The explicit quadratic constitutive relation

By adding various forms of the scalar product of second-order tensors (i.e., mean strain-rate tensor and mean vorticity tensor), the Menter et al.'s nonlinear stress-strain constitutive relation is regarded as an explicit extension of the Boussinesq constitutive relation, which is shown as follows (Menter et al., 2018):

where τijEV, S, and Ω denote the Reynolds stress tensor computed using the Boussinesq hypothesis, the strain-rate magnitude, and the vorticity magnitude, respectively. CCORNER is an adjustable constant coefficient that controls the strength of turbulence anisotropy, with a recommended range between 0 and 1.5 and the default value as 1.0. As equation (1) shows, the proposed constitutive relation was designed in combination with the k−ω type models to close the RANS governing equations. And here the Menter's hybrid k−ω/k−ε model is adopted, which avoids the model sensitivities to freestream turbulent variables (Wilcox, 1988).

The Menter's hybrid k−ω/k−ε model from 2003 (BSL-2003)

The Menter's hybrid k−ω/k−ε model along with the eddy viscosity definition read as follows (Menter et al., 2003):

where the production of TKE Pk~ is defined as:

The turbulent eddy viscosity is computed as:

Flow solver

In the current research, all validation cases are conducted using the ANSYS FLUENT solver, in which the above explicit quadratic constitutive relation is activated and modified via the user-defined macros. The numerical scheme adopted is the pressure-based pressure-velocity segregated algorithm, with the second-order upwind scheme used for the discretization of convection terms in all transport equations. For pressure interpolation the second-order scheme is adopted, and the gradients are approximated with the use of the node-based Green-Gauss method.

The LMFA-NACA65 cascade

A linear compressor cascade which was experimentally investigated by Ma et al. (2011) and Zambonini et al. (2016) at Laboratoire de Mécanique des Fluides et d’Acoustique (LMFA) is used here for the numerical investigation of corner separation. The LMFA-NACA65 compressor cascade consists of 13 NACA65-009 blades mounted at the end of an open circuit subsonic wind tunnel. In the experiment, two strips of sandpaper with 3.0 mm width by 0.3 mm thickness were stuck at an arc length of 6 mm from the leading edge of both the blade suction and pressure surfaces. The aim is to trigger the laminar-to-turbulent transition process at the early stage of the blade surface boundary layer development, whereby removing the impact of boundary layer transition on the downstream corner separation. This contributes to reducing the difficulties of RANS simulations, as no transitional model is needed in addition to the fully turbulent RANS models. The main geometrical parameters are presented in Table 1 (Zambonini et al., 2016).

Mesh

The computational domain for the LMFA-NACA65 cascade is shown in Figure 1. The inlet plane is located at 2.16cx in the axial direction upstream of the leading edge, the same as the location of hot-wire measurements for the incoming endwall boundary layer (Ma et al., 2011; Zambonini et al., 2016). The outlet plane is placed at 1.36cx in the axial direction downstream of the trailing edge, the same as the location of the outlet of cascade test rig.

Figure 1.

Computational domain (1 every 4 grid points shown for visual clarity).

As the pitchwise periodicity of the time-mean flow was ensured throughout the measurement process (Ma et al., 2011), one single blade passage is simulated, with the pitchwise boundaries set as translational periodicity. Furthermore, as the surface oil flow measurements indicate the flow symmetry with respect to the mid-span (Ma et al., 2011), one half of the blade span is simulated, with the mid-span boundary set as symmetry.

As for the mesh topology, an O4H topology is used, and hexahedral meshes are generated using NUMECA Autogrid^v5. 376 points are wrapped around the blade and in the near-wall region above the blade surface (O block), 41 points are distributed in the wall-normal direction, with the average y+ less than 1.0 and the near-wall grid expansion ratio about 1.15. In the spanwise direction, 97 points are distributed, with the maximum y+ around 1.0 and the near-wall grid expansion ratio less than 1.2. Overall, the mesh consists of about 3.04 million hexahedral cells.

