3.1. Mean-flow topology

Figure 5.

Visualisation of the mean velocity field. (a) Mean radial velocity $\langle u_r \rangle /U_b$ ; (b) mean axial velocity $\langle u_z \rangle /U_b$ . Panels (c) and (d) depict streamlines of the mean velocity field. Streamlines are colour coded to identify regions of positive (orange) and negative (blue) values of the mean radial velocity $\langle u_r \rangle$ , panel (c), and of the mean axial velocity $\langle u_z \rangle$ , panel (d).

The mean-flow field is visualised in figure 5 through colour maps of the mean radial and axial velocity components, respectively, $\langle u_r \rangle$ and $\langle u_z \rangle$ , together with their streamlines. Panels (a) and (b) in the figure highlight the free-jet region, which is characterised by a strong negative axial velocity, and the wall jet region, in which the mean flow develops radially along the impingement plate. Structurally, the mean flow is characterised by a large-scale recirculating cell which can be observed in the mean streamline patterns reported in figures 5(c) and 5(d). This recirculating structure has its centre at $r\approx 5.4D$ and $z\approx D$ and originates from the continuous momentum supply through the wall jet and the presence of the no-slip boundary condition on the confinement and impingement plates. In figure 6(a), the presence and the centre of such a recirculating cell is confirmed by comparing streamlines of the mean flow with particle image velocimetry measurements from Chen et al. (Reference Chen, Yang, Huang, Zhou, Zhang and Wang2024). A second recirculating zone is observed close to the open boundary of the computational domain. Due to its presence, inflow and outflow occur across the bottom and top halves of the open boundary, respectively. Nevertheless, the velocity magnitude in this region only reaches a few per cent, $\lesssim 5\,\%$ , of the mean bulk velocity in the free jet.

Figure 6.

(a) Streamlines of the mean velocity field. Solid black lines, present DNS; red dots, data from Chen et al. (Reference Chen, Yang, Huang, Zhou, Zhang and Wang2024). (b) Mean radial velocity and temperature profiles at different radial locations. , $\langle u_r \rangle /U_b$ ; , $\langle \theta \rangle$ ; symbols indicate the maxima and minima of, respectively, the mean radial velocity and temperature profiles. Vertical black dashed lines indicate the radial location at which the profiles are sampled. (c) Boundary-layer thicknesses. , $z_m/D$ ; , $z_{\theta m}/D$ ; , $z_{\varTheta m}/D$ . (d) Radial distribution of the maxima of the mean radial velocity profiles. , present DNS; red squares, Chen et al. (Reference Chen, Yang, Huang, Zhou, Zhang and Wang2024); , 1.12/(r/D).

Figure 6. Long description

Panel A: A contour plot showing streamlines of the mean velocity field. The x-axis is labeled r/D and ranges from 0 to 8. The y-axis is labeled z/D and ranges from 0 to 2. Solid black lines represent present DNS data, and red dots represent data from Chen et al. (2024). Panel B: A line graph showing mean radial velocity and temperature profiles at different radial locations. The x-axis is labeled r/D and ranges from 0 to 8. The y-axis is labeled z/D and ranges from 10^-2 to 10^0. Different colored lines represent different profiles, with symbols indicating the maxima and minima. Vertical black dashed lines indicate the radial location at which the profiles are sampled. Panel C: A line graph showing boundary-layer thicknesses. The x-axis is labeled r/D and ranges from 0 to 8. The y-axis is labeled z/D and ranges from 0 to 0.2. Different colored lines represent different boundary-layer thicknesses. Panel D: A line graph showing the radial distribution of the maxima of the mean radial velocity profiles. The x-axis is labeled r/D and ranges from 1 to 8. The y-axis is labeled um/Ub and ranges from 10^-2 to 10^0. Black lines represent present DNS data, red squares represent data from Chen et al. (2024), and a dashed line represents the equation 1.12/(r/D).

The development of the mean flow along the radial direction is further analysed in figure 6(b), which shows mean radial velocity and temperature profiles at several radial stations. The mean profiles represent well the characteristic shape of a wall jet that is generated by the injection of high-momentum fluid through a narrow gap between a solid wall and a quiescent fluid. Such a momentum inflow determines a two-layered flow in which the maximum velocity is attained in a region of the flow bounded by an inner wall-boundary layer and an outer free-shear flow. In the present case, it must be noted that the outer region of the flow is not characterised by a perfectly quiescent fluid, but rather, by a mild backflow which is induced, in the upper part of the domain, by the primary recirculating cell that was observed in the analysis of figure 5(c, d). Due to the recirculating flow, a wall jet also develops on the confinement plate, whose direction opposes that developing along the impingement plate, and with comparatively little streamwise momentum.

Due to the different boundary values that $\langle u_r \rangle$ and $\langle \theta \rangle$ have on the impingement plate, their mean profiles differ significantly from each other everywhere in the domain. In the wall jet along the bottom plate, high-momentum cold fluid, coming from the jet, determines that the mean temperature profiles have their minima over the same region where the mean radial velocity profiles are maximal. In the upper part of the domain, the recirculating flow carries warm fluid to flow over the confinement plate. Figure 6(b) shows that, in the bottom wall jet, the mean radial velocity maxima, $u_m$ and the mean temperature minima, $\theta _m$ , occur at significantly different wall-normal heights, $z_m$ and $z_{\theta m}$ , respectively. In particular, the difference $z_{\theta m}-z_m \gt 0$ increases for increasing the radial distance from the jet axis, despite the fact that, for a unit Prandtl number, the thicknesses of the velocity and temperature boundary layers are expected to be approximately equal.

For a negligibly small mean radial pressure gradient, the radial convection of momentum and of internal energy over a distance $r$ are balanced by the wall-normal diffusion of, respectively, radial momentum and internal energy. Due to mass conservation, the outer velocity scale, $u_m$ , is expected to vary radially as $u_m \sim u_0/r$ , for a constant $u_0$ , whereas such a constraint does not affect the outer temperature scale, $\theta _m$ . With these assumptions, an order of magnitude analysis of (2.1), (2.2) and (2.3) leads to the following estimates for the velocity and temperature boundary-layer thicknesses:

(3.1) \begin{align} \frac {z_m}{r} \sim \left ( \frac {\nu }{u_0} \right )^{\tfrac {1}{2}}, \quad \frac {z_{\theta m}}{r} \sim \left ( \frac {\alpha }{u_0} \right )^{\tfrac {1}{2}}. \end{align}

According to these estimates, the two boundary-layer thicknesses should grow linearly with the radial distance and, provided that the fluid has similar momentum and thermal diffusivities, $z_m$ and $z_{\theta m}$ should be approximately equal. Figure 6(c) displays the radial distributions of $z_m/r$ and $z_{\theta m}/r$ for $r/D \le 5.83$ . At $r=5.83D$ , the boundary layer over the impingement plate separates and backflow is observed in the near-wall region at larger radial distances. It can be seen that, for $3D\lesssim r \lesssim 5D$ , $z_m/r$ and $z_{\theta m}/r$ remain approximately constant with the radial distance, in accordance with the estimates (3.1). At smaller radial distances, the strong radial pressure gradient that accelerates the flow from the impingement region invalidates the assumption of a negligibly small mean radial pressure gradient for the estimation of the boundary-layer thickness. A similar scenario occurs near the separation point, where the flow develops an adverse pressure gradient.

