UFR 4-19 Test Case

Converging-diverging transonic diffuser

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Test Case Study

Brief Description of the Study Test Case

The geometry of the Sajben transonic converging-diverging diffuser and the flow characteristics have been obtained from the Alliance CFD Verification and Validation Archive-NASA website (http://www.grc.nasa.gov/WWW/wind/valid/archive.html). The diffuser geometry and the dimensions are given in fig.4. The flow starts subsonic with M=0.46 at the inlet of the diffuser, accelerates through the throat up to the supersonic region where a shock-wave is formed which abruptly decelerates the flow which is then subsonic in the remaining diverging part of the diffuser downstream of the shock-wave.

The test case geometry configuration has an entrance-to-throat ratio equal to 1.4 and an exit-to-throat ratio equal to 1.5. In the current study, two flow configurations were examined. Based on the Mach number that is reached in the diffuser, the configurations are named as the "weak" and the "strong" case. For the "weak" case, the flow accelerates up to Mach~1.25 and for the "strong" case up to Mach~1.35, is then decelerated by a weak or a strong shock-wave and leaves the diffuser at Mach=0.52 and Mach=0.65 respectively. The different flow development for the two cases is guided by the imposed static pressure at the outlet boundary of the Sajben diffuser and it is characterized by the ratio R, of the outlet static pressure to the inlet total pressure of the flow. The ratio R takes the values of 0.82 and 0.72 for the "weak" and the "strong" Mach number cases respectively. For the weak case, the boundary layer stays attached, while it separates from the upper wall of the diffuser for the "strong" case leading to different flow development in the diverging part of the diffuser. The position of the "weak" and the "strong" shock-waves together with the corresponding recirculation region for the "strong" Mach number case are indicated in fig.4.

Figure 4: Geometry of the Sajben diffuser and schematic representation of the shock-waves position and the recirculation region

The two test case setups regarding all the flow boundary conditions and the general characteristics e.g. Mach number values, total pressure, total temperature, outlet boundary static pressure, are provided in table 2.

The principal measured quantities that are available for the validation of the various computational CFD codes, are basically mean averaged quantities of the pressure distributions along the walls of the diffuser and the velocity distributions at various stations after the shock-wave. Unfortunately, there are no experimental data regarding the turbulent quantities in order to have a more qualitative assessment of the behavior of the adopted turbulent models.

Test Case Experiments

Experimental setup

The transonic diffuser experimental studies were conducted at McDonell Douglas research laboratories, St. Louis, Missouri, USA. Regarding the wind tunnel operation a brief description is given below based on Bogar et al. (1982). Dry, filtered air is supplied to the test section from a 51cm diameter plenum chamber which provides highly uniform low- turbulence air flow. The flow at the exit of the test section is ventilated to the atmosphere in order to provide a constant pressure boundary condition. The supplied air temperature variations provided a Reynolds number within the range of 6.4 – 8.4 × 10⁵, based on diffuser throat height and conditions.

A characteristic sketch of the converging-diverging experimental test-section in which the measurements were carried out is shown in fig.5. The sidewalls of the experimental test section were made of 25.4mm thick schlieren quality glass, for optical diagnostic methods to be more easily applied. The diffuser configuration had an entrance- to-throat area ratio of 1.4, an exit-to-throat area ratio equal to 1.5 and the width-to-throat ratio of the area at the throat was 4.03. In order to have minimum reflectivity, all the optical surfaces were covered with dielectric coatings.

Figure 5: Side and top sketch view of the experimental test section. Percentages indicate area decrease at slots. Slots sizes are enlarged. Dimensions are in mm. (The sketch and the comments were reproduced from Mohler (2005))

During the experiments it was observed that for maximum Mach numbers less than 1.27, the flow was fully attached. In cases where the Mach number exeeded the above limiting value, shock induced separation was observed in the diverging part of the diffuser. Two test cases were examined in the measurements, one having a "weak" and the other a "strong" Mach number that it is reached after the diffuser throat. The first one has Mach~1.25 and attached flow on both walls and the second one has Mach~1.35 resulting in shock induced boundary layer separation on the upper wall and a fully attached flow on the bottom wall. The boundary layer was tripped in the inlet at the top wall of the diffuser as indicated in fig.5. The top wall boundary layer at the throat was turbulent for all cases and the sidewall and bottom boundary layers were expected to be laminar up to the shock- wave. The experimental setup configuration provided highly uniform flow with low turbulence.

