1. Introduction
On the surfaces of high-speed flying vehicles, the generated compressible wall-bounded turbulence would bring about non-negligible heat transfer, which is as crucial as the aerodynamic force for safe flight. To achieve a precise design of future high-speed aircraft, it is thus essential to accurately predict both the thermal and aerodynamic force effects in compressible wall turbulence. One research approach is to delineate the similarities between the temperature and velocity fields, and further seek the velocity–temperature relationship in compressible wall turbulence, which is generally called the Reynolds analogy. This type of research has undergone a long period of development and achieved a satisfactory answer in the past decade with the help of direct numerical simulations (DNS). The deduced relations have also been deployed for modelling purposes (Griffin, Fu & Moin Reference Griffin, Fu and Moin2023; Song, Zhang & Xia Reference Song, Zhang and Xia2023; Chen, Gan & Fu Reference Chen, Gan and Fu2024; Cheng & Fu Reference Cheng and Fu2024a; Cheng et al. Reference Cheng, Chen, Zhu, Shyy and Fu2024; Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2024). As early as the 1930s, Busemann (Reference Busemann1931) and Crocco (Reference Crocco1932) independently deduced the first velocity–temperature relation for laminar flows by assuming that the Prandtl number (
$Pr$) of fluid is unity. This relation was then extended by van Driest (Reference van Driest1951) for turbulent boundary layers. Within their frameworks, a very strong analogy between these two instantaneous fluctuating fields is made, which reads as
\begin{equation} H'=U_wu', \end{equation}
where 
$H$ and 
$u$ denote the total enthalpy and the streamwise velocity, respectively, and 
$\xi '$ represents the fluctuating component of a variable 
$\xi$, which is the difference between the instantaneous value of 
$\xi$ and its Reynolds averaged statistic 
$\bar {\xi }$. Here 
$U_w$ is a constant for a boundary layer. For the mean fields, the relation is
\begin{equation} \bar{H}-\bar{H}_w=U_w\bar{u}, \end{equation}
where the subscript ‘
$w$’ denotes the wall quantity. Regarding an adiabatic boundary layer, the exact relationship between the instantaneous 
$u'$ and 
$T'$ can be deduced from (1.1), i.e. 
$T'=-\bar {u}u'/ C_p$, where 
$C_p$ and 
$T$ denote the specific heat at constant pressure and the temperature field, respectively. Based on this relation, Morkovin (Reference Morkovin1962) further proposed several derivative inferences, which are dubbed as the strong Reynolds analogy (SRA).
The assumption 
$Pr=1$ in the Crocco–Busemann relation is too idealized, because for air, 
$Pr\approx 0.7$. This deviation results in differences between the DNS results and the SRA predictions in later studies. To circumvent this deficiency, Walz (Reference Walz1962) hypothesized that a mixed Prandtl number, 
$Pr_m=C_p(\bar {\mu }+\bar {\mu }_t)/(\bar {k}+\bar {k}_t)$, is a constant, where 
$\mu$, 
$\mu _t$, 
$k$ and 
$k_t$ denote the viscosity, the eddy viscosity, the thermal conductivity and the eddy thermal conductivity, respectively. Meanwhile, the mean temperature is considered to be only a function of 
$\bar {u}$, and thus the Walz equation is derived. Its equivalent form is
\begin{equation} \bar{H}_r-\bar{H}_w=U_w\bar{u}, \end{equation}
where 
$\bar {H}_r=C_p[\bar {T}+r\bar {u}^2/(2C_p)]$ is called the local recover enthalpy, and 
$r$ is the recovery factor. Compared with (1.2), (1.3) suggests that the local excess recovery enthalpy is in direct proportion to the magnitude of mean velocity field. Walz's equation improves the Crocco–Busemann relation and agrees with DNS well in adiabatic turbulent boundary layers (Pirozzoli, Grasso & Gatski Reference Pirozzoli, Grasso and Gatski2004; Duan & Martin Reference Duan and Martin2011; Fu et al. Reference Fu, Karp, Bose, Moin and Urzay2021). However, for diabatic flows, its performance is not that good (Duan & Martin Reference Duan and Martin2011; Zhang, Duan & Choudhari Reference Zhang, Duan and Choudhari2018; Huang, Duan & Choudhari Reference Huang, Duan and Choudhari2022).
For the relationship between the fluctuating fields, the most vital modification is the celebrated Huang's strong Reynolds analogy (HSRA) (Huang, Coleman & Bradshaw Reference Huang, Coleman and Bradshaw1995), incorporating the phenomenological model of Gaviglio (Reference Gaviglio1987), which takes the form of
\begin{equation} Pr_tT^{\prime}_{rms}/\partial_y\bar{T}=u^{\prime}_{rms}/\partial_y\bar{u}, \end{equation}
where 
$Pr_{t}\equiv (\overline {{\rho v^{\prime }u^{\prime }}}/\overline {{\rho v^{\prime }T^{\prime }}})(\partial \bar {T}/\partial \bar {u})$ is the turbulent Prandtl number (
$\rho$ and 
$v$ denote the fluid density and the wall-normal velocity, respectively) and the subscript ‘
$rms$’ represents the root mean square value (r.m.s.). This relation works well for wall turbulence with different wall thermal conditions (Pirozzoli et al. Reference Pirozzoli, Grasso and Gatski2004; Duan & Martin Reference Duan and Martin2011; Huang et al. Reference Huang, Duan and Choudhari2022).
