In the past two decades, Computational Fluid Dynamics (CFD) simulation has advanced significantly and is now an essential segment in design for most engineering applications. (Müller, 2016) It is better defined as “a set of numerical methods and algorithms used to solve and analyse problems involving complex fluid flow behaviour.” (Schaschke, 2014)
This study involves the analysis of a stator blade aerofoil profile in an incompressible and inviscid flow. The blade profile is subject to a whirl angle of 42. Using STAR CCM+, the behaviour of the real flow is embodied into a physical model and is simulated. The model is based on assumptions, therefore imposing many limitations on its validity. (Müller, 2016)
The aim of this study is to use STAR CCM+’s visualisation package and post-processing tools to analyse the obtained discretisation. Using a mesh with the same number of cells, the obtained simulation will be compared against the provided simulation. Moreover, this study will compare the solutions obtained from different meshes, i.e. coarse, medium and fine mesh, to establish the influence of mesh refinement on the accuracy of the solution. The first order accurate discretisation will also be compared to the second order accurate discretisation to determine its influence on the accuracy of the solution. The simulations produce static pressure and velocity magnitude contours which is a numerical representation describing the fluid flow behaviour around the stator. Having evaluated the validity of each simulation, the most accurate value for circumferential and axial force coefficients can be calculated for the provided compressor stator blade model.
In this CFD simulation, one type of fluid behaviour (air flow) is considered – therefore one continuum model is defined. The solution obtained from the CFD model is to be computed based on Euler’s equations. Therefore, the simulation is formulated as a ‘segregated flow’. The time discretisation is defined as ‘steady’, and flow has a constant density (1.225kg/m^3 ) - the solution that will be obtained is known as the incompressible and inviscid steady state solution. Neglecting viscosity significantly reduces the validity of the solution as it cannot be exactly compared with the real flow. However, the solution can still reproduce some phenomena of the real flow.
The boundary at the stator surface is considered a wall, such that the fluid cannot penetrate the surface. As flow approaches the leading-edge region of the stator, it comes to rest due to the wall effect. This is known as the stagnation point. The kinetic energy of the fluid particles decreases as velocity decreases. From the inviscid Bernoulli’s equation, it is understood that total pressure, the sum of freestream static and dynamic pressure, must remain constant throughout. To compensate for this reduction, pressure increases. Since the stator is at -42 to the z-axis, the stagnation point is located beneath the leading tip. Similar, but the opposite effect will occur near the trailing edge as there is no boundary layer. A wake will not be seen as the velocity will increase further downstream of the trailing edge to maintain the constant total pressure. (Müller, 2016)
The flow is expected to accelerate considerably over the leading-edge. The flow follows the contour of the upper stator blade surface. Since the flow speed is higher, a greater pressure gradient is required for the flow to turn. It is proportional to the centripetal force. ∇P∝V^2/r
Where, ‘V’ and ‘r’ are ‘fluid velocity’ and ‘radius of curvature’, respectively. Compared to the freestream pressure, the pressure downstream of the stagnation point is lower, causing the streamline to curve towards the region of lower pressure. The point of minimum pressure is known as the suction peak. The flow is expected to have the highest velocities at the wall around the upper surface of the stator as the viscous effects are neglected in the simulation. The no-slip condition will not be obeyed. Since the upper surface region is subjected to lower pressure in comparison to the lower surface region, and that the stator profile has a relative angle to the flow field, the effect of this will cause the streamline to follow the curvature around the leading edge, whilst the trailing edge directs the flow downwards. The flow will adhere a clockwise, vorticial motion, therefore, is expected for a lift to be produced. (Müller, 2016)
Three different types of structured meshes were used: Coarse, Medium and Fine mesh. Fig.3, Fig.4 and Fig.5 shows the coarsest 128 x 32, medium 256 x 128 and finest 512 x 128 grid, respectively. Consider the trailing edges of each mesh, it can be seen that the vertices of the mesh are both tangential and perpendicular to the surface of the stator. By using a more refined mesh over the same profile view, the influences of mesh refinement can be observed. (Müller, 2016)
During first-order accurate discretisation, if there are twice as many mesh points, the time interval (t) and the mesh width (h) is halved. Therefore, the error is also reduced by a half. Overall, there are 2x as many mesh points between each mesh grid.
During second-order accurate discretisation, if there are twice as many mesh points, the mesh width (h) is halved and the error is reduced by a quarter. Overall, there are 4x as many mesh points between each mesh grid.
It is observed that around the trailing edge of the stator, the mesh remains orthogonal for all grid sizes. By refining the mesh, it is expected that there is a significant improvement in the resolution as it reduces the truncation error. To study this, the contour plots will be considered:
Table 1 exhibits the velocity contours for the stator profile produced by coarse, medium and fine meshes. It can be observed that over the upper surface of the stator, the contours bend back slightly illustrating there is a reduction in velocity near to the wall. This smearing behaviour suggests there is some boundary layer effect involved.
