3.1 Effect of eddy-viscosity model on turbulent stress estimates

The effect on turbulent stress estimates of the different eddy-viscosity models, including extended QCR, is studied using the DNS flows listed in Table 1. The estimated turbulent stress distributions are obtained by evaluating the relevant eddy-viscosity model equation using the mean velocities provided by the DNS data and the eddy viscosity, ${\mu _t}$, from Equation (12). The estimated turbulent stresses are then compared with the ‘true’ values from DNS.

Prior to evaluating the capabilities of extended QCR, conventional eddy-viscosity models are assessed for the cases of channel flow, boundary layers and pipe flow. The turbulent stresses are estimated for: an LEVM which doesn’t account for turbulent kinetic energy (Equation (2)); QCR-2000 (Equation (3)); QCR-2013 (Equation (5)); and an LEVM accounting for turbulent kinetic energy using the ${c_{cr2}}$ term from QCR-2013 (Equation (6)). Figure 2 shows how the estimated normal stresses (${\tau _{xx}}$, ${\tau _{yy}}$ and ${\tau _{zz}}$) from each model compare to the distributions from DNS^{(Reference Bernardini, Pirozzoli and Orlandi10,Reference Sillero, Jiménez and Moser12,Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson15)} . This comparison shows that an LEVM based on Equation (2) predicts all three normal stresses to be equal to zero everywhere, which is not an accurate representation of the true distribution.

Figure 2. Effect of different turbulent stress approximations on estimating the normal turbulent stresses (i. $\overline {{u^{\prime}}{u^{\prime}}} $, ii. $\overline {{v^{\prime}}{v^{\prime}}} $, iii. $\overline {{w^{\prime}}{w^{\prime}}} $), scaled by the friction velocity, ${u_\tau }$. The true stress from DNS (—) and estimated stress from an LEVM (Equation (2), -$ \cdot $-), an LEVM with the ${c_{cr2}}$ term (Equation (6), -$ \cdot $-), QCR-2000 (Equation (3), - -) and QCR-2013 (Equation (5), - -). Flowfields: (a) channel flow^{(Reference Bernardini, Pirozzoli and Orlandi10)}; (b) boundary layer^{(Reference Sillero, Jiménez and Moser12)}; (c) pipe flow^{(Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson15)}. In each case, the stress distributions are presented with the wall-normal coordinate in both outer units ($y$) and wall units (${y^ + }$).

From Fig. 2, it is also evident that the ${c_{cr1}}$ and ${c_{cr2}}$ terms in QCR-2013 serve two different purposes. QCR-2000, which contains only the ${c_{cr1}}$ term, better predicts the anisotropies between the turbulent stresses, capturing the fact that $\overline {{u^{\prime}}{u^{\prime}}} \gt \overline {{w^{\prime}}{w^{\prime}}} \gt \overline {{v^{\prime}}{v^{\prime}}} $. However, the estimated stresses remain much smaller than the true values. On the other hand, the estimates from Equation (6), which contains only the ${c_{cr2}}$ term, better estimate the size of the stresses but do not predict any variation between the different turbulent stresses. This observation suggests that the ${c_{cr1}}$ term improves prediction of the anisotropy between the normal turbulent stresses, while the ${c_{cr2}}$ term improves the estimate of the magnitudes. QCR-2013, which contains both ${c_{cr1}}$ and ${c_{cr2}}$ terms, therefore benefits from both improvements and features significantly better turbulent stress estimates than the other models. However, even QCR-2013 exhibits noticeable deviations from the true stress distributions, motivating the need for an extended QCR.

A comparison of the turbulent stress estimates from QCR-2013 and extended QCR against the true DNS values is shown in Fig. 3. This figure shows that extended QCR predicts the turbulent stresses substantially better than QCR-2013. Note that these improvements are achieved with coefficient values that are neither spatially dependent nor flow specific. However, in the near-wall region both models struggle to accurately predict the stresses. This observation is particularly evident when the spatial coordinate is scaled in wall units, ${y^ + }$. These plots show that extended QCR agrees reasonably well with the DNS data in the outer part of the flow but, below ${y^ + } \approx 100 - 500$, none of the eddy-viscosity models produce satisfactory estimates of the turbulent stresses. Nevertheless, even in this region, extended QCR still generally performs better than the other models.