The first empirical relation of mean fields which performs well for distinct wall thermal conditions was proposed by Duan & Martin (Reference Duan and Martin2011). They introduced a non-dimensional recovery enthalpy, which is defined as 
$\bar {H}_r^*=(\bar {H}_r-\bar {H}_w) /(\bar {H}_{r \delta }-\bar {H}_w)$, and found that 
$\bar {H}_r^*$ is a unique function of 
$\bar {u}/\bar {u}_{\delta }$ for all DNS data, where the subscript ‘
$\delta$’ refers to the quantities evaluated at the boundary edge. When calorically perfect gas is assumed, a new velocity–temperature relation can be obtained.
Thereafter, the most systematic model is the generalized Reynolds analogy theory (GRA) proposed by Zhang et al. (Reference Zhang, Bi, Hussain and She2014). They generalized the velocity–enthalpy relation by introducing a general recovery enthalpy 
$H_g$,
\begin{equation} \bar{H}_g-\bar{H}_w=U_w\bar{u}, \end{equation}
where 
$H_g=C_pT+r_gu^2/2$, and 
$r_g$ is an extended form of 
$r$. Simultaneously, a residual temperature 
$\phi '$ is also introduced to build a loose analogy between the fluctuating velocity and temperature fields, namely,
\begin{equation} H_g'+C_p\phi'=U_wu'. \end{equation}
Multiplying (1.6) by 
$(\rho v)'$ and averaging, 
$r_g$ can be solved as
\begin{equation} r_{g}=\frac{C_{p}}{\bar{u}}\left[\left.\frac{\partial\bar{T}}{\partial\bar{u}}\right|_{w}-\frac{1}{\overline{{Pr_{e}}}}\frac{\partial\bar{T}}{\partial\bar{u}}\right], \end{equation}
where 
$\overline {Pr_e}$ is called the effective Prandtl number by Zhang et al. (Reference Zhang, Bi, Hussain and She2014), whose definition is
\begin{equation} \overline{Pr_{e}}\equiv\frac{\overline{Pr_{t}}}{1+\varepsilon},\quad \overline{Pr_{t}}\equiv\frac{\overline{(\rho v)^{\prime}u^{\prime}}}{\overline{(\rho v)^{\prime}T^{\prime}}}\frac{\partial\bar{T}}{\partial\bar{u}}, \quad \varepsilon\equiv\frac{\overline{(\rho v)^{\prime}\phi'}}{\overline{(\rho v)^{\prime}T^{\prime}}}. \end{equation}
Herein 
$\overline {Pr_{t}}$ is a new definition of the turbulent Prandtl number, and its value is only a little different from that of 
$Pr_t$ in the outer layer of a boundary layer at a high Mach number (Zhang et al. Reference Zhang, Bi, Hussain and She2014). Substitution of (1.7) into (1.5) yields a differential equation for the mean fields,
\begin{equation} \bar{T}-\frac{\bar{u}}{2}\left[\left.\frac{\partial\bar{T}}{\partial\bar{u}}\right|_w+\frac{1}{\overline{Pr_e}}\frac{\partial\bar{T}}{\partial\bar{u}}\right]=\bar{T}_w. \end{equation}Simultaneously, for the fluctuating fields, their relation can be also deduced by substituting (1.7) into (1.6):
\begin{equation} T'+\phi'=\frac{1}{\overline{Pr_e}}\frac{\partial\bar{T}}{\partial\bar{u}}u'. \end{equation}
Zhang et al. (Reference Zhang, Bi, Hussain and She2014) further adopted the simplest model for 
$\overline {Pr_e}$ and assumed 
$\overline {Pr_e}=1$, and thus (1.9) becomes solvable. The deduced mean velocity–temperature relation is
\begin{equation} \frac{\bar{T}}{\bar{T}_\delta}=\frac{\bar{T}_w}{\bar{T}_\delta}+\frac{\bar{T}_{rg}-\bar{T}_w}{\bar{T}_\delta}\frac{\bar{u}}{\bar{u}_\delta}+\frac{\bar{T}_\delta-\bar{T}_{rg}}{\bar{T}_\delta}\left(\frac{\bar{u}}{\bar{u}_\delta}\right)^2, \end{equation}
where 
$\bar {T}_{rg}=\bar {T}_\delta +r_g\bar {u}_\delta ^2/(2C_p)$. Interestingly, through introducing the well-known Reynolds analogy factor and treating it as a constant, (1.11) can recover the empirical relation of Duan & Martin (Reference Duan and Martin2011). For the fluctuating fields, Zhang et al. (Reference Zhang, Bi, Hussain and She2014) modelled 
$\phi '$ as 
$(\overline {{(\rho v)^{\prime}\phi ^{\prime }}}/\overline {{(\rho v)^{\prime}u^{\prime }}})u^{\prime }$, and the HSRA can be restored with 
$Pr_t$ in (1.4) replaced by 
$\overline {Pr_{t}}$. Hereafter, we name this modified form as MHSRA.
Although the deductions of the GRA are successful in depicting the 
$\bar {u}$–
$\bar {T}$ relation, we want to remember some points that are not clearly clarified in previous work. Firstly, the GRA is an overdetermined system. For a given flow, the unknown functions are 
$r_g$ and 
$\phi '$. However, there are three related equations, i.e. (1.5), (1.6) and the assumption 
$\overline {Pr_{e}}=1$. Secondly, the physical meaning of 
$\phi '$ is not clearly identified. The core assumptions of the GRA are the existence of the residual temperature 
$\phi '$ and the constant value of 
$\overline {Pr_e}$. The latter is also associated with the existence of 
$\phi '$, because the extra wall-normal heat flux transfer originating from 
$\phi '$ in the definition of 
$\overline {Pr_e}$ (i.e. 
$\varepsilon$) acts as an effective correction to 
$\overline {Pr_t}$, making up its systematic deviation from unity. So we may ask two questions. What is the physical significance of 
$\phi '$? Is it a physically real field that exists independently of 
$T'$, or is it just a component of 
$T'$? Thirdly, if 
$-\phi '$ is a component of 
$T'$, (1.10) implies that the whole velocity fluctuating field shares a duality relation with just a portion of the temperature fluctuating field. This scenario is kind of counterintuitive.