Boundary conditions

As for the mean-flow boundary conditions, the inlet mean velocity profile, flow angle, density, static temperature, and the outlet mean static pressure are prescribed to match the experimental measurements. At inlet, the velocity profile is specified based on a precursor RANS simulation on a zero-pressure-gradient flat plate turbulent boundary layer, with the results of which matching the single hotwire measurements of the cascade inlet endwall boundary layer. Figure 2 compares the computed inflow velocity profile with the experimental measurements (Zambonini et al., 2016). It can be shown that the extracted velocity profile almost reproduces the measurements. The inlet freestream flow parameters are listed in Table 2 and the inlet boundary layer integral parameters are listed in Table 3.

Figure 2.

Inflow velocity profile in logarithmic plot.

As for the inlet turbulence specification, no detailed hotwire measurements of the fluctuating velocities inside the endwall boundary layer are available. Some uncertainties thus inevitably exist on the treatment of the inlet turbulence boundary conditions. Here the extracted boundary layer velocity profile, as well as the corresponding profiles of turbulent variables (TKE k and specific dissipation rate ω), were imposed on the cascade inlet plane.

Model validation

In this section, two flow cases for the LMFA-NACA65 compressor cascade are computed and discussed. The purpose is to demonstrate that compared with the BSL-2003 model with the Boussinesq constitutive relation (BSL-2003-L), BSL-2003 with the above-mentioned nonlinear stress-strain constitutive relation (BSL-2003-NL) yields improved predictions of the size of corner separation at various flow conditions.

Case 1 (i=0∘)

Figure 3 compares the surface limiting streamlines distribution by the BSL-2003-L and BSL-2003-NL models at the design flow condition. As shown on the endwall surface, the boundary layer predicted by the BSL-2003-NL model exhibits stronger resistance to the adverse pressure gradient when migrating to the blade suction surface, thus leading to the smaller size of corner separation and the delayed onset of separation. On the suction surface, the corner separation size predicted by BSL-2003-NL is also shown to be smaller in the chordwise direction than that by the BSL model with the linear Reynolds stress closure. The validation results confirm that the addition of the anisotropic effect of turbulence in the corner region acts to suppress the over-prediction of corner separation.

Figure 3.

Surface flow visualization (i=0∘). (a) BSL-2003-L, (b) BSL-2003-NL.

To see how the overpredicted blockage affects the blade loading, Figure 4 presents the distributions of static pressure coefficient C_ps on the blade surface. Above the endwall separation region (above 20%h), as shown in Figure 4a and 4b, no separation occurs on the blade surface, and the blade loading predicted by both models matches closely with the experimental measurements. This indicates that the spanwise extent of corner separation does not exceed 20% of the blade span. Furthermore, the original BSL-2003 model predicts the pressure loading a bit lower than measurements on the pressure surface, indicating a larger corner separation predicted in the pitchwise direction (see also in Figure 3). Due to the larger passage blockage predicted, more flow is pushed towards the pressure surface, leading to the pressure reduced on the pressure surface due to increased velocity.

Figure 4.

Blade spanwise Cps distributions (i=0∘).

In the endwall separation region, the original BSL-2003 model predicts premature corner separation, while the BSL-2003 model with the explicit nonlinear turbulence closure almost reflects the real flow field. As seen in Figure 4c and 4d, the BSL-2003-L model results in the early boundary layer separation on the suction surface. In contrast, a better pressure recovery is achieved by BSL-2003-NL in the rear part of the suction surface, which matches closely with the experimental measurements. However, in the region closest to the endwall (2.7%h), the BSL-2003-L model results in the overprediction of both the streamwise and pitchwise extent of corner separation, while the BSL-2003-NL model gives results that almost reproduce measurements. This indicates that the corner separation size can be predicted with high accuracy if the anisotropic properties of the flow in the corner region are reasonably modelled.

Overall, it is encouraging to see that the prediction results for corner separation are quite promising. This means that some of the corner separation physics is reasonably captured by addition of the quadratic strain-vorticity term to the Boussinesq constitutive relation.

Next, the validation results of a more challenging flow case are shown to highlight the generality of this nonlinear stress-strain relation for the corner separation prediction.

