Flow in a 3D diffuser

Introduction

The 3D (Stanford) Diffuser is a well documented case with complex internal corner flow and 3D separation while having a relatively simple geometry. It has an inlet section, an expansion section and an outlet section (see Figure 1). The flow at the inlet is assumed to be a fully developed rectangular channel flow. At the outlet, standard Dirichlet condition for the pressure is prescribed. An inflow Reynolds number of 10000 is considered based on the duct height and the flow is considered to be incompressible. Such configuration is also of engineering relevance. It represents a diffuser situated between the compressor and the combustor chamber of a jet engine, whose task is to decelerate the flow from the compressor to adapt for the combustor section. The following DNS data has been obtained using the in-house Finite Element Method (FEM) code Alya developed at BSC.

Review of previous studies

The physics of the Stanford diffuser were experimentally analyzed by Cherry et al. (2008). The sharp corners of the diffuser were smoothed with a fillet radius of 6.0. Experimental data was gathered in a recirculating water channel using the method of magnetic resonance velocimetry (MRV). In this turbulence causes a loss of net magnetization signal from the studied region. Such decrease in signal strength can be related to turbulent velocity statistics. This method was found to be accurate within 20% of the field of view and within 5% in in regions of high turbulence. Three scans were performed using three different magnetic field gradient intensities. For each one, 30 scans were completed and averaged. Then, the three averaged data sets were averaged to obtain a final data set. Such dataset provided detailed experimental data comprising the mean velocity field on its three components, the streamwise Reynolds stresses and the pressure distribution along the bottom wall of the diffuser. This provided information on the mean flow configuration, however, little insight was given on the more complex time-motions of this flow.

In the framework of two ERCOFTAC-SIG15 Workshops and in the European ATAAC project the flow configuration of the Stanford diffuser was thoroughly investigated. They studied both 3D diffuser configurations (denoted as SIG15 Case 13.2-1 and SIG15 Case 13.2-2, respectively) and were held in Austria (September, 2008) and Italy (September, 2009). Both in the workshops and in the ATAAC project a wide range of turbulence models in both LES and RANS frameworks as well as some novel Hybrid LES/RANS formulations have been employed. The workshop reports are published in the ERCOFTAC Bulletin Issues, see Steiner et al. (2009), Jakirlić et al. (2010) and the CFD methods section of UFR4-16. The ATAAC reports can be found through the links ATAAC_D3-2-36_excerpt3DDiffuser.pdf (excerpt from an ATAAC report) and ATAAC_finalWorkshop_ST04-Diffuser-ANSYS.pdf (PowerPoint presentation at ATAAC final workshop).

The only high-fidelity data available is the DNS performed by Ohlsson et al. (2010). The computational domain was designed to have a close agreement with the diffuser geometry, with the corners resulting from the diffuser expansion smoothly rounded with a radius of 6.0. As with the present case, this DNS had a Re=10000 and was computed using a spectral element code with 11th order polynomials. Their calculations were performed on the Blue Gene/P at ALCF, using 32768 cores and 8 million core hours. Another computation was performed on the cluster “Ekman” at PDC, Stockholm, Sweden, using 2048 cores and a total of 4 million core hours. The flow was computed for 13 flow-through-times (based on the bulk inlet velocity and diffuser length) before gathering statistics, which were gathered over an additional 21 flowthrough-times.

Description of the test case

The diffuser studied is the UFR4-16 Test Case, diffuser 1, provided in the ERCOFTAC database.

Geometry and flow parameters

The diffuser shape and dimensions as used in the experiment, and the coordinate system are shown in Fig. 2 (reproduced from UFR4-16 Test Case).

Figure 2: Geometry of the 3-D diffuser 1 considered (not to scale), Cherry et al. (2008); see also Jakirlić et al. (2010a).

Computational Domain

For the current diffuser, the upper-wall expansion angle is 11.3° and the side-wall expansion angle is 2.56°. The flow in the inlet duct (height h=1, width B=3.33) corresponds to fully-developed turbulent rectangular duct flow. The origin of the x-coordinates (x=0) is set at the entrance of the diffuser and z=0 at the left sidewall. The $L=15h$ long diffuser section is followed by a straight outlet part (12.5h long). Downstream of this the flow goes through a 10h long contraction followed by a 5h straight duct in order to minimize the effect of the outlet to the diffuser. A difference from the previous experimental and DNS works is that the geometry considered does keep the sharp angles on the walls transitioning between diffuser and the straight duct parts. The computational domain also includes a long inlet duct of 65h length in order to allow the flow in the inlet duct to fully develop. Before this, there is a section of 5h length with a small chevron placed 2h from the inlet in order to trigger the turbulent transition in the rectangular duct. An overview of the computational geometry with details of the rectangular duct is shown on Fig. 3.