Figure 3. Effect of different turbulent stress approximations on estimating the normal turbulent stresses (i. $\overline {{u^{\prime}}{u^{\prime}}} $, ii. $\overline {{v^{\prime}}{v^{\prime}}} $, iii. $\overline {{w^{\prime}}{w^{\prime}}} $), scaled by the friction velocity, ${u_\tau }$. The true stress from DNS (—) and estimated stress from QCR-2013 (Equation (5), - -) and extended QCR (Equation (11), —). Flowfields: (a) channel flow^{(Reference Bernardini, Pirozzoli and Orlandi10)}; (b) boundary layer^{(Reference Sillero, Jiménez and Moser12)}; (c) pipe flow^{(Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson15)}. In each case, the stress distributions are presented with the wall-normal coordinate in both outer units ($y$) and wall units (${y^ + }$).

Figure 4. Effect of Reynolds number on estimates of normal turbulent stresses (i. $\overline {{u^{\prime}}{u^{\prime}}} $, ii. $\overline {{v^{\prime}}{v^{\prime}}} $, iii. $\overline {{w^{\prime}}{w^{\prime}}} $), scaled by the friction velocity, ${u_\tau }$. The true stress from DNS (—) and estimated stress from QCR-2013 (Equation (5), - -) and extended QCR (Equation (11), —). Reynolds numbers: (a) Re$_\tau $ = 2,050; (b) Re$_\tau $ = 1,000; (c) Re$_\tau $ = 550; (d) Re$_\tau $ = 180^{(Reference Bernardini, Pirozzoli and Orlandi10)}. In each case, the stress distributions are presented with the wall-normal coordinate in wall units (${y^ + }$), with the data in outer units ($y$) only included for cases (a) and (d) for brevity.

The poor predictions in the near-wall region can be attributed to the fact that extended QCR has been developed in free shear flows and does not take advantage of any information regarding the proximity of the wall. Capturing this wall-normal dependence remains a challenge for the future, perhaps using the approach of QCR-2020, which allows the values of coefficients to vary with wall-normal distance. It is worth noting, however, that even QCR-2020 has trouble reproducing, in particular, the sharp near-wall peak of ${\tau _{xx}}$ at high Reynolds numbers^{(Reference Rumsey, Carlson, Pulliam and Spalart20)}.

In addition to evaluating the improvement in turbulent stress estimates by extended QCR for different flow fields, it is also useful to consider the effects of Reynolds number. This analysis is performed using the channel flow as an example, which was studied for Re$_\tau = 4100$ in Fig. 3(a). However, DNS simulations were also conducted by Bernardini et al. for the same flow field at Re$_\tau = 2050$, 1000, 550, and 180. Figure 4 presents the turbulent stresses from the velocity statistics for these additional simulations, alongside the estimated distributions from QCR-2013 and extended QCR. This comparison shows that extended QCR provides a better estimate of the normal stresses than QCR-2013, not just at Re$_\tau = 4100$ where the coefficient values were calibrated, but for the entire Reynolds-number range, Re$_\tau = 180$ – 4100.

A similar analysis to the parallel mean shear flows in Fig. 3 can also be extended to the more complex cases in Table 1, namely the separated boundary layer^{(Reference Coleman, Rumsey and Spalart16)} and the flow in a square duct^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)}. In contrast to the parallel shear flows from Fig. 3, the mean velocity in such flow fields (which have non-zero cross-flow and which vary in x or z) are no longer determined solely by ${\tau _{xy}}$. Therefore, it is not possible to use the eddy-viscosity distribution from Equation (12) to correctly relate shear stress with the mean flow. Instead, a different method is used to evaluate a relevant eddy-viscosity distribution, ${\mu _t}$, which provides the best fit for a weighted balance of all six turbulent stress components:

