2. Underlying equations and theory

2.1. Underlying equations

Passive scalar transport in incompressible Couette flow is governed by the Navier–Stokes equation for the velocity field $\boldsymbol {u}({{\boldsymbol {x}}},t) \equiv (u_x,u_y,u_z)$, the continuity equation and the transport equation for the scalar $\theta ({{\boldsymbol {x}}},t)$. In Cartesian coordinates $\boldsymbol {x}\equiv (x,y,z)\equiv (x_1,x_2,x_3)$, they read

(2.1)$$\begin{gather} {\partial_t u_i} + u_j {\partial_j u_i} ={-} {\partial_i p} + \nu {\partial^{2}_j u_i}, \end{gather}$$

(2.2)$$\begin{gather}{\partial_i u_i} = 0, \end{gather}$$

(2.3)$$\begin{gather}{\partial_t \theta} + u_j {\partial_j \theta} = \kappa {\partial^{2}_j \theta}, \end{gather}$$

where $p({{\boldsymbol {x}}},t)$ is the kinematic pressure and $\theta ({{\boldsymbol {x}}},t)$ the passive scalar with the arithmetic mean of the top and bottom wall values subtracted. We consider $x,y$ to be the wall-parallel directions and $z$ to be the wall-normal direction. The boundary conditions for the equations are

(2.4a–d)$$\begin{gather} \left. \theta \right|_{z=0} = \varDelta/2, \quad \left. \theta \right|_{z=H} ={-}\varDelta/2, \quad \left. u_x \right|_{z=0} ={-}U_b, \quad \left. u_x \right|_{z=H} = U_b, \end{gather}$$

(2.5a,b)$$\begin{gather}\left. u_y \right|_{z=0} = \left. u_y \right|_{z=H} = 0, \quad \left. u_z \right|_{z=0} = \left. u_z \right|_{z=H} = 0. \end{gather}$$

The scalar flux $Q$ from the bottom plate to the top plate is

(2.6)\begin{equation} Q= \left\langle u_z \theta \right\rangle_{A,t} - \kappa\left\langle \partial_z \theta \right\rangle_{A,t}. \end{equation}

2.2. Laminar case

In the laminar regime, steady uniform gradients of the streamwise velocity and temperature develop in the flow. Therefore, we have the profiles of the streamwise velocity and scalar given by

(2.7a,b)\begin{equation} u_x = U_b \left(\frac{2z}{H}-1\right), \quad \theta = \frac{\varDelta}{2} \left(1-\frac{2z}{H}\right), \end{equation}

with $z$ as the wall-normal coordinate which lies between $z=0$, corresponding to the bottom plate, and $z=H$, corresponding to the top plate. This gives the well-known and trivial laminar results ${\partial _z u_x} = 2U_b/H$, $\tau _w = 2\rho \nu {U_b}/{H}$, or in dimensionless form

(2.8a,b)\begin{equation} Re_{\tau} = \frac{1}{\sqrt{2} } Re_b^{1/2}, \quad C_f = 4 Re_b^{{-}1}, \end{equation}

and ${\partial \theta }/{\partial z} = -\varDelta /{H}$, $Q = \kappa \varDelta /{H}$, or in dimensionless form

(2.9)\begin{equation} Nu = 1. \end{equation}

2.3. Turbulent regime

From (2.1), we use the Reynolds (Reference Reynolds1895) averaging to split the velocity and scalar into the mean and fluctuating values as follows:

(2.10a,b)\begin{equation} u_i = \left\langle u_i\right\rangle_{t} + u_i^{\prime}, \quad \theta = \left\langle\theta\right\rangle_{t} + \theta^{\prime}, \end{equation}

with $\left \langle \dots \right \rangle _{t}$ indicating the Reynolds averaging. Assuming that $x$ is the streamwise direction and using (2.1) averaged in time and over the wall-parallel directions, we obtain

(2.11)\begin{equation} \partial_z \left(-\left\langle u_x^{\prime} u_z^{\prime} \right\rangle_{A,t} + \nu \partial_z \left\langle u_x\right\rangle_{A,t}\right) = 0, \end{equation}

which simply reflects momentum conservation in the $z$ direction. Integrating this relation $z$ between $z=0$ and $z$ gives the well-known Reynolds-averaged Navier–Stokes equation (Pope Reference Pope2000)

