A priori tests of eddy viscosity models in square duct flow

The numerical solution of Reynolds-averaged Navier–Stokes (RANS) equations is a standard approach to evaluate flows of industrial interest. Two methodologies for RANS closure are available in the literature, namely eddy viscosity transport models and Reynolds stress transport models. The former stem from the analogy between Reynolds stresses and viscous stresses and are often referred to as first-order closures, whereas the latter require the solution of transport equations for each component of the Reynolds stress tensor and are therefore second-order closures.

Eddy viscosity models are undoubtedly more popular in industry than Reynolds stress transport models, as they are easier to implement in existing flow solvers and they require less computational effort [37]. In their simplest form, eddy viscosity or explicit algebraic stress models (EASMs) rely on the assumption that the anisotropic stress tensor \(a_{ij}\equiv \tau _{ij} -2/3k\delta _{ij}\) (where \(\tau _{ij}\equiv \overline{u_i'u_j'}\) is the Reynolds stress tensor and \(k\equiv 1/2\tau _{ii}\) the turbulent kinetic energy) is linearly proportional to the local mean rate-of-strain:where \(\nu _t\) is the eddy viscosity, \(S_{ij}\) is the mean rate-of-strain:and \(u_i\) are the components of the velocity vector. The overline symbol is used to indicate ensemble averages, the prime symbol indicates turbulent fluctuations and \(x_i\) are the spatial coordinates in the three directions \(i=1,2,3\). Variables with tilde indicate modelled quantities in which the only modelling assumption is the constitutive relation (e.g. the anisotropic Reynolds stress tensor \({\widetilde{a}}_{ij}\), modelled with the linear eddy viscosity hypothesis in Eq. (1)). Despite their popularity, linear eddy viscosity models are inaccurate in predicting flows with large anisotropies and inequalities of the normal stresses [23]. A notable case in which linear eddy viscosity models fail to reproduce the correct flow physics is represented by turbulent secondary flows, as occurring in ducts of non-circular cross section [24], with the simplest prototype represented by the turbulent square duct flow [21]. The flow mechanism generating these secondary flows can be traced back to nonzero mean streamwise vorticity \({\overline{\omega }}_x\) [7, 36], which is related to the cross-stream velocity components through the stream function \(\psi \):The analysis of the mean streamwise vorticity equation,shows that the only sources of \({\overline{\omega }}_x\) are the derivatives of the secondary shear stress \(\overline{v'w'}\) and the anisotropy of the normal stresses \(\overline{v'}\) and \(\overline{w'}\) and they both have the same order of magnitude [21]. The linear eddy viscosity hypothesis \(\tau _{ij} = 2/3k\delta _{ij} - 2\nu _tS_{ij}\) inevitably returns \(\tau _{23}=0\) and \(\tau _{22}=\tau _{33}=2/3k\), leaving a homogeneous equation for \({\overline{\omega }}_x\), with homogeneous boundary conditions; therefore, the only possible solution is \({\overline{\omega }}_x=0\). Hence, the linear eddy viscosity (1) cannot predict secondary flows [37]. These limitations of the linear eddy viscosity hypothesis led [22] to derive a more general form for the constitutive relation (1), in which the anisotropic stress tensor of a three-dimensional flow is exactly represented as a nonlinear polynomial expansion of ten independent tensor bases and ten coefficients.

Moreover, Pope [22] provided analytical expressions for the tensor polynomial coefficients in the case of two-dimensional mean flow (two mean velocity components), for which exact representation of \(a_{ij}\) only requires three tensor bases. Using the same set of integrity bases, Gatski and Speziale [8] found the analytical expressions of the ten tensor polynomial coefficients for a general three-dimensional flow. Different authors [9, 12] showed that, except for some degenerate cases, five tensor bases are sufficient to represent the anisotropic stress tensor, as \(a_{ij}\) is traceless and symmetric; therefore, it has only five independent components. Additional bases are only necessary if \(S_{ij}\) has multiple eigenvalues (i.e. axisymmetric strain) or if the vorticity vector is aligned with one of the principal directions of strain [12]. After these seminal studies, many nonlinear RANS models have been proposed, the most common being quadratic [17, 19, 27, 31, 32, 37] and cubic models [2]. Nonlinear eddy viscosity models have been successfully used to reproduce turbulent secondary flows in different geometries such as non-circular ducts [13, 32, 33], wing–body junction [3, 28, 41], turbine–hub flow [38], and to recreate the spanwise rollers in Couette flow [34].

