Assessment of RANS Turbulence Models for Straight Cooling Ducts: Secondary Flow and Strong Property Variation Effects

For the RANS simulations the compressible Navier-Stokes equations (NSE) with the total energy equation are used as implemented in ANSYS CFX, see [1, 2]. The fluid properties are evaluated based on the IAPWS IF97 formulation. To close the equation system approximations for \(\rho \overline{u'_i u'_j}\) and \(\rho \overline{u'_i h'}\) are required. Reference [17] showed, that the \(\omega \)-based BSL RSM gives the overall best results for the HARCD setup. Additional PDEs are solved for each component of \(\rho \overline{u'_i u'_j}\) and the specific dissipation \(\omega \). At the walls the so-called automatic wall treatment functionality is employed.

For modelling \(\rho \overline{u'_i h'}\) we utilise the state of the art gradient approach with a constant \(Pr_t\), two algebraic and a second moment closure model. For the gradient approach the turbulent heat fluxes are proportional to the enthalpy gradients and the isotropic turbulent diffusivity \(\alpha _t\) with \(\alpha _t = \nu _t / Pr_t\). The algebraic Daly-Harlow and the improved Younis models employ an anisotropic \(\alpha _t\)-tensor as a function of the Reynolds stress tensor, see [7, 32]. For the second moment closure model an additional PDE is solved for each component of \(\rho \overline{u'_i h'}\). The latter is a beta feature within ANSYS CFX (CADFEM GmbH, personal communication, 2018).

For the LES database of [19] the incompressible NSE with the Boussinesq approximation are applied. The transport properties are evaluated using the IAPWS correlations. For time discretisation a third-order Runge-Kutta scheme with \(CFL = 1\) is utilised and for spatial discretisation a second-order finite-volume method. As an implicit LES is performed, the size of the subgrid scales (SGS) is determined by the chosen grid resolution. As SGS model the adaptive local deconvolution method (ALDM) is used, see [13].

The setup consists of two domains simulated independently, see Fig. 1. The adiabatic periodic section is \(50\times 25.8\times 6 \, \text {mm}^3\) and the heated section is \(600\times 25.8\times 6 \, \text {mm}^3\). The straight part is followed by a curved section, which is not part of the current study and analysed in [17]. The grid resolution for the well-resolved RANS has been determined with an extensive grid sensitivity study based on the periodic section and satisfies \(y^+ \approx 1\) for the adiabatic and heated walls. In total \(34/512\times 115\times 64\) nodes are used for the periodic, respectively the heated domain.

The periodic duct serves to generate a fully developed turbulent HARCD inflow profile. The simulation is performed with liquid water treated as incompressible with fixed fluid properties at \(T_b = 333.15\,\text {K}\). All walls are defined as smooth adiabatic walls. In streamwise direction a periodic boundary condition is set with a constant mass flow of \(\dot{m} = 0.8193 \, \text {kg/s}\) corresponding to \(u_b = 5.3833 \, \text {m/s}\) and \(Re_b = 110 \cdot 10^3\). Convergence is accelerated by using physical and local time stepping methods and lowering the pressure update multiplier, and is reached when a RMS target value of \(1 \cdot 10^{-6}\) is surpassed for the momentum and continuity equation residuals.

For the heated domain simulations the compressible NSE are used. All walls are treated as smooth walls with the automatic wall treatment option applied. The lower wall is an isothermal wall with a fixed \(T_W = 373.15\,\text {K}\) and the remaining are adiabatic walls. At the inlet velocity and turbulence fields from the periodic domain are prescribed, and at the outlet an average pressure of \(p_{out} = 101325\,\text {Pa}\) is set. Convergence is accelerated by using physical and local time stepping methods, and is reached when a RMS target value of \(1 \cdot 10^{-6}\) is surpassed for the momentum, continuity and total energy equation residuals.

In the following, the RANS results of the BSL RSM turbulence model in combination with different turbulent heat flux closures are compared to the LES of [19].

Figure 2 shows the cross-sectional flow and temperature field and Fig. 3 depicts the flow and temperature profiles along the duct midplane at \(x=300\,\text {mm}\) after the beginning of the heated straight HARCD. As the choice of turbulent heat flux closure has a negligible effect on the velocity field all RANS results coincide in Fig. 3a/b. We observe for the streamwise velocity of the adiabatic duct results, that the LES follows closely the analytical law of the wall, \(u^{+} = 1/0.41 \cdot \ln {y^{+}} + 5.2\), whereas in the RANS the velocity is underestimated in the viscous sublayer and buffer layer. The heating leads to an upwards shift in the log-law region, which is not represented by the RANS.

The secondary flow field in the HARCD consists of a counter-rotating vortex pair in each corner, a smaller vortex above the heated wall and a larger one along the lateral wall. Figure 3b depicts the heated wall-normal velocity with the maximum being the footprint of the small vortex and the minimum that of the large vortex in the duct midplane. The small vortex strength is significantly underestimated in the RANS, whereas the large vortex strength agrees well with the LES data. Using other turbulence models than the BSL RSM, the results deviate further from the LES, see [17]. When heating is applied the secondary flow strength becomes weaker along the duct, [19]. The RANS captures this behaviour only for the large vortex, whereas the small vortex strength slightly increases.