(2.12)\begin{equation} -\left\langle u_x^{\prime} u_z^{\prime} \right\rangle_{A,t} + \nu \partial_z \left\langle u_x\right\rangle_{A,t} = \left.\nu\partial_z \left\langle u_x\right\rangle_{A,t}\right|_{z=0} \equiv u_{\tau}^{2}. \end{equation}

Here, $-\left \langle u_x^{\prime } u_z^{\prime } \right \rangle _{A,t}$ represents the Reynolds shear stress and $\nu \partial _z \left \langle u_x\right \rangle _{A,t}$ represents the viscous shear stress. These quantities can be viewed as the convective and diffusive momentum fluxes. This equation means that the total momentum flux given by the sum of convective and diffusive fluxes at any height equals the diffusive momentum flux at the wall in the case of statistically steady flow, i.e. the diffusive flux at the wall is the bottleneck for the total flux through the system. At the wall, we introduce a characteristic length scale in terms of the kinetic boundary layer thickness $\lambda _u$ (Prandtl Reference Prandtl1904; Blasius Reference Blasius1908) such that

(2.13)\begin{equation} \left.\nu\partial_z \left\langle u_x\right\rangle_{A,t}\right|_{z=0} = u_{\tau}^{2} = \nu U_b/\lambda_u. \end{equation}

The diffusive time scale of the momentum flux at the wall is then $\lambda _u^{2} / \nu$. Since the convective heat transfer through turbulent motions in the bulk is much quicker than the diffusive heat transfer at the wall, this diffusive time scale represents the bottleneck for the for momentum transport through the system.

Similarly, the Reynolds-averaged form of (2.3) yields

(2.14)\begin{equation} \partial_z \left(-\left\langle \theta^{\prime} u_z^{\prime} \right\rangle_{A,t} + \kappa \partial_z\left\langle\theta\right\rangle_{A,t}\right) = 0, \end{equation}

which when integrated between the limits $z=0$ and $z$ gives

(2.15)\begin{equation} -\left\langle \theta^{\prime} u_z^{\prime} \right\rangle_{A,t} + \kappa \partial_z \left\langle \theta\right\rangle_{A,t} = \left.\kappa\partial_z \left\langle \theta\right\rangle_{A,t}\right|_{z=0} \equiv {\theta_Q}^{2}, \end{equation}

with ${\theta _Q}$ defined analogously to $u_{\tau }$ (Schlichting & Gersten Reference Schlichting and Gersten2016). This equation again reflects that the total scalar flux through the system is determined by the scalar flux at the wall. Therefore, we introduce another characteristic length scale in terms of the thermal boundary layer thickness $\lambda _{\theta }$ such that

(2.16)\begin{equation} {\theta_Q}^{2} = \kappa\varDelta/\lambda_{\theta} = (2\kappa\varDelta/H)Nu. \end{equation}

The diffusive time scale of heat flux at the wall is then $\lambda _{\theta }^{2}/\kappa$, which must be the bottleneck time scale of the system, as the diffusive scalar flux at the wall is the bottleneck for the total scalar flux through the system.

However, since the scalar is a passive quantity, the time scale of the system should be solely governed by the diffusive time scale of the momentum flux at the wall, in which case

(2.17)\begin{equation} \lambda_u^{2}/\nu \approx \lambda_{\theta}^{2}/\kappa. \end{equation}

Postulating this gives,

(2.18)\begin{equation} \lambda_u/\lambda_{\theta} \approx Pr^{1/2}. \end{equation}

Using (1.4), (1.5), (2.13) and (2.16) results in the relation

(2.19)\begin{equation} Nu \approx 2Pr^{1/2} Re_{\tau}^{2} Re_b^{{-}1} = \frac{C_f}{4} Pr^{1/2} Re_b. \end{equation}