Yet, figure 6(c) highlights a difference between the thicknesses of the velocity and temperature boundary layers which cannot be justified based on the assumptions that led to the estimates (3.1). This difference can be attributed to the dissimilar boundary values of the velocity and temperature fields on the impingement and confinement plates. The different temperature boundary values on the bottom and top plates drive a heat exchange, via molecular diffusion, which does not have an equivalent for the momentum transfer. This observation suggests decomposing the temperature field into $\theta = \theta _B + \varTheta$ ; where $\theta _B$ is the base steady-state temperature distribution realised when there is no flow. More precisely, it is the solution of ${\nabla} ^2\theta _{B} = 0$ with $\theta _B=1$ at the impingement plate and $\theta _B=0$ at the confinement plate. In particular, it is given by $\theta _B= 1 - z/(2D)$ . On the other hand, $\varTheta$ represents the local deviation of $\theta$ from the base solution, $\theta _B$ , and thus represents a measure of how strongly the flow affects the temperature field. Substituting $\theta = \theta _B + \varTheta$ into (2.3) gives

(3.2) \begin{align} \frac {\partial \varTheta }{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }(\boldsymbol{u} \varTheta ) = \alpha{\nabla} ^2 \varTheta + \frac {u_z}{2D}, \end{align}

with homogeneous boundary conditions for $\varTheta$ on the bottom and top plates. Equation (3.2) shows that local deviations from the base temperature distribution, $\theta _B$ , are driven by the wall-normal velocity $u_z$ .

Along the impingement plate, the boundary-layer thickness for $\langle \varTheta \rangle$ can be defined as the smallest wall-normal height, $z_{\varTheta m}$ , at which $\langle \varTheta \rangle$ has a stationary point (i.e. it has a minimum or a maximum). The radial distribution of $z_{\varTheta m}/r$ , reported in figure 6(c), displays a mild decrease in the range $3D\lesssim r \lesssim 5D$ , and shows close similarity, in magnitude, with $z_m/r$ at every radial location.

The estimates (3.1) have been obtained by assuming a radial decrease of the outer velocity as $u_m\sim u_0/r$ . Figure 6(d) shows that this assumption is approximately verified in the range $3D\lesssim r \lesssim 5D$ , where the radial distribution of $u_m$ aligns closely with $1.12/r$ . In the figure, the experimental data of Chen et al. (Reference Chen, Yang, Huang, Zhou, Zhang and Wang2024) are also reported and show good agreement with the present observations.

Figure 7.

Mean momentum and temperature budgets at different radial locations. Panels show (a) $r/D=0.25$ ; (b) $r/D=1$ ; (c) $r/D=2$ ; (d) $r/D=3$ . , $NL_r/(U_b^2/D)$ ; , $G_r/(U_b^2/D)$ ; , $V_r/(U_b^2/D)$ ; , $NL_z/(U_b^2/D)$ ; , $G_z/(U_b^2/D)$ ; , $NL_\theta /(U_b/D)$ ; , $V_\theta /(U_b/D)$ .

3.2. Mean momentum and temperature budgets

The flow development along the impingement plate is further analysed by considering the following balances for the mean radial and axial momentum components, and for the scalar temperature (a comma subscript is used to denote partial differentiation):

(3.3) \begin{align} 0 & \approx \underbrace {-\left ( \frac {1}{r}\left\langle r u_r^2 \right\rangle _{,_r} + \langle u_ru_z \rangle _{,_z} \right )}_{NL_r} \underbrace {- \frac {1}{\rho }\langle p \rangle _{,_r}}_{G_r} + \underbrace { \nu \langle u_r \rangle _{,_{zz}}}_{V_r}, \end{align}

(3.4) \begin{align} 0 & \approx \underbrace {- \left ( \frac {1}{r}\left\langle r u_r u_z \right\rangle _{,_r} + \left\langle u_z^2 \right\rangle _{,_z} \right )}_{NL_z} \underbrace {- \frac {1}{\rho }\langle p \rangle _{,_z}}_{G_z}, \end{align}

(3.5) \begin{align} 0 & \approx \underbrace {-\left ( \frac {1}{r} \langle r u_r\theta \rangle _{,_r} + \langle u_z\theta \rangle _{,_z} \right )}_{NL_\theta } + \underbrace { \alpha \langle \theta \rangle _{,_{zz}} }_{V_\theta }. \\[10pt] \nonumber \end{align}

Equation (3.3) is the Prandtl boundary-layer equation for the streamwise velocity component in a radial boundary layer. Unlike a canonical boundary layer, in the present case, equation (3.4) is necessary to take into account that a wall-normal mean pressure gradient can exist across the boundary layer that develops along the impingement plate. Equation (3.5) states the balance between the convection and the wall-normal molecular diffusion of the temperature field.

Figure 7 displays each term of equations (3.3), (3.4) and (3.5) as functions of the wall-normal distance at different radial stations. In each plot of the figure, the abscissa is scaled using the boundary-layer thickness $z_m$ to ease the distinction between inner and outer layers of the wall jet. At $r=0.25D$ (figure 7 a), well within the stagnation region of the flow, a strong, favourable radial pressure gradient, $G_r$ , exists which stays constant up to large distances from the wall, approximately up to $z=10 \, z_m$ . For $z\lt z_m$ , the radial pressure gradient is balanced by the combined action of the advection, $NL_r$ , and the wall-normal molecular diffusion of radial momentum, $V_r$ , even though, for $z\lesssim 0.3z_m$ , the advection term becomes negligible. Conversely, in the outer region of the flow, for $z\gt z_m$ , $V_r$ becomes quickly negligible and the radial pressure gradient is balanced by the radial momentum advection alone. This scenario is observed further downstream of the stagnation region, at $r=D$ (panel b in the figure), where the radial pressure gradient still makes a significant contribution to the momentum balance. However, with increasing radial distance from the jet axis (panels c and d in the figure display the budgets at $r=2D$ and $r=3D$ , respectively), the action of the radial pressure gradient becomes quickly negligible compared with the advection and molecular diffusion of radial momentum within the boundary layer. Hence, sufficiently downstream of the stagnation region, for $r\gtrsim 1.5D$ , the flow develops radially similarly to a zero-pressure-gradient boundary layer.

