2.1 Governing equations

For fluid flow with small density variations the incompressible Boussinesq approximation can be applied:

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}\boldsymbol{u})=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\frac{1}{Re}(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\text{T}})\right)-\frac{\unicode[STIX]{x1D70C}^{\ast }}{Fr^{2}}\boldsymbol{e}_{y}, & \displaystyle\end{eqnarray}$$

(2.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D70C}^{\ast }+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}^{\ast }\boldsymbol{u})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\frac{1}{Pr\,Re}\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}^{\ast }\right), & \displaystyle\end{eqnarray}$$

$\boldsymbol{u}=[u,v,w]$

$u_{b}$

$d_{h}$

$\unicode[STIX]{x1D70C}_{b}\,u_{b}^{2}$

$d_{h}/u_{b}$

$\unicode[STIX]{x1D70C}^{\ast }$

$\unicode[STIX]{x1D70C}_{b}$

$\boldsymbol{e}_{y}$

$Re$

$Fr$

$Pr$

(2.2a-c ) $$\begin{eqnarray}\displaystyle Re=\frac{u_{b}\,d_{h}}{\unicode[STIX]{x1D708}},\quad Fr=\frac{u_{b}}{\sqrt{g\,d_{h}}},\quad Pr=\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D6FC}}, & & \displaystyle\end{eqnarray}$$

where $g$ is the gravitational acceleration, $\unicode[STIX]{x1D708}$ the kinematic viscosity and $\unicode[STIX]{x1D6FC}$ the thermal diffusivity. The equation of state couples density variation and temperature variation

(2.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}^{\ast }=(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{b})/\unicode[STIX]{x1D70C}_{b}=-\unicode[STIX]{x1D6FD}(T-T_{b}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}$ is the thermal expansion coefficient of liquid water. For the present study, $\unicode[STIX]{x1D6FD}$ is approximated by averaging over the range of possible temperatures, from $T_{b}=333.15~\text{K}$ to $T_{w}=373.15~\text{K}$ , which yields

(2.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}T}\approx -\frac{1}{\unicode[STIX]{x1D70C}_{b}}\frac{\unicode[STIX]{x1D70C}(T_{w})-\unicode[STIX]{x1D70C}_{b}}{T_{w}-T_{b}}=6.32\times 10^{-4}~\text{K}^{-1}. & & \displaystyle\end{eqnarray}$$

The fluid temperature is calculated from the density variation $\unicode[STIX]{x1D70C}^{\ast }$ . The temperature and density dependent transport properties of the fluid are obtained using the IAPWS correlations (IAPWS 2008, 2011).

2.2 Numerical method

The equation system is discretised by a fractional step method on a block structured staggered Cartesian grid. As time advancement method, the explicit third-order Runge–Kutta scheme of Gottlieb & Shu (Reference Gottlieb and Shu1998) is applied, while the time step is adjusted dynamically to reach a Courant number of $1.0$ . A second-order finite volume method is used for spatial discretisation. The pressure Poisson equation is solved in every Runge–Kutta substep using a Krylov subspace solver with an algebraic-multigrid preconditioner.

In LES the turbulent large-scale structures are fully resolved, whereas the small scales are filtered out. The size of the small scales or subgrid scales (SGS) is determined by the chosen grid resolution. The influence of the SGS dynamics on the resolved scales is modelled with the adaptive local deconvolution method (ALDM), which has been developed by Hickel, Adams & Domaradzki (Reference Hickel, Adams and Domaradzki2006). ALDM is a nonlinear finite volume method, that provides a physically consistent subgrid-scale turbulence model for implicit LES. The basic concept of implicit LES is to use the discretisation error to model the dynamics of the SGS. Hickel, Adams & Mansour (Reference Hickel, Adams and Mansour2007) extended ALDM to passive scalar mixing and Remmler & Hickel (Reference Remmler and Hickel2012) to active scalars for turbulent flows governed by the Boussinesq equations. Extensive validation studies and applications to wall-bounded turbulence can be found in Hickel & Adams (Reference Hickel and Adams2007), Hickel & Adams (Reference Hickel and Adams2008), Grilli et al. (Reference Grilli, Schmid, Hickel and Adams2012), Quaatz et al. (Reference Quaatz, Giglmaier, Hickel and Adams2014) and Pasquariello, Hickel & Adams (Reference Pasquariello, Hickel and Adams2017).

3.1 Reference experiment configuration

Figure 1. Experimental duct set-up, reproduced from Rochlitz et al. (Reference Rochlitz, Scholz and Fuchs2015).

In cooperation with the researchers of the companion reference experiment Rochlitz et al. (Reference Rochlitz, Scholz and Fuchs2015), a generic cooling duct was defined with well-determined boundary conditions and water as the working fluid. Figure 1 shows the experimental set-up including the field of view (FOV), where particle image velocimetry (PIV) measurements were conducted.

The duct has a rectangular cross-section with a nominal width of $6.00~\text{mm}$ and height of $25.80~\text{mm}$ , resulting in an aspect ratio of 4.3 and a hydraulic diameter of $d_{h}=9.74~\text{mm}$ . Due to fabrication tolerances (H. Rochlitz and P. Scholz, personal communications, 2015–2018) the average width of the experimental duct is $6.23~\text{mm}$ and the height is $26.10~\text{mm}$ . The experimental aspect ratio thus reduces to $4.19$ and $d_{h}=10.06~\text{mm}$ . The duct length is $600~\text{mm}$ , i.e. 60 times the hydraulic diameter.

The side walls and upper wall are made from polymethyl methacrylate (PMMA) for optical accessibility. The lower wall is made from copper in the heated section and from aluminium in the feed line. All walls are hydraulically smooth with an average roughness of $R_{a}<0.1~\unicode[STIX]{x03BC}\text{m}$ . The temperature distribution of the heated wall is spatially uniform using a heat nozzle. The heat nozzle is a large cone-shaped block of copper, whose tip forms the lower heated wall. Inside the copper block are several cartridge heaters and temperature sensors. The heating of the block is regulated in a closed loop control system to ensure a constant wall temperature. Thus, the lower duct wall is isothermal and the wall temperature can be chosen independently from the heat flux into the coolant.

The experimental set-up is operated continuously in a closed loop. At the beginning of the cycle the water in the reservoir can be preheated or cooled down. The water is then pumped from this reservoir to the test section. The flow rate is controlled by an electromagnetic flowmeter mounted upstream of the pump. After pump and flow straightener the water enters a curved pipe followed by a smooth transition into the rectangular duct. The first part of the duct consists of a $600~\text{mm}$ unheated feed line to ensure fully developed turbulent duct flow. For verification, a test run with an additional $2~\text{m}$ feed line extension has been performed. Between feed line and test section a flow straightener is installed to generate homogeneous inflow conditions for the $600~\text{mm}$ heatable test section, after which the water flows back into the reservoir.

As optical measurement techniques particle image velocimetry (2C2D-PIV), stereo PIV (3C2D-PIV) and volumetric particle tracking velocimetry (3C3D-PTV) are employed using silver-coated hollow glass spheres with a diameter of $10~\unicode[STIX]{x03BC}\text{m}$ as tracer particles. The first two methods give two (2C) respectively three velocity components (3C) in a plane (2D), whereas PTV gives three components in a volume (3D). The laser sheet for both PIV methods is a $xy$ -plane located at the centre of the duct width ranging from the bottom to the top wall. The FOV is $50~\text{mm}$ long and extends from $350{-}400~\text{mm}$ with respect to the beginning of the test section. The laser sheet thickness is set to $1~\text{mm}$ for PIV. For PTV the laser sheet extends over the whole width of the duct.

3.2 Numerical setup

Figure 2. Numerical cooling duct set-up, reproduced from Kaller et al. (Reference Kaller, Pasquariello, Hickel and Adams2017).

The numerical set-up of the cooling duct is shown in figure 2. The isothermal feed line is modelled as an adiabatic periodic duct, denoted by $D_{per}$ . To resolve the large-scale turbulent structures, its streamwise domain length is chosen to $L_{x,per}=7.5\,d_{h}$ following numerical duct flow results by Vinuesa et al. (Reference Vinuesa, Noorani, Lozano-Duran, El Khoury, Schlatter, Fischer and Nagib2014) for an $AR=5$ case and experimental channel flow results by Monty et al. (Reference Monty, Stewart, Williams and Chong2007). The heated section, denoted by $D_{heat}$ , is spatially resolved, i.e. the experimental length of $L_{x,heat}=600~\text{mm}$ is fully simulated. Both duct simulations run simultaneously. For each time step the outflow velocity profile of $D_{per}$ is prescribed at the inlet of $D_{heat}$ . Thus, the periodic section generates a time-resolved fully developed turbulent inflow profile for the heated duct. At the outflow of the heated section a second-order Neumann boundary condition is applied for velocity and density fluctuations. All walls are treated as smooth walls and are defined adiabatic except the lower wall of the heated duct, where a fixed temperature of $T_{w}=373.15~\text{K}$ is prescribed by the corresponding $\unicode[STIX]{x1D70C}^{\ast }$ using equation (2.3).

The cooling duct simulation is initialised in several steps: first the velocity profile for a fully developed laminar duct flow (Shah & London Reference Shah and London1978), superimposed with white noise of amplitude $A\approx 5\,\%\,u_{b}$ , is defined as initial solution for the adiabatic domain $D_{per}$ on a coarse grid. When a fully developed turbulent duct flow is established, the solution is interpolated onto the fine grid and the simulation is continued for several flow-through times (FTT). The final flow state of $D_{per}$ forms the initial condition for the fully coupled set-up of both flow domains, where $D_{heat}$ is built as a sequence of periodic duct sections. After $1.33$ FTT with respect to $L_{x,heat}$ and $u_{b}$ (corresponding to 11 FTT of the periodic section), statistical sampling is started with a constant temporal sampling rate of $\unicode[STIX]{x0394}t_{sample}=0.025\,d_{h}/u_{b}$ . The sampling extends over $20$ FTT of the heated duct section.

