Mesh evaluation

The quality of the mesh affects the accuracy of the simulations. The shape and size of elements, the resolution of cells close to the wall, and the GCI analysis are utilized to evaluate uncertainties and errors related to the mesh.

Boundary layer modeling

Table 5.1 lists they⁺values obtained from the simulations. The averaged values are within the recommended range, ensuring that the first node is placed within the logarithmic region of the turbulent BL.

No evaluation of they⁺ values of the GV and runner blades are reported in this

6.2. Uncertainty and errors 73 thesis, but a previous evaluation of the mesh is presented by Iliev [11]. The quality of these meshes was assumed to be sufficient, without any further investigation.

However, this does not exclude the GVs and the runner as potential sources of errors.

The minimum values ofy⁺are found to be below the recommended limit of 30. As stated in subsection 2.2.6, the logarithmic BL is assumed to start aroundy⁺= 30.

y⁺ > 11.6 defines the lower bound of the buffer region, where the logarithmic relation is assumed to model the flow better than the linear relation in the viscous sub-layer. All minimum values ofy⁺ are found to be above 11.6, indicating no placement of nodes within the viscous sublayer. However, increasing the first layer thickness of the node might help to increase the minimum values ofy⁺further.

The maximum values ofy⁺for the mesh with 5 inflation layers in the elbow are in the range of about 200-300. High values like these may be an indication of the first node being placed far out in the logarithmic region of the BL, or in the worst case, outside of it. This may lead to a non-physical high thickness of the BL.

The average, minimum, and maximum values ofy⁺are decreased when increasing the number of inflation layers in the elbow. The first layer thickness of the inflation layers is constant, independent of the number of inflation layers in the elbow. As seen in section subsection 2.2.6,y⁺is a function of the first nodal placement close to the wall, kinematic viscosity, and the friction velocity. Since the former two are constant across the two approaches, the discrepancy must be attributed to the latter. This indicates that the friction velocity and the wall shear stress is modeled differently when the number of inflation layers in the elbow is changed. In general, the simulations performed with 10 inflation layers in the elbow result in more acceptabley⁺ values. This may indicate that the simulation performed with 10 inflation layers, undulating matters are more accurate. Considering they⁺values obtained and the upper limit of the logarithmic regiony⁺ ∼200−300, ANSYS’

recommendation of having 10 nodes in the logarithmic region is likely not achieved, which in turn may lead to poor modeling of the BL. Reducing the growth rate factor of the inflation layers may overcome this issue.

Correct modeling of the BL is crucial to obtain correct values of the shear stress and viscous losses. This will affect the calculation of pressure losses and be es-sential for findingCp. It should be noted that by using a wall function, the flow is not simulated, but modeled for the inner part of the wall BL. This may result in modeling errors, despite havingy⁺values within the acceptable limits.

74 6. Discussion

Mesh element quality

According to the mesh metrics listed in Table 5.2, the averaged values are within the recommended limits. However, the minimum values of the orthogonality angle of the mesh with 5 inflation layers, and the maximum volume expansion for both 5 and 10 inflation layers, are outside the recommended limits. This indicates that both meshes used in the simulations contained some elements of poor quality.

Both values for orthogonality angels and volume expansion are outside the recom-mended range. Mesh elements with these values are potential sources of discret-ization errors. Since the averaged values are within the acceptable limits, most elements are assumed to have sufficient quality. However, the elliptic nature of the governing equations might lead to a propagation of discretization error from the poor mesh elements.

Ansys Mechanical rates the averaged mesh quality as 60%, where 100% is ideal.

The exact consequence of this rating is unclear, but it suggests that it is possible to obtain higher quality meshes. The worst performing mesh element has a quality of 2%, highlighting the variation of mesh quality among the elements.

The number of elements and nodes affects the simulation time. Thus, increasing the mesh resolution for better accuracy must be traded off for a longer computational time. An unstructured mesh is utilized for the optimization study, but instead, con-sidering a structured mesh, computational time may be decreased as information is stored more efficiently. Additionally, more efficiently solvers exist for structured meshes.

GCI analysis

The results of the GCI analysis presented in Table 5.3, show considerable larger GCI for PL compared to BEP and HL. In general, a low GCI value indicates a well-converged result that is close to grid-independent.

A possible cause of the high GCI value obtained for PL might be the observed oscillation. The calculated values ofCpare lower for both the fine and the coarse meshes, compared to the medium resolution mesh. The oscillation can indicate instabilities, which makes it impossible to find a converged solution. A possible source of this issue may be that the simulation is badly posed with the BCs. Thus, performing the GCI study on an FS might give a lower GCI value.

For the optimization simulations in this work, the medium resolution mesh is util-ized. TheGCI_mediumvalues emphasize that numerical uncertainty is likely present in the results.

6.2. Uncertainty and errors 75