Motivated by these confusions, we try to propose a new Reynolds analogy theory. We find that this model might provide a good solution to each issue, and shed light on some subtle details of the 
$u$–
$T$ coupling. Meanwhile, the newly introduced residual quantities (temperature and velocity) in our model are of clear physical significance, which bridge the gap between the phenomenological model and the mathematical representation of the Reynolds analogy.
2. A Reynolds analogy model
We begin with the key point proposed by Zhang et al. (Reference Zhang, Bi, Hussain and She2014), that it is the local recovery enthalpy, rather than the total enthalpy, that remains constant in the shear region for air flow over an adiabatic wall. Also, we also adopt 
$r_g$ to account for the effects of 
$Pr\neq 1$ in flows over diabatic walls. However, we hypothesize that it is not 
$u$, but a new velocity 
$u_{e}$, named the local effective velocity, which constitutes the local effective recovery enthalpy (
$H_e$), i.e. 
$H_e=C_pT+r_gu_e^2/2$. The local effective velocity can be decomposed into two components,
\begin{equation} u_e=u+\chi', \end{equation}
where 
$\chi '$ can also be regarded as a residual velocity with zero mean value but non-negligible fluctuation, and thus 
$\bar {u}_e=\bar {u}$. On the other hand, the generalized mean enthalpy–velocity relation is
\begin{equation} \bar{H}_e-\bar{H}_w=U_w\bar{u}_e, \end{equation}
and 
$U_w$ can be determined to be 
$U_w=(C_p\partial \bar {T}/\partial \bar {u}_e)_w$. Equation (2.2) can be expanded as
\begin{equation} C_p\bar{T}+\tfrac{1}{2}r_g\overline{u^{2}}+\tfrac{1}{2}r_g\overline{\chi^{'2}}+r_g\overline{u'\chi^{\prime}}-C_p\bar{T}_w=U_w\bar{u}. \end{equation}For the relationship between the fluctuating local effective enthalpy and velocity, the following formula can be introduced:
\begin{equation} H_e'+C_p\phi'=U_wu_e', \end{equation}
where 
$H_e^{\prime }=C_pT^{\prime }+r_g\bar {u}_eu_e^{\prime }$. In our model, the residual temperature 
$\phi '$ is still included to describe the temperature fields of flows over adiabatic walls. Following Zhang et al. (Reference Zhang, Bi, Hussain and She2014), 
$r_g$ can be solved after multiplying (2.4) by 
$(\rho v)'$ and averaging, that is,
\begin{equation} r_g=\frac{C_p}{\bar{u}_e}\left[\frac{U_w}{C_p}-\frac{\overline{(\rho v)^{\prime}(T^{\prime}+\phi^{\prime})}}{\overline{(\rho v)^{\prime}(u^{\prime}+\chi^{\prime})}}\right]. \end{equation} Gaviglio (Reference Gaviglio1987) observed that, for a compressible boundary layer, the intensities of the velocity and temperature fluctuations carried by large-scale eddies are in direct proportion to the gradients of their mean quantities, respectively. And their corresponding ratios are positively related to the velocity length scale and the temperature length scale, respectively. In our model, we treat 
$T^{\prime }+\phi ^{\prime }$ and 
$u'+\chi '$ as the pure temperature and velocity fluctuations that are directly determined by the mean fields, and thus, carried by the large-scale eddies (see more detailed analyses in § 3). In this regard, combining our propositions and the phenomenological model of Gaviglio (Reference Gaviglio1987), we can directly link the ratio of the wall-normal momentum and heat flux transfers in the right-hand side of (2.5) with the gradients of the mean fields,
\begin{equation} \frac{\overline{(\rho v)^{\prime}(T^{\prime}+\phi^{\prime})}}{\overline{(\rho v)^{\prime} (u^{\prime}+\chi^{\prime})}}=\alpha\frac{\partial\bar{T}}{\partial\bar{u}}, \end{equation}
where 
$\alpha$ is a constant. Hence, (2.5) can further be deduced as
\begin{equation} r_{g}=\frac{C_{p}}{\bar{u}}\left[\left.\frac{\partial\bar{T}}{\partial\bar{u}}\right|_{w}- \alpha\frac{\partial\bar{T}}{\partial\bar{u}}\right], \end{equation}
where the relation 
$\bar {u}_e=\bar {u}$ is used. The differential equation for the mean temperature can be obtained by substituting (2.7) into (2.3) and using 
$\overline {u^{2}}\approx \bar {u}^{2}$, which reads as
\begin{equation} \bar{T}-\frac{\bar{u}}{2}\left[\left.\frac{\partial\bar{T}}{\partial\bar{u}}\right|_w+\alpha\frac{\partial\bar{T}}{ \partial\bar{u}}\right]+\frac{r_g\left(\overline{\chi^{'2}}+2\overline{u'\chi^{\prime}}\right)}{2C_p}=\bar{T}_w, \end{equation}
and the 
$u'$–
$T'$ relation can be derived by substituting (2.7) into (2.4), namely,
\begin{equation} T'+\phi'=\alpha\frac{\partial\bar{T}}{\partial\bar{u}}\left(u'+\chi'\right). \end{equation}
It can be seen that if we take 
$\alpha$ to be 
$1/\overline {Pr}_e(=1)$ and ignore the effect of 
$\chi '$, the present model can restore to the GRA. It is particularly noteworthy that the present model is well-posed, because the number of the unknown functions (
$\chi '$, 
$r_g$ and 
$\phi '$) is the same as that of the relevant equations ((2.2), (2.4) and (2.6)) for a given flow (
$\alpha$ is not an unknown quantity, see the related discussion in § 4). Furthermore, following the procedures outlined in Zhang et al. (Reference Zhang, Bi, Hussain and She2014), it can be mathematically proved that the similarity between the Reynolds averaged momentum and energy equations permits the solution expressed by (2.2) and (2.4).
