial-frame Reynolds analogy. But in three-dimensional flow, this symmetry is broken: additional, asymmetric partial derivative terms appear in the buoyancy and Coriolis force contribution terms. This mathematical difference under three-dimensional effects leads to distinct spatial actions of the two forces (as shown by the dashed lines in figure 6 a). The contribution of the Coriolis force to enstrophy is strictly concentrated in the near-wall region, whereas the contribution of buoyancy is distributed throughout the core region of the flow. Furthermore, as $ \textit{Ra}$ increases (figure 6 b), the vorticity generation mechanism dominated by buoyancy and acting in the flow core becomes increasingly important, causing the flow structure to further diverge from that under pure rotation, thereby exacerbating the deviation of transport coefficients.

Figure 6.

Contributions of buoyancy and Coriolis force in the enstrophy equation and wall-normal TKE equation for (a) $ \gamma =0$ and (b) $ \gamma =2$ , respectively, at $ \textit{Pr}=1$ , $ Re=10\,000$ , $ \textit{Ra}_\chi = 16 \times 10^6$ . Red denotes buoyancy contribution, blue denotes Coriolis force contribution, and magenta denotes the net contribution between buoyancy and Coriolis force. Black is the reference line for zero. Solid lines represent contributions in wall-normal TKE, and dashed lines represent contributions in enstrophy.

Figure 6. Long description

Two line graphs depict contributions of buoyancy and Coriolis force in enstrophy and wall-normal TKE equations. Panel A: The line graph shows the contributions of buoyancy and Coriolis force in the wall-normal TKE equation. The x-axis is labeled 'y' with a range from -0.50 to 0.50. The left y-axis is labeled 'TKE contribution' with a range from -2 x 10^-4 to 2 x 10^-4, and the right y-axis is labeled 'Enstrophy contribution' with a range from -0.2 to 0.2. The red line represents buoyancy contribution, the blue line represents Coriolis contribution, and the magenta line represents the total contribution. Solid lines indicate TKE contributions, while dashed lines indicate enstrophy contributions. Panel B: The line graph shows the contributions of buoyancy and Coriolis force in the enstrophy equation. The x-axis is labeled 'y' with a range from -0.50 to 0.50. The left y-axis is labeled 'TKE contribution' with a range from -2 x 10^-4 to 2 x 10^-4, and the right y-axis is labeled 'Enstrophy contribution' with a range from -0.2 to 0.2. The red line represents buoyancy contribution, the blue line represents Coriolis contribution, and the magenta line represents the total contribution. Solid lines indicate TKE contributions, while dashed lines indicate enstrophy contributions.

Differences in flow structure directly affect the coherence of transport, leading to the breakdown of the inertial-frame Reynolds analogy. The root cause lies in the different ways in which buoyancy and the Coriolis force influence vorticity generation in three-dimensional space. The Coriolis force acts primarily in the spanwise direction, tending to generate longitudinal vortices aligned with the flow, lacking the modes dominated by streamwise wavenumbers found in buoyancy-driven convection. Such highly ordered vortex structures induced by the Coriolis force can bring about stronger ejection and sweep events, thereby generating Reynolds stresses more efficiently. This ultimately leads to an asymmetry between momentum transport correlation ( $ R_{v,-u}$ ) and heat transport correlation ( $ R_{v,T}$ ). Here, the correlation coefficient is defined as

(4.4) \begin{align} R_{\xi _1,\xi _2} = \frac { \displaystyle \int _{-1/2}^{1/2} \big\langle \xi _1'\xi _2' \big\rangle \,\mathrm{d}y }{ \sqrt { \displaystyle \int _{-1/2}^{1/2} \big\langle \xi _1'^2 \big\rangle \,\mathrm{d}y \,\displaystyle \int _{-1/2}^{1/2} \big\langle \xi _2'^2 \big\rangle \,\mathrm{d}y } }. \end{align}

These correlation coefficients complement the Nusselt numbers by distinguishing changes in fluctuation amplitudes from changes in the correlation between the fluctuating fields. As shown in figure 7(a), in the Coriolis-dominated regime ( $\gamma \lt 0$ ), the correlation coefficient $R_{v,-u}$ is enhanced. Conversely, in the buoyancy-dominated regime ( $\gamma \gt 0$ ), $R_{v,T}$ becomes more prominent; this is because in an idealised state, the horizontally isotropic plume distribution statistically provides no net contribution to $ \langle u'v' \rangle$ .

Figure 7.

Comparison of correlation coefficients $ R_{v,-u}$ and $ R_{v,T}$ : (a) $ \textit{Pr}=1,\ Re=10\,000$ ; (b) $ \textit{Pr}=0.25,4$ and $ \textit{Re}=5000$ . Both cases have fixed $ Ri_\chi = 0.16$ .

This mechanism fundamentally elucidates the results reported by Zhang & Sun (Reference Zhang and Sun2024): specifically, under the condition $ \textit{Pr}=1$ , inertial-frame momentum transport is larger at $ \gamma = -2$ , while heat transport is larger at $ \gamma = 2$ .

4.2. Extended analogy at different Prandtl numbers: symmetry breaking and transport dissimilarity

When $ \textit{Pr} \neq 1$ , the mismatch between heat and momentum diffusion rates leads to corrections to the perfect analogy observed at $ \textit{Pr}=1$ .