However, very differently from a canonical boundary layer, across the boundary-layer thickness, the pressure does not remain constant. In fact, the mean wall-normal pressure gradient, $G_z$ , is, in magnitude, comparable to the terms appearing in the mean momentum equation (3.3) at every radial location, as shown in figure 7. Within the stagnation region, figure 7(a), the flow coming from the free-jet core experiences a strongly adverse pressure gradient which deflects the flow and aligns it parallel to the impingement wall. This process takes place essentially away from the boundary layer, so that the intrusion of the mean pressure gradient is only marginal for $z\lt z_m$ . Nevertheless, downstream of the stagnation region (panels a, b and c in the figure), the penetration of the mean wall-normal pressure gradient into the boundary layer becomes apparent. The rise of a wall-normal pressure gradient within the wall jet region is a direct consequence of the recirculating flow structure, which was shown with the mean streamlines in panels (c) and (d) of figure 5. The mean, wall-normal-directed flow, entrained in the outer region of the wall jet, aligns along the radial direction through the action of an adverse pressure gradient, relative to the local flow direction, which is balanced by the streamwise advection of wall-normal momentum.

The mean temperature equation (3.5) consists simply of the balance of two terms, $NL_\theta$ and $V_\theta$ , showing that, in the inner layer of the wall jet, the convection of heat flux balances the wall-normal molecular diffusion of the temperature field. Even if it is not shown in the plots of figure 7, at sufficiently large radial distances, $r\gtrsim 2D$ , the dominant term of $NL_\theta$ is that due to the wall-normal heat flux, namely $\langle u_z \theta \rangle _{,_z}$ , whereas $\langle r u_r \theta \rangle _{,_r}/r$ is negligible at the same radial locations. Thus, at large radial distances, the wall-normal molecular diffusion of the temperature is balanced by the wall-normal heat flux alone. The same scenario does not occur for the radial momentum equation (3.3), in which case all terms, except the radial pressure-gradient term, remain of comparable magnitude at large radial distances.

Figure 8.

Mean friction coefficient and Stanton number distributions. Panels show (a) , $C_f$ ; , $C_h$ . (b) , $C_f/C_{f0}$ ; , $C_h/C_{h0}$ ; dash-dot line, $\sim (r/D)^{-2}$ ; dash-dot-dot line, $\sim (r/D)^{-1}$ .

The structure of the flow in the vicinity of the impingement plate reflects on the distributions of the mean momentum and heat fluxes at the wall. These are visualised in figure 8(a) in terms of the friction coefficient and the Stanton number, $C_f=2\tau _w/(\rho U_b^2)$ and $C_h=q_w/(\rho c_p \Delta T U_b)$ , where $c_p$ indicates the fluid heat capacity at constant pressure, and $\tau _w$ denotes the wall-shear stress. Note that the Stanton number is directly related to the Nusselt number through the Péclet number, namely $C_h=Nu/Pe$ , where the Péclet number is $Pe=Pr Re$ . In the stagnation region of the flow, the heat flux at the wall is maximum, whereas the friction coefficient approaches zero for decreasing radial distance. In this region of the flow, the mean momentum and scalar behaviours deviate significantly from each other. The deflection of the flow, which corresponds to a redistribution of wall-normal momentum into radial momentum, takes place through the action of the pressure field, which does not directly affect the scalar temperature field.

As the flow accelerates radially, the friction coefficient increases quickly with the radial distance and reaches its maximum at $r\approx D/2$ , before decreasing further downstream as the flow expands and decelerates radially. Ultimately, at $r\approx 5.8D$ , flow separation occurs and the skin friction coefficient becomes negative. In contrast, the Stanton number decreases from the stagnation region onward, with the exception of the range $r\approx 1.5D \, - \, 2D$ , where a bump in the Stanton number distribution is evident. Note that, over the same region, also $C_f$ displays a similar behaviour. It is well established in the literature (for instance, Dairay et al. Reference Dairay, Fortuné, Lamballais and Brizzi2015) that over this region a secondary peak in the heat transfer rate occurs due to the unsteady interaction of coherent vortical structures with the wall. However, under the present conditions, it is evident that such a secondary peak is not yet well formed. Nevertheless, it is interesting to observe that, downstream of this region, the mean distributions of $C_f$ and $C_h$ become more similar to each other, even though, for $r\gtrsim 3D$ , $C_h$ noticeably exceeds $C_f$ , in magnitude. Further, close to and downstream of the flow separation, the two distributions differ significantly from each other.

The distributions of $C_f$ and $C_h$ , normalised by their maxima, $C_{f0}$ and $C_{h0}$ , respectively, are helpful for highlighting differences in the development of the mean radial momentum and heat wall fluxes along the impingement plate. These distributions, shown in figure 8(b), underline two main regions where a dissimilar behaviour is observed. In the stagnation region, for increasing radial distance, the wall heat transfer slowly decreases from its maximum value at the jet centreline, in contrast to a quickly increasing wall-normal momentum flux. In the wall jet region, for $2D \lesssim r \lesssim 5D$ , the skin friction and the Stanton number both decrease according to a power-law behaviour, in agreement with the experimental observations of Woodworth et al. (Reference Woodworth, Salazar and Liu2023). Despite the similar magnitude of $C_f$ and $C_h$ , as observed in figure 8(a), their rate of decay with the radial direction is significantly different. This difference can be readily explained based on the order of magnitude analysis of the radial momentum equation performed in § 3.1. Assuming $u_m\sim u_0/r$ and $\theta _m\sim const.$ , and using (3.1), the friction coefficient the Stanton number can be estimated to vary as $C_f\sim r^{-2}$ and $C_h\sim r^{-1}$ along the radial direction. As seen in figure 8(b), these estimates are approximately captured by the present data over the range $3D \lesssim r \lesssim 5D$ .