In Fig. 3c normalised temperature profiles are shown with \(T^+ = T/T_{\tau }\) and \(T_{\tau } = q_W/(\rho _W \, c_{p_W} \, u_{\tau })\). Kader’s law [15] is defined as \(T^+ = Pr\,y^+\) for the viscous sublayer and \(T^+ = 2.12 \ln (y^+) + (3.85Pr^{1/3} - 1.3)^2 + 2.12 \ln (Pr)\) for the log-law region, assuming Pr and \(Pr_t\) to be constant and pure channel flow. The LES follows Kader’s law in the sublayer and shows a significant upwards shift in the log-law region due to the secondary flow presence generating a local hot spot in the midplane. Strong differences between the RANS heat flux closure models become apparent in the log-law region, however, the temperature is underestimated for all models. One reason is the significantly weaker small vortex. From the BSL RSM with \(Pr_t = 0.9\) over \(Pr_t = 0.85\) and the algebraic models to the PDE-model, an increasing downwards shift of the \(T^+\)-profile is visible and a reduction of the profile-slope. Overall the \(T^+\)-deviation from the LES grows, accompanied by an increasing deviation of the local and global heat transfer. The integral wall heat flux over the first \(500\,\text {mm}\) increases from \(3.2\,\text {kW}\) in the LES over \(3.6\,\text {kW}\) for BSL RSM with \(Pr_t = 0.9\) and \(3.9\,\text {kW}\) for the Younis-model to \(4.35\,\text {kW}\) for the PDE model. Likewise, the lower wall shear stress is overestimated in the RANS with \(\tau _{W,RANS} \approx 51.0\,\text {Pa}\) for all closure models versus the LES value of \(45.7\,\text {Pa}\). Without heating the values show less deviation with \(\tau _{W,LES} = 53.2\,\text {Pa}\) and \(\tau _{W,RANS} = 54.8\,\text {Pa}\). The observed deviations are possibly due to the usage of the automatic wall treatment option of ANSYS CFX.

The turbulent heat flux comparison in Fig. 3d shows, that \(\overline{u'T'}\) is underestimated in the RANS unless the PDE model is employed. The Younis model offers an improvement over the simpler Daly-Harlow model and the \(Pr_t = const.\) models. For the latter \(\overline{u'T'} \approx 0\) due to the negligible streamwise temperature gradient. The \(\overline{u'T'}\)-maximum close to the heated wall cannot be represented by the RANS, see [17] for further details. The wall-normal turbulent heat flux is overestimated for all considered RANS models. A similar behaviour as for the temperature and the wall heat flux is observed: the deviation from the LES increases from the \(Pr_t = const.\) models over the algebraic models to the PDE model, providing a further explanation for the overestimated heat transfer in the RANS simulations.

The LES was performed solving the three-dimensional compressible continuity, momentum and total energy equations. The finite-volume method is applied in order to spatially discretise the governing equations on a block structured, curvilinear grid. The compact four cell stencil approach by [10] is used to compute the convective fluxes. A physically consistent subgrid-scale turbulence model based on ALDM [12] is implicitly included in the convective flux calculation. More information about the LES simulation can be found in [9].

The compressible NSE are solved for all transcritical RANS simulations using ANSYS FLUENT [3, 4]. The Reynolds stresses are modelled using the \(\omega \)-based BSL RSM showing the best results in our preliminary tests. The free stream sensitivity within the BSL RSM model is removed by scaling the baseline \(\kappa -\omega \) equations. The \(\omega \)-equation can be integrated throughout the viscous sublayer allowing for a blending between the viscous sublayer and logarithmic layer formulation. The turbulent heat flux is modelled establishing a relationship between the eddy diffusivity and turbulent Prandtl number. In this study we used \(Pr_t = 0.85\) (default in FLUENT) and an algebraic formulation by Kays and Crawford (KC) [21].

Thermodynamic and transport properties are obtained using an adaptive look-up table method, which is based on the REFPROP database [23]. This method has been used for the LES and RANS simulations extracting thermodynamic and transport properties from the tabulated look-up database via trilinear interpolation. The accuracy of the extracted values has been shown in [9].

A generic channel flow configuration is used to focus this study on transcritical heat transfer and on the impact of non-linear thermodynamic effects on turbulent flows. Periodic boundary conditions are imposed in stream- and spanwise direction, and isothermal no slip boundary conditions are applied at the top and bottom walls. The channel geometry is \(2\pi h \times 2h \times \pi h \) in the streamwise, wall-normal and spanwise direction, respectively, see Fig. 4. The channel half-height h is used as characteristic length. A hyperbolic stretching law is applied in wall-normal direction in order to fulfill the resolution requirements at walls, whereas a uniform grid spacing is used in the stream- and spanwise direction. Roughness and gravity effects are not considered in the simulations.