The main flow and simulation parameters are listed in table 1. The additional characteristic quantities with respect to § 2.1 are

(3.1a-c ) $$\begin{eqnarray}\displaystyle Nu(x,z)=\left.\frac{h\,d_{h}}{k}\right|_{w}=\frac{-d_{h}}{T_{w}-T_{b}}\left.\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}y}\right|_{w},\quad Gr_{b}=\frac{g\,\unicode[STIX]{x1D6FD}_{b}(T_{w}-T_{b})d_{h}^{3}}{\unicode[STIX]{x1D708}_{b}^{2}},\quad Re_{\unicode[STIX]{x1D70F}}=\left.\frac{u_{\unicode[STIX]{x1D70F}}\,d_{h}}{\unicode[STIX]{x1D708}}\right|_{w}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Reynolds, Prandtl and Grashof numbers are formed using bulk quantities for the adiabatic duct. All Reynolds numbers use the hydraulic diameter of the duct $d_{h}$ as reference length. The friction Reynolds numbers $Re_{\unicode[STIX]{x1D70F}}$ are measured in the centre of their respective side wall with the friction velocity $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}_{w}}$ , where $\unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D707}_{w}(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)|_{w}$ is the wall shear stress. When heating is applied to the lower wall, $Re_{\unicode[STIX]{x1D70F},y}$ increases to $7.25\times 10^{3}$ . The Nusselt number evaluation is based on equalising convective heat transfer and thermal conduction for the cells next to the heated wall with the heat transfer coefficient $h$ and the thermal conductivity $k$ . The resulting $Nu(x,z)$ distribution is then averaged in both directions to obtain the mean value $Nu_{xz}$ for $D_{heat}$ . Following Wardana et al. (Reference Wardana, Ueda and Mizomoto1994), buoyancy effects can be neglected if $Gr/Re^{2}\ll 1$ . As for the present study $Gr_{b}/Re_{b}^{2}=6.9\times 10^{-5}$ buoyancy effects are expected to be negligible. Nevertheless they are taken into account by the chosen equation system.

Figure 3. Computational grid and blocking (indicated in red) for the heated simulation in the $yz$ -plane, every second grid line is shown.

Table 1. Main flow and simulation parameters.

For the discretisation of the flow domain we use a block-structured Cartesian grid with $280\times 10^{6}$ cells. To avoid interpolation at the inlet of the heated section, a matching interface is used between $D_{heat}$ and $D_{per}$ . In order to reduce the numerical effort, we differentiate between boundary layer blocks with a finer and core blocks with a coarser grid resolution. At the block interfaces a 2:1 coarsening in the cross-sectional directions is applied, see figure 3. For the boundary layer blocks we use a hyperbolic grid stretching in the respective wall-normal direction. For the $y$ -direction follows

(3.2) $$\begin{eqnarray}\displaystyle y_{i}=l_{y}\,\tanh \left(\frac{\unicode[STIX]{x1D6FE}_{y}(i-1)}{N_{y}-1}\right)\bigg/\tanh (\unicode[STIX]{x1D6FE}_{y}). & & \displaystyle\end{eqnarray}$$

Herein $i$ is the grid point index, $\unicode[STIX]{x1D6FE}_{y}$ the stretching factor, $l_{y}$ the block edge length and $N_{y}$ the number of points. The same approach is used at all walls. The core block $B_{3}$ possesses an uniform cell distribution in the $yz$ -plane and $B_{2}$ is slightly stretched. In the streamwise direction a uniform discretisation is applied for all blocks. The grid parameters are summarised in table 3.

Table 2. Mesh parameters and wall shear stresses for the grid sensitivity study.

Table 3. Mesh parameters and wall shear stresses for the heated duct simulation resulting from the sensitivity study. For $D_{per}$ , $\unicode[STIX]{x1D70F}_{w}|_{z}=69.9~\text{Pa}$ and for $D_{heat}$ , $\unicode[STIX]{x1D70F}_{w}|_{z}=68.9~\text{Pa}$ .

3.3 Grid sensitivity analysis

A grid sensitivity study has been performed for the adiabatic periodic duct section $D_{per}$ , see figure 2. The main parameters for the considered grids are summarised in table 2 and the grid structure is the same as exemplarily shown for the heated set-up in figure 3, except that we use a symmetric grid with respect to the $y$ - and $z$ -axis as no heating is applied in this analysis.

The aim of the adiabatic grid sensitivity study is to determine the required resolution, that assures a well-resolved cooling duct LES under the given operating conditions at affordable numerical costs. As requirements, we define for the short as well as the large side walls $\unicode[STIX]{x0394}y_{min}^{+}\approx \unicode[STIX]{x0394}z_{min}^{+}\approx 1$ for the cells next to the wall, a velocity profile following the analytical law of the wall and a sufficient number of cells in the vicinity of the walls to correctly predict turbulence production within the turbulent boundary layers (TBL). We mainly focus on the streamwise Reynolds stress distribution. The dimensionless wall distance of the cell next to the wall is denoted as $\unicode[STIX]{x0394}y_{min}^{+}$ and $\unicode[STIX]{x0394}z_{min}^{+}$ respectively and is based on the respective cell height. The chosen requirements are evaluated in the respective centre of each of the four side walls.

For this study, we separately investigate three parameters and their influence on the adiabatic duct flow: the wall-tangential, the wall-normal and the streamwise cell size. The main focus lies on the TBL at the lower short side wall at $y=y_{min}$ , where the heat flux will be applied in the heated simulation. These results are representative for the large side walls, where the same effects are observed. The grid sensitivity with respect to the three parameters will be shown using the boundary layer velocity profiles and the Reynolds stress distributions along the duct centre line $z=0$ . Statistical quantities are sampled with a constant sampling rate of $\unicode[STIX]{x0394}t_{sample}=0.025\cdot d_{h}/u_{b}$ over at least $33$ FTT. Additionally, averaging in the homogeneous streamwise direction is applied and the grid symmetry is utilised by performing an averaging of lower and upper side wall statistics. Non-dimensionalisation for the velocity profile is performed using the inner length scale $l^{+}=\unicode[STIX]{x1D708}_{w}/u_{\unicode[STIX]{x1D70F}}$ and the friction velocity $u_{\unicode[STIX]{x1D70F}}$ . The Reynolds stresses are made non-dimensional using $u_{b}^{2}$ (and not $u_{\unicode[STIX]{x1D70F}}^{2}$ as often seen in the literature) to point out the respective effects more clearly.

First, the maximum wall-tangential cell size $\unicode[STIX]{x0394}z_{max}/\unicode[STIX]{x0394}z_{min}$ is varied from $23.6$ over $36.5$ to $47.4$ for $G_{1}$ , $G_{2}$ and $G_{3}$ respectively, while the size of the cell next to the wall $\unicode[STIX]{x0394}z_{min}$ is kept constant. The stretching factor increases from $\unicode[STIX]{x1D6FE}_{z}|_{G_{1}}=2.30$ over $\unicode[STIX]{x1D6FE}_{z}|_{G_{2}}=2.53$ to $\unicode[STIX]{x1D6FE}_{z}|_{G_{3}}=2.69$ and $\unicode[STIX]{x1D6FE}_{y}$ accordingly. Hence for this comparison the cross-section discretisation is modified. Figure 4(a,b) shows that the velocity profile for $G_{1}$ follows the analytical law of the wall. For $G_{2}$ and $G_{3}$ strong deviations in the log-law region as well as in the wake region are visible. This is accompanied by a significant drop of the wall shear stress from $\unicode[STIX]{x1D70F}_{w}|_{y,G_{1}}=58.4~\text{Pa}$ over $\unicode[STIX]{x1D70F}_{w}|_{y,G_{2}}=51.2~\text{Pa}$ to $\unicode[STIX]{x1D70F}_{w}|_{y,G_{3}}=45.5~\text{Pa}$ . Similarly the wall shear stress at the large side wall $\unicode[STIX]{x1D70F}_{w}|_{z}$ reduces with increasing tangential grid resolution, see table 2. The point of maximum streamwise Reynolds stress $\overline{u^{\prime }u^{\prime }}/u_{b}^{2}$ is moving closer to the wall and turbulence intensity increases slightly with falling $\unicode[STIX]{x0394}z_{max}/\unicode[STIX]{x0394}z_{min}$ . The same observation is made for $\overline{v^{\prime }v^{\prime }}/u_{b}^{2}$ and $\overline{w^{\prime }w^{\prime }}/u_{b}^{2}$ . For all three grids $9$ cells reside within the streamwise Reynolds stress maximum, respectively $8$ cells at the large side walls. The comparison of $G_{1}$ , $G_{2}$ and $G_{3}$ suggests for $\unicode[STIX]{x0394}y_{max}/\unicode[STIX]{x0394}y_{min}$ , respectively $\unicode[STIX]{x0394}z_{max}/\unicode[STIX]{x0394}z_{min}$ a value of ${\approx}25$ .

Figure 4. Grid sensitivity study with respect to (a,c,e) boundary layer velocity profile and (b,d,f) Reynolds stress distribution in the vicinity of the lower wall along the duct centre plane $z=0$ . The quantities are time- and streamwise-averaged for the adiabatic domain $D_{per}$ . (a,b) Show the influence of the maximum wall-tangential cell size using $G_{1}$  (——),  $G_{2}$  (– - – - –) and $G_{3}$  (- - - - - -), (c,d) the influence of the minimum cell size in the wall-normal direction using $G_{1}$  (——),  $G_{4}$  (– - – - –, blue) and $G_{5}$  ( $\cdots \cdots$ , blue) and (e,f) the influence of the streamwise cell size using $G_{1}$  (——),  $G_{6}$  (– - – - –, orange) and $G_{7}$  ( $\cdots \cdots$ , orange). See table 2 for reference. The classical law of the wall ( $u^{+}=(1/0.41)\ln y^{+}+5.2$ ) is represented by ( $\cdots \cdots$ , grey).

In figure 4(c,d) the minimum cell size in the wall-normal direction is modified, whereas the ratios of largest to smallest cell size $\unicode[STIX]{x0394}y_{max}/\unicode[STIX]{x0394}y_{min}$ and $\unicode[STIX]{x0394}z_{max}/\unicode[STIX]{x0394}z_{min}$ as well as the stretching factors $\unicode[STIX]{x1D6FE}_{y}$ and $\unicode[STIX]{x1D6FE}_{z}$ are approximately kept constant. As before, the cross-sectional discretisation is modified. The dimensionless wall distance $\unicode[STIX]{x0394}y_{min}^{+}$ varies from $\unicode[STIX]{x0394}y_{min}^{+}|_{G_{4}}=1.01$ over $\unicode[STIX]{x0394}y_{min}^{+}|_{G_{1}}=1.28$ to $\unicode[STIX]{x0394}y_{min}^{+}|_{G_{5}}=2.16$ . At the large side walls $\unicode[STIX]{x0394}z_{min}^{+}$ is altered accordingly, see table 2. The comparison of $G_{1}$ and $G_{4}$ verifies, that the resolution for $G_{1}$ is sufficient to perform wall-resolved LES as the results coincide. A further coarsening of the wall-normal cell size leads to an underprediction of turbulent fluctuations without significant influence on the velocity profile. The wall resolution of $G_{5}$ is too coarse to resolve the Reynolds stress maximum correctly. Only $6$ cells reside within the Reynolds stress maximum, for $G_{4}$ this number is $10$ and for $G_{1}$ 9.

