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Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics CFD Project 1 Introduction In this project the flow inside an expanded pipe is simulated and analyzed by ANSYS Fluent. Flow expansion is one of the important issues in fluid dynamics. In our case, the separation is occurs because of the sudden expansion of the pipe, but they’re not the only causes of the separation. For example, the flow over a cylinder can produce a large region of separated flow or an airfoil can produce a separated flow with high angle of attacks causing the stall, which is a quite important subject in aerodynamics and also flight dynamics. Thus we have to understand the flow separation and also the effects of the separation and flow expansion to have better engineering applications. Therefore, in this project we have studied on an expanded pipe flow to simplify the problem and the solution to observe the properties of separated flow and also what’s going on during the flow separation. As a result of this project it’s expected to demonstrate the flow expansion and vortex generation depending on the expansion of the flow. Since there will be a rapid change in diameter of the pipe, there will be a rapid pressure drop at the expanded region. Regarding the expansion of the pipe, there will be a flow separation just after the expansion and after a while flow will contact with the wall of the pipe again. In the region that the separation occurs, vortex or recirculation generation is expected to be observed, due to the rapid pressure drop. The aim of this project is to show especially the flow separation, vortex generation, pressure field and also the pressure drop and velocity profiles at different given locations, to compare them with previous studies and also the analytical solutions to validate the results. In addition to that, successful grid generation is one of the important aims of this project. In the project, ANSYS Fluent is used for geometry generation, meshing and also for the solution of the problem. Results were visualized by Fluent’s post processor. In this report, we will be discussing the mesh generation, mesh quality and grid convergence to understand the effects of the mesh on the solution. Also we will be observing the flow properties such as velocity field, pressure field, and temperature field for different expansion ratios, also the vorticity generation depending on different rapid expansion of the pipe and gradual expansion of the pipe. Also obtained results will be compared with analytical solutions and previous studies on the literature. Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics Description of the project In this project, subsonic flow inside an expanded pipe was studied and analyzed by ANSYS Fluent. In the analysis horizontal cross-section of the pipe was divided into two and used as the geometry for the analysis. Since the pipe is axisymmetric, the velocity profile and other related properties could be computed only for the described geometry to reduce the computational time. The sketch of the geometry used in analysis can be seen in Figure 1. Figure 1: Given geometry In the above Figure 1, R1 represents the smaller pipe’s radius, where R2 is the radius of the larger region of the pipe. V1 is the inlet velocity, L1 is the length of the smaller region of the pipe an L2 is the larger region of the pipe. In the geometry that is drawn in ANSYS, R1, R2, L1 and L2 was set to prescribed values as 0.5cm, 1cm, 10cm and 25 cm respectively for R1, R2, L1 and L2. We had 4 boundary conditions for this particular problem. Where the left edge of the geometry, AB, was defined as inlet with V1 = 0.554m/s, right edge, CD was defined as pressure outlet with 0 gauge pressure, since pressure at CD was given as 1 atm and the atmospheric pressure is assumed as 1 atm. BD line was defined as axis in order to have axisymmetric solution, and both AF, EF and EC lines were defined as wall. Also initial conditions, and , were set to 100 kg/m3 and 0.01 kg/(m∙s), respectively. Thus, it can be verified that the Re is 55.4 for this flow according to the relation shown in below Eq.1. = 21 ρ1 μ Since the Re of the flow is less than 2300, we can say that the flow is laminar. Thus for the solution of the given problem, ANSYS Fluent’s viscous laminar model is used. Actually it is defined as a state (regime) of the flow. Laminar, transitional and turbulence regimes obey to the same governing equations, the Navier-Stokes equations. The laminar option does not apply a turbulence model to the simulation and is only appropriate if the flow is laminar. This typically applies at low Reynolds number flows. Pressure based steady solver was used. And 2D space was set to axisymmetric. Least Squares Cell based solution was selected for discretization of Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics the gradient. Second Order and Second Order Upwind discretization was used for Pressure and Momentum equations. Simple scheme is used for the solution. Both analysis were made under the assumptions of compressible, steady, viscous Flow. Results and