Case 2 (i=2∘)

At the off-design flow condition, as with the design condition, the shrink of the size of corner separation is observed in both the chordwise and pitchwise directions (see in Figure 5). However, the spanwise extent of corner separation turns to be larger than that predicted by the original BSL model. This is due to the increased spanwise velocity close to the blade suction surface within the separation region, a resulting effect of the reconstruction of Reynolds stresses in the spanwise momentum transport equation.

Figure 5.

Surface flow visualization (i=2∘). (a) BSL-2003-L. (b) BSL-2003-NL.

Figure 6 shows the effect of the predicted corner separation on the blade loading distribution. Above the endwall separation (above 27%h), the loading predicted by both models matches closely with measurements. However, moving down into the endwall separation region, as shown in Figure 6c, a “pressure plateau” from 0.6cx towards the trailing edge is predicted by BSL-2003-L, indicating an overly large corner separation in the streamwise direction (0.06cx larger than measurements). In contrast, a pressure recovery that closely matches the measurements is predicted by BSL-2003-NL at 13.5%h. Furthermore, as seen from Figure 6d, the BSL-2003-NL model significantly improves the predictions of both the separation onset and blade loading. This indicates that a significantly improved prediction of the chordwise and pitchwise extent of corner separation is achieved by enhancement of turbulence anisotropy in the endwall flow region. This is achieved by a simple non-linear expansion of the Boussinesq eddy-viscosity hypothesis.

Figure 6.

Blade spanwise Cps distributions (i=2∘).

However, in the region closest to the endwall (see in Figure 6e), the BSL-2003-NL model predicts the corner separation onset earlier than measurements. One of the physical reasons is hypothesized to be the underestimation of the anisotropic effects of turbulence in the corner separation. As explained in the section “model formulation”, CCORNER controls the strength of turbulence anisotropy, with the default value of which being calibrated in the outer layer of a simple wall shear layer by requiring a fair level of the normal stress anisotropy (u′)2¯>(w′)2¯>(v′)2¯. Here it shall be pointed out that the chosen CCORNER value is preliminary for the 3D complex separated flows. For further improvement of Menter et al.'s explicit anisotropic correction model, it would be beneficial if systematic optimization of the CCORNER was carried out based on the high-fidelity large eddy simulation (LES) database of the compressor cascade flows at various operating conditions.

In the next section, the flow field around the corner region will be analysed to reveal the effect of the anisotropic modification on the mean flow field.

Flow field analysis

Figure 7 compares the size of corner separation via the distribution of the eddy-to-molecular viscosity ratio (vt/v) around the trailing edge. The cut plane is placed just downstream of the blade trailing edge (x=1.01cx) and is normal to the tangential direction of the blade camber line at the trailing edge. At the design flow condition, the original linear turbulence closure predicts larger corner separation than the quadratic turbulence closure does, with the turbulence intensity core (maximum turbulence intensity) for both being centred around 9.4% chord and 7.54% chord from the blade suction surface, respectively. Similar trend can be seen in the case i=2∘, and the results are not shown here due to space limitations.

Figure 7.

Comparison of the eddy-to-molecular viscosity ratio (i=0∘). (a) BSL-2003-L, (b) BSL-2003-NL.

In order to more intuitively compare the mean vorticity field and Reynolds stress field below, a local coordinate system is setup with its three axes s, y, and n respectively representing the streamwise, blade-spanwise, and blade-surface-normal directions (see in Figure 8). The velocity vector, vorticity vector and the Reynolds stress tensor that are computed in the Cartesian coordinate system are projected onto the new local coordinate system.

Figure 8.

Notation for the s-y-n local coordinate system.

The generation of vortices in the corner can be described with the use of the streamwise (s axis) vorticity transport equation. With the assumption of incompressible and steady, this equation can be written as (Perkins, 1970):

where (ε,η,ζ)T denotes the vorticity components in the local streamwise, wall-normal, and spanwise directions. (us¯,un¯,uy¯)T and (u′s,u′n,u′y)T denote the Reynolds-averaged velocities and Reynolds fluctuating components of velocity, respectively. In the right-hand side, let P1, P2, P3, P4 denote each part as:

According to Perkins, P1 describes the production of streamwise vorticity through the deflexion, or shewing, of the mean shear by a transverse pressure gradient and/or body force, P2 the production by primary shear stresses, P3 the production by normal stress anisotropy, P4 the production by secondary shear stress, and overall P2+P3+P4 represents the production of streamwise vorticity by Reynolds stress gradients. For detailed expression of each source term on the new local coordinate system s-y-n, see the Appendix.