(15) \begin{equation} {{\mu _t} = \frac{{\tau _{ij}}{{\tilde \sigma }_{ij}}}{{{\tilde \sigma }_{kl}}{{\tilde \sigma }_{kl}}}}. \end{equation}

In this equation, ${\tau _{ij}}$ is the true turbulent stress tensor from DNS and ${\tilde \sigma _{ij}} = {\tilde \tau _{ij}}$ / ${\mu _t}$, where the definition of ${\tilde \tau _{ij}}$ is taken from Equations (5) or (11) for QCR-2013 and extended QCR, respectively. Note that ${\tilde \tau _{ij}}$ is proportional to ${\mu _t}$ for all three definitions, and so ${\tilde \sigma _{ij}}$ is independent of ${\mu _t}$.

Figure 5 shows the estimated normal turbulent stresses (${\tau _{xx}}$, ${\tau _{yy}}$ and ${\tau _{zz}}$) for these flows. This figure exhibits similar behaviour to that from the simpler flow cases (Fig. 3). In the outer part of the flow, extended QCR provides substantial improvements compared to QCR-2013. However, whilst the turbulent stresses appear to be reasonably predicted down to ${y^ + } = 10$ for the separated boundary layer, neither of the eddy-viscosity models provide accurate estimates for the turbulent stresses in the near-wall region (${y^ + } \mathbin{\lower.3ex\hbox{$\buildrel \lt \over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} 60$) of the corner bisector for the square duct flow. This finding further highlights the importance of capturing the wall-normal dependency of the QCR coefficients in future improvements to the current model.

Figure 5. Effect of different turbulent stress approximations on estimating the normal turbulent stresses (i. $\overline {{u^{\prime}}{u^{\prime}}} $, ii. $\overline {{v^{\prime}}{v^{\prime}}} $, iii. $\overline {{w^{\prime}}{w^{\prime}}} $), scaled by the bulk velocity, ${u_b}$. The true stress from DNS (—) and estimated stress from QCR-2013 (Equation (5), - -) and extended QCR (Equation (11), —). Flowfields: (a) separated boundary layer^{(Reference Coleman, Rumsey and Spalart16)}; (b) corner bisector of duct flow^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)}. In each case, the stress distributions are presented with the wall-normal coordinate in both outer units ($y$) and wall units (${y^ + }$).

Figures 3 and 5, therefore, suggest that extended QCR improves the predictions of turbulent stresses, which might be expected to improve the calculation of mean flow fields. In order to test the effect on mean flow behaviour, it would be necessary to assess the accuracy of RANS simulations conducted using extended QCR. Such an assessment is outside the scope of the present paper but would be the ultimate test of the proposed quadratic eddy-viscosity model.

4.1 Analysis of turbulent stresses

Since corner flows are dominated by the presence of a counter-rotating pair of streamwise vortices^{(Reference Gessner and Jones1)}, we consider the mean streamwise vorticity equation:

(16) \begin{align} v{{\partial {\omega _x}} \over {\partial y}} + w{{\partial {\omega _x}} \over {\partial z}} = \underbrace {{\omega _y}{{\partial u} \over {\partial y}} + {\omega _z}{{\partial u} \over {\partial z}}}_{\rm{A}} & + \underbrace {\nu \left( {{{{\partial ^2}} \over {\partial {y^2}}} + {{{\partial ^2}} \over {\partial {z^2}}}} \right){\omega _x}}_{\rm{B}} \nonumber\\ & + \underbrace {\left( {{{{\partial ^2}} \over {\partial {y^2}}} - {{{\partial ^2}} \over {\partial {z^2}}}} \right)\left( { - \overline {v^{\prime}w^{\prime}} } \right)}_{\rm{C}} + \underbrace {{{{\partial ^2}} \over {\partial y\partial z}}\left( {\overline {{{v^{\prime}}^2}} - \overline {{{w^{\prime}}^2}} } \right)}_{\rm{D}}, \end{align}

where ${\omega _x}$, ${\omega _y}$ and ${\omega _z}$ are the vorticity components in the streamwise, wall-normal and spanwise directions, while $\nu $ is the kinematic viscosity of the flow. This equation shows the balance of the convection of vorticity (the left side) with production and diffusion on the right side. Term A on the right side corresponds to the generation of Prandtl vortices of the first kind, corresponding to a bending of vortex lines by the mean shear. The second term (B) represents the viscous diffusion of vorticity. The third and fourth terms (C and D) correspond to the production of Prandtl vortices of the second kind, which is related to turbulent stress anisotropy in the flow.