Discussion Task1 According to the dimensions stated in the problem, geometry was generated as shown in Figure 2. At this point, geometry was ready to be meshed. Above geometry was transferred to the meshing tool of the Fluent. Figure 2: Geometry Generated by ANSYS At first the mesh was generated for the whole domain, as quadrilateral dominant with 0.005 element size. The generated Mesh can be seen in Figure 3. Figure 3: Quadrilateral Mesh with 0.005 element size The solution was prepared using mesh shown in above figure. As a result of the analysis. Axial velocity, streamlines and velocity profiles at x = 0.005cm, x = 0.01cm, x = 0.015cm, x = 0.02cm and x = 0.03cm which are located at the smaller region of the pipe and also x = 0.105cm, x = 0.11cm, x = 0.115cm, x = 0.12cm, x = 0.13cm, x = 0.15cm and x = 0.17cm located at the Mech 301 Fluid Dynamics Deniz Sayınbaş Vildan Yurt larger region of the pipe were plotted. To observe the effect of the expansion and also for the validation of the analysis. Same process were repeated using different mesh element sizes, which are 0.002m and 0.001m, to observe the grid convergence. The results of both simulations can be seen below figures. Figure 4: Velocity profile at different locations (Mesh element size 0.005) Figure 5: Velocity profile at different locations (Mesh element size 0.002) When velocity profiles shown in Figure 4-5 were compared, there are significant differences between them. Since there are much more data points when mesh element size is set to 0.002m, the velocity profiles are quite smooth and curved compared to the velocity profiles obtained, when the mesh size is 0.005m. As we can see in Figure 4, there are sharp edges in the velocity profiles, since there are less data points. And also the value of maximum velocity which is obtained in symmetry axis, where y = 0. In Figure 4, maximum velocity for purple curve, which is the velocity profile at x = 0.03 (smaller radius region of the pipe), is Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics obtained as approximately 0.9 m/s, where in Figure 5 maximum velocity for the same location is around 1.05m/s. Since there is almost %20 error between two solutions, we can say that grid convergence couldn’t be obtained. Figure 6: (Velocity profile at different locations (Mesh element size 0.002) Therefore, the mesh element size is reduced to 0.001 and velocity profiles are obtained as shown in Figure 6. As we can see in Figure 5-6, the velocity profiles are quite similar and also maximum velocities at any location is almost the same. Now we can say that the grid convergence has reached and the mesh was verified. Thus, for the following analysis, mesh element size 0.001 is used, both for task 1 and the task 2 since there are no huge dimensional differences between geometries used in this project. Figure 7: Mesh with 0.001 element size For the sake of the quality visualization, mesh size 0.001 is used for the problem, since the grid convergence is achieved using this element size. Minimum orthogonal quality of the Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics mesh is quite important for the solution quality and for the convergence. Typically, minimum orthogonal quality that is higher than 0.1 is acceptable for simple problems such as pipe flow. Mesh shown in Figure 7, has minimum orthogonal quality of 1, which is quite enough for this analysis, and maximum aspect ratio of 1.4, where maximum aspect ratio is directly related to orthogonal quality. Analysis were repeated and residuals are plotted with the number of iterations as shown in Figure 8. Figure 8: Residuals vs iterations As we can see from the above figure. Residuals decreased in the order of 103 and the solution converged in 180 iterations. When the mesh size is considered, we can say convergence was achieved in a considerably short time. After that axial velocity plot, streamlines, velocity vectors, pressure contour for the whole domain and also velocity profiles at previously stated locations were plotted. Figure 9: Axial velocity As shown in Figure 9, once we run the analysis, initial velocity is 0.554m/s, then flow develops until x = 0.05m where it reaches fully developed state, then it remains constant until x = 0.1m, right at the x = 0.1 flow is expanding, thus the axial velocity decreases to Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics approximately 0.3m/s and again it is fully developed around x = 0.15, then remains constant as expected. Figure 10: Streamlines (1000 stream line is defined) Figure 11: Streamlines and vorticity generation in expansion region As shown in Figure 10 and Figure 11, streamlines were drawn for the flow with initial velocity of 0.554m/s. Depending on the expansion, rapid pressure drop leads to vorticity or recirculation right at the corner of the expanded region. Recirculation can be clearly seen in Figure 11. As expected flow velocity is higher at the center of the pipe. And also it is higher for smaller region of the pipe (red region in Figure 10). Velocity at the centerline decreases as flow expands as expected. Also the velocity distribution on the domain can