Converging-diverging transonic diffuser

Confined flows

Underlying Flow Regime 4-19

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Key Physics

Although the geometry of the Sajben converging-diverging diffuser is relatively simple and can be treated as two dimensional, there are interesting flow features present in the flow and their capturing is a considerable challenge for turbulence models. The key flow characteristics are summarized below:

The primary structure that governs the examined transonic converging-diverging diffuser is the shock-wave that is formed in the transonic diffuser throat.

The shock-wave position and strength strongly depend on the static pressure that is imposed at the outlet of the diffuser as a boundary condition. As the boundary pressure outlet decreases, (by keeping constant the total inlet conditions), a stronger shock-wave is formed in the diffuser throat resulting in an increased Mach number value. In the current UFR study, the two boundary conditions that were examined led to the formation of a “weak” and a “strong” shock-wave, associated with a lower and a higher maximum Mach number.

The pressure after the shock-wave increases and the Mach number decreases. This may lead to boundary layer separation and to the formation of a recirculation region right after the shock-wave in the diverging part of the diffuser near the upper wall. In the current study, this behavior was measured for Mach~1.35 which is the “strong” case. For the “weak” case, Mach~1.25, the boundary layer remains attached along the diverging part of the diffuser.

There is a strong shock-wave/turbulent boundary layer interaction that has to be precisely described by the adopted turbulence models in order to compute with accuracy the flow development and the possible recirculation region after the shock-wave depending on the boundary conditions.

Numerical Modelling

In order to describe the compressibility effects and capture with accuracy the shock-wave when a pressure based solver is used, a compressibility correction algorithm has to be adopted in order to take into consideration the mean density variations.

In case there are stability problems due to high Mach number caused by the low level pressure at outlet, initially a higher static pressure value can be used as a boundary condition, then gradually reducing the pressure at outlet until the desirable value is reached.

As reported in NASA study#2, for obtaining supersonic flow through the diffuser, the pressure initially was kept low at the boundary outlet. After the supersonic flow was achieved, the pressure was raised to the desired value in order to be consistent with the experimental value. This was reported for calculations with fairly simple turbulence models (BL and SA). In the current study such a measure was not necessary.

For the studies in the literature where an unstructured grid was used, as in the NASA study#5, it was shown that for the “strong” Mach number case the recirculation region extended down to the outlet and remained during the iterative procedure in the whole computational area of the upper diffuser wall. This numerical problem was solved by extending the outlet region and fixing the outlet pressure to such a value that the desired pressure at the original outlet position of the diffuser was established. For the structured grid and also the grids that were used in the current study, this problem did not occur.

Grid dependency studies from the current and previous studies showed that there is no specific fine grid resolution needed for capturing the shock-wave and the velocity distributions. However, it is necessary to have a fine grid for stability reasons when using more advanced turbulence models such as non-linear eddy-viscosity models and RSM.

Physical Modelling

In order to capture all the compressibility phenomena due to the increased Mach number, it is recommended that temperature dependent relations should be used for the viscosity and the thermophysical properties of air. However, the use of mean values instead of the temperature functions, increases the solver stability.

The more advanced turbulence models, i.e. NLEVM and RSM, had to be initialized with the LEVM in order to achieve convergence.

Regarding the turbulence modelling of the “weak” Mach number case for the current UFR, all the models (LEVM, NLEVM and RSM) provided almost similar results for the pressure and velocity distributions but different Reynolds-stresses distributions since the LEVM was not able to calculate the Reynolds-stresses anisotropy. Additionally, the RSM predicted slightly more accurately the boundary layer development on the upper wall of the diffuser.

The literature studies that investigated the “weak” Mach number case, as presented in the Description section, provided results which were in close agreement for the pressure, the velocity and the Mach number distributions. Among the turbulence models used in these studies (i.e. eddy-viscosity two-equation, one-equation and algebraic turbulence models) the two- equation EVMs, e.g. Chien k-ε and SST k-ω, seem to have minor superiority since they provided velocity profiles closer to the experimental data.