Note that the relation (2.19) is similar to the Reynolds (Reference Reynolds1874) analogy for turbulent boundary layer over a heated flat plate (Schlichting & Gersten Reference Schlichting and Gersten2016) and turbulent flow in heated pipes (Kays Reference Kays1994; McEligot & Taylor Reference McEligot and Taylor1996). According to the Reynolds analogy, the shear stress and the heat flux are analogous if the average value of the turbulent Prandtl number $\left \langle Pr_t\right \rangle _L \approx 1$ in the log-layer. Here, the value of the $Pr_t$ is defined at any given point in the flow by

(2.20)\begin{equation} Pr_t = \left. \frac{\left\langle u_x^{\prime} u_z^{\prime} \right\rangle_{t}}{\partial_z \left\langle u_x\right\rangle_{t}} \right/ \frac{\left\langle u_z^{\prime} \theta^{\prime} \right\rangle_{t}}{\partial_z \left\langle \theta\right\rangle_{t}} , \end{equation}

and $\left \langle \dots \right \rangle _L$ indicates the average value in the log-layer. For alternate definitions used to compute $Pr_t$ from the numerical simulations, we refer the reader to §4. Therefore, from the Reynolds analogy,

(2.21)\begin{equation} Nu \approx f(Pr) C_f Re_b. \end{equation}

The comparison of (2.19) and (2.21) thus confirms that the Reynolds analogy is also applicable to the case of passive transport in Couette flow with the value of $f(Pr) \approx Pr^{1/2}/8$.

For the scaling of the shear Reynolds number with the bulk Reynolds number in the turbulent regime, we use the empirical relation $C_f\sim Re_b^{-1/4}$. This relation was first suggested by Blasius (Reference Blasius1913) for pipe flows in an intermediate range of $3 \times 10^{3} \leqslant Re_b \leqslant 10^{5}$ and is widely accepted as a good representation of the friction coefficient of pipe flows for the aforementioned range of bulk Reynolds numbers (McKeon, Zargola & Smits Reference McKeon, Zargola and Smits2005). This scaling relation has since been well established through experiments of flows in non-circular pipes (Nikuradse Reference Nikuradse1930, Reference Nikuradse1950), rectangular ducts (Dean Reference Dean1978) and channel flows (Schultz & Flack Reference Schultz and Flack2013). Also, the DNS results by Bernardini, Pirozzoli & Orlandi (Reference Bernardini, Pirozzoli and Orlandi2014) and by Orlandi, Bernardini & Pirozzoli (Reference Orlandi, Bernardini and Pirozzoli2015) show that this power law is a good description of the friction coefficient in turbulent Poiseuille and Couette flows for the considered $Re_b$ range.

Using the scaling relation $C_f\sim Re_b^{-1/4}$ with relation (1.5), we get

(2.22)\begin{equation} Re_{\tau} \sim Re_b^{7/8}. \end{equation}

Since $C_f$ is independent of $Pr$, combining relations (2.22) and (2.19) gives

(2.23)\begin{equation} Nu \sim Pr^{1/2} Re_b^{3/4}. \end{equation}

The scaling relation $C_f\sim Re_b^{-1/4}$ (i.e. $Re_{\tau } \sim Re_b^{7/8}$) holds well for the range of $Re_b$ considered here but it is not valid at high bulk Reynolds numbers. The value of the friction coefficient deviates more and more from the empirical power law for $Re \geqslant 10^{5}$ (McKeon et al. Reference McKeon, Zargola and Smits2005; Scheel, Emran & Schumacher Reference Scheel, Emran and Schumacher2013). We note that, for extremely large $Re_b$, it might be more appropriate to use the logarithmic law to obtain the friction coefficient (similar to the Prandtl (Reference Prandtl1932) turbulent friction law). However, for the range of $Re_b$ for which we claim the validity of the scaling laws derived in this work, the Blasius law and the Prandtl turbulent friction law show minimal differences and therefore the use of the Blasius scaling is sufficient. At higher $Re_b$, we still expect the scaling given by (2.19) to hold, however with a different scaling for $C_f$.

