I+ D
U
∂(J−1U )
∂t +∂F
∂ξ +∂G
∂η +∂H
∂ζ = ∂FV
∂ξ +∂GV
∂η +∂HV
∂ζ (75)
The second-order numerical dissipation terms were set so that the variables decrease exponentially as a function of time. This technique allows a computation to damp out high-frequency errors at the initial stage of iterations and, therefore, avoids the second-order numerical dissipation terms to dissipate when the computation approaches a converged state. A finite-volume algorithm based on a Runge–Kutta time-stepping scheme [15] is used to obtain the steady-state solutions by solving the Navier–Stokes equations. The turbulence model used in this study is a Baldwin–
Lomax turbulence model [32]. The meshes generated in this study are hexagonal meshes; the grid lines near the solid wall are almost normal or parallel to the solid walls. In order to simplify the programming effort, the turbulence equations were applied along grid lines rather than in the direction normal to the solid boundaries.
This avoids flow calculations in all the normal directions to the solid wall and the necessary interpolations of the flow variables. During the calculations, it is difficult to keep all the y+values within a small level, especially close to the corner region. Therefore, the wall function was used in the calculation. The static pressure coefficient is defined as:
Cps= P− P01
p01− ps2
(76) The total pressure loss is defined as:
Cp0= P01− P0
p01− ps2
(77) where ps2represents the mass average static pressure of the blade exit plane.
3.9 Applications of Three-Dimensional Method
3.9.1 Analysis of pitch-width effects on the secondary flows of turbine blades
The inlet and outlet boundary conditions used in the computations were taken from measurement. The total pressure at inlet was equal to the test [38] for all the
calculations. At the exit plane, one of the most critical conditions was to specify the static pressure. The average exit pressure in the experiment [38] was used in the calculations. The asymptotic boundary conditions with second-order streamwise derivatives of variables were set to zero. The overall mass conservation through the blade flow passage is imposed by correcting the outlet velocity components with the calculated velocity profiles on the plane next to the outlet and inlet flow mass during iterations. The no-slip wall boundary conditions are used for all the solid walls.
Grid-converged solutions for the high-speed viscous flows over a turbine blade should be obtained with sufficiently fine meshes. However, a computation with fine meshes is very time consuming and difficult to assess the numerical accuracy of the solution. In fact Inoue and Furukawa [23] reported that it was difficult to opti-mize the artificial dissipation coefficients by using a common method that impaired accuracy and reliability of the schemes to predict a cascade performance. To pre-dict a high Reynolds number turbulent flow, highly clustered grids are required toward the wall. The grid refinement study was conducted through the computa-tional uncertainty study. The relationship between the calculated mass residual, 100× |min− mout|/mout %, and various grid sizes was tested to identify the point where the mass residual reaches its asymptotic value. In this study, the systematic grid refinement studies are performed. The grid refinement study was conducted by using Richardson extrapolation method [39]. The mass factor, f = |1 − min/mout|, is used to evaluate the grid refine study. Using the fine grid calculation with a magnification factor of 3, we obtain f1= 0.18 × 10−4. Then we change the grid with the magnification factor to 2, and obtain f2= 0.85 × 10−4. In this case, the error is ε= 100 × (f2− f1)/f1= 0.67%. For a grid increase rate, IN = 1.5, the fine grid value of the grid convergence index (GCI) for the present second order is GCI= 3ε/(IN2− 1) = 1.61%. Although the confidence in the GCI as error band is not justifiable, it shows the current grid structure to be conservative. It also shows that the uncertainty in the current calculation for the mass conservation is within 1.61%. The distance of the first mesh from the solid surface was chosen so that a maximum y+value becomes less than 15, which is considered to be fine enough for the present calculations. The mesh has 45× 41 × 115 node points in the pitchwise, spanwise, and streamwise directions, respectively, which was identified to an opti-mal situation as a grid-independent state. The blade-to-blade computational mesh is shown in Figure 3.13. The mesh is formed consisting of three zones: upstream of the blade, inside the blade, and downstream of the blade sections. For comparison with the test results [38], the outlet of the calculation plane was selected to be perpendic-ular to the x-direction (axial direction), which corresponds to the tested exit plane.
Most CFD studies [40–42] on turbomachines were mainly based on a linear cascade of high-turning turbine blades where the secondary flow was very strong.