The streamwise vortices which exist in corner flows are believed to be generated by mechanisms relating to Prandtl vortices of the second kind^{(Reference Gessner and Jones1)}, corresponding to terms C and D above. We therefore restrict our attention to these two terms, which are referred to as the ‘shear stress term’ (term C) and the ‘normal stress term’ (term D). These two terms are evaluated using the turbulent stresses estimated using an LEVM, QCR-2013 and extended QCR, and are then compared with the equivalent ‘correct’ distribution calculated from the DNS turbulent stresses.

For this analysis, we assume that in corner flows the streamwise velocities are much greater than any transverse velocity components ($v \approx w \approx 0$) and the flow is fully developed ($\partial u/\partial x = 0$). For a linear eddy-viscosity model, defined by Equation (6), we find that ${\tau _{yz}} = 0$ and ${\tau _{yy}} = {\tau _{zz}}$, which gives:

(17) \begin{equation}{\rm{shear\ stress\ term}}:\quad {1 \over \rho }\left( {{{{\partial ^2}} \over {\partial {y^2}}} - {{{\partial ^2}} \over {\partial {z^2}}}} \right){\tau _{yz}} = 0,\end{equation}

(18) \begin{equation}{\rm{normal\ stress\ term}}:\quad {1 \over \rho }{{{\partial ^2}} \over {\partial y\partial z}}\left( {{\tau _{zz}} - {\tau _{yy}}} \right) = 0.\end{equation}

Since the vorticity production terms equate to zero, RANS simulations with an LEVM are unable to generate the quasi-streamwise vortices that exist in corners, resulting in a poor prediction of these flows.

The situation is different when quadratic modifications are introduced, however. Rather than repeating the entire analysis separately for QCR-2013 and extended QCR, both models can be studied simultaneously using Equation (11). As determined in Section 3.0, the proposed values of ${c_{cr1}} = 0.7$, ${c_{cr2}} = 2.5$ and ${\tilde c_{cr3}} = 0.8$ give extended QCR. On the other hand, QCR-2013, which does not contain the ${\tilde c_{cr3}}$ term, can be replicated by using Equation (11) with the coefficient combination ${c_{cr1}} = 0.3$, ${c_{cr2}} = 2.5$ and ${\tilde c_{cr3}} = 0$. Evaluating the relevant turbulent stress combinations for the shear stress term and the normal stress term in Equation (16) for a corner flow, $u\left( {y,z} \right)$, we find that:

(19) \begin{equation}\tau _{zz} - \tau _{yy} = \frac{\left( {2{c_{cr1}} - {{\tilde c}_{cr3}}} \right){\mu _t}}{\sqrt{{\left( \frac{\partial u}{\partial y} \right)}^2 + {\Big( \frac{\partial u}{\partial z} \Big)}^2 }} \left[ {{{\left( {{\frac{\partial u}{\partial z}}} \right)}^2} - {{\left( {{\frac{\partial u}{\partial y}}} \right)}^2}} \right],\end{equation}

(20) \begin{equation}{\tau _{yz}} = {\frac{\left( {2{c_{cr1}} - {{\tilde c}_{cr3}}} \right){\mu _t}}{\sqrt {{{\left( {{\frac{\partial u}{\partial y}}} \right)}^2} + {\Big( \frac{\partial u}{\partial z} \Big)}^2} }} \left[ {\left( {{\frac{\partial u}{\partial y}}} \right)\left( {{\frac{\partial u}{\partial z}}} \right)} \right].\end{equation}