Various DNS results indicate that the transport properties systematically depend on $ \textit{Pr}$ (Blass et al. Reference Blass, Tabak, Verzicco, Stevens and Lohse2021; Zhang & Sun Reference Zhang and Sun2024). For $ \textit{Pr} \gt 1$ , the momentum boundary layer is thicker than the thermal boundary layer, making the flow more sensitive to the dynamical effects of rotation; consequently, both heat and momentum transport are enhanced as unstable rotation strengthens (decreasing $ \gamma$ ). Conversely, for $ \textit{Pr} \lt 1$ , the thermal boundary layer is thicker, rendering the flow more sensitive to the thermodynamic effects of buoyancy; thus transport efficiency increases with strengthening unstable stratification (increasing $ \gamma$ ). Although the individual transport coefficients diverge, the modified global Nusselt number $ \textit{Nu}_\chi$ still exhibits a weak dependence within a certain range of $ \gamma$ . This symmetry breaking originates from the mismatch between the thermal and momentum boundary layer thicknesses at different $ \textit{Pr}$ , which dictates the dominant spatial extent of buoyancy and Coriolis forces, respectively.

To quantitatively elucidate the turbulence structures and the underlying dynamic mechanisms, we combine quadrant analysis of the Reynolds stress-weighted joint probability density functions (at $y=-0.4$ ) with energy budget analysis. At $ \textit{Pr}=0.25$ , the energy budget analysis (figure 8) reveals that buoyancy makes significant contributions to both wall-normal TKE and enstrophy throughout the channel, whereas rotational effects are largely confined to the near-wall momentum boundary layer. This buoyancy-dominated driving force, acting over a broader spatial range, results in a more complex flow structure. The Reynolds stress quadrant analysis (Wallace Reference Wallace2016) (figure 9) provides direct evidence for this: for instance, under stable rotation ( $ \gamma =1.5$ ; figure 9 b), the negative contributions of inward/outward interaction events (Q1, Q3) to the Reynolds stress are significantly enhanced. This disrupts the coherence of momentum transport, leading to a reduction in $ R_{v,-u}$ (figure 7 b). In contrast, at $ \gamma =-2$ (figure 8 a), where the system is in the regime of unstable rotation and stable stratification, the thermal boundary layer is significantly thicker than the momentum boundary layer; consequently, the net contribution to enstrophy is negative, which also suppresses transport. The relatively high normalised correlations at $\gamma =-2$ do not contradict this suppression, because the turbulent fluxes also depend on the corresponding fluctuation amplitudes.

Figure 8.

Contributions of buoyancy and Coriolis forces to the budgets of enstrophy and wall-normal TKE, for (a,b) $ \textit{Pr}=0.25$ and (c,d) $ \textit{Pr}=4$ . The rotation ratios are (a,c) $ \gamma = -2$ and (b,d) $ \gamma = 1.5$ . Red and blue lines denote the buoyancy and Coriolis force contributions, respectively. Solid and dashed lines represent contributions in wall-normal TKE and enstrophy, respectively.

Figure 9.

Reynolds-stress-weighted joint probability density functions $ u' v'\, P(u', v')$ at $y=-0.4$ , where $P(u',v')$ denotes the joint probability density function of the streamwise and wall-normal velocity fluctuations $u'$ and $v'$ , respectively. The plotted variables are normalised by their respective root mean square (rms) values, for (a,b) $ \textit{Pr}=0.25$ and (c,d) $ \textit{Pr}=4$ . The relative strength parameter is (a,c) $ \gamma = -2$ and (b,d) $ \gamma = 1.5$ .

Figure 9. Long description

Panel A: A heat map showing the joint probability density function of streamwise and wall-normal velocity fluctuations normalized by their root mean square values. The x-axis represents the streamwise velocity fluctuation normalized by its root mean square value, ranging from -3.0 to 3.0. The y-axis represents the wall-normal velocity fluctuation normalized by its root mean square value, also ranging from -3.0 to 3.0. The color scale ranges from -0.35 to 0.35, with blue indicating lower values and red indicating higher values. The heat map shows two prominent blue regions diagonally opposite each other and two smaller red regions. Panel B: A heat map similar to Panel A but with a different relative strength parameter. The color scale ranges from -0.08 to 0.08. The heat map shows four distinct regions, two blue and two red, arranged in a symmetric pattern. Panel C: A heat map similar to Panel A but with a different relative strength parameter. The color scale ranges from -0.14 to 0.14. The heat map shows two prominent blue regions diagonally opposite each other and two smaller red regions. Panel D: A heat map similar to Panel B but with a different relative strength parameter. The color scale ranges from -0.09 to 0.09. The heat map shows four distinct regions, two blue and two red, arranged in a symmetric pattern.

Conversely, at $ \textit{Pr}=4$ , the momentum boundary layer is thicker than the thermal one. At $\gamma =1.5$ (figure 8 d), energy budget analysis clearly demonstrates that the Coriolis contributions to wall-normal TKE and enstrophy far exceed those of buoyancy, with buoyancy effects confined to the extremely thin thermal boundary layer. Rotation acts to stabilise the flow; in the core region, it makes a significant negative contribution to both wall-normal TKE and enstrophy, leading to a drop in transport coefficients, with the flow tending towards a steady state at $ \gamma =2$ (Zhang & Sun Reference Zhang and Sun2024). The quadrant analysis for $ \gamma =-2$ (figure 9 c) confirms a flow field dominated by a single mechanical effect with a more organised structure, where ejection (Q2) and sweep (Q4) events dominate the Reynolds stress contribution. This organised Reynolds stress mechanism explains why the momentum–transport correlation is stronger than the heat–transport correlation across all $ \textit{Pr}=4$ cases.

Therefore, when $ \textit{Pr} \neq 1$ , the fundamental reason for the breakdown of the inertial-frame Reynolds analogy is that the Prandtl number, by controlling the relative thicknesses of thermal and momentum boundary layers, determines whether buoyancy or the Coriolis force dominates the flow field. Because these two forces differ significantly in the magnitude and spatial location of vorticity generation in three-dimensional space, they induce distinct turbulent structures, preventing a simple linear superposition of their effects.