3.3. Energetics

While the distribution of the mean heat transfer coefficient is instrumental in determining the total thermal power exchanged at the target plate, the distribution of the mean friction coefficient is less informative in this respect, as it does not directly relate to the mechanical power input that is required to drive the system. To investigate this matter, we consider the balance equation for the total mechanical energy, $\mathcal{B}=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{u}/2+p/\rho$ , which can be obtained upon taking the inner product of equation (2.2) with the velocity vector and by applying standard vectorial relationships. By further applying time and spatial averaging, the mean balance equation reads (Secchi et al. Reference Secchi, Gatti, Piomelli and Frohnapfel2025)

(3.6) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\langle \boldsymbol{u}\mathcal{B} \rangle = \nu {\nabla} ^2 \langle \mathcal{B} \rangle - \nu \langle \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\omega }\rangle , \end{align}

where $\boldsymbol \omega$ indicates the vorticity vector, and $\nu \langle \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\omega }\rangle$ represents the mean dissipation rate of the mean mechanical energy. The integration of (3.6) over the entire flow domain allows the mechanical energy dissipation to be related to the power input of the system. Provided that the domain of integration is sufficiently large, the integral balance of mechanical energy states that the mechanical power input to the system equals the dissipation of mechanical energy within the domain. For a smaller integration volume, the integral balance links the bulk dissipation of mechanical energy to the mechanical energy fluxes across the volume boundaries. To evaluate the integral mechanical energy dissipation as a function of the radial coordinate, the integral balance of mechanical energy is performed on several cylindrical control volumes, $\mathcal{V}(r)$ , of increasing radius, whose bottom and top surfaces correspond, respectively, to the impingement and top plates of the flow domain, and whose axis of symmetry coincides with the jet axis. The resulting integral balance reads

(3.7) \begin{align} 0 &= \underbrace {-\int _0^{D/2} 2\pi \xi \langle u_z B \rangle \Bigr \rvert _{\xi , \, z=H} \, \mathrm{d}\xi }_{\varPi _{in}} \nonumber \\ &\quad - \underbrace {\int _0^{H} 2\pi r\langle B u_r \rangle \Bigr \rvert _{\xi =r, \, z} \, \mathrm{d}z}_{\varPi _{out}(r)} - \underbrace {\int _0^{H} \int _0^r 2\pi \xi \, \nu \langle \boldsymbol{\omega } \boldsymbol{\cdot }\boldsymbol{\omega } \rangle \, \mathrm{d}\xi \mathrm{d}z}_{\mathcal{E}(r)} \nonumber \\ &\quad + \underbrace {2\pi \nu \left [ \int _0^{H} r\langle B \rangle _{,\xi } \Bigr \rvert _{\xi =r, \, z} \, \mathrm{d}z \, +\, \int _0^r \xi \left ( \langle \mathcal{B} \rangle _{,z}\Bigr \rvert _{\xi , \, z=H} \, - \, \langle \mathcal{B} \rangle _{,z}\Bigr \rvert _{\xi , \, z=0}\right ) \, \mathrm{d}\xi \right ]}_{\mathcal{D}(r)}, \end{align}

where $\xi$ denotes the dummy integration variable replacing the radial coordinate $r$ . The convection of mechanical energy through the top boundary of the control volume, i.e. through the jet nozzle, is denoted by $\varPi _{in}$ and represents the driving power input of the system, whereas $\varPi _{out}$ corresponds to the convection of mechanical energy across the lateral surface of the control volume. According to (3.7), the loss of mechanical energy, $\Delta \varPi = \varPi _{out} - \varPi _{in}$ , that occurs across the control volume $\mathcal{V}(r)$ , balances the bulk dissipation, $\mathcal{E}(r)$ , plus the molecular flux of mechanical energy across the boundary of the control volume, $\mathcal{D}(r)$ . Figure 9 reports the integral budget terms of equation (3.7) computed for control volumes of increasing radial size, starting from a minimum radius of $r=D/2$ . The molecular diffusion of mechanical energy through the boundary of the control volume, $\mathcal{D}$ , is always negligible in comparison with the other terms of the equation and, therefore, is not reported in the figure. Hence, the integral balance states that the convection of mechanical energy through the control volume equals the bulk energy dissipation, $\Delta \varPi = -\mathcal{E}$ . The power input to the system, $\varPi _{in}$ is fixed by the prescribed inflow condition at the nozzle exit section. For increasing $r$ , the power input gets consumed by the bulk energy dissipation, $\mathcal{E}(r)$ , thus determining the decrease of the output power, $\varPi _{out}$ , which represents the available mechanical power for the downstream flow. For $3D\lesssim r \lesssim 5D$ , $\varPi _{out}$ clearly exhibits a power-law decay $\varPi _{out} \sim r^{-2}$ . This behaviour can be explained by assuming that the mean convection of mechanical energy, $\langle B u_r \rangle$ , scales with $u_m^3 \sim r^{-3}$ , based on the same arguments as in § 3.1. Downstream of the separation point, for $r\gtrsim 5.8D$ , the radial rate of decay of $\varPi _{out}$ increases further, as $\varPi _{out}$ tends to vanish with the radial distance as $\sim r^{-7}$ . The independence of these results from the size of the computational domain, specifically from the location of the open boundary, is evidenced in the figure by the distributions of $\varPi _{out}$ and $\mathcal{E}$ for the simulation on a larger computational box having $L_x=20D$ and $L_y=20D$ .

Figure 9.

Mechanical energy integral budgets. Results on the computational domain with $L_x=L_y=16D$ : $\varPi _{in}/(U_b^3D^2)$ ; $\varPi _{out}/(U_b^3D^2)$ ; $\mathcal{E}/(U_b^3D^2)$ . Results on the computational domain with $L_x=L_y=20D$ : $\varPi _{out}/(U_b^3D^2)$ ; $\mathcal{E}/(U_b^3D^2)$ A vertical black-dashed line indicates the radial location of flow separation on the impingement plate.

The equivalent of (3.6) for the temperature is given by (2.3), which embodies the first law of thermodynamics, assuming that the increase in internal energy associated with viscous dissipation can be neglected. As such, the internal energy of a fluid element varies only due to heat conduction, with the result that the integral balance of (2.3) across the control volume $\mathcal{V}(r)$ simply states the balance between the total convected internal energy across the lateral surface and the sum of the thermal power (per unit volumetric heat capacity, $\rho c_p$ , and temperature difference, $\Delta T$ ) exchanged through the bottom and top plates, respectively, $\varPi _\theta (r) = -\int _0^r 2\pi \xi \, \alpha \langle \theta \rangle _{,_z}\Bigr \rvert _{\xi , \, z=0} \, \mathrm{d}\xi$ and $\varPi _{\theta T} = -\int _0^r 2\pi \xi \, \alpha \langle \theta \rangle _{,_z}\Bigr \rvert _{\xi , \, z=H} \, \mathrm{d}\xi$ . Note that, in arriving at this result, the molecular conduction across the lateral surface of the control volume has been neglected. The integral balance of internal energy delivers little information on the links between thermal power exchanged at the impingement plate and the flow. Secchi et al. (Reference Secchi, Gatti, Piomelli and Frohnapfel2025) suggested considering the balance equation for the squared-scalar field, $\varGamma =\theta ^2/2$ , which, for a statistically stationary flow, has a form analogous to (3.6). Namely,

(3.8) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\langle \boldsymbol{u} \varGamma \rangle = \alpha {\nabla} ^2 \langle \varGamma \rangle - \alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle . \end{align}

Similarly to the mean mechanical energy, $\langle \varGamma \rangle$ is governed by an advection–diffusion equation with the inclusion of a sink term, $\alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle$ , which represents the local dissipation rate of $\langle \varGamma \rangle$ . Physically, $\alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle$ is directly related to the mean entropy production due to heat conduction, $\mathcal{P}_s$ , through $\mathcal{P}_s = \alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle /\theta ^2$ . As the passive scalar equation (2.3) states the first law of thermodynamics, equation (3.8) is a reflection of the second law, as the local destruction of $\varGamma$ corresponds to a local entropy increase. In fact, from the well-known Gibbs relationship, it follows that the material change of $\varGamma$ is $\mathrm{D} \varGamma / \mathrm{D} t = 2 \varGamma \mathrm{D} s / \mathrm{D} t$ , where $s$ indicates the entropy, and the material change is denoted by the operator $\mathrm{D} (\boldsymbol{\cdot }) / \mathrm{D} t = (\boldsymbol{\cdot }),_t + \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }(\boldsymbol{\cdot })$ .