These turbulent stress combinations only depend on the constant combination $2{c_{cr1}} - {\tilde c_{cr3}}$. This suggests that the generation of streamwise vorticity depends only on the value of $2{c_{cr1}} - {\tilde c_{cr3}}$ (i.e. some measure of the strength of the quadratic modification) rather than the individual contributions from the different terms. Therefore, while corner flows do show the importance of quadratic terms in turbulence modelling, they are poor at discriminating between them. The dependence of vorticity production on $2{c_{cr1}} - {\tilde c_{cr3}}$ suggests that, if the combined use of both (or all three) quadratic terms can generate accurate stresses, the streamwise vorticity in corner flows can be produced using just one of these terms, such as in conventional QCR. For this to be the case, it is necessary that the model which provides accurate stress estimates and conventional QCR both share a common value of $2{c_{cr1}} - {\tilde c_{cr3}}$.

Indeed, it is interesting to note that $2{c_{cr1}} - {\tilde c_{cr3}} = 0.6$ both for QCR-2013 and for extended QCR. This suggests that vorticity production in corner flows is essentially equivalent between the two forms and provides an explanation for the success of conventional QCR in such flows despite somewhat poor turbulent stress prediction. An additional consequence of this finding is that extending QCR to its more general form is unlikely to detrimentally impact the key success of conventional QCR, i.e. the prediction of the mean flow along streamwise corners.

4.2 Analysis of velocity statistics from DNS

Data from relevant direct numerical simulations are used to illustrate the findings from Section 4.1. A fully developed, incompressible square duct flow was computed by Pirozzoli et al. for a friction Reynolds number, Re$_\tau = 1100$^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)}. Figures 6(a) and (b) show the streamwise velocity and vorticity distributions, respectively, for a quarter of the duct. As well as capturing the shear in the boundary layers, the vorticity distribution displays the pair of counter-rotating vortices (labelled V⁺ and V⁻) either side of the corner bisector. The distorted boundary-layer shape, denoted by the dotted line in Fig. 6(a), demonstrates how the vortices affect the streamwise velocity profile through momentum transfer between the core flow and the boundary layers.

Figure 6. Direct numerical simulation of the flow in a quarter of a square duct^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)} with half-height, $h$, and bulk velocity, ${u_b}$: (a) the mean streamwise velocity component, $u$, with the highlighted contour at $u$/${u_b} = 1$, representative of the boundary-layer edge shape; (b) the mean streamwise vorticity, ${\omega _x}$. Contours of positive values are shown with solid lines and negative values are marked by dashed lines.

Figure 7(a) presents the turbulent stresses (${\tau _{yz}}$, ${\tau _{yy}}$ and ${\tau _{zz}}$) that appear in the streamwise vorticity equation (Equation (16)), extracted directly from DNS. The turbulent shear stress, $\overline {{u^{\prime}}{v^{\prime}}} $, is negative along the corner bisector and is roughly zero on either side. The normal stresses are generally larger in magnitude than the shear stress, with $\overline {{v^{\prime}}{v^{\prime}}} $ concentrated towards the floor and $\overline {{w^{\prime}}{w^{\prime}}} $ towards the sidewall. These distributions are compared with equivalent estimates using Equations (6), (5) and (11), which correspond to an LEVM, QCR-2013 and extended QCR, respectively.

Figure 7. Turbulent stresses relevant for streamwise vorticity production, in the square duct^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)}: i. $\overline {{u^{\prime}}{v^{\prime}}} $, ii. $\overline {{v^{\prime}}{v^{\prime}}} $ and iii. $\overline {{w^{\prime}}{w^{\prime}}} $. These stresses are (a) directly extracted from the DNS turbulence statistics, or are estimated using (b) a linear eddy-viscosity model, (c) QCR-2013, and (d) extended QCR. Contours of positive values are shown with solid lines and negative values are marked by dashed lines.

A comparison of Fig. 7(a) with (b) shows that linear eddy-viscosity models cannot accurately predict the distribution of relevant turbulent stresses. When the QCR-2013 modification is used, there is an improvement in ${\tau _{yz}}$ (Fig. 7(c–i)), which now more closely reflects the distribution in Fig. 7(a–i). However, the estimates of ${\tau _{yy}}$ (Fig. 7(c–ii)) and ${\tau _{zz}}$ (Fig. 7(c–iii)) are still inaccurate in both magnitude and topology. Figure 7(d) shows that extended QCR appears to provide a reasonable prediction of all three stresses, although the true stress distributions from Fig. 7(a) are still not perfectly replicated.