In this study, an annular turbine blade was focused, which is similar to actual turbine stator vanes. The airfoil model investigated by Sieverding [38], where the blade tests were cited from NACA TN D-3751, is used for validation of the code. The tip and root radii are 0.355 and 0.283 m, respectively. The stagger angle is 42.5 degrees with respect to axial direction and the blade axial width length, Ca= 0.087 m. The blade profile is constant along the blade height and is untwisted. Three root pitch-to-width
(a) Blade-to-blade mesh at middle span (45 x 117) (b) Meridional mesh (41 x 117)
Figure 3.13. Calculation meshes.
0 10−1
10−2 100
Mass error
No. of steps
1000 2000 3000 4000 5000 6000 7000
Figure 3.14. Convergence history.
ratios, b/Ca= 0.639, 0.973, and 1.460, are studied here. The root pitch-to-width ratio, b/Ca, was varied by altering the number of blades. Three blade numbers were used in the calculation: 26, 21, and 14. These calculations were compared with the experimental data of the annular turbine blades [38].
The computations were assumed to be converged when the mass flow error became less than 0.1% of the inlet mass flow rate. The mesh used in this study and typical convergence history are shown in Figure 3.14. The convergence speed became slow after the mass flow error became close to 0.1%. In all the plots shown in this section, the viewpoint is set to look in the downstream direction from the
0 10 20 30 40 50 60 70
Beta
−40
−50
−60
−70
−80
−90
Blade Height
Figure 3.15. Average spanwise flow angle distribution at x/Ca= 0.9 (dot is exper-imental results).
(a) b/Ca = 0.639 (left is leading edge and top is casing).
(b) b/Ca = 0.986. (c) b/Ca = 1.460.
Figure 3.16. Static pressure coefficient distributions on the suction surface of the blade.
upstream location, with the pressure surface on the left-hand side and the suction surface on the right-hand side.
Figure 3.15 shows the comparison of the experimental results [38] with the present computations for the average spanwise flow angle distribution located at x/Ca= 0.9. The computations are in agreement with the experiment. Figure 3.16 shows the static pressure coefficient distributions near the suction surface of blades for different root pitch-to-width rations, b/Ca. The inception and transport of the passage vortex and its impact on the surface boundary layers present different
b /Ca= 0.639 b /Ca= 0.973 b /Ca= 1.460
Sa1 Sa2
Sa1 Sa2
Figure 3.17. Velocity vectors at the leading edge of the blade at 0.28% height from root.
characteristics for different b/Ca. There is a region near the wall corner where the chordwise and spanwise pressure gradients are small. It is also clear from Figure 3.16 that the possible boundary layer separation lines occur for cases b/Ca= 0.639 and 0.986, and they are located at the corners of the hub and trailing edge. The separation region for the case b/Ca= 1.460 is located at 50% of Caposition. The separation location seems to move forward as b/Caincreases. This feature confirms that the inception of the passage vortex developed near the airfoil leading edge and grew fast when the pitch b/Cawas increased. Figure 3.16 also shows that the pressure coefficient on the suction surface is lower for larger b/Ca. This is because, as b/Caincreases, the flow constraint from the blade becomes small, and therefore, the blade’s local curvature plays an important role. For the case of b/Ca= 0.639 and 0.986, a vortex develops faster near the blade tip and moves toward the casing because the local pitch size at the tip is much larger than that at the hub. In the case of b/Ca= 1.460, the casing and the hub vortex have similar characteristics because b/Cais large enough, thus causing the blockage to be small enough so that it does not restrict the flow. The flow characteristics are dominated by local curvature for b/Calarger than 1.46.
The experimental data [43,44] indicated that one of the most important features in a turbine blade boundary layer is the occurrence of the separation and saddle point on the end wall near the leading edge. The predicted boundary layer flow patterns for different b/Canear the hub leading edge at 0.28% height are shown in Figure 3.17. The calculations show that there is a saddle point for all b/Ca. Two attachment lines, Sa1 and Sa2, and two separation lines, Ss1 and Ss2, can be identified from the velocity vectors. The separation lines correspond to the two legs of the horseshoe vortices formed near the leading edge. The separation line Ss2 is pushed away from the leading edge. The first separation line Ss1 moves around the leading edge and moves against the freestream flow direction and it merges with the flow coming from the upstream direction to form a suction-side horseshoe vortex. The separation line Ss1 is also developed along the suction side of the blade, which can be seen from the static pressure coefficient distribution in Figure 3.17.
b /Ca = 0.639 b /Ca = 0.973 b /Ca = 1.460
Figure 3.18. Velocity vectors in the meridional surface before the leading edge.