Nevertheless, only few studies used DNS data to assess the accuracy of eddy viscosity models for predicting turbulent secondary flows. Mompean et al. [16] carried out a priori tests of nonlinear k‐\(\varepsilon \) models using direct numerical simulation (DNS) data at friction Reynolds number, \({\text {Re}}_\tau =u_\tau h/\nu \approx 150\) (where \(u_\tau =\sqrt{\tau _w/\rho }\) is the mean friction velocity, \(\tau _w\) the mean wall shear stress, h the duct half side, \(\nu \) and \(\rho \) the kinematic viscosity and density of the fluid, respectively). Mompean [15] also carried out additional tests on the nonlinear model proposed by Speziale [37] showing that it underpredicts the intensity of the secondary flows. Rice et al. [25] carried out particle image velocimetry of the flow along supersonic streamwise corners and reported good agreement with RANS simulations using the quadratic correction of Spalart [32].

In this study, we aim at assessing the accuracy of popular RANS models using the square duct flow as benchmark. The accuracy of RANS models can be quantified using two tools, namely a priori and a posteriori tests, both with advantages and disadvantages. In a priori tests, modelled quantities such as eddy viscosity, Reynolds stresses and model coefficients are compared to reference high-fidelity data, without advancing the RANS equations or additional transport equations. The main drawback of this approach is that it only provides information on the accuracy of the modelled Reynolds stresses and not on the mean flow, which is the final goal of RANS. An advantage of this approach is that it gives us information on the model accuracy in the best-case scenario. Hence, if a priori tests show inaccurate prediction of the Reynolds stress tensor, solving additional transport equations can only worsen the results. On the other hand, a posteriori tests have the advantage of providing mean flow data, but solving the equations we introduce the effect of the numerical scheme. Hence, to carry out fully consistent a posteriori tests one should use the same solver and the same mesh to generate reference high-fidelity data and RANS results. In this study, we limit ourself to a priori tests; therefore, we are unable to inspect the mean flow velocity but we can only assess the accuracy of modelling assumptions on the Reynolds stress tensor.

We carry out a priori tests using recent DNS data of square duct flow at friction Reynolds number \({\text {Re}}_\tau =150\)–1055 [14, 21], thus extending the Reynolds number range of previous studies. We show that results become independent from Reynolds number as \({\text {Re}}_\tau \) increases and therefore we mainly focus on the highest Reynolds number data. We carry out two types of a priori tests, the first on nonlinear constitutive relations for the anisotropic stress tensor \(a_{ij}\) to determine the maximum accuracy that one should expect for a given number of tensor bases, and the second on popular nonlinear RANS models to determine the validity of their assumptions and whether they approach the maximum accuracy.

Pope [22] generalized the concept of linear eddy viscosity (1) modelling the anisotropic stress tensor as a polynomial expansion of tensor bases. The Cayley–Hamilton theorem guarantees that a finite number of bases are sufficient to recover exact representation of the stress tensor and \(\mathbf {a}\) can be expressed as a polynomial of \(\mathbf {S}\) and \(\varvec{\Omega }\) [12, 35]:where the coefficients \(G^{(n)}\) are scalar functions and \(\mathbf {T}^{(n)}\) is a set of ten independent bases:where \(\mathbf {S}\) is the mean rate-of-strain tensor (2) and \(\varvec{\Omega }\) the mean rate-of-rotation tensor:\(A_{ij}\) indicates the matrix associated with the tensor \(\mathbf {A}\) and the notation \(\mathbf {AB}=A_{ik}B_{kj}\) is used for tensor multiplication, whereas \(\left\{ \mathbf {AB}\right\} = A_{ik}B_{ki}\) indicates the trace operation.

An exact representation of the anisotropic stress tensor (\(\widetilde{\mathbf {a}}=\mathbf {a}\)) can be recovered for \(N=5\), apart from the degenerate cases of axisymmetric flow and vorticity vector aligned with an eigenvector of \(\mathbf {S}\), for which additional bases are required. Truncated versions of the tensor polynomial (5) constitute the theoretical framework for the development of nonlinear eddy viscosity models [2, 32], machine learning algorithms for the optimization of EASMs [10, 39] and uncertainty quantification of EASMs [4, 40].