Figure 4(e,f) depicts the influence of the streamwise cell size with $\unicode[STIX]{x0394}x^{+}|_{G_{1}}=98.6$ , $\unicode[STIX]{x0394}x^{+}|_{G_{6}}=63.2$ and $\unicode[STIX]{x0394}x^{+}|_{G_{7}}=47.1$ with an identical discretisation in the $yz$ -plane. We observe slight differences in the logarithmic region of the velocity profile, which get larger in the outer region. The finer discretisation in the streamwise direction leads to a reduction of the wall shear stress, which drops from $\unicode[STIX]{x1D70F}_{w}|_{y,G_{1}}=58.4~\text{Pa}$ to $\unicode[STIX]{x1D70F}_{w}|_{y,G_{6}}=55.8~\text{Pa}$ and $\unicode[STIX]{x1D70F}_{w}|_{y,G_{7}}=53.4~\text{Pa}$ , $u^{+}$ hence increases. Likewise, the Reynolds stresses are reduced uniformly. The maximum streamwise turbulence intensity $\overline{u^{\prime }u^{\prime }}$ expects a significant drop, $\overline{v^{\prime }v^{\prime }}$ and $\overline{w^{\prime }w^{\prime }}$ just drop slightly over the whole interval. The maximum streamwise Reynolds stress $\overline{u^{\prime }u^{\prime }}/u_{b}^{2}$ increases based on $1.64\times 10^{-2}$ for the finest grid $G_{7}$ by 5.0 % for $G_{6}$ and $13.8\,\%$ for $G_{1}$ . The location of the maximum turbulence intensity is not affected and is therefore only controlled by the cross-sectional grid resolution. Even though the streamwise turbulence intensity is slightly overpredicted, $\unicode[STIX]{x0394}x/\unicode[STIX]{x0394}y_{min}=50$ offers overall a good compromise between accuracy and numerical costs of the simulation.

Based on the grid sensitivity analysis for the adiabatic duct the grid for the heated duct set-up is generated. $G_{6}$ serves as source grid, as the study has shown that it satisfies the aforementioned requirements for a well-resolved LES of the adiabatic duct at affordable numerical costs. The numerical parameters for the final grid are shown in table 3. For comparison with the sensitivity study, parameters for the adiabatic section $D_{per}$ as well as the heated section $D_{heat}$ are listed. Both parameters at the refined heated wall at $y=y_{min}$ and the adiabatic wall at $y=y_{max}$ are included. Note, that the evaluation of the wall shear stresses $\unicode[STIX]{x1D70F}_{w}$ and the inner length scale $l^{+}$ for $D_{heat}$ is based on the streamwise-averaged flow condition over the last $7.5\,d_{h}$ of the heated duct.

For flows with $Pr>1$ , the thermal length scales are smaller than the momentum length scales and the temperature boundary layer is completely contained inside the momentum boundary layer. To resolve the wall-normal temperature gradient, the grid for the heated simulation is deduced from the adiabatic grid by increasing the resolution in the wall-normal direction at the heated wall, that is $y=y_{min}$ . The upper half of the duct as well as the blocking is left unaltered, see figure 3. The grid is only symmetric with respect to the $y$ -axis. The minimum cell size $\unicode[STIX]{x0394}y_{min}$ at the heated wall is reduced by the ratio of the smallest scales of the temperature field and the Kolmogorov scales following Monin, Yaglom & Lumley (Reference Monin, Yaglom and Lumley2007)

(3.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x0394}y_{min}|_{heated}}{\unicode[STIX]{x0394}y_{min}|_{adiabatic}}=\frac{\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x1D702}_{k}}=\left(\frac{1}{Pr}\right)^{1/2}, & & \displaystyle\end{eqnarray}$$

with $Pr=3.0$ , the value for water at $T_{b}$ . As the Prandtl number drops with rising temperature, the resolution in the wall-normal direction is slightly finer than required. In contrast to the sensitivity analysis, we also apply for block $B_{2}$ a slight stretching in the $y$ -direction.

Figure 5. Comparison of (a) boundary layer velocity profile and (b) Reynolds stress distribution for the source grid $G_{6}$ (– - – - –, orange) and the final grid for the heated simulation at the $y=y_{min}$ wall (——) and the $y=y_{max}$ wall ( $\cdots \cdots$ ) along the duct centre plane $z=0$ . The quantities are time- and streamwise-averaged and evaluated for the adiabatic domain $D_{per}$ . See also tables 2 and 3 for reference. The classical law of the wall ( $u^{+}=(1/0.41)\ln y^{+}+5.2$ ) is represented by ( $\cdots \cdots$ , grey).

Figure 5 shows the comparison of mean velocity and Reynolds stress profiles for the source grid $G_{6}$ and the finally used grid for the heated simulation. For the latter both walls are shown, as a finer resolution is applied at the $y=y_{min}$ wall, but both walls are modelled as adiabatic. The three velocity profiles coincide over a wide range, only in the outer layer a slight deviation is visible. The Reynolds stresses $\overline{w^{\prime }w^{\prime }}$ match, and only slight deviations in $\overline{u^{\prime }u^{\prime }}$ and $\overline{v^{\prime }v^{\prime }}$ are present. At the large side walls eight cells reside within the streamwise Reynolds stress maximum, at the upper short side wall nine and at the lower short side wall $13$ for the adiabatic section and $11$ for the heated duct. As listed in table 3, $\unicode[STIX]{x0394}y_{min}^{+}$ drops to $0.73$ at the lower short side wall for the unheated section and to $1.09$ when heating is applied. For the large side walls, $\unicode[STIX]{x0394}z_{min}^{+}=1.42$ remains unchanged compared to the original grid $G_{6}$ . Note, that $\unicode[STIX]{x0394}y_{min}^{+}$ and $\unicode[STIX]{x0394}z_{min}^{+}$ are calculated with respect to the whole cell height of the respective first cell, whereas the flow variables are located and evaluated at the cell centre corresponding to $z_{min}^{+}=0.71$ and $y_{min}^{+}=0.37$ , respectively $y_{min}^{+}=0.55$ , in a finite difference sense. A comparison of the adiabatic duct wall shear stresses $\unicode[STIX]{x1D70F}_{w}|_{y}$ for the upper and lower short wall shows, that the unequal meshes have a negligible effect.

5.1 Mean flow field of the adiabatic and heated duct

In the following we analyse the turbulent heat transfer in the asymmetrically heated duct based on the LES results. The main focus lies on investigating the differences of the adiabatic and the heated duct flow field, i.e. on the influence of the wall heating along the duct. Due to the heating, the temperature in the vicinity of the lower wall increases with streamwise distance, reducing the local viscosity, which may drop up to $\unicode[STIX]{x1D708}(T_{w})=0.62\,\unicode[STIX]{x1D708}(T_{b})$ .

Figure 8. Streamwise flow field in the duct cross-section of the adiabatic duct with (a) instantaneous streamwise velocity, (b) mean streamwise velocity and (c) mean streamwise velocity with additional averaging of the four quadrants. The contour lines are drawn in steps of $0.1$ .

Figure 8 displays the cross-section streamwise flow field for the adiabatic duct: an instantaneous snapshot in (a) and the mean solution in (b,c). The instantaneous velocity field shows the highly turbulent flow with the highest velocities in the duct core and smaller low-velocity structures along the side walls and in the duct corners. For the mean solution a temporal averaging over $164$ FTT with respect to the periodic section at a constant sampling rate of $\unicode[STIX]{x0394}t_{sample}=0.025\,d_{h}/u_{b}$ is performed, resulting in ${\approx}50\times 10^{3}$ snapshots. Additionally, a spatial averaging in the streamwise direction is applied. For the quadrant-averaged solution a further averaging over the four quadrants is done exploiting the duct symmetry in order to reduce the number of samples required for a statistically convergent result. As Vinuesa et al. (Reference Vinuesa, Noorani, Lozano-Duran, El Khoury, Schlatter, Fischer and Nagib2014) pointed out, the number of required samples is not reduced by a factor of four as the flow in the quadrants is not independent, in particular the corner vortices in the vicinity of the short side walls are strongly correlated. As a convergence measure we utilise the symmetry of the flow field. The comparison of figure 8(b,c) shows that the streamwise velocity field is sufficiently converged as hardly any difference is visible. The L2 norm of the streamwise velocity deviation between the not-quadrant-averaged and the quadrant-averaged result in the $yz$ -plane is $0.5\,\%$ .

Figure 9. Secondary flow field in the duct cross-section of the adiabatic duct with (a) mean cross-flow velocity magnitude $\overline{u}_{cf}=\sqrt{\overline{v}^{2}+\overline{w}^{2}}$ , (b) mean cross-flow velocity magnitude $\overline{u}_{cf}$ with an additional averaging of the four quadrants and (c) counter-rotating corner vortices represented by streamlines. The contour lines are drawn in steps of $0.15\times 10^{-2}$ .

In general, the convergence rate of the turbulence-induced secondary flow is slower than that of the streamwise velocity. Figure 9 depicts the secondary flow field represented by the cross-flow velocity magnitude $\overline{u}_{cf}=\sqrt{\overline{v}^{2}+\overline{w}^{2}}$ and streamlines. The comparison with the quadrant-averaged solution in figure 9(b) shows that the result is not perfectly converged as the secondary flow field is slightly asymmetric. The deviation from the symmetric state is higher than for the streamwise velocity, but sufficiently small. A reason for the larger deviation is the presence of very weak vortices in the duct centre, see figure 9(c). These persist over very long averaging times. Such a formation of an array of secondary vortices along the long side walls of high aspect ducts has been observed previously by Vinuesa et al. (Reference Vinuesa, Noorani, Lozano-Duran, El Khoury, Schlatter, Fischer and Nagib2014).

Figure 10. Cross-sectional temperature distribution and its streamwise development in (a–f) and development of the secondary flow velocity in the vicinity of the heated wall, exemplarily shown for the heated wall-normal component in (g–l). (a,g) Depicts the adiabatic duct, (b,h) the heated duct after $50~\text{mm}$ , (c,i) after $100~\text{mm}$ , (d,j) after $200~\text{mm}$ , (e,k) after $400~\text{mm}$ and (f,l) after $600~\text{mm}$ . In (a–f), the cross-flow velocity vectors indicate the influence of the secondary flow motions on the temperature distribution and the contour lines are drawn in steps of $2~\text{K}$ . In (g–l), the wall-normal velocity $\overline{v}$ is depicted on the left of the duct centre and on the right the change in $\overline{v}$ with respect to the unheated periodic duct, $\unicode[STIX]{x0394}\overline{v}=\overline{v}-\overline{v}_{per}$ , is shown. The contour lines are drawn in steps of $\overline{v}/u_{b}=0.002$ .