As the temperature fluctuation can be recognized as a passive scalar even in compressible wall turbulence (Cheng & Fu Reference Cheng and Fu2024b), we can consider the 
$u$–
$T$ coupling through the lens of a control system. The fluctuating velocity and temperature fields can be treated as the input and output, respectively, whereas the related transfer function is determined by the mean profiles of these two fields. The fluctuating temperature field is essentially the response of the temperature field to the velocity field. If this response is perfect, their coherence, which can be appraised by their correlated function, would be tight. Under these circumstances, the duality relation between 
$u'$ and 
$T'$ is well-established. In the parlance of this view, within the framework of the SRA, the magnitude of the transfer function is 
$|\bar {u}/C_p|$. However, the relation is built in an instantaneous sense, and thus the SRA is generally poorly agreed with the real flows. For the HSRA, the corresponding transfer function is 
$|({1}/{P r_t})({\partial \bar {T}}/{\partial \bar {u}})|$, which is only valid for the r.m.s. values of 
$T'$ and 
$u'$. For the MHSRA, the counterpart is 
$|({1}/{\overline {P r_t}})({\partial \bar {T}}/{\partial \bar {u}})|$, if 
$\phi '$ is modelled as 
$(\overline {{(\rho v)^{\prime}\phi ^{\prime }}}/\overline {{(\rho v)^{\prime}u^{\prime }}})u^{\prime }$. As can be seen, the duality relation between 
$u'$ and 
$T'$ is partially broken in the GRA due to the inclusion of 
$\phi '$.
However, the present model suggests that the input–output relation cannot be completely built between 
$u'$ and 
$T'$ without considering the influences of 
$\phi '$ and 
$\chi '$. In other words, the duality relation between 
$u'$ and 
$T'$ is more seriously broken in our model than in the GRA. Besides, the differential equation (2.8) cannot be explicitly solved, as the statistical characteristic of the term 
${r_g(\overline {\chi ^{'2}}+2\overline {u'\chi ^{\prime}})}/{2C_p}$ is unknown. In the next section, we will analyse the physical significance of 
$\chi '$, and assess its role in determining the distributions of mean fields.
3. Physical significance of 
$\chi '$ and $\phi '$
Up to now, it is natural to ask two questions. What is the physical significance of 
$\chi '$ and 
$\phi '$? Are they introduced as temporary variables to alleviate the theoretical crisis of the SRA, or are they physically meaningful quantities with genuine physical implications? We conjecture that the answers are hidden in the physical meanings of (2.6) and (2.9). These two equations suggest that 
$T^{\prime }+\phi ^{\prime }$ and 
$u'+\chi '$ are the pure temperature and velocity fluctuations that are directly controlled by the mean fields, and carried by the large-scale eddies. That is to say, 
$-\phi ^{\prime }$ and 
$-\chi '$ are the components of 
$T'$ and 
$u'$ that are not manipulated by the mean gradients. Because the mean shear is typically believed to be the energy source of the large-scale motions in shear flows (Jiménez Reference Jiménez2018), it implies that 
$-\phi ^{\prime }$ and 
$-\chi '$ must be two small-scale quantities which keep away from the influences of the mean gradients and are not directly related to the surrounding energy-containing motions. These analyses are reminiscent of the work of Cheng & Fu (Reference Cheng and Fu2023), who reported that the 
$u$–
$T$ coupling in compressible wall turbulence is sustained by energy-containing motions, just as the description of the phenomenological model proposed by Gaviglio (Reference Gaviglio1987). It can be seen that, compared with the GRA, the current model better captures the dominance of the large-scale eddies described phenomenologically by Gaviglio (Reference Gaviglio1987), with the mean fields as the connecting factors. Therefore, within the current framework, the assumption (2.6) is natural and reasonable.
Next, we combine the DNS data of compressible wall turbulence with a mature mathematical tool to verify our propositions. The DNS database of channel flows used in the present study for detailed analysis is built in our previous study (Cheng & Fu Reference Cheng and Fu2022). These cases are performed in a computational domain of 
$4{\rm \pi} h\times 2{\rm \pi} h\times 2 h$ in the streamwise (
$x$), spanwise (
$z$) and wall-normal (
$y$) directions, respectively. Herein, 
$h$ denotes the channel half-height. Three cases at bulk Mach numbers (
$M_b$) 0.8, 1.5 and 3.0 are selected, and their bulk Reynolds numbers (
$Re_b$) are 17 000, 20 020 and 15 000, respectively. They are of the highest friction Reynolds numbers in our dataset at the corresponding 
$M_b$, and denoted as Ma08Re17K, Ma15Re20K and Ma30Re15K, respectively. The details of each simulation and its validation are provided at length in Cheng & Fu (Reference Cheng and Fu2022). We will not repeat them here for the sake of brevity.