Velocity and pressure measurements

For the measurements of the streamwise velocity the Laser Doppler Velocimetry (LDV) technique was used. The laser light source was a 4-W argon-ion laser which was operated at 514.5nm. The selected stations were located after the shock-wave which is formed just after the diffuser throat. The optical axis of the laser beam was normal to the diffuser sidewalls. The probe volume that was generated had a streamwise resolution of 0.46mm and a fringe spacing of 14μm. The velocity sampling ranged from approximately 500Hz near the walls up to more than 1.5kHz in the core flow. The mean velocity distributions were obtained by simple averaging the measured data. The measurement stations were aligned to positions after the shock-wave (positions selected for comparison with calculations are given in fig.9). For the "weak" Mach number case, 13 stations were selected and for the "strong" Mach number case 11 since, for the strong case the shock- wave was closer to the exit and the available space towards the exit of the diffuser was limited. The vertical measurement positions from the bottom up to the top wall for all axial stations were 29, lying on the symmetry plane of the diffuser. The vertical spacing of the measurement positions was smaller near the walls and larger near the main core of the flow. The relative positions between the vertical measurement stations had an accuracy of 0.08mm. Regarding the mean static pressure distribution measurements, more than 100 orifices were placed on the bottom and top wall of the diffuser. Finally, for the visualization of the time-dependent shock location, a line-scan television-type camera combined with a shadowgraph optical system was used.

The measured velocity flow development is shown in figs.6 and 7. Figure 6 presents characteristic spark schlieren photographs, Bogar et. al. (1982), while fig.7 presents the measured mean axial velocity for the "weak" and the "strong" Mach number cases, where the differences between the sub-sonic boundary layers between the two examined cases are observed. The schlieren pictures were made of the flow field with capturing 5000 frames/sec and the flow field view for the photographs was extending from X/h_throat= -1 to 8. For the "weak" Mach number case no recirculation is present and both boundary layers on top and bottom walls are attached being relatively thin. Furthermore, an inviscid region in in the core flow of constant total pressure is present. On the other hand for the "strong" Mach number case, there is a separation observed on the top wall which rapidly thickens while the top and bottom boundary layers merge before the diffuser exit. The recirculation region is denoted with the shaded area in fig.7.

Figure 6: Spark schlieren photographs of attached and separated flow. Top: "weak" Mach number case. Bottom: "strong" Mach number case. "From Bogar et al. (1982), doi: 10.2514/3.8234; reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc."

Figure 7: Measured Velocity contours. Top: "weak" Mach number case. Bottom: "strong" Mach number case. "From Salmon et al. (1982), doi: 10.2514/3.8311; reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc."

Two-dimensionality of the flow

The boundary layer growth was limited by three sets of suction slots placed on the side and bottom walls, fig.5. More specifically, two slots upstream of the measurement area, where the flow was choked, and one downstream where the flow was nearly choked. This configuration obtained good approximation of a two-dimensional flow since the suction slots minimized the three-dimensional effects. The percentages that are shown in fig.5 are equal to the mass fraction that was removed from the diffuser, Bogar et al. (1982).

General remarks on the experimental studies

• The available experimental measurements used in the current UFR study were acquired from the NPARC Alliance Verification and Validation Archive of NASA (http://www.grc.nasa.gov/WWW/wind/valid/archive.html).

• Apart from the measurements with no excitation at the outlet of the diffuser, measurements were also conducted for an excitation outlet frequency of 330Hz. The outlet excitation experiments were conducted in order to simulate the ramjet burner pressure oscillations. In the current work only the experimental cases with no outlet excitation were investigated.