For extremely large $Re_b$, we expect that the scaling law given by relation (2.23) may not be valid and further studies are required to determine the scaling relations for $Nu$ at very high $Re_b$. Here, we draw an analogy from the ultimate regime of turbulent thermal convection, where the scaling relations of $Nu$ change for extremely large thermal forcing (i.e. Rayleigh Number) (Kraichnan Reference Kraichnan1962; Spiegel Reference Spiegel1971; Chavanne et al. Reference Chavanne, Chilla, Castaing, Hebral, Chabaud and Chaussy1997; He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012; Zhu et al. Reference Zhu, Verschoof, Bakhuis, Huisman, Verzicco, Sun and Lohse2018b).

3. Numerical method

We performed numerical simulations in a domain of dimensions $8H \times 4H \times H$ in the streamwise (along the $x$ axis), spanwise (along the $y$ axis) and wall-normal (along the $z$ axis) directions, respectively. We impose periodic boundary conditions in the wall-parallel directions and no-slip boundary conditions at the top and bottom plates.

The non-dimensional form of the incompressible Navier–Stokes equations (2.1) and (2.3) are integrated numerically using the AFiD GPU package (Zhu et al. Reference Zhu2018a) which uses a second-order finite-difference scheme (van der Poel et al. Reference van der Poel, Ostilla-Mónico, Donners and Verzicco2015). The code has been validated and verified many times (Verzicco & Orlandi Reference Verzicco and Orlandi1996; Verzicco & Camussi Reference Verzicco and Camussi1997, Reference Verzicco and Camussi2003; Stevens, Verzicco & Lohse Reference Stevens, Verzicco and Lohse2010; Stevens, Lohse & Verzicco Reference Stevens, Lohse and Verzicco2011; Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014; Kooij et al. Reference Kooij, Botchev, Frederix, Geurts, Horn, Lohse, van der Poel, Shishkina, Stevens and Verzicco2018). We use a uniform discretisation in the wall-parallel periodic directions and a non-uniform grid, with a clipped Chebyshev-like clustering of nodes in the wall-normal direction.

The grid for all cases except for ($Re_b=14\,142, Pr=1$) and ($Re_b=22\,360, Pr=1$) consists of 1536, 768 and 256 nodes in the streamwise, spanwise and wall-normal directions, respectively, whereas the grid for ($Re_b=14\,142, Pr=1$) and ($Re_b=22\,360, Pr=1$) consists of 2048, 1024 and 384 nodes in the streamwise, spanwise and wall-normal directions, respectively, to ensure sufficient resolution. We note that the computational domain used in the current study is smaller than those typically used in studies of Couette flow (Avsarkisov et al. Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014; Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014; Lee & Moser Reference Lee and Moser2018). However, here, we focus on the global quantities $Nu$ and $Re_{\tau }$ (and thus also $C_f$), for which a smaller domain is sufficient. The domain used is, however, larger than the minimal size mentioned in Sekimoto, Atkinson & Soria (Reference Sekimoto, Atkinson and Soria2018) to ensure ‘healthy’ turbulence in the log-layer region. To further confirm that the domain size is sufficient, we have verified that these integral properties obtained from an $8H \times 4H \times H$ domain agree to within 1 % of the corresponding values obtained from test domain of size $48H \times 24H \times H$.

4. Boundary layer thickness and turbulent Prandtl number

Figure 2(a) shows that the ratio of the kinetic and scalar boundary layer thicknesses in the turbulent flow simulations indeed scales as $Pr^{1/2}$ while figures 2(b) and 2(c) show the wall-normal location of the kinetic and scalar boundary layers. It is interesting to note that, for $Pr > 0.1$, the scalar boundary layer lies completely inside the buffer region whereas, for $Pr = 0.1$, the scalar boundary layer overlaps with the log-layer. This results in the minor disagreement with the $Pr$ dependence that can be seen in 2(a) for $Re_b = 10\,000$ and $Pr = 0.1$, suggesting that the predictions from the theory are not accurate for $Pr \ll 1$.

Figure 2. (a) The ratio of the kinetic and scalar boundary layer thicknesses normalised with $Pr^{1/2}$ plotted against bulk Reynolds number; (b) the thickness of the kinetic boundary layer in wall units; (c) the thickness of the scalar boundary layer in wall units.