be seen in Figure 12 and 13 as velocity vectors. When we zoom in the recirculation zone shown in Figure 13, we can clearly see that the velocity vectors recirculated around this region. Velocity vectors in negative x direction Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics turns out the vectors in –y direction as we go from the wall of the geometry to the centerline, indicates that recirculation occurs in the corner, where the pipe expands as shown in Figure 13. Figure 12: Velocity vectors around expanded region Figure 13: Velocity vectors around recirculation zone (zoom in) Also the pressure contours are plotted as shown in below Figure 14. Figure 14: Pressure Contour Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics Also the velocity profiles at x = 0.005cm, x = 0.01cm, x = 0.015cm, x = 0.02cm and x = 0.03cm which are located at the smaller region of the pipe and also x = 0.105cm, x = 0.11cm, x = 0.115cm, x = 0.12cm, x = 0.13cm, x = 0.15cm and x = 0.17cm located at larger region of the pipe were plotted as shown in Figure 15 and Figure 16. In order to validate our results, obtained results were compared with the Poiseuille-Hagen solution, which is an analytical solution for the straight part of the pipe. To validate our results we use the velocity profile at x = 0.03m, which is located at the smaller part of the pipe and where the flow is almost fully developed for the smaller region. Poiseuille-Hagen solution states that; 2 = 20 (1 − 2 ) 1 Where V0 is 0.554m/s, R1 is 0.005m and r is the distance from the centerline. Figure 15 Velocity profile at x = 0.005cm, x = 0.01cm, x = 0.015cm, x = 0.02cm and x = 0.03cm When we consider the velocity profile at x = 0.03m (purple curve in Figure 15), the velocity at r = 0.004m is 0.4m/s, from Eq. 2, u(velocity) can be calculated as 0.3988m/s as r is taken as 0.004m. Thus we can say that the error between analytical solution and the analysis is %2.5 which is quite acceptable. Thus we can say that our analysis is validated by the analytical solution. In addition, in the expanded region we can see that for x = 0.105cm, x = 11cm, x = 0.115cm x = 0.12cm and x = 0.13cm, shown in Figure 16, velocity has negative direction for the upper region of the expansion part, which are caused by the recirculation. After the flow is developed, negative values of velocity profiles are not present as it is in x = 15cm and x = 17cm. Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics After the expansion the axial velocity starts to decrease until the flow reaches the steady state. However the value of the axial velocity is around 0.275m/s at x = 0.17cm. Since the mass flow is conserved at every cross section of the pipe it’s reasonable to have such a results because of the ratio of 2 1 is 2, which means the area of the large region of the pipe is 4 times of the area of the small region. In the smaller region axial velocity of the steady state flow was around 1.1 m/s where it’s approximately 4 times of the steady state velocity of the flow inside the larger pipe, which is 0.275. Figure 16: x = 0.105cm, x = 11cm, x = 0.115cm x = 0.12cm and x = 0.13cm Triangular Mesh Figure 17: Triangular mesh (zoom in) Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics In order to see the effects type, the mesh was redefined as triangular with the element size of 0.001m and analysis was made again using triangular mesh. This time the minimum orthogonal quality of the mesh was 0.6 and the maximum aspect ratio was 3.9. Compared to quadrilateral mesh with rectangular elements, the minimum orthogonal quality was low and aspect ratio were high, however it’s still acceptable for the solution quality. We can say that the using triangular mesh with same element size, we actually reduced the mesh quality. The reason is clearly the shape of the flow domain. Since the flow domain is defined as a 2D rectangular shape, quadrilateral elements can be uniformly distributed along the flow domain, where triangular mesh fails to do so. In order to interpret the results, analysis were repeated. Figure 18: Residuals vs iterations (triangular mesh) As shown in Figure 18, Residuals were converged in almost 300 iterations. It took only 180 iterations for quadrilateral mesh. Since the orthogonal quality of the mesh was lower for triangular mesh, this result was quite expected. Figure 19 : x = 0.105cm, x = 11cm, x = 0.115cm x = 0.12cm and x = 0.13cm In above Figure 19, velocity profiles for x = 0.105cm, x = 11cm, x = 0.115cm x = 0.12cm and x = 0.13cm were plotted. Actually, it was quite similar with Figure 16, in terms of Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics velocity profiles at the same locations. There are some small differences especially between axial velocities at x = 0.105cm and 0.11cm, however axial velocities were almost the same for x = 0.17. Also the vorticity generation can be clearly seen again in Figure 19, due to the presence of negative velocity profile at top region. In general, it can be concluded as, visually both velocity profiles for triangular mesh and quadrilateral mesh are almost the same with small differences depending on the