0.7 0.8
b/Ca= 0.986 (experiment)
−0.85
Figure 3.19. Static pressure coefficient contour at x/Ca= 0.9.
The locations of the saddle points are closer to the leading edge for larger pitch sizes. This is why some of the flow visualization [42] did not find a saddle point if the experiment did not extend sufficiently far upstream. The velocity vectors in the meridional plane upstream of the leading edge are shown in Figure 3.18. The vector distributions show that there is a separation region occurring upstream of the leading edge. It is shown that the region of the separation depends on the pitch–
width ratio, b/Ca. A lower b/Cagenerates a relatively larger separation region. This characteristic determines the location of the saddle points on the blade-to-blade plane depending on the pitch–width ratio b/Ca. The present computations show that there is a saddle point upstream of the leading edge attributed to the blockage of the blades. With increasing blockage, the secondary saddle vortices move more forward and the separation in the front of the blade becomes smaller. The location of the saddle point depends on the b/Caas well as the local curvature of the blade.
Figures 3.19, 20, and 21 show the contours of the static pressure coefficient Cps distributions at 90%, 100%, and 110% chord positions, respectively, from the leading edge for different values of b/Ca. The measurement results for –Cps
at 90% chord position is plotted for comparison with the calculations. It should be pointed out that the measurement plane in the experiments [44] is defined to
−0.80
Figure 3.20. Static pressure coefficient contour at x/Ca= 1.
−1.1 −1.35 −1.3−1.25 (contour value from −1.1 to −0.65 with increment 0.05)
b/Ca = 0.973 (contour value from −1.35 to −0.75 with increment 0.05)
b/Ca = 1.460 (contour value from −1.35 to −0.8 with increment 0.05)
−1.3
Figure 3.21. Static pressure coefficient contour at x/Ca= 1.1.
be perpendicular to the axial direction while the computations are along the mesh direction, which causes a slight discrepancy. Figure 3.19 shows that the compu-tation and measurement are in good agreement with each other. The results for different b/Cashow that for smaller b/Ca, the lowest pressure becomes higher than those with larger pitch cases. The core of the secondary vortex occurs at different locations for all three cases. For the smaller b/Ca, the secondary vortex appears closer to the corner of the hub on the suction side. For larger b/Ca, the vortex core moves away from the suction-side toward the pressure side. When compared to the different axial location of Cps, it is shown that the vortex core is developed and moves away from the suction side, to the middle point between the suction and pressure-side surfaces. As the fluid moves out from the blade passage, the vor-tices become very small. This is because the blade-to-blade pressure gradient is reduced. This vortex core is part of the leading edge vortex. Initially, the vortex occurs very close to the suction-side wall and then it develops toward the exit plane.
At the same time, the horseshoe vortex at the pressure-side leg, Ss2, mixes with the secondary flow and rapidly decays. This phenomenon is in good agreement with the measurements for the pitch–width influence in the linear cascade [40].
b/Ca = 0.639 b/Ca = 0.973 b/Ca = 0.973 b/Ca= 1.460 (experiment)
TIF
Figure 3.22. Total pressure loss contour at x/Ca= 0.9.
0.55 0.60
b/Ca= 0.639 b/Ca= 0.973 b/Ca= 1.460
0.60 0.80
0.75 0.75
0.70
0.65 0.55
Figure 3.23. Total pressure loss contour at x/Ca= 1.0.
This figure also shows that the corner vortex at the tip of the blade is not very strong and cannot be clearly observed in the computations.
The total pressure losses Cpo in Figures. 19, 20, and 21 further confirm the aforementioned conclusions that corner vortices occur on both the hub and the casing suction-side corners. The computations for the case of b/Ca= 0.973 are compared with the experiment [44] and are shown in Figure 3.19. The mea-sured Cpovalues of the first line in the experiment are the same as the computations.
It is shown that the computations are in fair agreement with the experiments, except at tip and hub regions where the computations show a larger vortex region than that shown by the experiment.