Figure 3 shows $Pr_t$ obtained from our simulations. The average value of $Pr_t$ at a given distance from the wall can be computed from the numerical simulations using either

Figure 3. Turbulent Prandtl number $Pr_t^{(1)}$ computed using (4.1) and averaged over wall-parallel directions for (a) $Pr=0.1$, (b) $Pr=0.3$, (c) $Pr=1.0$, (d) $Pr=3.0$, (e) $Pr=10.0$. ( f) Turbulent Prandtl number $Pr_t^{(1)}$ and $Pr_t^{(2)}$ computed using (4.1) and (4.2), respectively, and averaged over wall-parallel directions in the log-layer i.e. for $z^{+} \geqslant 30$.

(4.1)\begin{equation} Pr_t^{(1)} = \left. \frac{\left\langle u_x^{\prime} u_z^{\prime} \right\rangle_{A,t}}{\partial_z \left\langle u_x\right\rangle_{A,t}} \right/ \frac{\left\langle u_z^{\prime} \theta^{\prime} \right\rangle_{A,t}}{\partial_z \left\langle \theta\right\rangle_{A,t}} , \end{equation}

or

(4.2)\begin{equation} Pr_t^{(2)} = \left. \left\langle \frac{\left\langle u_x^{\prime} u_z^{\prime} \right\rangle_{t}}{\partial_z \left\langle u_x\right\rangle_{t}}\right\rangle_{A} \right/ \left\langle \frac{\left\langle u_z^{\prime} \theta^{\prime} \right\rangle_{t}}{\partial_z \left\langle \theta\right\rangle_{t}}\right\rangle_{A} . \end{equation}

To obtain the profile for $Pr_t^{(1)}$, we first average the time-averaged profiles of $u_x^{\prime } u_z^{\prime }, \partial _z u_x, u_z^{\prime } \theta ^{\prime }$ and $\partial _z \theta$ over the wall-parallel directions and then compute the $Pr_t$ using (4.1). On the other hand, to obtain the profile for $Pr_t^{(2)}$, we use time-averaged three-dimensional flow fields to compute the ratios $\left \langle u_x^{\prime } u_z^{\prime } \right \rangle _{t} / \partial _z \left \langle u_x\right \rangle _{t}$ and $\left \langle u_z^{\prime } \theta ^{\prime } \right \rangle _{t}/\partial _z \left \langle \theta \right \rangle _{t}$ at each grid point, average them over the wall-parallel directions and then compute the $Pr_t$ using (4.2). For a statistically steady flow, the difference between the values computed using the equations (4.1) and (4.2) should be very small.

The wall-normal variation of $Pr_t^{(1)}$ is shown in figures 3(a)–3(e). From figure 3( f), we see that the mean value of the turbulent Prandtl number in the log-layer ($z^{+} \geqslant 30$) is quite sensitive to the definition used in the computation. The range of $z^{+}$ over which $Pr_t(z^{+})$ is averaged also affects the value of the mean $Pr_t$. The average $Pr_t^{(1)}$ obtained by using (4.1) approaches unity for larger $Re_b$ while for larger viscous Prandtl number ($Pr=3.0, Pr=10.0$), the average $Pr_t^{(2)}$ obtained by using (4.2) drops to lower than unity and approaches 0.85, as suggested in the literature (Kader Reference Kader1981; Malhotra & Kang Reference Malhotra and Kang1984; Kays Reference Kays1994; McEligot & Taylor Reference McEligot and Taylor1996). Once again, it is noteworthy that the data for $Pr = 0.1$ show a slower convergence of $Pr_t$ than the cases for larger $Pr$, suggesting that the predictions of the theory would become inaccurate for $Pr \ll 0.1$. However, for the control parameters of this paper, and for a large range of wall-normal distances $z^{+}$, we can say that, indeed, $Pr_t \approx 1$ is a reasonable assumption consistent with the numerical data.