The diagnostic tool deployed in the present study is the spectral linear stochastic estimation (SLSE). The SLSE is a data-driven tool for investigating the multiphysics couplings in both incompressible and compressible wall-bounded turbulence. It is capable of measuring the degree of the linear coherence between two variables in the flow field, and estimating the instantaneous signal of a target physical quantity with another quantity as input. Note that the estimated signal is linearly coherent with the input signal, due to the linear nature of the SLSE. In the present study, we first resort to the SLSE to extract the component of 
$T'$ that bears a linear coherence with the streamwise velocity fluctuation 
$u'$ at a given wall-normal position 
$y$ in turbulent channel flows. It takes the form of
\begin{equation} T_{l}^{\prime}\left(y\right)=F_{x,z}^{{-}1}\left\{H_{T u}\left(\lambda_{x},\lambda_{z}; y\right) F_{x,z}\left[u^{\prime}\left(y\right)\right]\right\}, \end{equation}
where 
$T_{l}^{\prime}$ is the component of 
$T'$ sharing a linear coherence with 
$u'$ at the wall-normal position 
$y$. In this regard, the uncorrelated component can be calculated as 
$T_{nl}^{\prime}=T^{\prime}-T_{l}^{\prime}$. In the expression, 
$F_{x,z}$ and 
$F_{x,z}^{-1}$ denote the two-dimensional (2-D) fast Fourier transform and the inverse 2-D fast Fourier transform in the streamwise and the spanwise directions, respectively. Here 
$H_{T u}$ is the transfer kernel, which gauges the correlation between 
$\widehat {T_{}^{\prime}}(y)$ and 
$\widehat {u'}(y)$ at streamwise length scale 
$\lambda _{x}$ and spanwise length scale 
$\lambda _{z}$, and can be calculated as
\begin{equation} H_{T u}\left(\lambda_{x},\lambda_{z};y\right)=\frac{\left\langle\widehat{T'}\left(\lambda_{x},\lambda_{z}; y\right) \breve{\widehat{u'}}\left(\lambda_{x},\lambda_{z}; y_{}\right)\right\rangle}{\left\langle\widehat{u'}\left(\lambda_{x},\lambda_{z}; y_{}\right) \breve{\widehat{u'}}\left(\lambda_{x},\lambda_{z}; y_{}\right)\right\rangle}, \end{equation}
where 
$\langle {\cdot }\rangle$ denotes the ensemble averaging, 
$\widehat {u'}$ and 
$\widehat {T^{\prime}}$ are the Fourier coefficients of 
$u'$ and 
$T'$, respectively, and 
$\breve {\widehat {u^{\prime}}}$ represents the complex conjugate of 
$\widehat {u^{\prime}}$. Similarly, we can also extract the component of 
$u'$ that holds a linear relationship with 
$T'$, that is,
\begin{equation} u_{l}^{\prime}\left(y\right)=F_{x,z}^{{-}1}\left\{H_{u T}\left(\lambda_{x},\lambda_{z}; y\right) F_{x,z}\left[T^{\prime}\left(y\right)\right]\right\}, \end{equation}
where the kernel function 
$H_{u T}$ can be constructed by analogy. The uncorrelated component can be calculated as 
$u_{nl}^{\prime}=u^{\prime}-u_{l}^{\prime}$. On the other hand, we also can measure the overall response intensity by comparing the r.m.s. values of 
$T'$ and 
$u'$, namely, 
$T_{rms}'/u_{rms}'$. The ratios between 
$T_l'$ and 
$u'$, and 
$T_l'$ and 
$u_l'$, can be defined by analogy. They can be compared with the following response functions, i.e.
\begin{equation} f_1(y)=\left|\frac{1}{P r_t} \frac{\partial \bar{T}}{\partial \bar{u}}\right|, \quad f_2(y)=\left| \frac{\partial \bar{T}}{\partial \bar{u}}\right|. \end{equation}
Note that, the 
$P r_t$ in 
$f_1(y)$ is computed from the DNS data, and we observe that the difference between the values of 
$Pr_t$ and 
$\overline {Pr_t}$ is negligible for the DNS data of channel flows considered in the present study, hence we will not make a distinction between the HSRA and the MHSRA. Specifically, 
$f_1(y)$ is the transfer function suggested by the HSRA (MHSRA) and 
$f_2(y)$ is the transfer function indicated by the GRA and the present model. If the HSRA (MHSRA) is true, the distribution of 
$f_1(y)$ should match well with that of 
$T_{rms}'/u_{rms}'$ for each case (see (1.4)). If the present model is established, the distribution of 
$f_2(y)$ should agree with that of 
$T_{l,rms}'/u_{l,rms}'$ (see (2.9)), whereas for the establishment of the GRA, the profile of 
$f_2(y)$ should be in close agreement with that of 
$T_{l,rms}'/u_{rms}'$ (see (1.10)). Additionally, to facilitate comparison, three ratio functions are defined as 
$r_1=(|({1}/{Pr_t})({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{rms})/T_{rms}'$, 
$r_2=(|({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{l,rms})/T_{l,rms}'$ and 
$r_3=(|({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{rms})/T_{l,rms}'$, which are the quantitative metrics corresponding to the HSRA, the present model, and the GRA.
Figure 1(a) shows the variations of 
$T_{rms}'/u_{rms}'$ for all cases of channel flows mentioned above, and the distributions of 
$f_1$ are added for comparison. It can be found that the magnitudes of 
$T_{rms}'/u_{rms}'$ agree tolerably well with those of 
$f_1$ within 
$0< y/h<0.6$, below the core region of a channel. This observation is consistent with the finding of Huang et al. (Reference Huang, Coleman and Bradshaw1995) for compressible channel flows. It suggests that the overall response of 
$T'$ follows the prediction of the HSRA, as well as the MHSRA.
Variations of (
$a$) 
$T_{rms}'/u_{rms}'$ and (
$b$) 
$T_{l,rms}'/u_{rms}'$ and 
$T_{l,rms}'/u_{l,rms}'$ for all cases of channel flows. The response functions given by different Reynolds analogy models are represented by solid lines in each panel: 
$f_1(y)$ is the transfer function suggested by the HSRA (MHSRA) and 
$f_2(y)$ is the transfer function indicated by the GRA and the present model.