Now, consider the leading-edge velocity and pressure contours shown in Table 2. It can be observed that as the flow approaches the leading-edge region of the stator, the velocity contours become bluer. The flow velocity decreases dramatically from freestream velocity to 0.11066 m/s. As this happens, the static pressure contours in this region can be seen to be red – a high-pressure value. The pressure in this region increases dramatically to 978.69 Pa. This is known as the stagnation point. This can be explained using flow physics - the kinetic energy of the fluid particles decreases as velocity decreases. From the inviscid Bernoulli’s equation, it is understood that total pressure, the sum of freestream static and dynamic pressure, must remain constant throughout. The particles bunch up, but since the flow is incompressible, the static pressure increases to maintain continuity. If the velocity in the freestream is increasing, the pressure gradient is larger because the dynamic pressure is greater.
After the stagnation point, the flow accelerates considerably over the leading-edge following the curvature of the upper stator blade surface. The flow speed is observed in being higher, a greater pressure gradient is therefore also seen. The higher pressure is required for the flow to turn. It is proportional to the centripetal force as explained by Equation 1. At the wall around the upper surface of the stator, it can be seen from the yellow regions that the pressure downstream of the stagnation point is much lower. Although the no-slip condition has been satisfied, the flow does not have the highest velocities but instead shows relatively high velocities. This should not be the case as the viscous effects are neglected in the simulation. It causes the streamline to curve towards the region of lower pressure. The point of minimum pressure is known as the suction peak. Referring to Fig.1 and Fig.5, the scales of the pressure coefficient axis are reversed such that the upper curve and lower curve represent the static pressures on the upper and lower regions of the stator, respectively. The solution from the fine second-order discretisation shows that at the position x = 0.016 on the upper surface, the suction peak has a minimum static pressure coefficient of -2.81(2s.f.). Whereas, the coarse first-order discretisation shows a much lower magnitude of static pressure. The magnitude of static pressure at the suction peak increases. There are significant improvements in static pressure with mesh refinement and using higher order accuracy. The magnitude of static pressure is seen to increase with better accuracy.
This smearing behaviour shown in the contours proves that there is viscosity present that should not be there. The boundary layer effects should not exist as the simulation model is chosen such that the flow is inviscid – the effects of viscosity are negligible. At the boundary of the stator, the flow should be a tangent to the wall such that the fluid particles slide freely past. The wall should only cause redirection of the flow. This is seen to be an inaccuracy in the solution and suggests the presence of artificial viscosity. The stagnation point is located beneath the leading tip as the stator has a relative angle with the fluid flow.
These effects are more apparent in the coarse mesh discretisation compared to the fine mesh discretisation. Similarly, by considering a specific mesh, the second-order discretisation velocity contour shows less smearing behaviour compared to the first-order discretisation velocity contour - as the growth of this boundary layer effect is less pronounced. The accuracy is slightly improved as the mesh is more refined and/or higher order accurate discretisation methods are used. Using more accurate solutions reduce the presence of artificial viscosity.
Upward lift is linked to the turning of the flow at the trailing edge. From Table 2, consider the fine second-order accurate discretisation. The upper surface region is seen to be subjected to lower pressure in comparison to the lower surface region, and that the stator profile has a relative angle to the flow field. This causes the streamline to follow the curvature around the leading edge, whilst the trailing edge directs the flow downwards. The flow will adhere to a clockwise, vorticial motion, therefore, is expected for an upward lift to be produced.
The flow solution for the second-order discretisation was carried out on a medium-mesh with a maximum stopping criterion of 3000 for second-order. A minimum limit was initially defined where the case was said to converge if the residual is 1x10-4. If the residual falls below this value, the solver will stop. This was changed during the second-order case as the solver stopped at this threshold, and the solution was not steady enough. The stopping criterion was set so that if the case does not converge, this will avoid running the case infinitely. Whereas for the provided case, the stopping criterion was 1x10-7, and the maximum step was 2500.
It was ensured that the case was correctly executed and sufficiently converged by considering the number of iterations and the residual values. If the maximum number of iterations was reached before reaching the minimum limit, then it was deemed unsuccessful and the maximum criterion was increased to that closer to provided cases.
Furthermore, the mesh diagram between the two cases slightly differ. In the provided mesh, it can be seen that the vertices of the mesh are more orthogonal to the surface of the stator compared to that of our own. By refining the mesh as described, it is expected that there is a significant improvement in the resolution as it reduces the truncation error. However, it can also be observed that the contour lines for the provided case mesh are jagged giving rise to unsteadiness in the solution. Consider the velocity contour for the provided simulation and own simulation on Table and Table 3, respectively. If we focus on the trailing edge of the stator, the region of blue (lower) velocity is smaller for the provided simulation. Therefore, means the presence of artificial viscosity is less. Also, consider Fig.7 for the static pressure coefficient for both cases. It can be seen that both cases are similar. However, as the simulation model is chosen such that the flow is inviscid – the effects of viscosity are negligible. At the surface of the stator, the flow should be a tangent to the wall such that the fluid particles slide freely past. Hence, static pressure should be lower (minimum) whilst velocity magnitude is at maximum. It is also known that, for total pressure to remain constant, the pressure contour line should be the same as the velocity contour lines. Therefore, it is concluded that the second-order discretisation using medium mesh is more accurate.