Integrating (3.8) over the control-volume $\mathcal{V}(r)$ leads to the following integral balance for $\langle \varGamma \rangle$ :

(3.9) \begin{align} 0 &= \underbrace {-\int _0^r 2\pi \xi \theta _w \alpha \langle \theta \rangle _{,_z}\Bigr \rvert _{\xi , \, z=0} \, \mathrm{d}\xi }_{\varPi _{\varGamma \, in}= \theta _w\varPi _\theta (r)} - \underbrace {\int _0^{H} 2\pi r\langle \varGamma u_r \rangle \Bigr \rvert _{\xi =r, \, z} \, \mathrm{d}z}_{\varPi _{\varGamma \, out}(r)} \nonumber \\ &\quad - \underbrace {\int _0^{H} \int _0^r 2\pi \xi \, \alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle \, \mathrm{d}\xi \, \mathrm{d}z}_{\mathcal{E}_\varGamma (r)} + \underbrace {\int _0^{H} 2\pi r \, \alpha \langle \varGamma \rangle _{,\xi }\Bigr \rvert _{\xi =r, \, z} \, \mathrm{d}z}_{\mathcal{D}_\varGamma (r)}. \end{align}

Figure 10.

The $\varGamma$ integral budgets: , $\theta _w\varPi _{\theta }/(U_b D^2)$ ; , $\varPi _{\varGamma \, out}/(U_b D^2)$ ; , $\mathcal{E}_\varGamma /(U_b D^2)$ . A black-dashed line indicates the radial location of flow separation on the impingement plate.

Due to the homogeneous boundary condition for $\theta$ on the confinement plate, the net thermal power exchanged across it does not contribute to the integral balance of $\langle \varGamma \rangle$ ; in fact, since $\theta =0$ on the confinement plate, $\langle \varGamma _{,z}\rangle \Bigr \rvert _{\xi , \, z=H} = (\theta \langle \theta \rangle _{,z} )\Bigr \rvert _{\xi , \, z=H} = 0$ . Conversely, at the impingement plate, the exchanged thermal power represents an input term for the balance of $\langle \varGamma \rangle$ , namely through the term $\varPi _{\varGamma \, in}= -\theta _w \varPi _\theta (r)$ . The molecular flux of $\langle \varGamma \rangle$ across the lateral surface of the control volume, $\mathcal{D}_\varGamma (r)$ , is always negligible compared with the other terms in the budget, due to the small radial gradients. Therefore, in analogy with the mechanical energy balance, the integral budget equation (3.9) reduces to $\Delta \varPi _\varGamma = - \mathcal{E}_\varGamma (r)$ , where $\Delta \varPi _\varGamma = \varPi _{\varGamma \, out} - \theta _w \varPi _\theta$ .

Figure 10 shows the radial distribution of $\varPi _{\varGamma \, out}$ , $\theta _w \varPi _\theta$ and $\mathcal{E}_\varGamma (r)$ , obtained by applying equation (3.9) to control volumes of increasing radius, starting from $r=D/2$ . The term $\theta _w \varPi _\theta$ , numerically equal to the exchanged thermal power, increases monotonically with the radial distance, as $\varPi _\theta$ naturally increases with increasing surface area over which heat transfer occurs. Interestingly, at small radial distances, for $r\lesssim 2D$ , the convection of $\langle \varGamma \rangle$ across the lateral surface of $\mathcal{V}(r)$ is significantly smaller than the other terms, and $\theta _w \varPi _\theta \approx \mathcal{E}_\varGamma (r)$ . Namely, near the impingement region, all the thermal power extracted at the impingement plate goes into the bulk dissipation of $\varGamma$ . For larger control volumes, the flow becomes increasingly less capable of dissipating the net thermal power input across the impingement plate, and the convection of $\langle \varGamma \rangle$ across the lateral surface, $\varPi _{\varGamma \, out}$ , becomes relevant for large $r/D$ .

The thermal transfer efficiency of the system can be defined as the ratio between the net heat flux, exchanged at the boundary of the control volume, and the mechanical energy expenditure occurring within the control volume. Assuming a negligible molecular heat flux across the lateral surface of the control volume, this is

(3.10) \begin{align} \eta & = \left [ \int _0^r c_p \Delta T \alpha \left ( \!\langle \theta \rangle _{,z}\Bigr \rvert _{\xi , \, z=H} - \langle \theta \rangle _{,z}\Bigr \rvert _{\xi , \, z=0} \right )\xi \, \mathrm{d}\xi \!\right ] \times \left ( \int _0^{H}\!\!\int _0^r \!\nu \langle \boldsymbol{\omega } \boldsymbol{\cdot }\boldsymbol{\omega } \rangle \xi \, \mathrm{d}\xi \mathrm{d}z\!\right )^{-1} \nonumber \\ & = \frac {c_p \Delta T \left [ \varPi _{\theta }(r) - \varPi _{\theta T}(r) \right ]}{\mathcal{E}(r)}. \end{align}

By virtue of the integral balances of internal and mechanical energy over the control-volume $\mathcal{V}(r)$ , the thermal transfer efficiency defined in (3.10) expresses the ratio between the net convection of internal and mechanical energy across the control volume.

The radial distribution of $\eta \, U_b^2/(c_p \Delta T)$ is reported in figure 10(a). For small radial distances, the net heat flux across the confinement plate is negligible in comparison with the heat flux at the impingement plate, thus the heat transfer efficiency can be well approximated via $\eta \approx c_p \Delta T\varPi _\theta /\mathcal{E}(r)$ . This is verified in figure 10(a), which shows that this approximation delivers very similar results to equation (3.10) for $r\lesssim 4D$ . For greater radii, the mean heat transfer at the impingement plate is comparable, in magnitude, to the heat flux at the confinement plate, and the approximated heat transfer efficiency deviates significantly from (3.10).