In figure 1(b), we concentrate on the components of 
$T'$ and 
$u'$ that are linearly coherent with each other, namely, 
$T_l'$ and 
$u_l'$. One can notice that the distributions of 
$f_2$ are consistent with those of 
$T_{l,rms}'/u_{l,rms}'$ for 
$0< y/h<0.6$, at least within the Mach numbers under consideration. The accuracy is rather similar to that of the HSRA (MHSRA) (see the variations of 
$r_1$, 
$r_2$ and 
$r_3$ for each case shown in figure 2). This is an interesting finding, because it implies that it is essentially 
$T_l'$, rather than 
$T'$, that would response to 
$u_l'$, rather than 
$u'$, if the transfer function is 
$f_2$; that is to say,
\begin{equation} T'-T_{nl}'=\frac{\partial\bar{T}}{\partial\bar{u}}(u'-u_{nl}'). \end{equation}It is noted that (3.5) may be not fully accurate in an instantaneous sense, but it suggests the key input–output relation between them, and is valid in a statistical sense. Compared with (2.9), the establishment of (3.5) underscores the fact that
\begin{equation} -\phi'=T_{nl}', \quad -\chi'=u_{nl}'. \end{equation}
That is to say, 
$-\phi '$ and 
$-\chi '$ are not physically real fields that exist independently of 
$T'$ and 
$u'$, respectively, but just the components of 
$T'$ and 
$u'$ that are not linearly correlated with 
$u'$ and 
$T'$, respectively. Furthermore, this result also suggests 
$\alpha =1$. Interestingly, it is precisely the assumed value of 
$\overline {Pr_e}$. More discussion regarding the value of 
$\alpha$ is given in § 4. On the other hand, if the input–output relation follows (1.10) predicted by the GRA, it is 
$u'$, rather than 
$u_l'$, that would be linearly coherent with the 
$T_l'$, and the response intensity should be 
$f_2$. Figure 1(b) also shows the variations of 
$T_{l,rms}'/u_{rms}'$ for comparison. Obviously, it deviates from the prediction of the GRA ( 
$f_2$) for each case. It suggests the existence of 
$\chi '$, and thus demonstrating the superiority of the present model in depicting the 
$u'$–
$T'$ relation and providing direct evidence of the phenomenological model proposed by Gaviglio (Reference Gaviglio1987). Moreover, Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000) pointed out that the 
$u'-T'$ relation (2.4) implies the 
$\bar {u}$–
$\bar {T}$ relation (2.2). In this regard, the improvement of the present model over the GRA is fundamental.
Variations of 
$r_1=(|({1}/{Pr_t})({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{rms})/T_{rms}'$, 
$r_2=(|({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{l,rms})/T_{l,rms}'$ and 
$r_3=(|({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{rms})/T_{l,rms}'$ for three channel flows at different 
$M_b$.
Another fact supports our judgement is that for a compressible wall boundary layer with a cold wall, the fluctuating velocity and temperature fields show a strong decorrelation in the proximity of the wall-normal region where 
$\partial \bar {T}/\partial \bar {y}\approx 0$, (see figure 14a of Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023)). According to (3.5) and (3.6a,b), near the extreme point of a mean temperature profile where 
$\partial \bar {T}/\partial \bar {y}\approx 0$, the right-hand side of (3.5) should be zero. In this way, we can deduce that 
$T_l'\approx 0$ and 
$\phi '=-T_{nl}'\approx -(T'-T_l')\approx -T'$. As can be seen, 
$\phi '$ happens to be 
$-T'$ that decorrelates with 
$u'$, i.e. the component of 
$T'$ that is not linearly correlated with 
$u'$. Moreover, figure 3 shows the premultiplied streamwise spectra of 
$T_{nl}'$ and 
$u_{nl}'$ for the case Ma15Re20K with the streamwise wavelength scaled by the semilocal units at 
$y^*\approx 20$. One can notice that their spectral peaks are located at 
$\lambda _x^* \approx 250$ and 
$\lambda _x^* \approx 400$, respectively. Here, we use the superscript ‘
$*$’ to represent the normalization in semilocal units. It can be seen that their streamwise length scales are shorter than those of the temperature and velocity streaks populating the near-wall region (
$\lambda _x^* \approx 1000$) (Cheng & Fu Reference Cheng and Fu2022). This observation is consistent with the analyses conducted above, which assert that 
$-\phi ^{\prime }$ and 
$-\chi '$ ought to be two small-scale quantities which are not directly related to the surrounding energy-containing motions controlled by the mean fields.
(a) Premultiplied streamwise spectrum of 
$T_{nl}'$; (b) premultiplied streamwise spectrum of 
$u_{nl}'$. The case Ma15Re20K is taken into consideration. Each spectrum is normalized by its maximum value, and the streamwise wavelength is scaled by the semilocal units at 
$y^*\approx 20$.
We also verify our model in supersonic and hypersonic turbulent boundary layers. The DNS database of compressible turbulent boundary layers used here is built by Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022). For each case, a selected domain is adopted for analysing, and the parameters are provided in table 1. It can be observed that the Reynolds number effects are negligible because the corresponding variation of 
$Re_{\tau }$ is small. Under the circumstances, the 2-D SLSE introduced above can be deployed. More details about the computational set-ups and the numerical methods regarding the DNS of this dataset can be found in Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022).
The parameters of compressible turbulent boundary layers. Here 
$M_{\infty }$, 
$Re_{\infty }$ and 
$T_{\infty }$ are the freestream Mach number, Reynolds number and temperature, respectively; 
$T_w/T_r$ denotes the ratio of isothermal wall temperature and recovery temperature; 
$T_w/T_e$ denotes the ratio of isothermal wall temperature and boundary edge temperature; 
$L_x$, 
$L_y$ and 
$L_z$ are the lengths of selected domain in the streamwise, wall-normal and spanwise directions, respectively; 
$\delta _i$ and 
$Re_{\tau }$ are the inlet boundary layer thickness and the range of the friction Reynolds number of the selected domain, respectively; and 
$t_su_{\infty }/\delta _i$ is the time period used to accumulate statistics.