As well as investigating the individual turbulent stresses, Fig. 8 shows the equivalent comparison for the stress combination ${\tau _{zz}} - {\tau _{yy}}$, which appears in the normal stress term of Equation (16). Linear eddy-viscosity models that obey Equation (6) assume that ${\tau _{yy}} = {\tau _{zz}}$, and so the difference between the two stress components evaluates to zero. Since this distribution would correspond to an empty plot, it is not shown in Fig. 8. More interestingly, Fig. 8(b) and (c) appear to be equivalent, which shows that QCR-2013 exactly emulates the estimates from extended QCR. This is a direct consequence of the finding in Section 4.1 that, for these quadratic modifications, ${\tau _{zz}} - {\tau _{yy}}$ depends only on $2{c_{cr1}} - {\tilde c_{cr3}}$. This coefficient combination evaluates to 0.6 for both QCR-2013 and extended QCR. For the same reason, the distribution of ${\tau _{yz}}$ in Fig. 7(c–i) (QCR-2013) and Fig. 7(d–i) (extended QCR) appear to be equivalent.

Figure 8. Turbulent stress combination $\overline {{w^{\prime}}{w^{\prime}}} - \overline {{v^{\prime}}{v^{\prime}}} $, in the square duct^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)}. These terms are (a) directly extracted from the DNS turbulence statistics, or are estimated using (b) QCR-2013 and (c) extended QCR. Contours of positive values are shown with solid lines and negative values are marked by dashed lines. Note that the equivalent plot for an LEVM is blank and so has not been presented here.

From the turbulent stress distributions, the shear stress and the normal stress terms from Equation (16) are evaluated, both individually and in combination (Fig. 9). Figure 9(a) presents these vorticity production terms for the DNS stress distributions. The overall vorticity production (Fig. 9(a–iii)) takes the form of one lobe each side of the corner bisector (P$^ + $ and P$^ - $), with opposite sign. Both production terms affect these lobes, with the shear stress term contributing more towards the corner bisector and the normal stress term influencing the behaviour close to the walls.

Figure 9. The streamwise vorticity production terms from Equation (16), in the square duct^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)}: i. shear stress term $\left( {{h^2}/u_b^2\left( {{\partial ^2}/\partial {y^2} - {\partial ^2}/\partial {z^2}} \right)\left( { - \overline {v^{\prime}w^{\prime}} } \right)} \right)$, ii. normal stress term $\left( {{h^2}/u_b^2\left( {{\partial ^2}/\partial y\partial z} \right)\left( {\overline {{{v^{\prime}}^2}} - \overline {{{w^{\prime}}^2}} } \right)} \right)$, and iii. sum of shear stress term and normal stress term. These terms are (a) directly extracted from the DNS turbulence statistics, or are estimated using (b) QCR-2013 and (c) extended QCR. Contours of positive values are shown with solid lines and negative values are marked by dashed lines. Note that the equivalent plot for an LEVM is blank and so has not been presented here.

As expected from the analysis of the streamwise vorticity equation, the vorticity production terms for LEVMs evaluate to zero, so such models cannot produce streamwise vorticity using this mechanism. Since this distribution would produce an empty contour plot, it is not included in Fig. 9. The production terms for QCR-2013 (Fig. 9(c)) and extended QCR (Fig. 9(d)) produce very similar distributions to one another. This is a consequence of these two quadratic modifications displaying equivalent spatial distributions for the relevant turbulent stress combinations ${\tau _{yz}}$ (Figs. 7(c–i) and (d–i)) and ${\tau _{zz}} - {\tau _{yy}}$ (Fig. 8(b) and (c)). Therefore, the following discussion does not distinguish between QCR-2013 and extended QCR.