In the discussions below, the effect of the explicit anisotropic modification on the mean vorticity field is first illustrated by checking the streamwise vorticity source terms P1, P2, P3, P4, and the streamwise vorticity distribution upstream of the corner separation origin.

Figure 9 shows one selected cut plane that is normal to the blade camber line, i.e., 40%cx (upstream of the corner separation origin). The contour plots of vorticity and vorticity source terms P1, P2, P3, and P4 are shown on this plane to illustrate the effect of the anisotropic modification on the generation of streamwise vortices in the corner region.

Figure 9.

Endwall surface friction lines, BSL-2003-NL, i=0∘, with location of cut plane normal to the blade camber line.

Figure 10 shows the distribution of P1 change caused by the anisotropic modification of Reynolds stresses, i.e. ((P1)NEV). The given anisotropic modification results in the generation of counter-rotating corner vortices around the corner. This is understandable as based on the RANS governing equations, the mean flow field is determined by the way the Reynolds stresses are modelled. This means that the change in P1 is a resulting effect of the change in Reynolds stresses by the anisotropic modification given in this paper. Judging from the rotating direction of the corner vortices, higher-momentum fluid in the main passage flow is entrained into the corner area, making the corner flow more resistant to the adverse pressure gradient and thus delaying the onset of corner separation.

Figure 10.

The net streamwise vorticity source term (P1)NEV at 40%cx (i=0∘).

Figures 11, 12, and 13 show the distributions of the non-linear part of the source terms (P2)NEV, (P3)NEV, (P4)NEV on the cut plane 40%cx, respectively. The linear part (P2)EV, (P3)EV, (P4)EV and the non-linear part (P2)NEV, (P3)NEV, (P4)NEV are computed based on the solution from the BSL-2003-NL simulation. They are defined as the source terms P2, P3, and P4 being solved by the linear and quadratic parts of the Reynolds stresses, respectively.

Figure 11.

The non-linear part of the streamwise vorticity source term (P2)NEV at 40%cx (i=0∘).

Figure 12.

The non-linear part of the streamwise vorticity source term (P3)NEV at 40%cx (i=0∘).

Figure 13.

The non-linear part of the streamwise vorticity source term (P4)NEV at 40%cx (i=0∘).

For example, the linear term ⟨u′u′⟩EV and the quadratic term ⟨u′u′⟩NEV of the axial normal stress ⟨u′u′⟩ are expressed as (see in equation (1) and (2)):

Similarly, each component of the Reynolds stress tensor can be split into its linear and quadratic parts. The footnotes 1, 2, 3 in the components of the strain/vorticity tensor denote axial, spanwise and pitchwise directions, respectively.

As seen from Figures 10, 11, 12, and 13, in the corner region just upstream of the origin of separation (see in Figure 9), most of the streamwise vorticity production comes from P1, P3 and P4, and those are of two orders of magnitude larger than that due to P2. In Figures 10, 12 and 13, judging from the rotating direction of corner vortices, it can be concluded that the change in skewing of the mean shear, the anisotropic modification of the normal stress anisotropy ⟨uy′uy′⟩−⟨un′un′⟩ and that of the secondary shear stress ⟨un′uy′⟩ all contribute to the counter-rotating corner vortices produced in the corner region.

Figures 14 and 15 show the effect of the given anisotropic modification on the streamwise vorticity distribution on the cut plane at 40%cx. The incorporation of higher-order terms in the stress-strain relation results in the generation of counter-rotating corner vortices. This is consistent with the distribution of streamwise vorticity source terms P1, P3 and P4 at 40%cx (see in Figures 10, 12, and 13). Judging from the rotating direction of these two corner vortices, the higher-momentum fluid in the free space is entrained into the corner region, which energizes the corner flow and in turn delays the onset of separation.

Figure 14.