The developing temperature boundary layer along the heated duct length is highly affected by the secondary flow structures. Figure 9(c) shows the pairs of counter-rotating vortices forming in each of the duct corners. In the left upper corner ( $y<0$ , $z>0$ ), a small counter-clockwise (CCW) rotating vortex forms along the short side wall and a large clockwise (CW) rotating vortex along the long side wall (mirror inverted for the opposite half of the duct). Each vortex extends to the respective symmetry plane, where it encounters the neighbouring vortex from the opposite side. The vortex strength is relatively weak. The maximum cross-flow velocity for the adiabatic duct is $\overline{u}_{cf}/u_{b}=1.93\,\%$ , which lies within the 1–3 % range reported in the literature, see for example Salinas-Vásquez & Métais (Reference Salinas-Vásquez and Métais2002). Figure 10(a–f) depict the axial development of the temperature boundary layer at different positions along the heated duct section for the lower duct quarter. The thermal boundary layer thickness increases in the streamwise direction due to conduction, turbulent mixing and through transport by the mean secondary flow. The latter is responsible for the characteristic bent shape of the temperature profile. In the left half of the duct, the CW vortex transports hot fluid away from the heated wall along the long side wall into the duct core and cold fluid downwards along the centre line. The CCW vortex conveys hot fluid from the corner along the heated wall to its centre at $z=0$ and then upwards along the symmetry line until it mixes with the cold fluid transported downwards. Both vortices push cold fluid into the left corner, whereby the flow vectors follow a slightly flatter path than the corners bisecting line.

Even though the temperature increase and the accompanying viscosity decrease are overall relatively moderate, we observe a significant weakening of the secondary flow strength in figure 10(g–l), where the heated wall-normal secondary flow component $\overline{v}$ is shown at the same spatial positions along the heated duct section as the temperature boundary layer in figure 10(a–f). In the left half of each picture the mean vertical velocity $\overline{v}$ is depicted and in the right half the difference of the $\overline{v}$ -field with respect to the adiabatic case, $\unicode[STIX]{x0394}\overline{v}=\overline{v}-\overline{v}_{per}$ . We observe a significant reduction of the vortex strength. The upward transport of hot fluid in the vicinity of the lateral wall is slowed down increasingly, in the end cross-section the maximum $\unicode[STIX]{x0394}\overline{v}/u_{b}$ is ${\approx}-0.004$ with a velocity of $\overline{v}/u_{b}\approx 0.015$ , which corresponds to a reduction of more than $25\,\%$ . The positions of the corner vortices change only a little, thus they are mainly defined by the duct geometry. The small CCW vortex centre moves from $(2y/L_{y},2z/L_{z})=(-0.947,0.414)$ for the adiabatic duct to ( $-0.948$ , $0.394$ ) at position $600~\text{mm}$ and the large CW vortex centre from ( $-0.752$ , $0.569$ ) to ( $-0.763$ , $0.601$ ). Hence, we observe for the large vortex a slight shift towards both side walls and for the small one a slight shift towards the midplane.

Figure 11. Profiles of (a) mean streamwise velocity and (b) Reynolds stresses along the duct midplane at $z=0$ for the adiabatic (——) and the heated duct (——, orange). The law of the wall is represented by ( $\cdots \cdots$ , grey).

As the corner vortices are turbulence-induced secondary flow structures, we further analyse the influence of the reduced wall viscosity on the mean turbulence and velocity profiles. In figure 11, we investigate the influence of the wall heating on the turbulent boundary layer in the duct centre at $z=0$ by comparing the spatially averaged solutions over the adiabatic domain $D_{per}$ and the last $7.5\,d_{h}$ of the heated duct. For both sections good agreement with the classical law of the wall velocity profile is obtained, $u^{+}=y^{+}$ for the viscous sublayer and $u^{+}=(1/\unicode[STIX]{x1D705})\ln y^{+}+B$ for the log-law region. Like Lee et al. (Reference Lee, Jung, Sung and Zaki2013), we observe in figure 11 (a) that the heating leads to an upwards shift in the log-law region of the velocity profile, the integration constant increases from $B=5.2$ to $B=6.0$ for the heated case. The slope, i.e. the von Kármán constant remains unchanged at $\unicode[STIX]{x1D705}=0.41$ . Figure 11(b) depicts the change in the Reynolds stress profiles. The peak in $\overline{u^{\prime }u^{\prime }}$ shifts slightly closer to the wall, whereas the maximum value remains unaltered. Similarly to Zonta, Marchioli & Soldati (Reference Zonta, Marchioli and Soldati2012), we observe that the turbulence intensities in all directions are reduced, when heating is applied to the flow. Although counterintuitive, as one would expect an increase in turbulent fluctuations with lower viscosity, this observation is in agreement with previous studies showing that the heating of the fluid accompanied by a drop in viscosity has a stabilising effect on the boundary layer, see Lee et al. (Reference Lee, Jung, Sung and Zaki2013) and Zonta et al. (Reference Zonta, Marchioli and Soldati2012).

Figure 12. Distribution of the mean velocity components and mean temperature difference along the heated wall-normal direction at the spanwise positions of (a) $2z/L_{z}=0$ , (b) $2z/L_{z}=0.5$ and (c) $2z/L_{z}=0.9$ for the adiabatic duct (——) and the heated duct at a streamwise position of $100~\text{mm}$  ( $\cdots \cdots$ , orange), respectively $595~\text{mm}$  (– - – - –, orange) after the beginning of the heated section. Streamwise averaging has been performed over $10~\text{mm}$ for the heated duct, respectively over $L_{x,per}$ for the periodic duct, and the $y$ -symmetry is utilised.

Figure 13. Distribution of the diagonal components of the Reynolds stress tensor along the heated wall-normal direction at the spanwise positions of (a) $2z/L_{z}=0$ , (b) $2z/L_{z}=0.5$ and (c) $2z/L_{z}=0.9$ for the adiabatic duct (——) and the heated duct at a streamwise position of $100~\text{mm}$  ( $\cdots \cdots$ , orange), respectively $595~\text{mm}$  (– - – - –, orange) after the beginning of the heated section. Streamwise averaging has been performed over $10~\text{mm}$ for the heated duct, respectively over $L_{x,per}$ for the periodic duct, and the $y$ -symmetry is utilised.

Figure 14. Distribution of the off-diagonal components of the Reynolds stress tensor along the heated wall-normal direction at the spanwise positions of (a) $2z/L_{z}=0$ , (b) $2z/L_{z}=0.5$ and (c) $2z/L_{z}=0.9$ for the adiabatic duct (——) and the heated duct at a streamwise position of $100~\text{mm}$  ( $\cdots \cdots$ , orange), respectively $595~\text{mm}$  (– - – - –, orange) after the beginning of the heated section. Streamwise averaging has been performed over $10~\text{mm}$ for the heated duct, respectively over $L_{x,per}$ for the periodic duct, and the $y$ -symmetry is utilised.

In figures 12–14 we present the mean velocity, temperature and Reynolds stress distributions for three different $z$ -locations, for the duct centre at $2z/L_{z}=0$ , at $2z/L_{z}=0.5$ and at $2z/L_{z}=0.9$ . All figures compare the results for the adiabatic duct with those for the heated duct at $100~\text{mm}$ and $595~\text{mm}$ after beginning of the heated section. For the adiabatic duct streamwise averaging is performed over $L_{x,per}$ and for the heated duct over a $10~\text{mm}$ interval from $-5~\text{mm}$ to $+5~\text{mm}$ of the respective location.

Moving outwards from the duct centre towards the lateral wall, i.e. from figure 12(a–c), we observe for the adiabatic case a broadening of the $\overline{u}$ -profiles shoulder section, which is formed as a consequence of the corner vortex pair. The asymmetric heating applied to the duct leads to a mass flux redistribution. The flow in the lower quarter of the duct below $2y/L_{y}\approx 0.5$ is accelerated leading to a thickening of the near-wall profile. This behaviour is qualitatively consistent with previous channel flow and TBL studies by Sameen & Govindarajan (Reference Sameen and Govindarajan2007) and Lee et al. (Reference Lee, Jung, Sung and Zaki2013). Moreover, the heated duct $\overline{u}$ -profile exhibits a more pronounced shoulder section due to the weaker secondary flow and the accompanying reduced vertical momentum transport. In the duct core, the streamwise velocity drops slightly compared to the adiabatic case, for example at position $2z/L_{z}=0$ and $x=595~\text{mm}$ by $-0.5\,\%$ . Due to the duct symmetry, the secondary flow in the centre at $z=0$ has only a $y$ -component. The maximum of the $\overline{v}$ -velocity close to the lower wall is the signature of the two smaller corner vortices pushing fluid upwards and the following $\overline{v}$ -minimum is the signature of the two larger corner vortices pushing fluid downwards. For the second cut at $2z/L_{z}=0.5$ , we observe in the $\overline{w}$ -profile close to the wall the effect of the small corner vortex transporting fluid from the duct corner to the midplane. The coincidence of the $\overline{v}$ -minimum and the $\overline{w}$ -maximum at $2y/L_{y}\approx -0.875$ marks the area, where both the small CW vortex and the large CCW vortex push fluid into the duct corner. The region close to the lateral wall at $2z/L_{z}=0.9$ is then dominated by the large corner vortex transporting fluid upwards into the duct core. As seen before, the viscosity modulation leads to a weakened secondary flow. This effect is especially visible in the $\overline{v}$ -profiles, where the strength of both the small vortices as well as the large vortices represented by the $\overline{v}$ -minima and $\overline{v}$ -maxima is getting weaker, particularly at $2z/L_{z}=0.9$ . In contrast, the $\overline{w}$ -profiles remain nearly unaltered and only a slight reduction at the off-centre positions is observable. The locations of the extrema remain approximately constant signifying an only slight shift of the vortex positions. The last column of figure 12 depicts the temperature increase. The influence of the secondary flow on the temperature distribution is clearly visible, especially for the $595~\text{mm}$ -lines. The corner vortices affect the heat transport significantly. We observe a non-uniform distribution in the spanwise direction and kinks in the $\overline{T}$ -profile, which coincide with the secondary flow extrema.