Regarding the turbulence modelling of the “strong” Mach number case for the current UFR, the pressure distributions were calculated sufficiently well by all the turbulence models (LEVM, NLEVM and RSM). On the other hand, regarding the velocity distributions, only the EVMs were able to provide computational results being in close agreement with the velocity measurements. The model with the best performance was found to be the NLEVM since it predicted the recirculation region after the shock-wave closer to the experiments. In the RSM used, a coefficient related to the “fast” pressure strain correlation term was slightly modified based on suggestions found in the literature. This modification was implemented because the adoption of the original expression of the coefficient provided non-physical results. Although this modification gave numerical stability, the resulting RSM was not able to capture the boundary-layer separation on the upper wall since it calculated fully attached boundary layers on both diffuser walls. As a result, the RSM in the adopted modified form, although it was able to compute the strength of the shock-wave and the pressure distributions with reasonable accuracy, failed to compute the recirculation region and hence, it is not recommended for modelling supersonic flows similar to the current UFR.

In the presented literature studies that investigated the “strong” Mach number case, including the VY contribution which is presented in the current UFR, the pressure distributions were calculated with good accuracy in comparison to the experiment. However not all turbulence models provided qualitative results regarding the computed mean velocities compared with the measured velocity data. In particular, among the turbulence models presented in the Description section, the Chien k-ε, the k-ω SST, the LEVM and the NLEVM (the last two presented in the current UFR) are suggested for modelling the “strong” Mach number case since they provided velocity profiles with the best qualitative agreement with the experiment, with a small superiority observed for the NLEVM computational results.

The dilatation-dissipation expression, which was adopted in the current study, proved to have an impact only on the LEVM for the “weak” Mach number case, as it is also presented in the VY publication. For the more advanced turbulence models (NLEVM and RSM), it seems to have no effect for both “weak” and “strong” Mach number cases. This behavior is possibly related to the ability of the turbulence models to capture the mean density variations, as the Mach number of the flow increases, which enhances the compressible turbulence dissipation (dilatation-dissipation). Since the density variations can be described by the turbulence models, the compressible flow is more accurately predicted. In the transonic Mach regime, the LEVM is not able to compute the mean density variations and the dilatation-dissipation expression needs to be adopted. As the Mach number increases towards the supersonic regime, the LEVM is also capable of computing the mean density variations and the adopted dilatation-dissipation expression has no effect.

In the literature the dilatation-dissipation expression of Sarkar, together with a variable eddy-viscosity Cμ coefficient has been adopted by NASA#2 study for modelling the “strong” Mach number case. It was proven that the best results for the velocity and pressure distributions were provided by the combination of both Sarkar and variable Cμ for the Chien k-ԑ turbulence model.

As an overall assessment for both “weak” and “strong” Mach number cases among all the turbulence models used in the literature (the Description section) and the current UFR, the NLEVM seems to provide the best computational results in comparison to the experimental pressure and velocity data. Additionally, the superiority of the NLEVM is also related to the fact that it is able to calculate the anisotropic Reynolds-stresses distributions near the walls, even though the damping of the wall-normal fluctuations is still too weak. However, regarding the accuracy of the turbulent quantities calculations, no experimental data are available for a more thorough analysis and validation.

Application Uncertainties

The flow development is governed by the static pressure value at the outlet which controls the intensity of the formed shock-wave. In general, there are no uncertainty issues since the boundary conditions and the geometry are both accurately defined. However, for the “strong” Mach number case a special treatment to the outlet boundary position or a proper treatment of the outlet static pressure value needed, depending on the adopted turbulence model and the type of the used computational grid.

Two dimensionality of the flow, regarding the experimental setup and the measurements, was excellent for the cases with no separation and acceptable for the cases with separation, as reported by Bogar et al. (1982) and Sajben et al. (1983).

Recommendations for Future Work

Additional studies are needed regarding the modelling of the “fast” term of the non-linear pressure strain correlation expression of RSM for supersonic flows in order for physically realistic solutions to be obtained.

The impact of dilatation-dissipation on turbulence modelling should be further investigated for a variety of high speed flow regimes, from high subsonic up to hypersonic area.

Experimental data of turbulence quantities for supersonic flows, e.g. turbulent kinetic energy and Reynolds-stresses would be very helpful for further assessing the performance of turbulence models.

Contributed by: Z. Vlahostergios, K. Yakinthos — Dept. of Mechanical Engineering, Aristotle University of Thessaloniki, Greece