### 6. Assessment of hybrid RANS / LES

** 6.1. Test-case assignment**– TP B3 (Hybrid RANS / LES): 3.11, ??, 4.3

*6.2. TP B3 (Hybrid RANS / LES)*

*6.2.1. Flat plate boundary layer 2**Flow features*

The flow is characterized by a highly unsteady behavior of the turbulent shear layer (Re_{θ}≈ 9000) shedding from the forebody and subsequently reattaching on the splitter plate, which leads to pronounced wall pressure oscillations and consequently, structural loads.

*Numerical methods and discretization scheme*

The three-dimensional unsteady compressible Navier-Stokes equations are solved based on a large-eddy simulation (LES) using the MILES (monotone integrated LES) approach. The vertex-centered finite-volume flow solver is block-structured. A modified advection-upstream-splitting method (AUSM) is used for the Euler terms which are discretized to second-order accuracy by an upwind-biased approximation. For the non-Euler terms a centered approximation of second-order is used. The temporal integration is done by a second-order accurate explicit 5-stage Runge-Kutta method, the coefficients of which are optimized for maximum stability. The RANS simulations use the one-equation turbulence model of Fares and Schröder [34] to close the averaged equations.

The zonal RANS-LES method, referred to as STGMR, is based on the work of Jarrin et al. [32] and Pamiès et al. [35], which describes turbulence as a superposition of coherent structures. The structures are generated over the LES inlet plane by superimposing the influence of N virtual eddy cores that are defined in a specified volume around the inlet plane. The influence of the virtual eddy cores on the velocity field is defined by a shape function, which describes the spatial and temporal characteristics of the turbulent structure. The velocity signal at the LES inflow plane is composed of an averaged velocity component which is provided from the upstream RANS solution and the normalized stochastic fluctuation that is subjected to a Cholesky decomposition to assign the values of the Reynolds-stress tensor. Depending on the distance from the wall, the inflow plane is divided in several domains, each of which is characterized by specific shape factors, turbulent length-, and time scales. Morkovin’ s hypothesis is applied at the inlet to relate density and velocity fluctuations and to enforce the strong Reynolds analogy. The density field is obtained by enforcing a constant-pressure condition at the inflow.

*Computational grid*

The numerical details of each simulation are presented in Table 12. The grids are clustered in the wall-normal direction using a hyperbolic tangent stretching function such that the minimum grid spacing near the surface in wall units is approximately one and a stretching factor of 1.05 is not exceeded.

*TABLE 12. Computational domain, grid resolution, and number of mesh points for pure LES, and zonal RANS-LES configurations of a turbulent boundary layer simulation at M = 2.0 and Re _{θ }= 870. The zonal RANS-LES configuration consists of the RANS domains upstream (Zo-RANS-u) and downstream (Zo-RANS-d) and of the embedded LES domain (Zo-LES).*

_{} |
L_{x}/δ_{0} |
L_{y}/δ_{0} |
L_{z}/δ_{0} |
Δx^{+} |
Δy_{wall}^{+} |
Δz^{+} |
N_{x } |
N_{y} |
N_{z} |
total |

pure LES | 14.1 | 10.0 | 1.6 | 15.1 | 1.0 | 10.2 | 245 | 110 | 45 | 1.21⋅10^{6} |

Zo-RANS-u | 8 | 10.0 | 1.6 | 34.3 | 1.0 | 150 | 76 | 110 | 3 | 25,080 |

Zo-RANS-d | 11 | 10.0 | 1.6 | 60.1 | 1.0 | 150 | 49 | 110 | 3 | 25,080 |

Zo-LES | 8.7 | 10.0 | 1.6 | 15.0 | 1.0 | 10.1 | 151 | 110 | 45 | 7.47⋅10^{5} |

*Boundary and initial conditions*

Supersonic outflow boundary conditions are used at the upper and downstream boundaries. The no-slip boundary condition is imposed at the adiabatic wall for all configurations, and periodic boundary conditions are used in the spanwise direction to exploit homogeneity. The inflow distribution of the pure LES is determined using the rescaling method of El-Askary et al. [31]. The recycling station is located seven inflow boundary-layer thicknesses downstream of the inlet of the computational domain. A sponge layer is applied at the upper- and outflow boundary to damp spurious pressure fluctuations.*Results*