A closer analysis of Fig. 9 shows that the estimated vorticity production for QCR exhibits primary lobes of the correct sign. These are labelled P$_1^ + $ and P$_1^ - $, corresponding to P$^ + $ and P$^ - $ from Fig. 9(a). This rough agreement in the structure, sign and magnitude of the production terms explains the improvement in simulation of corner flows when QCR is used.

However, the topology of the production terms for the quadratic models do exhibit deviations from the ‘true’ DNS distribution. A direct comparison of Fig. 9(b–i) and (c-i) to Fig. 9(a–i) shows that QCR underestimates the contribution from the shear stress term. Instead, the primary lobes in the quadratic models (P$_1$) are determined mainly from the normal stress term. As a result, the estimate of overall vorticity production by QCR (Fig. 9(b-iii) and (c–iii)) does not correspond well with the true distribution (Fig. 9(a–iii)). In particular, QCR predicts the P$_1$ lobes to be elongated close to the corner bisector whilst the true production distribution (P$^ + $ and P$^ - $) is concentrated towards the walls. This suggests that, whereas QCR is able to predict the presence of corner vortices and the resultant boundary-layer shape, the properties of the vortices (such as their position) may not quite correspond to those in the physical flow.

Another reported limitation of QCR in corner flows occurs when simulations are run with ${c_{cr1}}$ greater than the recommended value of 0.3, which may occur, for example, in an attempt to better match the mean flow. In such cases, additional non-physical vortices are generated^{(Reference Leger, Bisek and Poggie8)}. To investigate this phenomenon further, the vorticity production terms are estimated for ${c_{cr1}} = 0.5$ and ${\tilde c_{cr3}} = 0$, i.e. $2{c_{cr1}} - {\tilde c_{cr3}} = 1.0$ rather than 0.6. The primary effect is to strengthen all the turbulent stresses and thus increase the overall vorticity production, shown in Fig. 10(c). This figure also reveals the appearance of additional lobes away from the corner bisector, labelled P$_2^ + $ and P$_2^ - $. These lobes are of opposite sign to the primary P$_1$ lobes. The P$_2$ lobes also appear in Fig. 10(a), and so seem to be associated with the shear stress term in Equation (16). Note that weak analogues to these P$_2$ lobes can be identified in Fig. 9(b–iii) and (c–iii), where $2{c_{cr1}} - {\tilde c_{cr3}} = 0.6$. The lobes in Fig. 10, however, are strong enough to be prominent in the distribution of overall vorticity production. The presence of the P$_2$ lobes may correspond to the non-physical vortices reported in RANS simulations of these geometries^{(Reference Leger, Bisek and Poggie8)}.

Figure 10. The streamwise vorticity production terms from Equation (16), in the square duct^{(Reference Pirozzoli, Modesti, Orlandi and Grasso17)}: (a) shear stress term $\left( {{h^2}/u_b^2\left( {{\partial ^2}/\partial {y^2} - {\partial ^2}/\partial {z^2}} \right)\left( { - \overline {v^{\prime}w^{\prime}} } \right)} \right)$, (b) normal stress term $\left( {{h^2}/u_b^2\left( {{\partial ^2}/\partial y\partial z} \right)\left( {\overline {{{v^{\prime}}^2}} - \overline {{{w^{\prime}}^2}} } \right)} \right)$, and (c) sum of shear stress term and normal stress term. These terms are obtained with ${c_{cr1}} = 0.5$, ${c_{cr2}} = 2.5$ and ${\tilde c_{cr3}} = 0$ in Equation (11). Contours of positive values are shown with solid lines and negative values are marked by dashed lines.

In this way, the appearance of the spurious vortices can be linked to the shear stress term in Equation (16) and, in turn, the inaccuracy of estimating the turbulent shear stress component, ${\tau _{yz}}$. However, the dependence of the production terms only on the combination $2{c_{cr1}} - {\tilde c_{cr3}}$ suggests that any simple quadratic modification would face similar issues with corner flow prediction. To overcome this, it is necessary to add complexity either by capturing the wall-normal dependence of the coefficients or by estimating the turbulent stresses using a different, more involved technique.