The influence of the losses due to the pitch–width variation can be seen by comparing Figures 3.22, 23, and 24 for different b/Ca. For the cases of b/Ca= 0.973 and 1.46, the high-loss regions from both tip and hub corners near the suction side are developed. These losses are caused due to the corner vortices and the suction-side pressure characteristics. The end wall boundary layer is very thin near the pressure side within the blade passage. The pressure-side boundary layer and the suction-side boundary layer meet after the flow passes the blade trailing edge, as shown in Figure 3. 20. It is also shown that the total pressure losses Cpoincreases with the increase in b/Ca. The loss contour shows that the secondary flow is stronger for larger b/Ca, and it develops faster. This is because for large b/Ca, the flow control capacity become worse than that for small b/Ca. The secondary vortex is easy to
b/Ca= 0.639 b/Ca= 0.973 b/Ca= 1.460
Figure 3.24. Total pressure loss contour at x/Ca= 1.1.
b/Ca= 0.639 b/Ca= 0.973 b/Ca= 1.460
Figure 3.25. Blade-to-blade Mach number contour at 0.28% height.
develop in the blade channel. For the smallest b/Ca, the two passage vortices occur in the upper and lower parts of the blade passage closer to the suction side of the blade without strong interactions. With the increase in the b/Ca, the two vortices start interacting with each other. Owing to the occurrence of the passage vortices, and their interactions for the large b/Cacases, the secondary flow is very strong.
The strong three-dimensional effects dominate the whole passage flow for large b/Ca. However, for smaller b/Ca, it has been found that the flow configuration is different from those for larger b/Cacases. The two vortices occur from lower and upper parts of the suction side, and they are more confined in the end wall region, and there remains a two-dimensional flow pattern for a wide blade height range.
The blade-to-blade Mach number contours for the hub boundary layer region and the pitch and tip boundary layer regions are shown in Figures 3.25, 26, and 27, respectively, where the Mach number contours present different characteristics for the boundary region and the main flow region. In the near-hub boundary region, a smaller b/Cadoes not show a high Mach number region near the suction surface.
It seems that the flow accelerates smoothly along the suction side for smaller b/Ca. For the cases of b/Ca= 0.973 and 1.460, a small region of a high Mach number zone is observed close to the middle of the width in the suction side of the blade.
It is shown that a separation zone occurs and expands with the increase in b/Ca.
b/Ca= 0.639 b/Ca= 0.973 b/Ca= 1.460
Figure 3.26. Blade-to-blade Mach number contour at 50% height.
b/Ca= 0.639 b/Ca= 0.973 b/Ca= 1.460
Figure 3.27. Blade-to-blade Mach number contour at 99.72% height.
The results show that the blade-to-blade Mach number gradient along the turbine axial direction increases with the increase in b/Ca.
The blade-to-blade static pressure coefficient contours near the hub, pitch, and tip are shown in Figures 3.28, 29, and 30, respectively. It is shown that, due to the strong secondary-flow effect, the minimum pressure occurs in the passage away from the blade surface. Also note that, with the increase in b/Ca, the flow accelerates faster in the leading edge region. A clear low-pressure zone occurs on the suction side of the blade for all the different pitch–width cases when a flow separation zone on the blade suction surface.
Figures 3.31, 32, and 33 show the contours of the blade-to-blade total pressure coefficient distributions in the hub boundary (0.28% height), pitch (50% height), and tip boundary (99.72% height) regions, respectively. The losses for different b/Ca, the plots start with the same level at the same blade location. The total pressure losses are much larger in hub and tip boundary regions than in the midspan for all cases. In general, small b/Cahas small losses for all flow regions. For the smaller b/Ca, a small zone of high loss extending from trailing edge can be observed only
b/Ca= 0.639 b/Ca= 0.973 b/Ca= 1.460
Figure 3.28. Blade-to-blade static pressure coefficient contour at 0.28% height.
−0.8
Figure 3.29. Blade-to-blade static pressure coefficient contour at 50% height.
b/Ca= 0.973
Figure 3.30. Blade-to-blade static pressure coefficient contour at 97.72 height.
1.4
Figure 3.31. Blade-to-blade total pressure loss at 0.28% height.
1.4
Figure 3.32. Blade-to-blade total pressure loss at 50% height.
b/Ca= 0.639 b/Ca= 0.973 b/Ca= 1.460
Figure 3.33. Blade-to-blade total pressure loss at 99.72% height.
on the tip boundary. For a larger b/Ca, there appears a high-loss zone starting from the trailing edge extending to the upstream of the suction side of the blade.