In order to carry out a priori tests on the constitutive relation (5), we compute the coefficients \(G^{(n)}\), using the anisotropic stress tensor from DNS data of square duct flow [21]. \(G^{(n)}\) can be obtained by multiplying equation (5) by the basis \(T^{(m)}\) and taking the trace operation [9], thus obtaining a linear system of N equations for the unknown coefficients \(G^{(n)}\):For \(N=1\), this approach leads to the classical definition of linear eddy viscosity:In the more general case, we solve Eq. (8) to find the coefficients \(G^{(n)}\) for different \(1\le N\le 5\) and then evaluate the modelled anisotropic stress tensor \(\widetilde{\mathbf {a}}\) from (5). By solving the algebraic system (8), we let the coefficients \(G^{(n)}\) vary in space to approximate the “exact” anisotropic stress tensor from DNS \(\mathbf {a}\). Hence, the coefficients \(G^{(n)}\) are the ones that bring the most accurate representation of the anisotropic stress tensor (5), for a given number of bases.

This kind of analysis was proposed by Jongen and Gatski [9], who used experimental data to assess the validity of the eddy viscosity hypothesis. Schmitt [29, 30] also carried out the same analysis for linear and nonlinear constitutive relations highlighting the limitations of the eddy viscosity hypothesis. Nevertheless, this approach has never been applied systematically for different truncations of Pope’s polynomial expansion. In the following analysis, only three components of the Reynolds stress tensor are reported, namely \(a_{12}\), \(a_{22}\) and \(a_{23}\), as in square duct flow \({a}_{12}(y,z) =- {a}_{13}(z,y)\), \({a}_{22}(y,z)={a}_{33}(z,y)\) (where y and z are the cross-stream coordinates) and \(a_{11}\) does not appear in Eq. (3) (i.e. it does not influence the secondary flows). In the following, we use the term primary shear stress to denote \(a_{12}\), as it appears in the mean streamwise momentum equation and it directly influences the skin friction coefficient, whereas we use secondary shear stress for \(a_{23}\), as it appears in the cross-stream mean momentum equation.

Moreover, we only report in detail the anisotropic stresses for selected N, corresponding to popular choices for the development of eddy viscosity models. In particular, \(N=1\) corresponds to the linear eddy viscosity hypothesis, \(N=2\) the nonlinear correction proposed by Spalart [32], \(N=4\) has been used to develop quadratic eddy viscosity models, and for \(N=5\) we recover the exact representation \(\widetilde{\mathbf {a}}=\mathbf {a}\). Note that the exact representation can only be recovered when the ideal coefficients obtained from (8) are used; therefore, using five bases alone, with arbitrary coefficients, is not sufficient to recover \(\widetilde{\mathbf {a}}=\mathbf {a}\).

Figure 1 shows the scatter plots of the modelled anisotropic stress tensor components, \({\widetilde{a}}_{12}\), \({\widetilde{a}}_{22}\), \({\widetilde{a}}_{23}\) as a function of the ones from DNS \({a}_{12}\), \({a}_{22}\), \({a}_{23}\) at \({\text {Re}}_\tau =1055\), normalized with the friction velocity from DNS (\(\mathbf {a}^+=\mathbf {a}/u_\tau \)). Figure 1a–c shows that the linear eddy viscosity hypothesis (\(N=1\)) returns accurate representation of \({\widetilde{a}}_{12}\), the stress component directly affecting the streamwise velocity. The success of the linear eddy viscosity hypothesis in predicting the primary shear stress can be traced back to the strong similarity between square duct and canonical wall-bounded flows, namely to the fact that this flow can be seen as superposition of four concurrent walls [21]. If we assume that \(S_{12}\) and \(S_{13}\) are the most relevant components of \(S_{ij}\), the eddy viscosity (9) becomeswhich reduces to \(-\overline{u'v'}/(2\partial {\overline{u}}/\partial y)\) of canonical wall-bounded flows, away from the side wall. On the other hand, \({\widetilde{a}}_{22}\approx 0\) and \({\widetilde{a}}_{23}\) are uncorrelated with respect to reference data, in agreement with the fact that the linear eddy viscosity hypothesis cannot predict the occurrence of secondary flows (Fig. 1b, c).

Figure 1d–f shows that using \(N=2\) improves the prediction of both the normal stress and secondary shear stress. Nevertheless, \({\widetilde{a}}_{22}\) is consistently overpredicted, as well as the most intense values of \({\widetilde{a}}_{23}\). For \(N=4\), exact representation of \({\widetilde{a}}_{22}\) is recovered, together with a very good prediction of \({\widetilde{a}}_{23}\) (Fig. 1g–i). As predicted from theory, five bases allow us to recover the stresses from DNS (\(\widetilde{\mathbf {a}}\equiv \mathbf {a}\), Fig. 1j–l).