5. Global scalar transport and wall shear

Now we investigate the variation of $Nu$ and $Re_{\tau }$ (and thus also $C_f$, or vice versa) with increasing $Re_b$. From figures 4(a) and 4(b) in the turbulent regime, we indeed find the Blasius scaling $C_f \sim Re_b^{-1/4}$ (Blasius Reference Blasius1913; Nikuradse Reference Nikuradse1930, Reference Nikuradse1950; Orlandi et al. Reference Orlandi, Bernardini and Pirozzoli2015) (indicated with the black dashed line) from the present numerical simulations as well as from the data taken from Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014), Orlandi et al. (Reference Orlandi, Bernardini and Pirozzoli2015), Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014) and Lee & Moser (Reference Lee and Moser2018), closely approximating the Prandtl (Reference Prandtl1932) turbulent friction law (von Kármán Reference von Kármán1934)

(5.1)\begin{equation} \sqrt{\frac{2}{C_f}} = \frac{1}{k} \ln \left(Re_b \sqrt{\frac{C_f}{8}}\right)+B. \end{equation}

It is shown as black dash dotted line where $k = 0.41$ (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014) is the von Kármán (Reference von Kármán1934) constant and $B = 5$.

Figure 4. (a) Value of $C_f$ vs $Re_b$ for various $Pr$. The $C_f \sim Re_b^{-1/4}$ scaling in the turbulent regime (represented with black dashed line) is consistent with the Prandtl (Reference Prandtl1932) turbulent friction law (shown as the purple dash dot line). (b) Value of $Re_{\tau }$ vs $Re_b$.

In figures 4(c) and 4(d), the scaling for $Re_{\tau }$ changes sharply with the transition from the laminar to the turbulent regime. In the laminar regime, the effect of steady linear gradients can be seen from the $Re_{\tau } \sim Re_b^{1/2}$ scaling. In the turbulent flow regime, we see a good agreement with the suggested scaling law given by (2.22).

A similar sharp transition is also observed for $Nu$ in figure 5(c). Figure 5(d) shows that $Nu$ varies as $Pr^{1/2}$ for the range of $Pr$ considered in the numerical simulations, which reflects the analogy between turbulent transport of the passive scalar in Couette flow and turbulent transport of heat in Rayleigh–Bénard flow in the limit of zero thermal driving. It can be observed that the values of $Nu$ obtained from the numerical simulations follow the scaling laws given, respectively, by (2.9) and (2.23) quite well. We note that Kays & Crawford (Reference Kays and Crawford1993) give an empirical relation $Nu = 0.022 Pr^{0.5} Re^{0.8}$ for turbulent pipe flows with gases ($0.5< Pr<1$) which is very close to our fit of $Nu = 0.015 Pr^{0.5} Re^{0.75}$.

Figure 5. (a) Value of $Nu$ vs $Re_b$ for different $Pr$. (b) Value of $Nu$ compensated with $Pr^{1/2}$ vs $Re_b$.

The sharpness of the transition from the laminar regime to the turbulent one in the Couette-type shear flow under consideration here is in vast contrast to the very smooth transition from the laminar regime to the turbulent one in Rayleigh–Bénard flow (Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). The reason for this difference lies in the different type of flow instability: whereas in Rayleigh–Bénard flow linear instabilities are very crucial (Landau & Lifshitz Reference Landau and Lifshitz1987), in Couette-type shear flow, the onset of turbulence is of nonlinear non-normal type (Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscol1993; Grossmann Reference Grossmann2000; Barkley Reference Barkley2016; Lemoult et al. Reference Lemoult, Shi, Avila, Jalikop, Avila and Hof2016; Shi, Avila & Hof Reference Shi, Avila and Hof2013).

6. Mean velocity and scalar profiles

We now come to the local flow properties. The existence of a logarithmic inertial sublayer in turbulent wall-bounded flows with passive scalar transport has been studied extensively (Yaglom Reference Yaglom1979; Jischa & Rieke Reference Jischa and Rieke1979a; Kader Reference Kader1981; Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2016). The log-laws for the mean streamwise velocity and the mean temperature may be written as

(6.1a,b)\begin{equation} u^{+} = \frac{1}{k} \ln (z^{+})+B, \quad \theta^{+} = \frac{Pr_t}{k} \ln (z^{+})+D(Pr,Pr_t). \end{equation}

The values of $z^{+}$, $u^{+}$ and $\theta ^{+}$ are defined as usual as

(6.2a,b)$$\begin{gather} z^{+} = {zu_{\tau}}/{\nu}, \quad u^{+} = u/u_{\tau}, \end{gather}$$

(6.3a,b)$$\begin{gather}\theta^{+} = {\left|\theta_w-\theta\right|}/{\theta_{\tau}}, \quad \theta_{\tau} = Q/u_{\tau}. \end{gather}$$

Here, $\theta _{\tau }$ is a quantity analogous to the friction velocity, $\theta _w$ is the mean value of $\theta$ at the wall and $D(Pr,Pr_t)$ is an a priori unknown offset temperature, which in general, depends on $Pr$ and $Pr_t$.

