Capífalo
2. COMPLEJOS METÁLICOS EN ESTUDIO
6.4
Tube Bundle
The flow across a bank of staggered tubes is an example of turbulent flow calculated using the q−ζ turbulence model developed by Gibsonet al.[53]. This case is a part of the ERCOFTAC database [43]. The geometry of the computational domain and the boundary conditions employed here are shown in Fig. 6.17. Even though the flow is periodic, see Fig. 6.19(a), periodic boundary conditions were not employed for the inlet and the outlet boundary. This is because the refinement procedure cannot guarantee that the adapted mesh will have same distribution of faces at the left and the right boundary which is required by the flow solver.
Symmetry plane
Symmetry plane Symmetry plane
Wall Wall
Wall
Outlet Developed inlet profile
H H Y X 0.5D 0.5D
Figure 6.17: Geometry and boundary conditions for the tube bundle case
The ratio between H and D is: H
D = 1.0369.
The Reynolds number of the flow is Re = 18000 based on the tube diameter D and the average inlet velocity Uavg. A developed inlet profile is obtained by fitting
a cubic spline through the data obtained by a separate calculation using periodic boundary conditions. The data sets for velocity,q andζ are taken at the location of the inlet boundary while the data set for the pressure field is taken at the location of the outlet boundary.
Figure 6.18: Starting mesh for the tube bundle case (640 cells)
The calculation was started on the coarse mesh with 640 cells, shown in Fig. 6.18. Second-order accurate Central Differencing was used for approximations of convec- tive terms for all fields. Gradients were evaluated by using the Gauss Theorem, Eqn. (3.24).
Figs. 6.19(a) and 6.19(b) show the flow field for this problem. Fig. 6.20 shows the pressure field which is given as a pressure coefficient defined as:
Cp =
P −Patm
0.5ρ U2
avg
(6.1) where Patm is the atmospheric pressure. The flow is periodic with high gradients
of velocity, q and ζ present near the tube walls, as shown in Figs. 6.19(a), 6.21 and 6.22. Separations on the downstream sides of the tubes are a consequence of the adverse pressure gradients on the leeward sides which slow the fluid down and eventually cause separation, Fig. 6.20. The turbulence variables q and ζ vary strongly in the near-wall region, while their variation in most parts of the domain is slow, see Figs. 6.21 and 6.22.
The required accuracy of the adapted solution was: E = 1 %
for velocity, q and ζ fields based on the maximums Umax = 1.689Uavg, qmax =
0.599Uavg and ζmax = 5.83 Uavg2
D , respectively. The accuracy of the adapted solution
6.4 Tube Bundle 155
(a) Velocity field
(b) Streamlines
Figure 6.19: Velocity field for the tube bundle case (17505 cells)(given as U
Figure 6.20: Pressure coefficient for the tube bundle case (17505 cells)
Figure 6.21: q field for the tube bundle case (17505 cells)(given as
q Uavg)
Figure 6.22: ζ field for the tube bundle case (17505 cells)(given as
ζ D U avg2)
6.4 Tube Bundle 157 benchmark. The error in the benchmark is estimated to 0.2%, 1.3% and 0.51% for the velocity, q and ζ fields, respectively, using the FREE. The solution is also compared with the measured data of Simonin and Barcouda [43].
The final adapted mesh has 17505 cells and is highly refined near the walls in the direction parallel to the wall, see Figs. 6.23(a) and 6.23(b), which is the direction of steepest variations of the velocity, q and ζ fields. The regions of fine mesh downstream of the tubes were needed to resolve the shear layers near the separation zones. The refined mesh regions also show periodic behaviour of the flow, except near the outlet because of the zero gradient boundary condition imposed on the velocity, q and ζ fields is not accurate on the coarse mesh.
Figs. 6.24(a), 6.24(c) and 6.24(e) show good agreement between the estimated and exact maximum errors for the first five cycles of refinement. It is believed the under-estimation of these errors during the last cycles of refinement, see Figs. 6.24(a), 6.24(c), 6.24(e), 6.24(b) and 6.24(d), is caused by the fact that the errors on the adapted mesh become comparable to the error in the benchmark solution. The ζ error is over-estimated in the near-wall region dominated by diffusion for which the estimator may predict higher errors because its truncation error, Eqn. (4.45), is not consistent with the truncation error for the diffusion term Eqn. (3.54).