The changes in total pressure can be investigated to quantify the effects of artificial viscosity. The inflow has a constant total pressure of 896 Pa. This is equal to the dynamic pressure since the freestream static pressure at the inlet is zero. This is because the inlet and outlet pressure are taken as zero-gauge pressure. In the figure below, the actual total pressure along the profile is illustrated.
From Fig.9, it can be observed that there is an unusual behaviour of total pressure for the first-order solutions. Each symbol is a mesh point, there is more mesh concentration near the leading edge and trailing edges. Near to the stagnation point, there is a severe overshoot in total pressure. This is due to the presence of the defined boundary condition where the fluid cannot penetrate through the surface of the stator. It is a solid boundary. It can also be seen that the second-order discretisation corrects this error. Downstream, it is clearly seen that the total pressure reduces significantly, where the total pressures for the upper and lower surface of the stator are identical. This can be explained using Table 1 - the contour plots show streamlines from both upper and lower surface joining; therefore, the total pressure is averaged.
In the region on top of the stator, a loss in total pressure can be observed. This can be explained by the flow behaviour due to the suction peak. The upper streamline close to the wall experiences stronger acceleration and curvature compared to the streamline on the lower side. Therefore, there is greater change (more residual) involved resulting in greater artificial viscosity. There are 7 curves above the fine first-order accurate discretisation for the upper surface. The error is significant due to the viscous boundary layer effects. But by refining the mesh and changing from first to second-order, this loss of total pressure is considerably reduced.
The plot for the force coefficient values is displayed on Fig.7. The second-order (own) simulation has x number of cells compared to the second-order provided simulation having 2448 cells. These are both below the stopping criterion indicating the residuals are within a minimum range. Both cases are, therefore, stable.
Overall, the effect of refining the mesh and switching to 2nd order has improved the solution by reducing the errors. Both methods have contributed to reducing the artificial viscosity and maintaining the constant total pressure. In the light of the results, the residual is the least for fine second-order discretisation, therefore it is the most stable solution. It has produced the largest magnitude of the coefficient of axial and circumferential force illustrated in Table 4. As explained above, at the surface of the stator fluid particles slide freely past. Hence, static pressure should be the lowest (minimum) whilst velocity magnitude is at maximum. Also, when the pressure contour lines are the same as the velocity contour lines, it suggests that total pressure has been kept constant. Therefore, it is concluded that the second-order discretisation using medium mesh is more accurate and stable producing the best values for force coefficients.
Artificial viscosity scales with the product of mesh width, h or h¬2, and the first gradient (du/dx) or second gradient (d2u/dx2) of the first or second order, respectively. To reduce the errors from artificial viscosity, a smaller mesh-width can be used i.e. mesh refinement. An example of this is seen when there is greater loss/change in total pressure in the region after the leading edge, indicating that an improvement can be made. By concentrating more mesh points in these regions of large change, the appearance artificial viscosity present is reduced. But it becomes expensive very quickly. Another approach is to reduce the mesh density in regions where little happen and maintain good mesh quality in areas of strong variation. Otherwise, it is easier to use the second order scheme to obtain more accurate solutions.
The variant of medium mesh used for the provided case is better than the medium mesh used for ‘own’ case. The mesh contours for both cases is located in the appendix. The vertices of the mesh are more orthogonal to the surface of the stator compared to that of our own. By using more accurate discretisation and refining the mesh, the involvement of artificial viscosity in the solution is reduced. The second-order fine mesh discretisation has significantly reduced the boundary layer effects the most in comparison to the other cases and the model for inviscid and incompressible flow is better simulated. This was clearly shown by contours provided. At the upper wall of the stator, this solution showed higher velocity as the no-slip condition is not obeyed. The velocity streamlines are more tangential, which results in lower static pressure. Since it is known that lift is related to this static pressure, a reduction in the static pressure produces higher lift prediction. The suction peak is also seen to be highest compared to all of the cases. The solution obtained for the axial and circumferential forces are better also predicted. It can be said the solution has mesh converged and is stable as the change (residual) is below the defined minimum.
At times, there are modelling errors that are bigger than artificial viscosity errors. i.e. running the stator case as turbulent flow. Once artificial viscosity is less than the modelling error, it is considered negligible and the solver can be stopped. In practice, for CFD engineering there is never a situation when refining is completed, but whether they are significantly affecting the solution has to be evaluated as presented by this report.
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