The streamwise development of the skin-friction coefficient c_{f} is presented in Figure 40. The c_{f}^{-}distribution for the zonal RANS-LES result with STGMR are in good agreement with the pure LES solution and the DNS results of Pirozzoli and Bernardini [28]. Within the RANS-LES transition region the skin-friction coefficient converges to the pure LES values within one boundary-layer thickness. The zonal RANS-LES result with SEM, where the turbulent structures generated by the inflow method of Jarrin et al. [32], would too strongly dissipate such that a much larger streamwise extent would be necessary for the LES to recover the correct c_{f}^{-} level. A small overshoot in the c_{f}^{-} value is observed near the inlet of the RANS-regime at x ∕ δ_{0} ≈ 5.2. It converges to the pure LES c_{f}^{-} distribution immediately downstream of the overlapping zone in case of zonal RANS-LES results with STGMR. Such a transition behavior of cf characterized by an overshoot and an immediate recovery is also reported for incompressible boundary-layer flow by König et al. [33]. However, in case of the zonal RANS-LES with SEM the cf distribution downstream of the embedded LES part does not converge to the correct pure LES and DNS values.

FIGURE 40. Streamwise distribution of the skin-friction coefficient of the supersonic flat-plate boundary layer (M = 2.0, Re_{θ} = 870) for several numerical configurations. Grey shaded areas indicate the RANS-LES overlapping zones.

The van-Driest velocity profile of the zonal RANS-LES simulation with STGMR in Figure 41 shows a good agreement with the pure LES data. There are minor discrepancies between the the velocity profiles in the wake region of the pure LES, the zonal RANS-LES with STGMR and reference DNS data [28] which are caused by the applied grid resolution. The velocity profile of the zonal RANS-LES with SEM does not match the other results since the flow is still in a transitional state at this streamwise position.

FIGURE 41. Van-Driest-velocity distributions at x ∕ δ_{0} = 3 for several numerical configurations for a supersonic flat-plate boundary layer (M = 2.0, Re_{θ} = 870)

The streamwise development of the distribution of the Reynolds-stress components of the pure LES and the zonal RANS-LES configurations are compared with the DNS results at M = 2.0 and Re_{δ2} = 670 of Pirozzoli and Bernardini [28] in Figure 42. A good agreement with the reference DNS data is obtained at x ∕ δ_{0} = 4. However, the acceptable agreement of the pure LES distributions and zonal RANS-LES with STGMR results at x ∕ δ_{0} = 3 corroborate that the inflow generation method STGMR for the zonal RANS-LES configuration is capable of generating physically meaningful Reynolds stresses within a short transition length, i.e., in less than three boundary-layer thicknesses δ_{0}. The distrbutions of the Reynolds stresses of the zonal RANS-LES with SEM do not converge to the reference DNS values within four boundary-layer thicknesses δ_{0}.

FIGURE 42. Streamwise development of Reynolds stresses of the supersonic flat-plate boundary layer (M = 2.0, Re_{θ} = 870) for several numerical configurations.

Original authors: Statnikov et al. [71].

*6.2.2. Hypersonic launcher wake*

*Flow features*

The wake flow at Ma_{∞} = 6 is determined by strong compressibility effects. On the base shoulder a strong expansion of the incoming supersonic turbulent boundary layer takes place, which results in a strong radial deflection of the shed shear layer towards the models longitudinal axes. Further downstream of the base, the formed shear layer subsequently interacts with either the recirculating flow in the blunt base case or the solid surface in the nozzle dummy case.

*Numerical methods and discretization scheme*

The same as described in Sec. 6.2.1

*Computational grid*

According to the applied zonal approach, the RANS domain covers the main rocket geometry and the LES region encompasses the wake. Since the main body geometry is identical for both considered cases, a modular setup is used for the computational grids that consists of using the same RANS mesh for the main body and two different LES meshes for the applied aft-body extensions. To illustrate the applied grid topologies, the meshes for the RANS and LES zones used for the configuration with nozzle dummy are shown in Figs. 43(a) and 43(b). To reduce computational costs, the grids span a region of 180◦ with a mirror boundary condition in the symmetry plane. The RANS grid for the main body ranges from −8D to 0D in the streamwise and up to 4D in the radial direction with D being the diameter of the cylindrical main body part and the origin of ordinates lying in the center of the rocket base. The RANS/LES transition plane is positioned at x = −0.25D upstream of the base shoulder with the boundary layer thickness being δ ∼ 0.1D which yields a satisfactory transition length of at least 2 boundary layer thicknesses as expected by the RSTG method. The LES grid spans between x = −0.2D and x = 3.2D in the streamwise and up to 3.6D in the radial direction in the above mentioned coordinate system. The maximum grid resolution for the RANS and LES zones in inner and outer coordinates with their total grid sizes is given in Tab. 13. In the initial stage of the computations, a coarser grid with a resolution of y^{+} = 2, x^{+} = 60, ϕ^{+} = 40 was used and the same wake flow topology including the shock positions and reattachment lengths was obtained as for the final grid. To enhance the capturing of smaller scales, a refined grid with the resolution of y^{+} = 0.7 , x^{+} = 30, ϕ^{+} = 20 was used.