The scatter plots allow us to easily compare the modelled stresses to DNS data, but they do not provide information on the spatial distribution of the stress components and therefore it is not possible to understand how the error is spatially distributed. For this reason, we also report \({\widetilde{a}}_{12}\), \({\widetilde{a}}_{22}\), \({\widetilde{a}}_{23}\) at \({\text {Re}}_\tau =1055\) in a duct quadrant, for \(N=1\), 2, 4 (Fig. 2a–i) compared to \(N=5\) (DNS data) \(a_{12}\), \(a_{22}\) and \(a_{23}\), (Fig. 2j–l). The spatial distribution of the primary shear stress \({\widetilde{a}}_{12}\) is well predicted by the linear model, and only minor differences are observed between the linear and nonlinear constitutive relations. As observed from the scatter plots, the linear eddy viscosity hypothesis returns \({\widetilde{a}}_{22}\approx 0\) (Fig. 2b). The spatial distribution of \({\widetilde{a}}_{23}\) (Fig. 2c) is similar to DNS only in the core region of the duct, where a large patch of negative stress approximately matches DNS values, whereas an incorrect stress sign is predicted towards the walls, where the flow is more anisotropic. Using \(N=2\) leads to slightly less accurate \({\widetilde{a}}_{12}\) with a small tail close to the side wall, which is absent in the linear case and in DNS (Fig. 2d).

A significant improvement in the prediction of both \({\widetilde{a}}_{22}\) and \({\widetilde{a}}_{23}\) is observed for \(N=2\). We find that the largest error in \({\widetilde{a}}_{22}\) is localized towards the side wall, where the sharp variation of \({\widetilde{a}}_{22}\) is not captured (Fig. 2e). The spatial organization of the secondary shear stress also improves compared to the linear model, but the values of \({\widetilde{a}}_{23}\) are larger than DNS (Fig. 2f). Similarly to the linear case, the sign and intensity of \({\widetilde{a}}_{23}\) are correct in the duct core, where the flow is more isotropic, whereas the error with respect to DNS becomes larger close to the corner and the walls, where very high stress values are predicted. Using four tensor bases allows us to obtain a spatial distribution of \({\widetilde{a}}_{22}\) which is undistinguishable from DNS data, whereas differences are still observed for \({\widetilde{a}}_{23}\), although the main stress topology is captured (Fig. 2h, i). Also in this case the greatest error on \({\widetilde{a}}_{23}\) is localized in proximity of the corner and at the walls, where the stress is overpredicted.

In order to quantify the error with respect to DNS, we also define the averaged correlation coefficient between \(a_{ij}\) and \({\widetilde{a}}_{ij}\):where the angle brackets denote average over the duct cross section:By construction, \({\widetilde{C}}\in [-1,1]\), where \({\widetilde{C}}=1\) indicates the perfect correlation \({\widetilde{C}}=-1\) the negative correlation.

In Table 1 and Fig. 3a, we report the mean correlation coefficient \({\widetilde{C}}_{ij}\) at \({\text {Re}}_\tau =1055\) for increasing number of tensor bases. As first gauged from qualitative inspection of the scatter plots and mean flow fields, we find that correlation with respect to DNS increases with N. The primary shear stress has a correlation coefficient \({\widetilde{C}}_{12}\approx 1\) already for \(N=1\); therefore, the linear eddy viscosity hypothesis can accurately predict \({\widetilde{a}}_{12}\), whereas \({\widetilde{a}}_{22}\) and \({\widetilde{a}}_{23}\) are essentially uncorrelated with respect to DNS. For \(N=2\), we find a good correlation for the normal stress component \({\widetilde{C}}_{22}\approx 0.9\) and relatively good also for \({\widetilde{C}}_{23}\approx 0.74\). On the other hand, passing from \(N=2\) to \(N=4\) leads to perfect correlation on the normal stress \({\widetilde{C}}_{22}=1\), but it does not increase the accuracy on \({\widetilde{a}}_{23}\).

Figure 3b, c shows \({\widetilde{C}}_{ij}\) as a function of the friction Reynolds number \({\text {Re}}_\tau =\left[ 150,227,519,1055\right] \) for fixed number of bases \(N=2\) and \(N=4\), respectively. We observe a minor Reynolds number effect, which tends to saturate for increasing Reynolds number, suggesting that Reynolds number independence is achieved for \({\text {Re}}_\tau \gtrsim 500\).