The bulk velocity in the inflow duct is ${U_{\textrm {bulk}}=U_{\textrm {inflow}}=1\ m/s}$ in the x-direction resulting in the Reynolds number based on the inlet channel height ( ${h}$ ) of ${Re={\frac {U_{\textrm {bulk}}h}{\nu }}=10000}$ .

Figure 3: Computational domain used for this test case sideview top with detail of the inlet roughness element, topview bottom.

Boundary conditions

The inflow is set to $U=1$ at the inlet of the inlet duct (x=-70h), setting a Reynolds number of 10000, matching with the previous DNS and experimental data. Turbulence is triggered by creating a small discontinuity in the form of a small chevron in the duct. This method is preferred over using a precursor calculation of rectangular duct flow with streamwise periodicity conditions as Nikitin (2008) argued that such conditions might not be suitable since they cause a spatial periodicity, which is not physical for turbulent flows. As a matter of fact, Ohlsson et al. (2010) use a 63h long inflow development duct, accounting even for the transition of the initially laminar inflow. The chevron method of this work is aimed to accelerate such transition. At the outlet (x=47.5h) $dU/dn=0$ and standard Dirichlet condition for the pressure is prescribed. The walls of the duct and the diffuser are set to no slip.

Figure 4: Profile of the fully-developed turbulent inlet flow compared with the DNS of Ohlsson et al. (2010) and the experimental data of Cherry et al. (2008).

Figure 5: Axial velocity profile in semi-log coordinates corresponding to the fully-developed flow in the inflow duct compared with the DNS of Ohlsson et al. (2010). For the log law B is taken as 5.2.

The development of turbulence along the long inlet rectangular duct is shown in Fig. 6 by plotting the evolution of $Re_{\tau }$ defined as ${Re_{\tau }={\frac {u_{\tau }h/2}{\mu }}}$ for the top and bottom walls and ${Re_{\tau }={\frac {u_{\tau }B/2}{\mu }}}$ for the side walls. The straight line refers to ${Re_{\tau }}$ of a fully developed rectangular duct flow. Eventually, the entrance to the diffuser section at x=0h corresponds to that of a fully developed rectangular duct flow.

Figure 6: Evolution of the Reynolds number based on the wall friction velocity for the long inlet duct.

As specified in UFR4-16 Evaluation, the flow in a duct with rectangular cross-section is no longer unidirectional. It is characterized by a secondary motion with velocity components perpendicular to the axial direction, as shown in Fig. 22 for the UFR4-16 and Fig. 7 for the present case. Briefly, this secondary flow transporting momentum into the duct corners is characterized by jets directed towards the duct walls bisecting each corner with associated vortices at both sides of each jet. They have an influence on the development of the flow in the diffuser.

Figure 7: Velocity vectors in the y-z plane in the inflow duct at x/h = -2. Points have been sampled at each 10th grid node.

References

Ohlsson, J., Schlatter, P., Fischer P.F. and Henningson, D.S. (2009): DNS of three-dimensional separation in turbulent diffuser flows. In Advances in Turbulence XII, Proceedings of the 12th EUROMECH European Turbulence Conference, Marburg. Springer Proceedings in Physics, Vol. 132, ISBN 978-3-642-03084-0
Ohlsson, J., Schlatter, P., Fischer P.F. and Henningson, D.S. (2010): DNS of separated flow in a three-dimensional diffuser by the spectral-element method. J. Fluid Mech., Vol. 650, pp. 307–318
Steiner, H., Jakirlić, S., Kadavelil, G., Šarić, S., Manceau, R. and Brenn. G. (2009): Report on 13^th ERCOFTAC Workshop on Refined Turbulence Modelling. September 25–26, 2008, Graz University of Technology, ERCOFTAC Bulletin, No. 79, pp. 24–29
Jakirlić, S., Kadavelil, G., Sirbubalo, E., von Terzi, D., Breuer, M. and Borello, D. (2010): 14th ERCOFTAC SIG15 Workshop on Turbulence Modelling: Turbulent Flow Separation in a 3-D Diffuser. "Sapienza" University of Rome, September 18, 2009, ERCOFTAC Bulletin, December Issue, No. 85, pp. 5–13
Cherry, E.M., Elkins, C.J. and Eaton, J.K. (2008): Geometric sensitivity of three-dimensional separated flows. Int. J. of Heat and Fluid Flow, Vol. 29(3), pp. 803–811
Nikitin, N. (2008): On the rate of spatial predictability in near-wall turbulence. J. of Fluid Mechanics, Vol. 614, pp. 495–507