The diagonal elements of the Reynolds stress tensor are depicted in figure 13 for the same positions as before. The results in the midplane are similar to those presented in figure 11(b), the main differences are the larger streamwise averaging interval and the logarithmic scaling for the $y$ -axis. When the lower wall is heated, we observe in the $\overline{u^{\prime }u^{\prime }}$ -profiles a shift of the turbulent production peak slightly closer to the wall. These results are in agreement with Salinas-Vásquez & Métais (Reference Salinas-Vásquez and Métais2002), who observed the inverse trend of the peak shifting further away from the heated wall for air as working fluid. At $x=100~\text{mm}$ the maximum value is reduced compared to the adiabatic case, whereas at $x=595~\text{mm}$ it increases slightly. Due to the shift towards the short side wall, the streamwise Reynolds stress component is lowered over a large area in the heated case until the adiabatic and heated duct results coincide in the bulk flow. Moving from the centre in the direction of the lateral wall, from figure 13(a) to (c), this coinciding point moves closer to the heated wall from $2y/L_{y}\approx -0.75$ over $2y/L_{y}\approx -0.84$ to $2y/L_{y}\approx -0.97$ . In contrast to $\overline{u^{\prime }u^{\prime }}$ , no shift of the peak position occurs for the $\overline{v^{\prime }v^{\prime }}$ -Reynolds stress profile, but likewise it experiences a drop of the maximum value in the midplane and at $2z/L_{z}=0.5$ . However at $2z/L_{z}=0.9$ , for the position strongly influenced by the large vortex, the profile shape changes entirely and no heating influence is visible. For the $\overline{w^{\prime }w^{\prime }}$ -profile we observe also a profile shift closer to the lower wall like for $\overline{u^{\prime }u^{\prime }}$ . Also the maximum values are reduced, regardless of the shape of the respective $\overline{w^{\prime }w^{\prime }}$ -profile, which changes from a plateau-like maximum in the midplane to a smaller sized maximum closer to the lateral wall.

In figure 14 the off-diagonal components of the Reynolds stress tensor are depicted. Due to the $y$ -symmetry, the off-diagonal components of the Reynolds stress tensor including the $z$ -component $\overline{u^{\prime }w^{\prime }}$ and $\overline{v^{\prime }w^{\prime }}$ vanish in the duct midplane, see the second and third column of figure 14. The off-centre profiles for $\overline{u^{\prime }w^{\prime }}$ and $\overline{v^{\prime }w^{\prime }}$ are not significantly affected by the heating. Similar to the $\overline{u^{\prime }u^{\prime }}$ -profiles and $\overline{w^{\prime }w^{\prime }}$ -profiles in figure 13, a shift of the $\overline{u^{\prime }w^{\prime }}$ -minimum towards the heated wall is visible at $2z/L_{z}=0.9$ , but not at $2z/L_{z}=0.5$ . The viscosity modulation has a strong effect on the $\overline{u^{\prime }v^{\prime }}$ -component. For all three $z$ -positions with their different profile shapes, from a plateau-like maximum in the centre to a smaller sized peak close to the lateral wall, we observe a significant reduction of the respective maximum. We again note a slight shift towards the heated wall, however less clear as for the $\overline{u^{\prime }u^{\prime }}$ -profiles. The $\overline{u^{\prime }v^{\prime }}$ -component describes turbulent ejection and sweeping motions. As Huser & Biringen (Reference Huser and Biringen1993) have stated, the dominant turbulent mechanism generating the secondary flow is the ejections from the wall, we hence will discuss the $\overline{u^{\prime }v^{\prime }}$ -component in more detail in § 5.2 using the Reynolds stress quadrant analysis technique.

5.2 Turbulent sweeping and ejection motions

Figure 15. Quadrant analysis of the Reynolds shear stress component $\overline{u^{\prime }v^{\prime }}$ along the duct centre line $2z/L_{z}=0$ for the adiabatic (——) and the heated duct (——, orange) as well as along $2z/L_{z}=0.5$ marked by ( $\cdots \cdots$ ) and ( $\cdots \cdots$ , orange), respectively.

First utilised by Wallace, Eckelmann & Brodkey (Reference Wallace, Eckelmann and Brodkey1972), the quadrant analysis of the Reynolds stress tensor allows us to identify the main contributions to turbulence (Wallace Reference Wallace2016). The Reynolds shear stress $\overline{u^{\prime }v^{\prime }}$ is split into four quadrants depending on the sign of the streamwise, $u^{\prime }$ , and the heatable wall-normal velocity fluctuation, $v^{\prime }$ . The first quadrant $Q_{1}(u^{\prime }>0/v^{\prime }>0)$ comprises outward motion of high-velocity fluid, the second quadrant $Q_{2}(u^{\prime }<0/v^{\prime }>0)$ outward motion of low-velocity fluid, the third quadrant $Q_{3}(u^{\prime }<0/v^{\prime }<0)$ inward motion of low-velocity fluid and the fourth quadrant $Q_{4}(u^{\prime }>0/v^{\prime }<0)$ inward motion of high-velocity fluid. Willmarth & Lu (Reference Willmarth and Lu1972) have shown for a TBL that $Q_{2}$ is connected to turbulent ejection events and $Q_{4}$ to turbulent sweeping motions.

Figure 15 depicts the quadrant analysis for the Reynolds stress component $\overline{u^{\prime }v^{\prime }}$ for the adiabatic duct and the end section of the heated duct at $2z/L_{z}=0$ and $2z/L_{z}=0.5$ . The duct symmetry is exploited for the latter. For the quadrant analysis the conditional sampling has been performed over a shorter period of $8.5$ FTT with respect to $L_{x,heat}$ with the same sampling rate as in the rest of the investigation, the results are therefore somewhat noisier. The discussion concentrates on the $Q_{2}$ - and $Q_{4}$ -distributions, for completeness the ones for $Q_{1}$ and $Q_{3}$ are also included. First, the focus is set on the adiabatic case. In the midplane all four quadrants show a maximum in the vicinity of the heatable wall and fall to an approximately constant value in the duct centre. In contrast to a TBL, this constant value is non-zero as the boundary layers originating from all side walls influence the flow field of the duct core. Similarly to Salinas-Vásquez & Métais (Reference Salinas-Vásquez and Métais2002) the size of the ejections is slightly larger than that of the sweeping motions indicated by the location of the respective maximum. The ejection size is $l_{ejec}=0.115\,L_{y}/2$ compared to $l_{sweep}=0.085\,L_{y}/2$ for the sweeping motions. We define this size as the distance from the wall to the location, where the intensity has dropped to $90\,\%$ of the respective maximum. At $2z/L_{z}=0.5$ , the peak intensity of the ejections is slightly larger and that of the sweeps smaller than in the centre plane and the sizes of both the ejection and the sweeping motions are reduced significantly. Moreover, the constant duct centre value is larger due to the stronger lateral wall influence than in the midplane.

For the heated section, the maximum values for all four quadrants are reduced at both considered spanwise positions. In the duct centre, the $Q_{2}$ - and $Q_{3}$ -extrema drop significantly stronger than the ones for $Q_{1}$ and $Q_{4}$ . At $2z/L_{z}=0.5$ , we observe a similar reduction of $Q_{2}$ and $Q_{4}$ . Overall the intensity reduction of the ejections is more sensible to the viscosity modulation than the one of the sweeping motions. This result is in accordance with the heated TBL investigation by Lee et al. (Reference Lee, Jung, Sung and Zaki2013), who also observed an intensity reduction of all four quadrants due to the stabilising effect of the viscosity modulation. The differences between their and our results are attributed to the influence of the lateral walls on the duct centre plane profiles. Moving from the midplane to the lateral wall, we observe that the viscosity effect on the intensity drop weakens for the ejections and intensifies for the sweeping motions. The $Q_{2}$ -maximum in the midplane drops by $-17.3\,\%$ , at $2z/L_{z}=0.5$ by $-7.4\,\%$ and at $2z/L_{z}=0.75$ by $-7.4\,\%$ (plot not shown). In the centre the drop of the sweeping motion intensity is significantly lower than the one of the ejections, but increasing towards the lateral wall, whereas the ejection intensity drop decreases. The $Q_{4}$ -maximum intensity drop increases from $-1.4\,\%$ in the midplane over $-8.9\,\%$ at $2z/L_{z}=0.5$ to $-10.6\,\%$ at $2z/L_{z}=0.75$ . Moreover, we observe a heating-induced change in the size of the turbulent structures. The effect is strongest in the centre plane, where the ejection size is reduced by $-23.0\,\%$ from $l_{ejec}=0.115\,L_{y}/2$ to $l_{ejec}=0.089\,L_{y}/2$ . The sweep size increases by $11.7\,\%$ from $l_{sweep}=0.085\,L_{y}/2$ to $l_{sweep}=0.096\,L_{y}/2$ . At $2z/L_{z}=0.5$ the effect is significantly weaker, so that the ejection as well as the sweeping motion sizes are reduced only slightly.

In contrast to Salinas-Vásquez & Métais (Reference Salinas-Vásquez and Métais2002), we use liquid water as working fluid, which leads to a viscosity reduction at the heated wall, whereas the viscosity of air increases when heated. By observing the opposite effect on size and intensity of turbulent ejections as Salinas-Vásquez & Métais (Reference Salinas-Vásquez and Métais2002), we can therefore confirm that the secondary flow modulation is a viscosity effect. In our case, the viscous length scale at the centre plane decreases from $l^{+}=2.02\times 10^{-6}~\text{m}$ for the adiabatic case to $l^{+}=1.34\times 10^{-6}~\text{m}$ corresponding to a drop of $-39.1\,\%$ . As streaky structures scale with the viscous thickness, the $l^{+}$ drop leads to a reduction of their size. This is indicated in figure 13 with the $\overline{u^{\prime }u^{\prime }}$ -maximum moving closer to the heated wall. Likewise the size and the intensity of the ejections is reduced significantly, which in turn leads to the observed weakening of the secondary flow along the duct length.

5.3 Turbulent heat transfer

In this section we discuss the influence of the secondary flow on the turbulent heat transfer analysing the Nusselt number development and the turbulent Prandtl number distribution along the heated duct length.

Figure 16. Nusselt number and spanwise velocity distribution between 200 and $250~\text{mm}$ in (a) and Nusselt number distribution along the duct length at different spanwise locations in (b). The location $2z/L_{z}=0$ is shown by (——), $2z/L_{z}=0.33$ by (– – – –), $2z/L_{z}=0.75$ by ( $\cdots \cdots$ ) and $2z/L_{z}=0.9$ by (– - – - –). The respective grey-coloured lines represent the function $Nu(x,z)=-220\,x^{0.1}+c(z)$ , where $c(z)$ varies from $727$ in the centre over $739$ and $748$ to $789$ close to the lateral wall.

