UFR 4-16 Test Case

Flow in a 3D diffuser

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Test Case Study

Brief description of the test case studied

The diffuser shapes, dimensions and the coordinate system are shown in Fig. 3 and Fig. 4. Both diffuser configurations considered have the same fully‐developed flow at channel inlet but slightly different expansion geometries: the upper-wall expansion angle is reduced from 11.3° (Diffuser 1) to 9° (Diffuser 2) and the side-wall expansion angle is increased from 2.56° (Diffuser 1) to 4° (Diffuser2). The flow in the inlet duct (height h=1 cm, width B=3.33 cm) corresponds to fully-developed turbulent channel flow (enabled experimentally by a development channel being 62.9 channel heights long). 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 into a 1 inch diameter tube. The curvature radius at the walls transitioning between diffuser and the straight duct parts are 6 cm (Diffuser 1) and 2.8 cm (Diffuser 2). The bulk velocity in the inflow duct is ${U_{\textrm {bulk}}=U_{\textrm {inflow}}=1m/s}$ in the x-direction resulting in the Reynolds number based on the inlet channel height of 10000. The origin of the coordinates (y=0, z=0) coincides with the intersection of the two non-expanding walls at the beginning of the diffuser's expansion (x=0). The working fluid is water (ρ=1000 kg/m³ and μ=0.001 Pas).

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

Figure 4: Geometry of the 3-D diffuser 2 considered (not to scale), Cherry et al. (2008).

Experimental investigation

Brief description of the experimental setup

The measurements were performed in a recirculating water channel using the method of magnetic resonance velocimetry (MRV), Fig. 5. MRV makes use of a technique very similar to that used in conventional medical magnetic resonance imaging (MRI), Fig. 6. Experiments were performed on a 1.5 Tesla magnet with resolution of 0.9 x 0.9 x 0.9 mm and a 7 Tesla magnet with resolution of 0.4 x 0.4 x 0.4 mm. Interested readers are referred to Cherry et al. (2008, 2009) for more details about the measurement technique.

Figure 5: Schematic of the experimental flow system (upper) and design of the 3D diffuser. Courtesy of J. Eaton (Stanford University)

Figure 6: 3D diffuser arrangement in a medical magnetic resonance imaging device. Courtesy of J. Eaton (Stanford University)

Mean velocity and Reynolds stress measurements

Cherry et al. provided a detailed reference database comprising the three- component mean velocity field and the streamwise Reynolds stress component field within the entire diffuser section. Both diffuser configurations considered are characterized by a three-dimensional boundary-layer separation, but the slightly different expansion geometries caused the size and shape of the separation bubble exhibiting a high degree of geometric sensitivity to the dimensions of the diffuser as illustrated in Figs. 7, 8 and 9

Figure 7: Streamwise velocity contours in a plane parallel to the top wall, from Cherry et al. (2008)

Figure 8: Measured streamwise velocity contours in the central plane (upper) of the Diffuser 1 and in the three selected cross-sectional slices positioned at different distances from the diffuser inlet (lower; their locations are denoted by thick white lines in the upper figure). Note that the black line indicates zero streamwise velocity and the purple and pink regions are reverse flow. The velocity values are normalized by the bulk inlet velocity being Vref=1 m/s. Courtesy of J. Eaton (Stanford University)

Pressure measurements

In addition Cherry et al. (2009) provided the pressure distribution along the bottom non-deflected wall of diffuser 1 at different Reynolds numbers. Complementary to the Reynolds number 10000 (for which the entire flow field was measured), two higher Reynolds numbers — 20000 and 30000 — were also considered, Fig. 11. The surface pressure distribution was evaluated to yield the coefficient $C_{p}=(p-p_{\textrm {ref}})/(0.5\rho U_{\textrm {bulk}}^{2})$; the reference pressure was taken at the position x/L = 0.05. The pressure curve exhibits a development typical of flow in diverging ducts. The pressure decrease in the inflow duct is followed by a steep pressure increase already at the very end of the inflow duct and especially at the beginning of the diffuser section. The transition from the initial strong pressure rise to its moderate increase occurs at x/L ≈ 0.3, (x/h=4.5) corresponding to the position where about 5% of the entire cross-section is occupied by the flow reversal (see e.g., Fig. 19. The onset of separation causes a certain contraction of the flow cross‐section, leading to a weakening of the deceleration intensity and, accordingly, to a slower pressure increase. The region characterized by a monotonic pressure rise was reached in the remainder of the diffuser section.