FIGURE 43. Computational grids for the RANS and LES zones. Every 4^{th} grid point is shown [71].

*Boundary and initial conditions*

Characteristic boundary conditions are used at the upper, RANS-inflow, and LESdownstream boundaries. A mirroring boundary condition is applied in the symmetry plane of the model.

*Results*

Fig. 44 shows the time-averaged numerical schlieren pictures (weighted absolute density gradient) in the symmetry plane of the investigated wind tunnel models. In addition, the time-averaged positions of the recompression shocks detected experimentally by Saile et al. [70] using high-speed schlieren measurements are drawn in Fig. 44 by red stars for the blunt base and by blue crosses for the nozzle dummy configuration illustrating a good agreement between the numerical and the experimental results for both configurations.

FIGURE 44. Time-averaged numerical schlieren pictures in the wake of the configurations in the symmetry plane. The positions of the recompression shocks detected in the experiments are indicated by the red stars (blunt base) and blue crosses (nozzle dummy) [71].

Fig. 45 shows the comparison of the wall pressure fluctuations on the base plate between the computation and the experiment for both the blunt base and the nozzle dummy configuration. Dominant peaks around Sr_{D} ≈ 0.2 − 0.27 clearly appear in the corresponding spectra for both configurations. Additionally, the numerical and experimental base pressure spectra feature also clearly pronounced peaks in the range of Sr_{D} ≈ 0.05 which are presumably associated with the oscillatory movement of the large scale toroidal vortex within the separation bubble as known from the former investigations performed by the authors on a similar generic configuration [72].

The dynamic behavior changes crucially when moving downstream along the nozzle dummy as indicated in Fig. 46 showing the spectra of the pressure fluctuations at different axial positions on the wall of the nozzle dummy. First, it can be noted that, compared to their LES counterparts shown in Fig. 46(a), the spectra in Fig. 46(b) obtained from the experiment are smoother due to significantly longer wind tunnel runs at the H2K facility and consequently, superior signal lengths improving the statistical quality of the spectral analysis compared to the simulations. The analyzed time interval of the performed wind tunnel runs is 7.1 sec and 1200 times longer than the LES one. While in the base area the available length of the LES data is satisfactory, as was shown in Fig. 16, it becomes more critical in a statistical sense when moving downstream of the base area into the region near the reattachment point. Unlike the near wake, this region is strongly affected by the variety of involved turbulent length and time scales determined by the boundary layer. This fact results in rather wide peaks in the experimental spectra, as can be observed in Fig. 46(b) showing the Fourier transforms of 1200 times longer wall pressure signals which were measured at several discrete positions on the nozzle dummy in the experiment. On the whole, the presented wall pressure spectra along the nozzle dummy show that the signals in the corner at x/D = 0.0 are consistent with the finding on the fluctuations on the base seen in Fig. 45, meaning periodic oscillations in the low frequency range at about Sr_{D} ≈ 0.05 and at ≈ 0.2 − 0.27. Additionally, another higher dominant frequency can be found, which centers at Sr_{D} ≈ 0.7. Especially the oscillations taking place at 0.05 and 0.27 can also be found for the two other transducers in the recirculation region, but the frequencies are less distinct and widened over a larger range. The sensors at x/D = 0.0, 0.25, and 0.5 still indicate periodic mechanisms inside the recirculation region which, however, become less dominant when moving downstream to the zone where the shed shear layer reattaches on the nozzle dummy. The spectrum of the signal at x/D = 0.75 reveals that the region around the reattachment point exhibits no distinct frequencies and can be rather characterized as broadband.

FIGURE 45. Comparison of the numerical and experimental pressure spectra on the base wall [71].

FIGURE 46. Wall pressure spectra along the nozzle dummy [71].

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