The secondary flow structures enhance the mixing of hot and cold fluid and consequently increase the heat transport away from the heated wall into the duct core. Figure 16 depicts the heat transfer distribution via the Nusselt number varying in the streamwise and spanwise directions due to the effect of the corner vortices. The two small vortices above the bottom wall, indicated in figure 16(a) with their respective rotating direction, produce a significant spanwise gradient in the heat transfer. By transporting hot fluid into the duct centre and together with the larger vortices pushing cold fluid into the corner, the secondary flow increases the temperature gradient in the corner area and reduces it at the duct centre. The heat transfer characterised by the Nusselt number $Nu(x,z)$ calculated following equation (3.1) varies for the considered section from ${\approx}410$ in the corner to ${\approx}350$ at the duct centre. Additionally a streamwise Nusselt number variation is present. The degradation of the heat transfer along the entire duct length can be seen in figure 16(b). The distribution is typical for a thermal entrance problem. After a short initial phase, where the highest temperature gradients are present and the strongest heat transfer occurs, the Nusselt number at all considered spanwise locations decreases following a power law of the form $Nu(x,z)=a\,x^{b}+c(z)$ , where in our case $a=-220$ , $b=0.1$ and $c(z)$ varies in the spanwise direction, see figure 16. Due to the continuous mixing of hot fluid at the heated wall with cold fluid in the duct core, the heat flux remains relatively high, however, it drops steadily in the streamwise direction. Averaging the local Nusselt number distribution in the spanwise and streamwise direction results in a mean value $Nu_{xz}$ for the investigated configuration of $Nu_{xz}=370.7$ .

The turbulent Prandtl number $Pr_{t}$ is defined as the ratio of turbulent eddy viscosity and turbulent eddy thermal diffusivity, $Pr_{t}=\unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D6FC}_{t}$ . Often a constant value is assumed for $Pr_{t}$ employing the Reynolds analogy (equal turbulent heat flux and momentum flux yielding a constant value of $Pr_{t}=1$ ) or based on experimental data, as $Pr_{t}$ depends on the molecular Prandtl number. The latter has been used, e.g. in the heated duct studies by Salinas-Vásquez & Métais (Reference Salinas-Vásquez and Métais2002) and Hébrard et al. (Reference Hébrard, Salinas-Vásquez and Métais2005), in which $Pr_{t}$ is set to $0.6$ . An extensive overview of available experimental data is given by Kays (Reference Kays1994).

Figure 17. Distribution of the angles $\unicode[STIX]{x1D711}_{M}$ and $\unicode[STIX]{x1D711}_{H}$ in the lower half of the heated duct cross-section at a streamwise location of $595~\text{mm}$ , see equation (5.2) for the angle definition. Streamwise averaging is performed over $10~\text{mm}$ and the duct symmetry is utilised. The contour lines are drawn in steps of $1^{\circ }$ .

For a TBL with the $x$ -axis marking the streamwise and the $y$ -axis the wall-normal direction, the eddy viscosity is defined as $\overline{u^{\prime }v^{\prime }}=-\unicode[STIX]{x1D708}_{t}\,(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)$ and the eddy thermal diffusivity as $\overline{T^{\prime }v^{\prime }}=-\unicode[STIX]{x1D6FC}_{t}\,(\unicode[STIX]{x2202}\overline{T}/\unicode[STIX]{x2202}y)$ . This definition has also been applied for more complex configurations, e.g. for a mixed convection set-up consisting of an asymmetrically heated channel and a heated cylinder slightly above the heated wall (Kang & Iaccarino Reference Kang and Iaccarino2010) and an symmetrically heated square duct flow (Hirota et al. Reference Hirota, Fujita, Yokosawa, Nakai and Itoh1997). For our case, however, we observed that this definition is unsuitable to analyse the cross-sectional $Pr_{t}$ -distribution due to the additional lateral walls and especially the secondary flow influence. Hence, we introduce a new turbulent Prandtl number formulation taking both the heated as well as the adiabatic lateral wall boundary layers into account at every point of the cross-section flow field. We define the vectors

(5.1a-d ) $$\begin{eqnarray}\displaystyle \boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},corr}=\left[\begin{array}{@{}c@{}}\overline{u^{\prime }v^{\prime }}\\ \overline{u^{\prime }w^{\prime }}\end{array}\right],\quad \boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},\unicode[STIX]{x1D735}}=-\!\left[\begin{array}{@{}c@{}}\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y\\ \unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}z\end{array}\right],\quad \boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},corr}=\left[\begin{array}{@{}c@{}}\overline{T^{\prime }v^{\prime }}\\ \overline{T^{\prime }w^{\prime }}\end{array}\right],\quad \boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},\unicode[STIX]{x1D735}}=-\!\left[\begin{array}{@{}c@{}}\unicode[STIX]{x2202}\overline{T}/\unicode[STIX]{x2202}y\\ \unicode[STIX]{x2202}\overline{T}/\unicode[STIX]{x2202}z\end{array}\right]. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The angles between the flux and gradient vectors are

(5.2a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{M}=\arccos \left(\frac{\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},corr}\boldsymbol{\cdot }\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},\unicode[STIX]{x1D735}}}{|\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},corr}|\boldsymbol{\cdot }|\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},\unicode[STIX]{x1D735}}|}\right),\quad \unicode[STIX]{x1D711}_{H}=\arccos \left(\frac{\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},corr}\boldsymbol{\cdot }\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},\unicode[STIX]{x1D735}}}{|\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},corr}|\boldsymbol{\cdot }|\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},\unicode[STIX]{x1D735}}|}\right). & & \displaystyle\end{eqnarray}$$

If the angles between the correlation vectors $\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},corr}$ and $\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},corr}$ , and the mean gradient vectors $\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},\unicode[STIX]{x1D735}}$ and $\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},\unicode[STIX]{x1D735}}$ are zero, $\unicode[STIX]{x1D708}_{t}$ and $\unicode[STIX]{x1D6FC}_{t}$ can be calculated using the TBL formulation. In this case, the Boussinesq turbulent viscosity hypothesis is valid. The cross-sectional distributions of the angles defined by equation (5.2) are shown in figure 17. We observe that in the regions influenced by the corner vortices both the values for $\unicode[STIX]{x1D711}_{M}$ as well as $\unicode[STIX]{x1D711}_{H}$ differ significantly from zero. Hence, turbulence models based on an isotropic eddy viscosity and diffusivity are invalid. Nevertheless, we can employ the least square method to determine the optimum eddy viscosity $\unicode[STIX]{x1D708}_{t}$ and eddy diffusivity $\unicode[STIX]{x1D6FC}_{t}$ at a specific location in the cross-section. This leads to the definitions

(5.3a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D708}_{t}=\frac{\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},corr}\boldsymbol{\cdot }\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},\unicode[STIX]{x1D735}}}{|\boldsymbol{v}_{\unicode[STIX]{x1D708}_{t},\unicode[STIX]{x1D735}}|^{2}},\quad \unicode[STIX]{x1D6FC}_{t}=\frac{\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},corr}\boldsymbol{\cdot }\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},\unicode[STIX]{x1D735}}}{|\boldsymbol{v}_{\unicode[STIX]{x1D6FC}_{t},\unicode[STIX]{x1D735}}|^{2}}. & & \displaystyle\end{eqnarray}$$

Figure 18. Turbulent Prandtl number distribution in the lower half of the heated duct cross-section at streamwise locations of (a) $x=25~\text{mm}$ , (b) $x=50~\text{mm}$ , (c) $x=100~\text{mm}$ and (d) $x=595~\text{mm}$ . Streamwise averaging is performed over $10~\text{mm}$ and the duct symmetry is utilised. $Pr_{t}$ is calculated where $\overline{T}-T_{b}>0.05~\text{K}$ and the contour lines are drawn in steps of $0.1$ .

Figure 18 shows the development of the cross-sectional turbulent Prandtl number distribution along the heated duct for the regions where the local heating surpasses a threshold value of $\overline{T}-T_{b}=0.05~\text{K}$ . The value range of $Pr_{t}$ is between approximately $0$ and $1.3$ in figure 18, which is in good agreement with data available in the literature, e.g. with Kang & Iaccarino (Reference Kang and Iaccarino2010) using liquid water at a lower temperature, but at a similar temperature difference $T_{w}-T_{b}$ .

Now we focus on the $Pr_{t}$ -distribution in the duct end cross-section, figure 18(d). In the heated wall centre directly at the wall $Pr_{t}$ is ${\approx}0.89$ . Above the heated wall, we observe a dome-shaped region of enhanced turbulent Prandtl number coinciding with the influence region of the two smaller corner vortices. Hence, we attribute this increase to the mixing by the secondary flow. A local maximum is located in the centre of this region, where the interaction of the small corner vortices leads to a strong upwards flow, with $Pr_{t}$ reaching ${\approx}1.06$ . This maximum is located slightly below the maximum $\overline{v}$ -velocity. The dome-shaped area is bordered by the interaction zone of the small and the respective large corner vortex both pushing fluid into the duct corner. There $Pr_{t}$ drops to values between ${\approx}0.9$ and ${\approx}0.95$ . The mixing of the large corner vortices also leads to an area of enhanced turbulent Prandtl number. However, due to the proximity of the adiabatic walls the $Pr_{t}$ -levels are lower than in the small vortex influence zone. In the core of the large vortex, $Pr_{t}$ reaches ${\approx}0.84$ versus $Pr_{t}\approx 0.94$ in the core of the small vortex. Along the lateral side walls $\unicode[STIX]{x1D6FC}_{t}$ is two orders of magnitude larger than $\unicode[STIX]{x1D708}_{t}$ leading to a turbulent Prandtl number of almost zero, which is a consequence of the adiabatic wall boundary condition. Depending on the $y$ -location, $Pr_{t}$ increases steadily in the spanwise direction towards the $z=0$ centre line until either the small or the large vortex influence region is reached, or for $2y/Ly>0.7$ a narrow maximum region located along the centre line. There, the global maximum is reached with $Pr_{t}=1.3$ located slightly above the minimum $\overline{v}$ -velocity in the interaction zone of the large vortices.

Along the duct length, i.e. from figure 18(a–d), the turbulent Prandtl number levels in the vicinity of the heated wall drop continuously. The maximum values within the dome-shaped region decrease from $1.28$ at $x=25~\text{mm}$ over $1.16$ at $x=50~\text{mm}$ and $1.11$ at $x=100~\text{mm}$ to $1.06$ at $x=595~\text{mm}$ . Also the maximum location moves slightly closer towards the heated wall. Close to the lateral wall, the variations in the streamwise direction are smaller compared to those in the centre as this location is strongly influenced by the adiabatic wall boundary layer.