Methane is used as working fluid with its critical pressure of \(p_{cr} = 4.5992\, \text {MPa}\) and critical temperature of \(T_{cr} = 190.564\, \text {K}\). The bulk pressure is \(p_b \approx 5.0\, \text {MPa}\), corresponding to a reduced pressure of \( p_r = p_b/p_{cr} = 1.09 \). The cold wall temperature is set to \(T_{w_{\text {c}}} = 180\, \text {K}\) (\(T_{w_{\text {c}}} < T_{cr} \)) and the hot wall temperature to \( T_{w_{\text {h}}} = 400\, \text {K} \) (\(T_{w_{\text {h}}} > T_{cr}\)), thus a temperature ratio of \( T_{w_{\text {h}}}/T_{w_{\text {c}}} = 2.22 \) is obtained. These boundary conditions encompass the pseudo-boiling temperature of \( T_{pb} \approx 193.6\, \text {K}\) at \(p_b\) and result in a density ratio of \( \rho _{w_{\text {c}}}/\rho _{w_{\text {h}}} = 12.0 \).

A body force in the momentum and energy equation is added to maintain a constant mass flux, which corresponds to a bulk velocity of \( u_b = 74\, \text {m/s}\). This results in a bulk Reynolds number of \( Re_{b} = ( u_b 2h \rho _b )/ \mu _b \approx 1.67 \cdot 10^4 \). The LES and RANS simulations are initialised with a parabolic velocity profile. A linear temperature distribution with a bulk pressure of \(5\, \text {MPa}\) is prescribed to accelerate the convergence and reduce high gradients at the beginning of the simulations. Results obtained with the SST model are used as initial guess for the BSL RSM simulations. Convergence is reached when a RMS target value of \(1 \cdot 10^6\) is surpassed for the momentum, continuity, total energy and RSM transport equation residuals.

In the following, the RANS simulations using the BSL RSM turbulence model together with a constant turbulent Prandtl number and the KC model are compared with the LES. The mean flow properties in the LES are generated by averaging in time and subsequently in streamwise and spanwise direction after reaching a quasi-stationary state.

Figure 5a shows the van Driest transformed mean velocity distribution at the cold and hot wall side over wall units. Since the turbulent heat flux model has a minor effect on the velocity similar results for the constant turbulent Prandtl number and the KC model are achieved. For this reason, the velocity profiles using the KC model have been excluded in Fig. 5a. A good agreement is observed between the LES and the RANS simulation at the hot wall. The profiles also follow the analytical log law with \(\kappa = 0.41\) and \(B = 5.2\), since the fluid exhibits an ideal gas like behavior towards the hot wall. The latter has been shown using the compressibility factor in [9]. The pseudo-boiling position is located at \(y^+ \approx 11\) in the vicinity of the cold wall, where strong property variations are present. As a consequence, the LES and RANS do not coincide and do not follow the law of the wall. Other studies [9, 20, 22, 24] showed, that no general transformation including the semi-local scaling [14] and the transformation by Trettel and Larsson [30] is able to collapse the mean velocity profiles for transcritical flows throughout the viscous sublayer and log law region onto the analytical law of the wall.

The temperature distribution is shown in Fig. 5b using \(T^+\), see Sect. 2.3. The distribution at both walls exhibits a viscous sublayer and log law with different slopes. The specific heat capacity peak at the pseudo-boiling position acting as a heat sink leads to a flattening at the cold wall. Only a small difference is observed between RANS and LES, which is slightly improved using the KC model for the turbulent Prandtl number. Due to the strongly varying molecular Prandtl number no analytical law of the wall formulation is included at the cold wall. The RANS and LES profiles at the hot wall diverge with increasing wall distance. The RANS profiles follow the empirical formulation by Kader, see Sect. 2.3, in the sublayer, but underestimate the temperature in the log layer. Using KC model for \(Pr_t\) slightly adjusts the temperature towards the Kader law.

The turbulent Prandtl number profiles are compared in Fig. 6. The turbulent Prandtl number in the LES is derived based on the enthalpy since the perfect gas relation \(h = c_p T \) is not valid in transcritical flows [8]. A good agreement between RANS and LES is achieved at the cold wall for \(y^+ > 10\), but the turbulent Prandtl number in the LES starts to increase closer to wall. This can also be observed at the hot wall, where the KC formulation increases earlier to the wall value of 1.70.

The evaluation of the integral wall values for the performed simulations in Table 1 shows, that the wall shear stress at the hot wall for the RANS with \(Pr_t = 0.85\) is close to the LES. The discrepancy in the velocity profiles at the cold wall can also be seen by means of the wall shear stress, which is approximately double as high in the LES. Higher heat flux values in the RANS simulations result in the observed smaller temperature values compared to the LES. The use of KC for \(Pr_t\) does not lead to major improvements in the integral values. Thus, the Reynolds stress modelling has to be analysed and improved as proposed by [20].