The value of $D(Pr,Pr_t)$ as a function of $Pr$ and $Pr_t$ as obtained from the numerical simulations is shown in table 1. Kader (Reference Kader1981) proposed the following empirical fitting relation:

(6.4)\begin{equation} D = \left( 3.85Pr^{1/3}-1.3 \right)^{2} + \frac{Pr_t}{\kappa} \ln (Pr), \end{equation}

along with $Pr_t = 0.85$ as a modelling parameter.

Table 1. Comparison between the value of the intercept $D$ for the log law obtained from the numerical simulations and the value computed from the empirical relation from Kader (Reference Kader1981).

Figure 6(a) shows the mean streamwise velocity profile normalised with the friction velocity vs height in wall units for the various $Re_b$ obtained from the numerical simulations. The laminar cases (shown by the blue lines) follow the linear profile whereas the turbulent cases (green–red lines) display the log-layer beyond the viscous sublayer. This is also seen when plotting the diagnostic function in figure 6(b) where we see an appreciably well developed log-layer for the turbulent cases. Figure 6(c) shows the mean scalar profiles where we see a similar family of lines, each associated with a particular value of $Pr$.

Figure 6. (a) Mean streamwise velocity normalised with the friction velocity $(u_{\tau })$ vs height in wall units. (b) The diagnostic function for the mean streamwise velocity. (c) Mean scalar value normalised with $\theta _{\tau }$ vs height in wall units. The value of the intercept of the log-law is obtained from the empirical relation (6.4) given by Kader (Reference Kader1981). (d) The diagnostic function for the mean temperature.

The dependence of the turbulent Prandtl number $Pr_t$ on $Pr$ has been well studied for pipe flows (Malhotra & Kang Reference Malhotra and Kang1984; McEligot & Taylor Reference McEligot and Taylor1996), channel flows (Kim & Moin Reference Kim and Moin1989) as well as for stratified Couette flows (Zhou, Taylor & Caulfield Reference Zhou, Taylor and Caulfield2017; Glazunov et al. Reference Glazunov, Mortikov, Barskov, Kadantsev and Zilitinkevich2019) with models suggested by Jischa & Rieke (Reference Jischa and Rieke1979a,Reference Jischa and Riekeb). It is considered that, for $Pr<1$, $Pr_t>1$ and for $Pr>1$, $Pr_t<1$ with $Pr_t \to 0.85$ for large $Pr$ and $Re_b$ (Kays Reference Kays1994). This observation is consistent with the numerical results shown in figure 3. We note that a constant value of $Pr_t = 0.85$ suggested by Kader (Reference Kader1981) seems to be a good fit for the mean temperature profiles of all the values of $Pr$ considered in this study, as shown by the diagnostic function in figure 6(d).

7. Passive scalar transport in Couette flow as limiting case of heat transport in sheared Rayleigh–Bénard

For weak temperature fluctuations and small thermal driving (i.e. small or even zero Rayleigh number), the transport of heat may be considered as passive transport of temperature (Subramanian & Antonia Reference Subramanian and Antonia1981). Therefore, we may view the passive scalar transport problem as a limiting case of the heat transport in the sheared Rayleigh–Bénard system (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Lohse & Xia Reference Lohse and Xia2010; Chilla & Schumacher Reference Chilla and Schumacher2012). The sheared Rayleigh–Bénard system consists of the standard Rayleigh–Bénard set-up with an additional Couette-type shear forcing. The strength of the thermal driving in the sheared Rayleigh–Bénard system, due to the temperature difference $\varDelta$ between the hot bottom plate and the cold top plate, is given by the Rayleigh number as

(7.1)\begin{equation} Ra \equiv \frac{\beta g H^{3} \varDelta}{\nu \kappa}. \end{equation}

Here, $\beta$ is the thermal expansion coefficient and $g$ is the gravitational acceleration.