Figure 4(a,c,e,g) show the distributions of 
$T_{rms}'/u_{rms}'$ and 
$f_1(y)$ within the boundary layers, and figure 4(b,d,f,h) display the variations of 
$T_{l,rms}'/u_{rms}'$, 
$T_{l,rms}'/u_{l,rms}'$ and 
$f_2(y)$ for all cases. It is noted that the 
$Pr_t$ in the definition of 
$f_1(y)$ is replaced by 
$\overline {P r_t}$ herein, as previous studies showed that this modified version of the turbulent Prandtl number is less sensitive to the freestream Mach number and wall temperature condition at the edge of the turbulent boundary layer (Zhang et al. Reference Zhang, Bi, Hussain and She2014; Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023). It can be observed that for all cases, the profiles of 
$f_1(y)$ agree roughly with those of 
$T_{rms}'/u_{rms}'$, irrespective of the Mach number and wall thermal condition. It implies the success of the MHSRA (HSRA), which is documented in numerous previous studies (Guarini et al. Reference Guarini, Moser, Shariff and Wray2000; Duan & Martin Reference Duan and Martin2011; Huang et al. Reference Huang, Duan and Choudhari2022). However, significant deviations can still be traced within the intermittent layers (
$y/\delta >0.8$). Similar inconsistency can also be found in previous studies, for example, figure 9(b) of Zhang et al. (Reference Zhang, Duan and Choudhari2018) and figure 18(c) of Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023).
Variations of (a,c,e,g) 
$T_{rms}'/u_{rms}'$ and (b,d,f,h) 
$T_{l,rms}'/u_{rms}'$ and 
$T_{l,rms}'/u_{l,rms}'$ for all the cases of turbulent boundary layers. The response functions given by different Reynolds analogy models are represented by solid lines in each panel: 
$f_1(y)$ is the transfer function suggested by the MHSRA (HSRA) and 
$f_2(y)$ is the transfer function indicated by the GRA and the present model. Here, 
$\delta$ is the mean boundary layer thickness of the selected domain for each case.
On the other hand, it is apparent that the distributions of 
$T_{l,rms}'/u_{l,rms}'$ match reasonably well with those of 
$f_2(y)$ in most regions of the boundary layers for all cases, even in the intermittent layers, see figure 4(b,d,f,h). Observable deviations can only be traced around the extreme points of the mean temperature profiles for the cases with cold walls. This is acceptable, because the gradient of the mean temperature is zero around this kind of point, which leads to the breakdown of the 
$u_l'$–
$T_l'$ relation (3.5). By contrast, for all cases, the distributions of 
$T_{l,rms}'/u_{rms}'$ are not in tune with those of 
$f_2(y)$ unequivocally. One can also refer to the distributions of 
$r_1$, 
$r_2$ and 
$r_3$ for all cases displayed in figure 5. Combining with the results of compressible channel flows reported above, we might claim that the present model is superior to the GRA in depicting the relationship between fluctuating fields of velocity and temperature in canonical compressible wall turbulence. Of course, it also needs more data for validation in the future.
Variations of 
$r_1=(|({1}/{\overline {Pr_t}})({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{rms})/T_{rms}'$, 
$r_2=(|({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{l,rms})/T_{l,rms}'$ and 
$r_3=(|({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{rms})/T_{l,rms}'$ for compressible turbulent boundary layers with different wall thermal conditions and freestream Mach numbers.
Moreover, the present model seems to work better than the MHSRA (HSRA) in the intermittent layers of compressible boundary layers, irrespective of the Mach number and wall thermal condition, for the cases under consideration (see figures 4 and 5). By the way, in the core region of an internal flow, the acoustic component of the fluctuating motions is dominant (Zhang et al. Reference Zhang, Bi, Hussain and She2014; Cheng & Fu Reference Cheng and Fu2024b), as a result, the basics of each Reynolds analogy model might not hold, leading to the significant deviations of the MHRSA (HSRA) and the present model illustrated in figure 1.
4. Implications for the $\bar {u}$–$\bar {T}$ relation
One may ask, why does the GRA fail to accurately characterize the relationship of the fluctuating fields, but is able to effectively capture the distributions of the mean fields in canonical flows? This is due to the negligible effect of the term 
${r_g(\overline {\chi ^{'2}}+2\overline {u'\chi ^{\prime}})}/{2C_p}$ in (2.8) in determining the 
$\bar {u}$–
$\bar {T}$ relation. According to (3.6a,b), this term can be rewritten as
\begin{equation} \frac{r_g\left(\overline{\chi^{'2}}+2\overline{u'\chi^{\prime}}\right)}{2C_p}= \frac{r_g\left[\overline{u_{nl}^{'2}}-2\overline{(u_{nl}^{\prime}+u_{l}^{\prime})u_{nl}^{\prime}}\right]}{2C_p} \approx{-}\frac{r_g\overline{u_{nl}^{'2}}}{2C_p}, \end{equation}
where the assumption 
$\overline {u_{l}^{\prime}u_{nl}^{\prime}}=0$ is adopted. Here 
$u_{nl}'$ is a small-scale fluctuating quantity, and the magnitude of 
$-{r_g\overline {u_{nl}^{'2}}}/{2C_p}$ is significantly smaller than those of other terms associated with mean fields in (2.8), and thus can be ignored. As a result, an analytic solution of (2.8) becomes available with 
$\alpha =1$, that is (1.11). We note in passing that, ignoring this small-scale fluctuating term is equivalent to adopting the approximation 
$\overline {u_e^{2}}\approx \bar {u}_e^{2}=\bar {u}^{2}$, which is widely deployed for modelling the canonical wall turbulence (Zhang et al. Reference Zhang, Bi, Hussain and She2014).