The agreement between the exact error fields shown in Figs. 6.25(a), 6.25(c) and 6.25(e) and the estimated error fields presented in Figs. 6.25(b), 6.25(d) and 6.25(f) is poor and the reasons for that are threefold:
1. The FREE which tends to over-estimate the error on faces at which dif-
fusion is dominant. This is the reason why the estimated error in veloc- ity, Fig. 6.25(b), is higher in the bulk of the domain than the exact error, Fig. 6.25(a).
2. Face error averaging, Eqn. (4.48), used for plotting the errors. For example,
the maximum of the estimated velocity error field is located in the boundary layer, as expected, but the Fig. 6.25(b) shows that the error is highest in the stagnation zones. This happens because the size of the cell face determines its weighting factor in Eqn. (4.48).
3. Benchmark solution which is not exact. It is believed that the main reason
(a) Final adapted mesh
(b) Detail near the bottom tube wall
Figure 6.23: Mesh after 7 cycles of refinement for the tube bundle case (17505 cells)
6.4 Tube Bundle 159 Velocity Error 102 103 104 105 10-1 100 101 102 no of cells m ax e rr o r [% ] Exact error Estimated error
(a) Maximum velocity errors
Velocity Error 102 103 104 105 10-1 100 101 no of cells m ea n e rr o r [% ] Exact error Estimated error
(b) Mean velocity errors
Q Error 102 103 104 105 10-1 100 101 102 no of cells m ax e rr o r [% ] Exact error Estimated error (c) Maximum q errors Q Error 102 103 104 105 10-1 100 101 no of cells m ea n e rr o r [% ] Exact error Estimated error (d) Mean q errors Zeta Error 102 103 104 105 10-1 100 101 102 no of cells m ax e rr o r [% ] Exact error Estimated error
(e) Maximumζ errors
Zeta Error 102 103 104 105 10-2 10-1 100 101 no of cells m ea n e rr o r [% ] Exact error Estimated error (f) Meanζ errors
Figure 6.24: Variation of errors with adaptive mesh refinement for the tube bundle case (errors given as percentage of Umax, qmax and ζmax respectively)
(a) Exact velocity error (b) Estimated velocity error
(c) Exact q error (d) Estimated q error
(e) Exactζ error (f) Estimatedζerror
Figure 6.25: Exact and estimated error fields after 7 cycles of refinement (17505 cells)(errors are given as percentage of Umax,qmax and ζmax respectively).
Exact errors are calculated as the difference from the benchmark solution. Estimated errors are plotted as a weighted average of face errors.
6.4 Tube Bundle 161 the remaining error in the benchmark solution because the estimated error in the benchmark solution is highest there.
shearStress 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 0.0 0.01 0.02 0.03 0.04 angle[deg] sh ea rS tr es s cycle1 cycle7 Benchmark solution
(a) Shear stress at bottom tube wall (given as τw
ρU2 avg) Ux at x = 0.489 H 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 y/H U x cycle1 cycle7 Benchmark solution Measurements (b)Ux profiles at x= 0.489H (Ux nor- malised byUavg) Uy at x = 0.489 H 0.0 0.2 0.4 0.6 0.8 1.0 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 y/H U y cycle1 cycle7 Benchmark solution Measurements (c) Uy profiles at x= 0.489H (Uy nor- malised byUavg)
Figure 6.26: A comparison of profiles for the tube bundle case
Fig. 6.26(a) shows the shear stress at the bottom tube wall which is in a very good agreement with the benchmark solution even though the adapted mesh has only 3.6% of the number of cells of the benchmark mesh. Figs. 6.26(b) and 6.26(c) show that velocity profiles after 7 cycles of refinement match very well with the benchmark solution but the agreement with the measured data by Simonin and Barcouda [43] is poor. This happens because their data is given for higher inflow velocity than stated in [43]. Another limiting factor is the turbulence model which is not the exact representation of flow physics.