The analysis has been carried out using sets of tensor bases ordered as they appear in the polynomial expansion (5). This is motivated by the fact that nonlinear models are typically built following this approach, but for a given number of tensor bases different combinations are possible and results can be affected by the choice of the subset. For this reason, we explore also this possibility and in Table 2 we report the mean correlation coefficient \({\widetilde{C}}_{ij}\) for all possible subsets with \(N=2,3,4\). The subsets are limited to the first five bases, as only using these ones it is possible recover exact representation of the anisotropic stress tensor (i.e. using \(T^{(1)}\), \(T^{(2)}\), \(T^{(3)}\), \(T^{(4)}\), \(T^{(5)}\), \(\widetilde{\mathbf {a}}=\mathbf {a}\), whereas using \(T^{(1)}\), \(T^{(2)}\), \(T^{(3)}\), \(T^{(4)}\), \(T^{(6)}\), \(\widetilde{\mathbf {a}}\ne \mathbf {a}\)).

Table 2 shows that \({\widetilde{a}}_{12}\) is barely affected by the choice of the subset, whereas the accuracy on \({\widetilde{a}}_{22}\) and \({\widetilde{a}}_{23}\) can have a large variation. The best subset for a given number of bases is highlighted in boldface, showing that \(T^{(2)}\) and \(T^{(5)}\) are essential to describe both \({\widetilde{a}}_{22}\) and \({\widetilde{a}}_{23}\), whereas bases \(T^{(3)}\) and \(T^{(4)}\) alone do not bring any improvement with respect to the linear case. The reason for this is that \(T^{(2)}\) and \(T^{(5)}\) have the same structure and contain both \(\mathbf {S}\) and \(\varvec{\Omega }\), contributing more to the flow anisotropy.

Moreover, in Fig. 4 we report the coefficient \(G^{(1)}\) in a duct quadrant, for different truncations of Pope’s polynomial expansion. Ideally, for a smooth flow, the tensor bases are smooth functions and so are the coefficients. Nevertheless, increasing N can give rise to sharp spatial variations of \(T^{(n)}\). This causes system (8) to be ill-conditioned and difficult to solve numerically, a problem which we overcome using a truncated singular value decomposition to invert the system matrix. The figure shows that for \(N>2\), \(G^{(1)}\) exhibits sharp spatial variations, and a similar result holds for the other coefficients (not shown). More aggressive matrix regularization techniques are possible and lead to smoother coefficients, but in this case we do not recover the exact representation of \(a_{ij}\) for \(N=5\). We recall that even though coefficients \(G^{(n)}\) are spiky, \({\widetilde{a}}_{ij}\) is always smooth (Fig. 2). The analysis reveals that, even for a smooth flow solution as incompressible duct flow, the exact coefficients can exhibit unphysical spatial variations and therefore modelling, or even using, the exact coefficients \(G^{(n)}\) for \(N>2\) might be impossible or at least very difficult in practical calculations.

The idea of generalized eddy viscosity hypothesis of Pope [22] has inspired several nonlinear eddy viscosity models using truncated forms of the tensor polynomial (5). Nevertheless, eddy viscosity models inevitably rely on additional assumptions to represent the coefficients \(G^{(n)}\) needed by the polynomial expansion and therefore they can only approach the accuracy of the nonlinear constitutive relations using the same number of tensor bases. In reality, it is not possible to tune the coefficients \(G^{(n)}\) to achieve the maximum accuracy for all variety of flows, and therefore some models try to account for this by using more than five bases. We carry out a priori analysis on different turbulence models and compare their accuracy to the one of the constitutive relations using exact coefficients \(G^{(n)}\). First, we focus on the popular k‐\(\varepsilon \) turbulence model [11] and then on nonlinear [32] and linear [20] corrections specifically developed to improve the prediction of turbulent secondary flows. In the following, we denote the anisotropic stress tensor of the RANS models with a hat symbol \({\hat{a}}_{ij}\) to distinguish it from \({\widetilde{a}}_{ij}\), obtained using the exact coefficients \(G^{(n)}\). We also introduce the correlation coefficient between \({\hat{a}}_{ij}\) and DNS data \(a_{ij}\):where the angle brackets denote averaging over the duct cross section (12). For each RANS model, we report the correlation coefficient with respect to DNS \({\widehat{C}}_{ij}\), Table 3. The value of \({\widehat{C}}_{ij}\) allows us to assess how the model performs with respect to DNS data, but this is an unfair comparison as RANS models are not supposed to match DNS data and can at most reach the ideal accuracy of the constitutive relation they are built on. Hence, in Fig. 5 we report \({\widehat{C}}_{ij}\) as a function of \({\widetilde{C}}_{ij}\), with the same number of tensor bases (e.g. \({\widehat{C}}_{ij}\) of the linear k–\(\varepsilon \) model as a function of \({\widetilde{C}}_{ij}\) for \(N=1\)); therefore, the axes bisector represents the maximum mean accuracy.