Finally, it is instructive to discuss the value of 
$\alpha$. Ignoring the fluctuating term and solving (2.8) yields the general solution without additional assumptions,
\begin{equation} \bar{T}=\bar{T}_w+A\left.\frac{\partial\bar{T}}{\partial\bar{u}}\right|_w\bar{u}+B\bar{u}^{2\alpha}, \end{equation}
where 
$A=\alpha /(2\alpha -1)$ and 
$B$ is a constant. Differentiating (4.2) with respect to 
$\bar {u}$ and taking the value on the wall boundary, one has 
$\alpha =1$. It indicates that 
$\alpha =1$ is a requirement for the mathematical self-consistency of the present model, which is fully consistent with the results shown in figures 1(b) and 4(b,d,f,h). As such, the well-established 
$\bar {u}$–
$\bar {T}$ relation (1.11) proposed by Zhang et al. (Reference Zhang, Bi, Hussain and She2014) can also be deduced under the framework of the present model by replacing the artificial assumption 
$\overline {Pr_{e}}=1$ with a stricter mathematical derivation of 
$\alpha =1$. On the other hand, because the 
$\bar {u}$–
$\bar {T}$ relation (1.11) has been broadly validated in numerous studies of compressible wall turbulence, including pipe flows (Zhang et al. Reference Zhang, Bi, Hussain and She2014), channel flows (Modesti & Pirozzoli Reference Modesti and Pirozzoli2016) and turbulent boundary layers (Huang et al. Reference Huang, Duan and Choudhari2022), it implies that 
$\alpha =1$ should be generic. Unlike the arbitrary assumption of 
$\overline {Pr_{e}}=1$ in the GRA, 
$\alpha =1$ carries distinct physical significance as per (2.6). It suggests that the momentum and heat transfers, which are controlled by the gradients of the mean fields, are dynamically similar. This similarity is not sensitive to the Mach number and the geometry of the flows. In simpler terms, within the regime controlled by the mean fields, the input–output relation between 
$u'$ and 
$T'$ holds a certain universality.
5. Discussion and concluding remarks
As the 
$u'$–
$T'$ relations are shown to be related to the multiscale eddies in the boundary layers, it is of significance to examine their Reynolds number dependence. A new set of DNS data of supersonic channel flows is used. Three cases at 
$Re_b=3000$, 9400 and 20 020 are selected, and their bulk Mach numbers are all 1.5. Validation of this dataset is also given in Cheng & Fu (Reference Cheng and Fu2022). They are denoted as Ma15Re3K, Ma15Re9K and Ma15Re20K, respectively. Figure 6(a) shows the variations of 
$r_1$ for all three cases. Remarkable Reynolds number dependence can be traced. Within 
$0.3< y/h<0.7$, the magnitude of 
$r_1$ approaches unity as the increase of the Reynolds number. It suggests that the HSRA (MHSRA) will be more accurate at higher Reynolds numbers. Figure 6(b) shows the counterparts of 
$r_2$. Interestingly, clear Reynolds number dependence can also be observed, and the values of 
$r_2$ are closer to unity for high Reynolds numbers in the outer region.
Variations of (a) 
$r_1=(|({1}/{Pr_t})({\partial \bar {T}}/{\partial \bar {u}})|u^{\prime }_{rms})/T_{rms}'$, (b) 
$r_2=(|{\partial \bar {T}}/{\partial \bar {u}}|u^{\prime }_{l,rms})/T_{l,rms}'$ for three channel flows at 
$M_b=1.5$ with different 
$Re_b$.
In conclusion, in the present study, we are dedicated to proposing a Reynolds analogy model for compressible wall turbulence. This model is demonstrated to be able to alleviate the defects of the GRA, and maintains its success in describing the 
$\bar {u}$–
$\bar {T}$ relation. Furthermore, the present model is superior to the GRA in depicting the 
$u'$–
$T'$ relation. The key points of the present model are validated by analysing the data of compressible wall-bounded turbulence with different Mach numbers, Reynolds numbers and wall thermal conditions.
The present study provides a new perspective of the velocity–temperature coupling in compressible wall turbulence, and may trigger new studies on this topic. For example, as the 
$u_l'-T_l'$ relation is universal for compressible wall turbulence, further work can concentrate on the relation between 
$u_l'$ and 
$u_{nl}'$, as well as that between 
$T_l'$ and 
$T_{nl}'$. This may open up a new research avenue for modelling the momentum and heat transfers in compressible wall-bounded turbulence, with the SLSE introduced in the present study serving as a helpful tool.
Acknowledgements
We are grateful to the owners of the DNS dataset listed in table 1 for kindly sharing their data. C.C. would like to thank Y. Zhao at SJTU for helpful discussions.
Funding
L.F. acknowledges the fund from the National Natural Science Foundation of China (no. 12422210), the fund from the Research Grants Council (RGC) of the Government of Hong Kong Special Administrative Region (HKSAR) with RGC/ECS Project (no. 26200222), RGC/GRF Project (no. 16201023) and RGC/STG Project (no. STG2/E-605/23-N), the fund from Guangdong Basic and Applied Basic Research Foundation (no. 2024A1515011798), the fund from Guangdong Province Science and Technology Plan Project (no. 2023A0505030005), the fund from CORE as a joint research centre for ocean research between Laoshan Laboratory and HKUST, and the fund from the Project of Hetao Shenzhen-Hong Kong Science and Technology Innovation Cooperation Zone (no. HZQB-KCZYB-2020083).
Declaration of interests
The authors report no conflict of interest.
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