Figure 11: Pressure recovery coefficients relative to the pressure on the bottom wall of the diffuser 1 inlet in a range of flow Reynolds number. L=15 cm represents the length of Diffuser 1, from Cherry et al. (2009)

Measurements uncertainties

(adopted from Cherry et al., 2008, IJHFF, Vol. 29(3))

Elkins et al. (2004) estimated the maximum relative uncertainty of individual mean velocity measurements to be about 10% of the measured value in a similar highly turbulent flow. However, comparisons to PIV in a backward facing step flow (Elkins et al., 2007) show that only a small percentage of MRV velocity samples deviate by that much and most are much more accurate. To test this, the streamwise velocity component was integrated over 250 cross-sections of the MRV data and the results were compared to the known volume flow rate. This indicated an uncertainty in the integral of less than 2% with a 95% confidence level.

Measurements of turbulent normal stresses in Diffuser 1 were also taken using the MR technique described by Elkins et al. (2007). This method is based on diffusion imaging principles in which the turbulence causes a loss of net magnetization signal from a voxel in the flow. This causes a decay in signal strength which can be related to turbulent velocity statistics. Elkins et al found this method to be accurate within 20% everywhere in the FOV and within 5% in regions of high turbulence. Three turbulence scans were completed using three different magnetic field gradient strengths. For each gradient strength, 30 scans were completed and averaged. The three averaged data sets were then averaged to obtain a final data set.

Experimental data available

Velocity, (and streamwise Reynolds stress components for diffuser 1) and coordinate data for both diffusers are available online at http://stanford.edu/~echerry/. The data are seven 3D matlab matrices. The x, y, and z matrices give the coordinates of each point in the coordinate system shown in Fig. 5. The units are meters. The Vx, Vy, and Vz matrices give the corresponding velocity components for each point in m/sec. The matrix mg gives the relative signal magnitude detected by the MRI machine.

Experimental data for the pressure coefficient in Diffuser 1 for the inflow Reynolds number Re=10000 are available here (cp_Re=10000.xls). The coordinate system is the same as the coordinate system described in the corresponding manuscript (see below). C_p is defined as $\left(p-p_{\textrm {ref}}\right)/\left({\frac {1}{2}}\rho V^{2}\right)$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbases.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\left. P_\textrm{ref}}} is the pressure at x=0.05 (see Fig. 11) at the midpoint (z/B=0.5) of the bottom flat wall (opposite the wall expanding at 11.3 degrees), ? is the density, and V is the bulk inlet velocity. The data were taken in a line along the bottom wall of Diffuser 1 at constant y and z coordinates. L (=15 cm) indicates the length of the diffuser.

Please acknowledge the authors of the experiment when using their database!

CFD Methods

Provide an overview of the methods used to analyze the test case. This should describe the codes employed together with the turbulence/physical models examined; the models need not be described in detail if good references are available but the treatment used at the walls should explained. Comment on how well the boundary conditions used replicate the conditions in the test rig, e.g. inflow conditions based on measured data at the rig measurement station or reconstructed based on well-defined estimates and assumptions.

Discuss the quality and accuracy of the CFD calculations. As before, it is recognized that the depth and extent of this discussion is dependent upon the amount and quality of information provided in the source documents. However the following points should be addressed:

• What numerical procedures were used (discretisation scheme and solver)?

• What grid resolution was used? Were grid sensitivity studies carried out?

• Did any of the analyses check or demonstrate numerical accuracy?

• Were sensitivity tests carried out to explore the effect of uncertainties in boundary conditions?

• If separate calculations of the assessment parameters using the same physical model have been performed and reported, do they agree with one another?

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Contributed by: Suad Jakirlić, Gisa John-Puthenveettil — Technische Universität Darmstadt

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