• In the studies of Bogar et al. (1982), Salmon et al. (1982), Bogar (1985) and Sajben et al. (1984) the experimental setup and the measurement techniques for the cases with and without forced oscillations are presented. The current test case description is mainly based on the studies of Bogar et al. (1982), who published the LDV measurements, and Salmon et  al. (1982), who published the pressure distributions along the top and bottom walls.

CFD Methods

For the CFD calculations of the current contribution, the "strong" and the "weak" Mach number cases of the Sajben transonic diffuser were investigated having a maximum Mach number ~1.35 and ~1.25 respectively, as measured by the experiments. The CFD code that was used for the study of the current UFR is an academic in-house flow solver that has been developed in the Laboratory of Fluid Mechanics and Turbomachinery (LFMT) at the Mechanical Engineering Department of the Aristotle University of Thessaloniki (AUTH). The code solves the Favre-averaged Navier-Stokes and energy equations governing compressible flow as will be discussed further below. The turbulence models for calculating the turbulent (Reynolds) stresses appearing in the averaged Navier-Stokes equations will be introduced first.

Turbulence modelling

Three turbulence models were implemented that follow different turbulence modelling philosophies: the linear eddy-viscosity (LEVM) of Launder and Sharma (1978), the non-linear eddy-viscosity model (NLEVM) of Craft, Launder and Suga (1996) and the non- linear Reynolds-stress model (RSM) of Craft (1998). The models are implemented in the solver in their low-Reynolds number formulations, in order to integrate the transport equations up to the wall, resolve the flow quantities in all boundary layer regions and capture in a better manner the turbulent boundary layer/shock-wave interaction. The details of all the turbulence models with all the equations defining them are given in VY.

• The eddy-viscosity models (EVM)

The eddy-viscosity models are two-equation turbulence models that solve a transport equation for the turbulent kinetic energy k and a transport equation for the isotropic part of the turbulent dissipation ε^*. For the modelling of the Reynolds-stresses, the LEVM adopts the linear Boussinesq approximation, while the NLEVM uses a non-linear cubic expression, which relates the Reynolds-stresses to products of strain and vorticity tensors. In order to fully describe the boundary layer behavior, appropriate near-wall damping functions for calculating the eddy-viscosity are used. Additionally, the NLEVM uses a strain- sensitized expression for calculating the C_μ coefficient of the eddy-viscosity. This equation enables the model to provide more accurate results in regions where turbulence is not in equilibrium.

• The Reynolds-stress model

The adopted Reynolds-stress model solves six different transport equations, one for each Reynolds-stress component, and an additional transport equation for the isotropic part of the turbulent dissipation rate. For the modeling of the pressure-strain correlation terms, the model uses non-linear expressions relating the terms to turbulent stresses and velocity gradients.

In the current work, some minor modifications were applied to the original model of Craft (1998) in order to ensure convergence stability and, only for the "strong" Mach number case, some additional modifications were made in order to obtain physically consistent results. More specifically, in the original RSM the triple correlations needed for the calculation of the turbulent diffusion terms of the Reynolds-stresses are derived from a simplification of transport equations for these correlations. The convection and diffusion terms of the triple correlations transport equations are neglected, and a set of algebraic equations is solved for their calculation. This approach which is quite sophisticated, incorporates more physical insight and should produce more realizable results, gave rise to instabilities during the calculations and often led to divergence of the solution. In order to overcome this numerical problem, two alternative expressions were adopted for the modeling of the triple correlations and hence the turbulent diffusion: the expression of Hanjalic and Launder (1972), which was also used in the work of Craft and Launder (1996), and the most widely used Generalized Gradient Diffusion Hypothesis (GGDH) of Daly and Harlow (1970). Both expressions gave similar results for the "weak" and the "strong" shock-wave cases. Hence, the final calculations were performed using the GGDH approach which was found to be more stable during the numerical calculations.

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

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