uniformity and orthogonal quality. Since the refined mesh has element size that is small enough to resolve the domain, there are no big difference between two solutions. In order to observe the effect of the Reynolds number in re-attachment length, number of analysis were made for flows having different Reynolds number. Reynolds number was adjusted for the analysis by keeping R1, and but changing V1 in Eq. 1. Velocities calculated for different Reynold’s numbers. Analysis were made using adjusted velocity values and reattachments lengths were estimated for each Reynold number. Streamlines obtained by analysis for different Reynolds number flows can be seen in App. 1. In below table, velocities adjusted according to the Reynolds number, approximate re-attachment length obtained from the analysis (Lr), diameter of the narrower region of the pipe (d) and Lr/d ratios can be seen according to different Reynolds numbers can be seen. Re V1 Lr D Lr/d 55.4 0.554m/s 0.026m 0.01m 2.6 100 1 m/s 0.04m 0.01m 4 150 1.5 m/s 0.06m 0.01m 6 200 2 m/s 0.09m 0.01m 9 Table 1 Data were transferred to MATLAB to obtain Re vs Lr/d graph. In order to see the linear relation between Re vs Lr/d, curve fit was applied. As shown in Figure 20 (left), the difference between the curve-fit and the obtained data can be seen. Since the Lr values were estimated using the legend below the streamlines in App. 1 and the obtained Lr values were rounded. In order to validate simulation results, obtained data was compared with previous studies. This relation have been studied by Hammad et al. (1999). When Figure 20 two figures are compared, in both figure, Lr /d ratios are quite similar for Reynolds numbers of 55.4, 100, 150, 200. Thus, we can conclude that the Reynolds number and re-attachment length has a linear relation, as Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics Reynolds number increases means that as flow velocity increases, re-attachment length in the expanded pipe is increases leading the larger vortices. Figure 20: Re vs Lr/d MATLAB (left), Experimental results (Hammad et al. (1999)[1]) (right) Task 2 Effect of Expansion Rate For the additional objective, the effect of the expansion rate on pressure loss was investigated. For this purpose 3 geometries are generated as keeping smaller region’s radius constant and adjusting the larger region’s radius. Thus, d/D ratios were set to, 0.6, 0.4 and 0.2 by adjusting D where dis the diameter of narrower region and D is the diameter of larger region. Pressure differences before and after the expansion were calculated using pressure contours and according to those pressure differences in each pipe, head loss coefficients are calculated using below formula. = ℎ 2 ⁄2 Where, head loss is given as below ℎ = In order to compare obtained head loss coefficients with analytical data head loss coefficients due to the sudden expansion for each pipe were recalculated by below equation. 2 = (1 − ⁄2 )2 Obtained Head loss coefficients and head loss depending on the d/D can be seen in below Table 2. Also the pressure contours and streamlines obtained from the simulations can be seen in App. 2. Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics d/D ΔP K K SE 0.2 23.9 Pa 0.89 0.92 0.4 20.3 Pa 0.73 0.706 0.6 11.9 0.43 0.41 Table 2 The errors between simulation data (K) and analytical data (KSE) for head loss coefficients were calculated according to the values shown in Table and for d/D= 0.2, d/D= 0.4 and d/D= 0.6 error between K and KSE were calculated as 3.2%, 3.4% and 5% respectively. The reason for the error is most probably the ΔP calculation. Since pressure difference was calculated using pressure contour legend, precision of the results were a little bit less precise compared to analytical solution. However, in order to visualize the difference of analytical and simulation results, d/D vs K and d/D vs KSE graphs were plotted in MATLAB and also curve-fit was applied to the data. Figure 21: d/D vs K, d/D vs Kse and curve-fit graph generated in MATLAB according to numerical and analytical results As shown in Figure 21, there are some small differences between K and KSE curve-fit data. However, shape of the both curves seems quite identical. Therefore we can say that the results obtained in the simulations were validated by the analytical solutions. In addition, Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics obtained data was compared with the d/D vs K graph located in the lecture book, shown in Figure 22. When Figure 21and Figure 22 for sudden expansion case, the graphs seem quite similar both in shape and the values. Thus, it is acceptable to say that effect of the sudden expansion rate of head loss coefficient was validated analytically and qualitatively. Figure 22: d/D vs K from the book [2] Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics References [1] Hammad, KJ, Otugun, MV and Arik, EB “A PIV study of the laminar axisymmetric sudden expansion flow”, Experiments in Fluids 26 (1999) pp. 266-272. [2] White, F. M. (2003). Fluid Mechanics, (2003). Chap, 6, p350. Appendix: Appendix 1 Deniz Sayınbaş Vildan Yurt Mech 301 Fluid Dynamics Appendix 2