While retaining the approximation $\eta \approx c_p \Delta T \varPi _\theta/\mathcal{E}(r)$ at small radial distances, it must be noted that the numerator, $\varPi _\theta (r)$ , also includes the heat flux, $\varPi _{\theta B}$ , induced by the temperature difference between the impingement and the confinement plates, which would be present even in the absence of a flow between the two plates. Without a flow, the mechanical energy consumption would be zero, rendering $\eta$ undefined. This issue can be overcome by simply removing the contribution $\varPi _{\theta B}$ from $\varPi _\theta (r)$ in the approximated expression for $\eta$ . The result of this exercise is shown in figure 10(a), where it can be seen that for the present configuration the contribution from $\varPi _{\theta B}$ remains limited at any radial location.

In either case, the radial distribution of $\eta U_b^2/(c_p \Delta T)$ , reported in figure 10(a), reveals that the thermal transfer efficiency of the system is significantly smaller than unity for any considered size of the control-volume $\mathcal{V}(r)$ , suggesting that jet impingement is not an efficient means for removing heat from a surface. This result is particularly interesting, given that jet impingement is usually referred to as an example of a flow that favourably breaks the Reynolds analogy, for a finite wall heat transfer is realised in the stagnation region in spite of a vanishingly small wall-shear stress (Bunker Reference Bunker2013; Rouhi et al. Reference Rouhi, Endrikat, Modesti, Sandberg, Oda, Tanimoto, Hutchins and Chung2022). Indeed, as reported in § 3.2, for $r\lesssim D/2$ the wall heat transfer significantly exceeds the wall momentum flux. However, such high heat transfer rates are realised at comparatively higher mechanical energy expenses. As a result, at small radial distances, a very low thermal efficiency is observed, with $\eta U_b^2/(c_p \Delta T)\lt 0.6$ for $r\lesssim 2.5D$ . For increasing radial extent of the control-volume $\mathcal{V}(r)$ , the thermal efficiency increases at different rates depending on the considered flow region and, particularly, the increase is approximately linear in the range $3D\lesssim r \lesssim 5D$ . Notably, in correspondence of the separation point, the efficiency is still appreciably less than unity.

Figure 11.

(a) Thermal transfer efficiency, $\eta U_b^2/(c_p \Delta T)$ . Dash-dot line, (3.10); solid line, $U_b^2(\varPi _\theta (r) - \varPi _{\theta B})/\mathcal{E}(r)$ ; dashed-line, $U_b^2\,\varPi _\theta (r)/\mathcal{E}(r)$ . (b) Local thermal transfer efficiency, $\tilde {\eta } U_b^2/(c_p\Delta T)$ . In both panels, a vertical black dashed line indicates the radial location of flow separation on the impingement plate.

To obtain local information on the integral balances at different radial locations, it is useful to apply the above analysis to control volumes shaped as hollow circular cylinders, coaxial with the jet axis and having the bottom and top bases on the impingement and confinement plates, respectively. The volume of such a cylinder is denoted by $ [ \mathcal{V}(r) ] = \mathcal{V}(r+\Delta r/2) - \mathcal{V}(r-\Delta r/2)$ , where $ [ \boldsymbol{\cdot }]$ indicates the differential of a quantity computed over the control volumes of the circular cylinders $\mathcal{V}(r+\Delta r/2)$ and $\mathcal{V}(r-\Delta r/2)$ , and $\Delta r$ indicates an arbitrary radial increment. It then follows that the balances of $\langle \mathcal{B} \rangle$ and $\langle \varGamma \rangle$ across $ [ \mathcal{V}(r) ]$ reduce to $ [ \varPi _{out}(r) ] = - [ \mathcal{E}(r) ]$ and $ [ \varPi _{\varGamma , out}(r) - \theta _w \varPi _\theta (r) ] = - [ \mathcal{E}_\varGamma (r) ]$ , respectively. In particular, localised information on the energetic efficiency of the heat transfer augmentation across $ [ \mathcal{V}(r) ]$ can be gained by considering the ratio $\tilde {\eta }=c_p \Delta T [ \varPi _\theta (r) - \varPi _{\theta _B}(r) ]/ [ \mathcal{E}(r) ]$ , which is reported in figure 11(b). Here, $\varPi _{\theta B}(r)$ denotes the net heat flux associated with the base solution, $\theta _B$ . For $r\lesssim 3D$ , the local heat transfer augmentation induced by the flow is particularly inefficient, as the mechanical power dissipated within $ [ \mathcal{V}(r) ]$ exceeds significantly the thermal power exchanged across the base of $ [ \mathcal{V}(r) ]$ . On the contrary, at greater radial distances, $\tilde {\eta }U_b^2/(c_p \Delta T)$ exceeds unity and reaches a local maximum of $\tilde {\eta } U_b^2/(c_p \Delta T)\approx 2.3$ at $r\approx 5D$ . Noticeably, the wall jet region of the flow, for $r\gtrsim 3D$ , appears to be characterised by little mechanical energy dissipation in comparison with the thermal power augmentation induced by the flow at the impingement plate. Downstream of the local maximum of $\tilde {\eta }U_b^2/(c_p \Delta T)$ , the flow approaches separation and, correspondingly, $\tilde {\eta }U_b^2/(c_p \Delta T)$ decreases until reaching a global minimum, $\tilde {\eta }U_b^2/(c_p \Delta T)\approx 1.3$ , downstream of the separation point. At larger radial distances, a vanishingly small amount of mechanical energy is left available to the flow, and yet a significantly larger wall heat transfer augmentation occurs at the base of $ [ \mathcal{V}(r) ]$ , as can be seen from the sharp increase of $\tilde {\eta }U_b^2/(c_p \Delta T)$ for $r\gtrsim 6.5D$ .

The joint analysis of figures 5(d) and 11(b) shows that radial locations in which near-wall fluid is moving on average towards the wall, i.e. regions where $\langle u_z \rangle \lt 0$ near the wall, are characterised by low values of the local heat transfer efficiency, $\tilde {\eta }$ . The wall-directed fluid produces high heat transfer rates, but at the same time, determines a higher dissipation of mechanical energy. Conversely, for $r\gtrsim 2D$ , where near-wall fluid moves on average away from the wall as a consequence of the growing boundary layer, i.e. near-wall regions where $\langle u_z \rangle \gt 0$ , the local heat transfer efficiency increases. The presence of warm fluid moving away from the wall penalises the heat transfer rates, which are relatively low in this region, yet determining a lower amount of mechanical energy dissipation.