The streamwise vorticity at 40%cx, BSL-2003-L, i=0∘.

Figure 15.

The streamwise vorticity at 40%cx, BSL-2003-NL, i=0∘.

The finding here shows consistency with that from the external flow community (Yamamoto et al., 2012; Mani et al., 2013; Dandois, 2014). In the numerical study of the turbulent square-duct flow (Mani et al., 2013), secondary corner vortices were observed around the corner of the duct by replacement of the Boussinesq closure by the quadratic constitutive relation (Spalart, 2000). The authors claimed that this is attributed to the capability of the quadratic constitutive relation to reproduce the physics of the normal stress anisotropy in the corner region, which leads to the appearance of corner vortices.

Next, the distributions of P1, P2, P3, P4 well within the corner separation are shown in relation to the flow behaviour within the corner separation region and in the separated shear layer between the mainstream flow and the corner separation flow.

Figure 16 shows the selected cut plane, and again it is normal to the blade camber line, i.e. 68%cx. The contour plots of streamwise vorticity and vorticity source terms are shown on this plane to illustrate the effect of the anisotropic modification on the generation of streamwise vortices within the corner separation region.

Figure 16.

Endwall surface friction lines, BSL-2003-NL, i=0∘, with location of cut plane normal to the blade camber line.

Within the corner separation region (68%cx), as shown from Figures 17, 18, 19, and 20, most of the streamwise vorticity production comes from P1, P3 and P4. The contribution of the anisotropic modification of the primary shear stresses ⟨us′uy′⟩, ⟨us′un′⟩ in P2 to the streamwise vorticity production (shown in Figure 18) is significantly larger than that upstream of the corner separation region (40%cx) (see in Figure 11). However, it is still significantly smaller compared to the change of other three terms.

Figure 17.

The net streamwise vorticity source term (P1)NEV at 68%cx (i=0∘).

Figure 18.

The non-linear part of the streamwise vorticity source term (P2)NEV at 68%cx (i=0∘).

Figure 19.

The non-linear part of the streamwise vorticity source term (P3)NEV at 68%cx (i=0∘).

Figure 20.

The non-linear part of the streamwise vorticity source term (P4)NEV at 68%cx (i=0∘).

As seen in Figure 17, the contribution of the mean shear skewing to the streamwise vorticity generation mainly concentrates in the separated shear layer between the corner separation region and the main flow where counter-rotating vortices coexist, which enhances the turbulent shear between the corner separation flow and the main passage flow. This contributes to the stronger momentum exchange between the flow outside the separation region and the corner separation flow, energizing the flow inside the corner separation and making it resistant to bulk adverse pressure gradients.

Similar trend can be seen in Figures 19 and 20. The anisotropic modification of the normal stress anisotropy in P3 and the secondary shear stress in P4 leads to more and stronger streamwise vortices generated both along the outer layer of corner separation and within the separation region. This indicates that strong shear shall exist both between the main flow and the corner separation flow and within the separation region. Again, this leads to stronger momentum exchange between the main passage flow and the corner separation, which energies the corner flow and makes it more resistant to separation.

In summary, it can be clearly seen that the inclusion of the Reynolds stress anisotropy contributes to the induction of the streamwise vorticities in the corner region. Next, the streamwise vorticity distributions are shown in Figures 21 and 22 to illustrate the effect of the Reynolds stress change on the streamwise vorticity field.

Figure 21.

The streamwise vorticity at 68%cx, BSL-2003-L, surface friction lines, i=0∘.

Figure 22.

The streamwise vorticity at 68%cx, BSL-2003-NL, surface friction lines, i=0∘.

Within the corner separation region, the streamwise vorticity contour lines of BSL-2003-NL concentrate in the separated shear layer between the corner region and the main flow region that are close to the endwall (see in Figure 22). This is consistent with the distribution of the streamwise vorticity source terms P3 and P4 at 68%cx (see in Figures 19 and 20). This indicates that stronger shear interaction occurs in the separated shear layer between the main flow and the corner separation flow, thus resulting in the higher-momentum fluid in the free space being entrained into the corner region. The corner flow is energized and in turn is more resistant to separation.

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