We use DNS data of square duct flow up to friction Reynolds number \({\text {Re}}_\tau =1055\) from Pirozzoli et al. [21] to carry out a priori tests of algebraic Reynolds stress models and assess their ability to predict turbulent secondary flows. We perform a priori analysis of constitutive relations for eddy viscosity models which gives us useful information on the maximum accuracy achievable for a given set of tensor bases.

According to this analysis, the linear eddy viscosity hypothesis brings accurate prediction of the shear stress component \({\widetilde{a}}_{12}\), with a mean correlation coefficient with respect to DNS \({\widetilde{C}}_{12}=0.998\), whereas the normal and secondary shear components are essentially decorrelated from reference data, \({\widetilde{C}}_{22}=-0.044\) \({\widetilde{C}}_{23}=0.313\). Moreover, we test all possible nonlinear bases combinations and show that optimal subsets always include \(T^{(2)}\) and \(T^{(5)}\), in addition to the linear term \(T^{(1)}\). Two tensor bases bring a fairly accurate representation of the normal and secondary shear stresses, \({\widetilde{C}}_{22}=0.911\), \({\widetilde{C}}_{23}=0.743\). With three bases (\({\widetilde{C}}_{22}=0.910\), \({\widetilde{C}}_{23}=0.871\)) and four bases (\({\widetilde{C}}_{22}=1.000\), \({\widetilde{C}}_{23}=0.938\)), we achieve exact or almost exact representation of the normal stress and secondary shear stress.

Unfortunately, RANS models rely on additional assumptions on the coefficients \(G^{(n)}\), leading to a reduced accuracy compared to the one achieved with exact coefficients. The analysis of popular k–\(\varepsilon \) models shows that only few of them are able to improve the prediction of \({\widehat{a}}_{22}\) and \({\widehat{a}}_{23}\) compared to the linear model, namely the model of Craft et al. [2] (\({\widehat{C}}_{22}=0.643\), \({\widehat{C}}_{23}=0.532\)) and of Nisizima and Yoshizawa [19] (\({\widehat{C}}_{22}=0.696\), \({\widehat{C}}_{23}=0.616\)).

Moreover, we assess the performance of the nonlinear correction of Spalart [32], based on two tensor bases. Interestingly, we find that the model is as accurate (\({\widehat{C}}_{22}=0.635\), \({\widehat{C}}_{23}=0.691\)) as the one by Nisizima and Yoshizawa [19] using four tensor bases, but the QCR is closer to its maximum accuracy. A priori analysis of nonlinear models based on Pope’s polynomial expansion shows that using three or four tensor bases could potentially lead to very accurate prediction of the turbulent stresses, but this is difficult to achieve in practice as the gain in tuning power is partially spoiled by the uncertainty on the additional model coefficients. Hence, using a quadratic constitutive relation as the one proposed by Spalart [32] leads de facto to the same accuracy of higher-order models.

We also carry out a priori analysis of the \(v^2\)–f model, originally developed to overcome some of the limitations of the k–\(\varepsilon \) model. The assumption \({\widehat{\nu }}_t=c_\mu ^v\overline{v^{'2}}T\) is more accurate than the one used by the k–\(\varepsilon \) model, and it returns a slightly better prediction of the primary shear stress (\({\widehat{C}}_{12}=0.997\) versus \({\widehat{C}}_{12}=0.988\) for k–\(\varepsilon \)).

Particularly interesting is the linear correction of Pecnik and Iaccarino [20] to the \(v^2\)–f model, as it shows that representation of the normal stress and secondary shear stress does not necessarily require nonlinear constitutive relations. The model is the most accurate in predicting the normal and secondary shear stress (\({\widehat{C}}_{22}=0.984\) and \({\widehat{C}}_{23}=0.844\)), but the correction does not use Pope’s theoretical framework and therefore direct comparison with other nonlinear models is not possible.

DNS data and exact coefficients from Pope’s polynomial expansion are available at http://​doi.​org/​10.​4121/​uuid:​3226d808-f354-47b1-b6f8-3a8048215193.