3.4. Mechanical and scalar dissipation rates

The integral dissipation rates, $ [ \mathcal{E} ]$ and $ [ \mathcal{E}_{\varGamma } ]$ , are relevant for assessing the efficiency characteristics of the system; the former determines the total mechanical power dissipated within $ [ \mathcal{V} ]$ , whereas the latter is directly linked to the thermal power input across the base of $ [ \mathcal{V} ]$ . It is thus instructive to analyse the time-averaged mechanical and scalar dissipation fields, $\nu \langle \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\omega }\rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle$ , respectively. The quantities $\boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\omega }$ and $\boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta$ are governed by similar equations, as noted by Corrsin (Reference Corrsin1953). Such similarities have since been reported in the literature for both homogeneous isotropic turbulence and fully developed wall-bounded flows (see, for example, Abe et al. Reference Abe, Antonia and Kawamura2009 and references therein). Recently, for Reynolds analogy breaking flows, Secchi et al. (Reference Secchi, Gatti, Piomelli and Frohnapfel2025) have shown how different momentum and scalar transports can be re-conducted to dissimilarities in the dissipation fields $\nu \langle \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\omega }\rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle$ . A similar approach is followed here. In particular, the mean-flow and turbulent contributions to $\nu \langle \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\omega }\rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle$ are considered. The decomposition into mean-flow and turbulent parts is carried out by means of the classical Reynolds decomposition of any flow variable.

Figure 12.

Mean and turbulent dissipation rates. Panels show (a) $\nu \langle \boldsymbol{\omega } \rangle \boldsymbol{\cdot }\langle \boldsymbol{\omega }\rangle D/U_b^3$ ; (b) $\alpha \langle \boldsymbol{\nabla }\theta \rangle \boldsymbol{\cdot }\langle \boldsymbol{\nabla }\theta \rangle D/U_b$ ; (c) $\nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle D/U_b^3$ ; (d) $\alpha \langle \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime \rangle D/U_b$ ; $z_m/D$ ; $z_{\theta \, m}/D$ .

Figure 12 shows the mean-flow, $\nu \langle \boldsymbol{\omega } \rangle \boldsymbol{\cdot }\langle \boldsymbol{\omega }\rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta \rangle \boldsymbol{\cdot }\langle \boldsymbol{\nabla }\theta \rangle$ , and the turbulent, $\nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime \rangle$ , contributions to the time- and space-averaged dissipation fields, $\nu \langle \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\omega }\rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta \boldsymbol{\cdot }\boldsymbol{\nabla }\theta \rangle$ . The plots in the figure use a logarithmic scale in the wall-normal direction and in the colour space to improve the visualisation of the near-wall region and to accommodate the wide range of intensities that are realised in different flow regions. Further, to distinguish between the boundary layer, along the impingement plate, and the outer flow regions, the velocity and temperature boundary-layer thicknesses are also shown in the figure.

The mean-flow contributions exhibit evident differences across the entire flow domain. The main difference occurs within the stagnation region of the flow, where the mean scalar dissipation reaches its global maximum in correspondence with the jet axis; conversely, $\nu \langle \boldsymbol{\omega } \rangle \boldsymbol{\cdot }\langle \boldsymbol{\omega }\rangle$ becomes maximum further downstream of it, around $r\approx D/2$ . This dissimilarity is driven by the fact that, in the stagnation region, momentum is predominantly affected by the action of the pressure which determines the deflection of the flow along the impingement plate. Consequently, the fluid near the wall is characterised by low streamwise momentum, and, in turn, the molecular momentum flux at the wall remains small close to the jet axis. In contrast, the fluid internal energy – i.e. the fluid temperature – is not directly affected by the action of the pressure field, thus allowing cold, low-momentum fluid to approach the impingement plate and to determine intense wall-normal temperature gradients, which result in high heat transfer rates. The strong radial acceleration of the mean flow produces high shear in the vicinity of the wall which reflects into large values of the mean mechanical and scalar dissipation rates. Note that $\nu \langle \boldsymbol{\omega } \rangle \boldsymbol{\cdot }\langle \boldsymbol{\omega }\rangle$ peaks in this region at $r\approx D/2$ . Further downstream, the flow decelerates in the radial direction and the mean mechanical and scalar dissipation rates consequently decrease. A region of apparent dissimilarity between the mean mechanical and scalar dissipation rates occurs where flow separation takes place, at $r\approx 5.8D$ . There, a vanishingly small $\nu \langle \boldsymbol{\omega } \rangle \boldsymbol{\cdot }\langle \boldsymbol{\omega }\rangle$ is observed, in contrast to $\alpha \langle \boldsymbol{\nabla }\theta \rangle \boldsymbol{\cdot }\langle \boldsymbol{\nabla }\theta \rangle$ which remains visibly larger over the same region. Away from the boundary layer, the mean mechanical and scalar dissipation rates are large in the region of the shear layer of the free jet and in the outer shear layer of the wall jet. It is interesting to observe that $\nu \langle \boldsymbol{\omega } \rangle \boldsymbol{\cdot }\langle \boldsymbol{\omega }\rangle$ appears more intense than $\alpha \langle \boldsymbol{\nabla }\theta \rangle \boldsymbol{\cdot }\langle \boldsymbol{\nabla }\theta \rangle$ in these flow regions, indicating that the mean scalar distribution tends to be more diffused across these shear layers, compared with the mean momentum components.

The distributions of the turbulent dissipation rates of mechanical energy and $\langle \varGamma \rangle$ , $\nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime \rangle$ are shown in figure 12(c, d). In both cases, the turbulent contribution to the dissipation fields is qualitatively similar, especially within the boundary-layer thickness, where the dissipation rates show a two-layered structure in which two regions of high values of $\nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime \rangle$ are separated by local minima. From the figure it can be appreciated how the wall-normal height of these local minima increases with the radial distance from the jet axis. The turbulent contributions to the dissipation rates are maxima at the wall and reach their highest values within the range $D\lesssim r \lesssim 2.5D$ . Over the same range, $\nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime \rangle$ show very large intensities also at large wall-normal distances, and, in the case of $\nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle$ , even outside of the boundary layer, in the outer region of the wall jet.

The region where $\nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle$ and $\alpha \langle \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime \rangle$ are particularly intensified, $D\lesssim r \lesssim 2.5D$ , approximately coincides with the region of the impingement plate where the secondary peak in the mean Nusselt typically occurs. Depending on the configuration, this local peak has been observed for $r\approx 1.5D - 2D$ (Jambunathan et al. Reference Jambunathan, Lai, Moss and Button1992). In the present case, the secondary peak is not yet fully developed, but a clear change of slope can be seen in the $C_f$ and $C_h$ distributions in figure 8(a). The appearance of the secondary peak has been attributed by Dairay et al. (Reference Dairay, Fortuné, Lamballais and Brizzi2015) to near-wall unsteady mechanisms which are driven by the generation, via Kelvin–Helmholtz instabilities, of large-scale coherent structures in the shear layers of the free and wall jets. As these vortical structures approach the impingement plate, they generate secondary, counter-rotating, large-scale coherent vortices near the wall, which promote a local heat transfer augmentation due to their convection of cold fluid towards the wall. A key aspect is that these secondary vortices are highly distorted in the azimuthal direction, so that local spots of extremely large heat flux are realised in the regions where these vortical structures are closer to the wall.