Based on the $Pr_{t}$ -distributions we conclude that the assumption of a constant value for the turbulent Prandtl number for asymmetrically heated duct flows is invalid.

5.4 Turbulence anisotropy

In the following we will further analyse the influence of wall heating on turbulence anisotropy as the secondary flow is a consequence of the Reynolds stress anisotropy. For this investigation we apply the anisotropy-invariant map (AIM) and the barycentric anisotropy map (BAM), see Emory & Iaccarino (Reference Emory and Iaccarino2014) for an overview.

Figure 19. Reynolds stress anisotropy-invariant map for the adiabatic duct (——) and the heated duct (——, orange) evaluated at (a,b) $2z/L_{z}=0$ and (c,d) $2z/L_{z}=0.95$ along the $y$ -line from the heated wall to the duct centre, and limiting states defined by the Lumley triangle. The wall location is represented by ● and ● (orange) respectively.

The Reynolds stress anisotropy tensor is defined as $\unicode[STIX]{x1D622}_{ij}=\overline{u_{i}^{\prime }u_{j}^{\prime }}/(2k)-\unicode[STIX]{x1D6FF}_{ij}/3$ with the turbulent kinetic energy being $k=\overline{u_{n}^{\prime }u_{n}^{\prime }}/2$ and $\unicode[STIX]{x1D6FF}_{ij}$ the Kronecker delta. Three limit states are defined forming the vertices of the so-called Lumley triangle: the state of 1-component turbulence, for which turbulent fluctuations in one direction are dominant, the state of 2-component turbulence, for which turbulent fluctuations in two directions are much higher than in the third direction, and the state of 3-component isotropic turbulence, where fluctuations in all directions are equally high (Lumley Reference Lumley1978; Choi & Lumley Reference Choi and Lumley2001). Any anisotropy state $\unicode[STIX]{x1D622}_{ij}$ can be described as convex combination of these three limiting states, i.e. every realisable state has to reside within the borders of the Lumley triangle. The construction of the AIM is based on the anisotropy tensor eigenvalues $\unicode[STIX]{x1D706}_{i}$ . The two axes are the second and third invariants of the anisotropy tensor with $I_{2}=\unicode[STIX]{x1D622}_{ij}\unicode[STIX]{x1D622}_{ji}/2=\unicode[STIX]{x1D706}_{1}^{2}+\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}+\unicode[STIX]{x1D706}_{2}^{2}$ and $I_{3}=\unicode[STIX]{x1D622}_{ij}\unicode[STIX]{x1D622}_{jn}\unicode[STIX]{x1D622}_{ni}/3=-\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}(\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D706}_{2})$ respectively. The location of a certain turbulence state in the AIM then describes the shape of the Reynolds stress tensor, see Simonsen & Krogstad (Reference Simonsen and Krogstad2005).

Figure 19 presents the AIM evaluated at two spanwise locations at $z=0$ in the duct midplane and at $2z/L_{z}=0.95$ close to the lateral wall. The AIM describes the evolution of the turbulence anisotropy along the $y$ -direction starting at the heated wall and ending at the duct centre line for the heated and the adiabatic case. For the duct midplane in figure 19(a,b) the anisotropy development resembles that of a plane channel or boundary layer flow, cf. Banerjee et al. (Reference Banerjee, Krahl, Durst and Zenger2007) and Pasquariello et al. (Reference Pasquariello, Grilli, Hickel and Adams2014). The trajectory starts at the 2-component limit edge and moves upwards in direction of the 1-component limit until the maximum anisotropy is reached in the buffer layer at $y^{+}=6.4$ for the adiabatic case. The trajectory then turns and follows a path parallel to the axisymmetric expansion limit until a kink in the log layer. Finally a state close to isotropic turbulence is reached at the duct symmetry line. For the second location at $2z/L_{z}=0.95$ , figure 19(c,d) shows an overall similar behaviour, but also the significant influence of the lateral wall. The trajectory starts closer to the 2-component axisymmetric limit and moves upwards to the 1-component limit. In contrast to the duct centre trajectory, the off-centre trajectory follows a steeper path than the axisymmetric expansion curve after the turning point, no kink exists in the log layer and in the duct centre the state of isotropic turbulence is not reached.

Figure 20. Distribution of (a) diagonal components and (b) off-diagonal component $\overline{a_{xy}}$ of the Reynolds stress anisotropy tensor in the duct midplane along the heated wall-normal direction for the adiabatic (——) and the heated duct (——, orange).

The effect of wall heating on turbulence anisotropy is overall relatively small, restricted to the vicinity of the heated wall and more pronounced close to the lateral wall than in the duct midplane. The start points of both trajectories lie at the 2-component limit and are shifted towards the 1-component limit compared to the adiabatic case. Within the buffer layer the heated duct trajectories follow a path slightly closer to the 2-component limit. The turning point of maximum anisotropy lies still within the buffer layer, however, it is located closer to the upper right corner of the Lumley triangle. Moreover, in figure 19(a,b) a slight change of the kink in the log layer area is visible.

Figure 21. Barycentric anisotropy map illustrating regions of 1-, 2- and 3-component turbulence with (a) lower left quadrant of the adiabatic duct, (b) zoom into the lower left corner of the adiabatic duct and (c) the same detail for the heated duct (opposite corner shown). The isolines denote a constant 3-component turbulence fraction.

As the changes due to the heating are limited to the near-wall area, we investigate the anisotropy tensor components there in detail and compare our findings qualitatively with a recent study by Patel, Boersma & Pecnik (Reference Patel, Boersma and Pecnik2016), who analysed the influence of viscosity gradients on the near-wall turbulence anisotropy of a channel flow. In figure 20 we observe the same trends as Patel et al. (Reference Patel, Boersma and Pecnik2016): the streamwise component of the Reynolds stress anisotropy tensor $\overline{\unicode[STIX]{x1D622}_{xx}}$ increases, and the spanwise component $\overline{a_{zz}}$ decreases with heating applied. Patel et al. (Reference Patel, Boersma and Pecnik2016) reported, that the wall-normal component $\overline{a_{yy}}$ remains unaffected, however for our case a decrease starting within the buffer layer is visible. The normalised turbulent shear stress $\overline{a_{xy}}$ in figure 20(b) increases, indicating an augmented momentum transfer. The off-diagonal components $\overline{a_{yz}}$ and $\overline{a_{xz}}$ are negligible in the duct midplane.

In contrast to the classical AIM, the so-called barycentric map proposed by Banerjee et al. (Reference Banerjee, Krahl, Durst and Zenger2007) provides a more intuitive tool to analyse the turbulence anisotropy. The construction again is based on the eigenvalues $\unicode[STIX]{x1D706}_{i}$ of the Reynolds stress anisotropy tensor and relies on the fact, that any realisable turbulence state can be represented as a combination of the three limiting states of 1-, 2- and 3-component turbulence. The limiting states are now defined as the corners of an equilateral triangle with $\boldsymbol{x}_{1c}=(1,0)$ , $\boldsymbol{x}_{2c}=(0,0)$ and $\boldsymbol{x}_{3c}=(1/2,\sqrt{3}/2)$ . The coordinates of a certain turbulent state are then computed as $\boldsymbol{x}=C_{1c}\boldsymbol{x}_{1c}+C_{2c}\boldsymbol{x}_{2c}+C_{3c}\boldsymbol{x}_{3c}$ . In contrast to the AIM, the BAM is a linear anisotropy-invariant map, where the coordinates depend linearly on the eigenvalues with $C_{1c}=\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{2}$ , $C_{2c}=2(\unicode[STIX]{x1D706}_{2}-\unicode[STIX]{x1D706}_{3})$ and $C_{3c}=3\unicode[STIX]{x1D706}_{3}+1$ ensuring $\sum C_{ic}=1$ . For visualisation the coefficient vector $C_{ic}$ is mapped to the RGB triplet. Red corresponds to 1-component, green to 2-component and blue to 3-component turbulence.

Figure 21(a) shows the application of the barycentric map combined with the RGB colouring for the lower left quadrant of the adiabatic duct. In the duct core the state of isotropic turbulence is almost reached, and in the vicinity of the walls a mixture of 2- and 1-component turbulence is found. As noted previously using the AIM, the 2-component turbulence transitions to 1-component turbulence in the buffer layer. In the duct corner 1-component turbulence is dominant as fluctuations perpendicular to the two walls are suppressed. The influence of the corner vortices on the turbulence anisotropy distribution is clearly visible in the $C_{3c}$ -isolines with the mean secondary flow transporting fluid from the isotropic core region into the duct corner following approximately the corner bisecting line.

Figure 21(b,c) show the influence of the wall heating for the duct corner region. In the vicinity of the heatable wall, we observe for the heated case a reduction of 2-component turbulence in favour of 1- and 3-component turbulence. At the adiabatic upper wall (not shown) the turbulence anisotropy remains identical for both cases. The $C_{3c}$ -isolines illustrate the anisotropy reduction in this region as an anisotropy measure can be defined as $C_{ani}=1-C_{3c}$ (Banerjee et al. Reference Banerjee, Krahl, Durst and Zenger2007). As $C_{3c}$ increases in the near-wall region, the flow becomes more isotropic leading to a weaker production term for streamwise vorticity and in turn to the observed weakening of the secondary flow over the duct length.

5.5 Length scales of turbulent structures

As discussed in § 5.2, the size and strength of the ejections and streaky structures reduce with increasing temperature and the associated viscosity reduction. Salinas-Vásquez & Métais (Reference Salinas-Vásquez and Métais2002) have shown, that the ejections and streaks as well as turbulent length scales grow with increasing viscosity. Hence, we expect in our case a reduction of the turbulent length scales when heating is applied.

Figure 22. Streamwise velocity fluctuations within the heatable wall-parallel plane at $2y/L_{y}=0.9975$ for the adiabatic and the heated duct.

Figure 23. Spanwise velocity fluctuations within the heatable wall-parallel plane at $2y/L_{y}=0.9975$ for the adiabatic and the heated duct.

As mainly the area close to the lower wall is affected by the heating, we first analyse the turbulent length scales qualitatively on a plane parallel to and directly above the heatable wall at $2y/L_{y}=0.9975$ . This location corresponds to $y^{+}=16.1$ for the adiabatic duct and to $y^{+}=24.2$ for the end section of the heated duct, where $y^{+}$ is evaluated at $2z/L_{z}=0$ . Figure 22 depicts the streamwise velocity fluctuations $u^{\prime }/u_{b}$ and figure 23 the spanwise velocity fluctuations $w^{\prime }/u_{b}$ within this plane.