The relative strength of thermal and shear driving is quantified by the Richardson number given by

(7.2)\begin{equation} Ri \equiv \frac{Ra}{Pr Re_b^{2}}. \end{equation}

Viewing the passive transport in Couette flow as the limiting case of the sheared Rayleigh–Bénard system with $Ra \to 0$ and $Ri \to 0$, the numerical simulations and results shown in this paper correspond to

(7.3a,b)\begin{equation} Nu(Re_b, Pr, Ra=0) \quad \mbox{and} \quad Re_\tau (Re_b, Pr, Ra=0) . \end{equation}

For the passive scalar transport in Couette flow, given that there is a temperature difference $\varDelta \ne 0$ between the bottom and top plates, the limit $Ra \to 0$ is achieved by setting the thermal expansion coefficient $\beta \to 0$, or alternatively the gravitational acceleration $g \to 0$. The standard Rayleigh–Bénard case of purely thermally driven convective flow (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Lohse & Xia Reference Lohse and Xia2010; Chilla & Schumacher Reference Chilla and Schumacher2012) is another limiting case, namely

(7.4a,b)\begin{equation} Nu(Re_b = 0 , Pr, Ra) \quad \mbox{and} \quad Re_\tau (Re_b=0, Pr, Ra) . \end{equation}

For that case, Grossmann & Lohse (Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001, Reference Grossmann and Lohse2002, Reference Grossmann and Lohse2004), Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013) and Shishkina et al. (Reference Shishkina, Emran, Grossmann and Lohse2017) have developed a unifying theory, based on the decomposition of the kinetic and thermal dissipation rates into boundary layer and bulk contribution, which very successfully describes the experimental and numerical data for the control parameter dependencies of the Nusselt and wind Reynolds numbers (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). Note that some other limiting cases of problem (1.6a,b) have already been analysed before, namely

(7.5a,b)\begin{equation} Nu(Re_b , Pr = 1, Ra) \quad \mbox{and} \quad Re_\tau (Re_b, Pr=1, Ra), \end{equation}

by Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) and

(7.6a,b)\begin{equation} Nu(Re_b , Pr, Ra= 10^{6}) \quad \mbox{and} \quad Re_\tau (Re_b, Pr, Ra = 10^{6}), \end{equation}

by Blass et al. (Reference Blass, Verzicco, Lohse, Stevens and Krug2021). In these two papers, the focus was on the interplay and the competition between thermal driving and shear driving, identifying the transitions from dominance of one to dominance of the other, in order to understand the dependencies (7.5a,b) and (7.6a,b).

With increasing shear forcing as compared with thermal forcing, i.e. $Pr Re_b^{2} \gg Ra$, one approaches $Ri \to 0$, which is the limiting case of passive scalar transport in Couette flows. For such large shear forcing (compared with thermal forcing), we expect the $Nu$ and $Re_{\tau }$ to follow the scaling dependencies derived in this work. From figure 7, it can indeed be observed that $Nu(Re_b , Pr, Ra)$ and $Re_\tau (Re_b, Pr, Ra)$ for the sheared Rayleigh–Bénard system as obtained by Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020, Reference Blass, Verzicco, Lohse, Stevens and Krug2021) seem to, in the limit of large shear forcing, asymptotically converge to the $Nu(Re_b , Pr, Ra=0)$ and $Re_\tau (Re_b, Pr, Ra=0)$ scaling laws discussed here in the context of passive scalar transport in Couette flows.

Figure 7. (a) Shear Reynolds number ($Re_{\tau }$) plotted against the bulk Reynolds number $Re_b$. (b) Nusselt number ($Nu$) plotted against the bulk Reynolds number ($Re_b$). The data are taken from by Blass et al. (Reference Blass, Zhu, Verzicco, Lohse and Stevens2020, Reference Blass, Verzicco, Lohse, Stevens and Krug2021). In the limiting case of large shear, the scaling relations assumed ($Re_{\tau } \sim Re_b^{7/8}$) or derived ($Nu \sim Re_b^{3/4}$) in this paper are recovered.