It can be suggested that these structures, which contribute favourably to the wall heat transfer augmentation, are also responsible for determining the regions where the turbulent contributions to the dissipation rates are maxima, for $D\lesssim r \lesssim 2.5D$ . This scenario is qualitatively assessed in figure 13, which shows instantaneous snapshots of the fluctuating radial velocity field on a wall-parallel plane near the impingement plate, at $z=0.005D$ . At small radial distances regions of high azimuthal coherence are manifest. In particular, regions of positive and negative radial velocity fluctuations alternate in the radial direction, representing the trace of downstream convecting secondary vortices. As they move towards greater radial distances, their coherence is partially lost, as smaller-scale structures are generated. The figure evidences that the regions of high azimuthal coherence give rise to intense fluctuations, $\nu \boldsymbol \omega ^\prime \boldsymbol{\cdot }\boldsymbol \omega ^\prime$ and $\alpha \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime$ , which also present a significant coherence in the azimuthal direction. Moving towards larger radial distances, as these structures break into finer ones, the spots of intense turbulent contributions to the dissipation rates become also more sparse.

Figure 13.

Instantaneous snapshots of the fluctuating radial velocity $u_r^\prime /U_b$ on a wall-parallel plane at $z=0.005D$ . Contour lines, colour coded from white, $0.2$ , to dark grey, $1.0$ , represent iso-contours of $\nu \boldsymbol \omega ^\prime \boldsymbol{\cdot }\boldsymbol \omega ^\prime D/U_b^3$ , in panel (a), and $\alpha \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime D/U_b$ , in panel (b).

Figure 14.

Mean and turbulent contributions to the bulk dissipation rates. Panels show (a) , $D \left[ \mathcal{E} \right] / (\left[ \mathcal{V} \right] U_b^3)$ ; , $D \left[ \mathcal{E}^m \right] / (\left[ \mathcal{V} \right] U_b^3)$ ; , $D \left[ \mathcal{E}^t \right] / (\left[ \mathcal{V} \right] U_b^3)$ . (b) , $D \left[ \mathcal{E}_\Gamma \right] / (\left[ \mathcal{V} \right] U_b)$ ; , $D \left[ \mathcal{E}_\Gamma^m \right] / (\left[ \mathcal{V} \right] U_b)$ ; , $D \left[ \mathcal{E}_\Gamma^t \right] / (\left[ \mathcal{V} \right] U_b)$ .

A more quantitative description of the mean-flow and the turbulent contributions to the integral mechanical and scalar energy dissipation is provided in figure 14, which displays the radial distributions of $ [ \mathcal{E} ] = [ \mathcal{E}^m ] + [ \mathcal{E}^t ]$ and $ [ \mathcal{E}_\varGamma ] = [ \mathcal{E}_\varGamma ^m ] + [ \mathcal{E}_\varGamma ^t ]$ , per unit volume, $ [ \mathcal{V}(r) ]$ . Here, the superscripts $m$ and $t$ denote, respectively, the mean-flow and turbulent contributions, i.e. $\mathcal{E}^m = \int _0^{H} \int _0^r 2\pi \xi \, \nu \langle \boldsymbol{\omega }\rangle \boldsymbol{\cdot }\langle \boldsymbol{\omega } \rangle \, \mathrm{d}\xi \mathrm{d}z$ , $\mathcal{E}_\varGamma ^m = \int _0^{H} \int _0^r 2\pi \xi \, \alpha \langle \boldsymbol{\nabla }\theta \rangle \boldsymbol{\cdot }\langle \boldsymbol{\nabla }\theta \rangle \, \mathrm{d}\xi \, \mathrm{d}z$ , $\mathcal{E}^t = \int _0^{H} \int _0^r 2\pi \xi \, \nu \langle \boldsymbol{\omega }^\prime \boldsymbol{\cdot }\boldsymbol{\omega }^\prime \rangle \, \mathrm{d}\xi \mathrm{d}z$ and $\mathcal{E}_\varGamma ^t = \int _0^{H} \int _0^r 2\pi \xi \, \alpha \langle \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime \rangle \, \mathrm{d}\xi \, \mathrm{d}z$ . From the figure it is evident that turbulent fluctuations provide the largest contribution to $ [ \mathcal{E} ]$ almost everywhere in the flow field, except at small radial distances, where a significant contribution to the dissipation comes from the intense mean shear near the impingement plate. However, the mean-flow contribution decreases rapidly with increasing the radial distance. In contrast, $ [ \mathcal{E}_\varGamma ]$ is determined, almost everywhere, by the joint contributions of the mean flow and of the turbulent dissipation rates. Indeed, both $ [ \mathcal{E}_\varGamma ^m ]$ , and $ [ \mathcal{E}_\varGamma ^t ]$ contribute significantly to $ [ \mathcal{E}_\varGamma ]$ and, in fact, the mean-flow contribution exceeds everywhere the turbulent one. Figure 14 shows that $ [ \mathcal{E}^t ]$ and $ [ \mathcal{E}_\varGamma ^t ]$ both increase rapidly for $r\gt D$ , until they reach their maxima at $r\approx 2D$ , before decreasing further downstream. This observation adds to the phenomenological scenario described in figure 13; the secondary vortices are generated at the wall at small radial distances, $D/2\lesssim r \lesssim D$ , they convect downstream and become more intense, thus generating localised patches of strong velocity and scalar gradient fluctuations. This process culminates at $r/D\approx 2D$ , where $ [ \mathcal{E}^t ]$ and $ [ \mathcal{E}_\varGamma ^t ]$ are maximal, and then deteriorates further downstream as the coherence of the secondary vortices is lost and dissipation occurs at increasingly smaller length scales.

While the described scenario affects both $\nu \boldsymbol \omega ^\prime \boldsymbol{\cdot }\boldsymbol \omega ^\prime$ and $\alpha \boldsymbol{\nabla }\theta ^\prime \boldsymbol{\cdot }\boldsymbol{\nabla }\theta ^\prime$ – in particular, their volume integrals, $ [ \mathcal{E}^t ]$ and $ [ \mathcal{E}_\varGamma ^t ]$ , increase and become maximal in the range $D\lesssim r \lesssim 2D$ - figure 14 evidences that $ [ \mathcal{E}^t ]$ is significantly more amplified compared with $ [ \mathcal{E}_\varGamma ^t ]$ , indicating a clear dissimilarity between the velocity and scalar gradients fluctuations. While the physical reason for such a dissimilarity still remains an open question, the present findings show how its effects can have important practical consequences, especially in determining the thermal efficiency of the system.