In figure 22, the typical streaky structures of the TBL with the darker regions of comparatively low-speed fluid surrounded by lighter regions of high-speed fluid are visible. Due to the restricting influence, the streaks close to the lateral wall, i.e. in the duct corner, are thinner and shorter compared to those in the centre. As the streak size scales with viscous thickness and based on previous studies, we expect a reduction of streak size as well as turbulent length scales in the heated duct (Salinas-Vásquez & Métais Reference Salinas-Vásquez and Métais2002; Zonta et al. Reference Zonta, Marchioli and Soldati2012; Lee et al. Reference Lee, Jung, Sung and Zaki2013). However, comparing the two snapshots, no qualitative difference in turbulent structures is visible. We make the same observation for the spanwise fluctuations in figure 23, which we attribute to the moderate heating.

Figure 24. Longitudinal autocorrelations of streamwise velocity $R_{uu}^{x}$ at different locations marked in the sketches on the respective right-hand side. In (a) the adiabatic duct results are shown and in (b) the change due to the viscosity modulation. Heated duct correlations are coloured orange and the ones at off-centre locations blue. The line $R_{uu}^{x}=1/\text{e}^{2}$ (——, grey) is used to determine $r_{lim,x}$ . For figure legend and line parameters see table 4.

The use of autocorrelations allows for a quantitative investigation of turbulent length scales and their respective changes. To reduce the number of samples required for a converged result, we assume local homogeneity in the streamwise direction for the definition of the longitudinal autocorrelation function of the $u$ -velocity $R_{uu}^{x}$ (Pirozzoli, Grasso & Gatski Reference Pirozzoli, Grasso and Gatski2004). The sampling rate is every $25$ time steps over a sampling period of $20$ FTT with respect to $L_{x,heat}$ , leading to a total of ${\approx}56.7\times 10^{3}$ samples. The integral length scale is generally defined as $L_{ii}^{j}=\int _{0}^{\infty }R_{ii}^{j}(r_{j})\,\text{d}r_{j}$ . For the upper bound of integration, a finite value $r_{lim,j}$ has to be specified. O’Neill et al. (Reference O’Neill, Nicolaides, Honnery, Soria, Behnia, Lin and McBain2004) proposed the integration up to the first zero crossing of the correlation or until the correlation function has fallen to a value of $1/\text{e}$ . As the latter cuts off a large part of the correlation function, it tends to underestimate the turbulent length scale. Hence we use a limit of $1/\text{e}^{2}$ , so that $r_{lim,j}$ is defined as $r_{lim,j}=r_{j}(R_{ii}^{j}=1/\text{e}^{2})$ .

Figure 24 shows the longitudinal autocorrelation functions $R_{uu}^{x}$ at different ( $y/z$ )-positions for the heated and adiabatic duct. All positions are located in the vicinity of the heatable wall, i.e. in the area influenced by the small corner vortices except $R_{uu,7}^{x}$ , which lies in the area influenced by the larger corner vortices, see table 4 for line parameters and turbulent length scales. Figure 24(a) allows for two observations: first the trivial result, that with increasing distance from the heatable wall the turbulent structures grow larger with diminishing growth rate for each $R_{uu}^{x}$ -curve, except for $R_{uu,7}^{x}$ at $2y/L_{y}\approx -0.8$ , where a reduction in turbulent length scale is apparent. Second, the structures become significantly smaller when moving in the direction of the duct corner due to the increasing influence of the lateral wall. Comparing the turbulent length scales in the plane shown in figure 22 for the duct centre position $R_{uu,1}^{x}$ with the two off-centre positions $R_{uu,1a}^{x}$ and $R_{uu,1b}^{x}$ , we measure a reduction of $-15.0\,\%$ and $-34.7\,\%$ respectively. Further away from the lower wall the $2y/L_{y}=\text{const.}$ correlation triplets follow the same trend.

In figure 24(b) we investigate the influence of the viscosity modulation on turbulent length scales $L_{uu}^{x}$ . We observe the expected heating-induced shortening for all considered locations. As the temperature increase is highest directly at the lower wall, we observe a strong reduction there with a maximum value of $-10.2\,\%$ for $y^{+}=16.1$ . Moving upwards along the duct midplane, the reduction in the small corner vortex region becomes increasingly weak, see table 4. For instance at $2y/L_{y}\approx -0.9$ , the shortening has fallen to just $-1.6\,\%$ . However, the viscosity effect on turbulent length scales does not only affect the immediate vicinity of the heated wall, but also the duct centre due to the modified secondary flow transport. This becomes apparent by comparison of $R_{uu,6}^{x}$ and $R_{uu,7}^{x}$ , where the first is in the influence region of the small corner vortices and the latter in that of the large corner vortices. At $2y/L_{y}\approx -0.9$ , the viscosity drop is approximately twice as much as at $2y/L_{y}\approx -0.8$ , but the length scale reduction increases from $-1.6\,\%$ for $R_{uu,6}^{x}$ to even $-14.8\,\%$ for $R_{uu,7}^{x}$ . Thus, the reduction of turbulent length scales is not a mere function of temperature increase, but depends on the specific location as the viscosity modulation affects turbulent transport as well as the transport by secondary flows and both influence $L_{uu}^{x}$ . The argument is supported by comparing the off-centre locations, exemplarily $R_{uu,1}^{x}$ in the centre with $R_{uu,1a}^{x}$ and $R_{uu,1b}^{x}$ . The relative shortening in the centre is with $-10.2\,\%$ significantly higher than the $-4.7\,\%$ and $-4.3\,\%$ . The correlations are influenced by the small CW vortex above the heatable wall, which leads to a non-uniform viscosity drop in the spanwise direction by transporting fluid from the duct corner into the wall centre. This motion produces an increased temperature in the midplane of $19.6~\text{K}$ versus $17.9~\text{K}$ , respectively $17.6~\text{K}$ . However, the viscosity drop differs too little in the spanwise direction as to explain the large difference in length scale shortening and thus it is a secondary flow effect.

Table 4. Parameters for the longitudinal autocorrelations of streamwise velocity shown in figure 24. Listed are ( $y/z$ )-positions, dimensionless height above the heatable wall, temperature increase, viscosity drop, integral length scales and their relative changes due to the viscosity modulation. The lengths $L_{uu}^{x}$ are normalised by a factor of ( $100\,d_{h}$ ).

Table 5. Parameters for the autocorrelations of streamwise and spanwise velocity in the spanwise direction shown in figure 25. $R_{ii,k,lw}^{z}$ denote the curves taken at the lateral wall and $R_{ii,k}^{z}$ the ones taken further away from it. Listed are the position of the correlations, temperature increase, viscosity drop, integral length scales and their relative changes due to the heating. All $L_{ii}^{z}$ are normalised by a factor of ( $100\,d_{h}$ ).

Figure 25. Transverse autocorrelations of streamwise velocity $R_{uu}^{z}$ in (a,b) and longitudinal autocorrelations of spanwise velocity $R_{ww}^{z}$ in (c,d) at different locations. In (a,c) the correlations at the lateral wall are shown and in (b,d) the ones at $2z/L_{z}=0.74$ . Results for the adiabatic duct are coloured black and for the heated duct orange. The line $R_{ii}^{z}=1/\text{e}^{2}$ used to determine $r_{lim,z}$ is marked by (——, grey). For figure legend and line parameters see table 5.

For the turbulent length scales in the spanwise direction, specifically for the transverse autocorrelations $R_{uu}^{z}$ and the longitudinal autocorrelations $R_{ww}^{z}$ , data are gathered along a $z$ -line with the same sampling parameters as before and evaluated following Pope (Reference Pope2000). Additionally, a quadrant averaging is performed to utilise the symmetry with respect to the $y$ -axis. The parameters for the correlations are listed in table 5. We first discuss the results for the adiabatic case and subsequently analyse the changes due to heat transfer at two spanwise locations. The first is directly at the lateral wall and the second further away from it at $2z/L_{z}=0.74$ . The latter position is chosen instead of the duct centre due to the higher $z$ -direction grid resolution.

In figure 25(a,b) transversal correlations of streamwise velocity in the spanwise direction $R_{uu}^{z}$ are depicted for different locations. Moving away from the lower wall, the structures at the lateral wall $L_{uu,k,lw}^{z}$ as well as the larger $L_{uu,k}^{z}$ at $2z/L_{z}=0.74$ increase. However, the $L_{uu,k,lw}^{z}$ become only slightly larger due to the strong influence of the lateral wall, see table 5. The longitudinal correlations $R_{ww}^{z}$ are shown in figure 25(c,d). The $L_{ww,k}^{z}$ are consistently larger than the $L_{uu,k}^{z}$ , whereas close to the lateral wall, the $L_{ww,k,lw}^{z}$ are shorter than the $L_{uu,k,lw}^{z}$ as the spanwise fluctuations are blocked by the wall. The integral length scales may serve to characterise quantitatively the structures visible in the velocity fluctuation plots at the beginning of this section, in figures 22 and 23. Qualitatively, the streaky structures are much more elongated than the structures visible in the $w^{\prime }$ -plot. Close to the heatable wall, corresponding to the plane shown in figures 22 and 23, and at $2z/L_{z}=0.74$ , the ratios of integral length scales are $L_{uu}^{x}/L_{uu}^{z}\approx 7.6$ and $L_{ww}^{x}/L_{ww}^{z}\approx 2.8$ , which supports the qualitative observation of the more elongated streaky structures.

When heating is applied, we observe close to the lateral wall a significant reduction of $L_{uu,k,lw}^{z}$ by up to $-31.1\,\%$ and of $L_{ww,k,lw}^{z}$ by up to $-10.6\,\%$ . At $2z/L_{z}=0.74$ however, no definite trend is visible. The $L_{uu,k}^{z}$ at all three considered positions are slightly shortened and the $L_{ww,k}^{z}$ increase slightly, showing again that the length scale change is not a mere function of the local viscosity drop. Additionally, the integral length scale ratios of the turbulent structures change. At $2y/L_{y}=-0.9975$ and $2z/L_{z}=0.74$ , the ratio of streamwise to spanwise length scale reduces from $L_{uu}^{x}/L_{uu}^{z}\approx 7.6$ to ${\approx}7.4$ for the streamwise velocity and that for the spanwise velocity from $L_{ww}^{x}/L_{ww}^{z}\approx 2.8$ to ${\approx}2.6$ .