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Table of Contents SUMMARY……………………………………………………………………………………………………………………………………………..2

RESULTS AND DISCUSSIONS

QUALITATIVE ANALYSIS ..................................................................................................................................... 3

Relative Pressure ........................................................................................................................................... 3

X-VELOCITY: .................................................................................................................................................... 3

QUANTITATIVE ANALYSES .................................................................................................................................. 4

Validation ....................................................................................................................................................... 4

Recirculation Length ...................................................................................................................................... 5

Reynolds Number .......................................................................................................................................... 5

Pressure Drop Coefficient .............................................................................................................................. 5

EFFECTS DUE TO CHANGES IN VISCOSITY .......................................................................................................... 6

Recirculation Length ...................................................................................................................................... 6

Non Dimensional Analysis .............................................................................................................................. 7

TURBULENCE MODELLING ................................................................................................................................. 7

Convergence .................................................................................................................................................. 7

Bulk Velocity, Reynolds Number and Kp ....................................................................................................... 7

Pressure and Velocity Contours ..................................................................................................................... 8

CONCLUSION

Accuracy ............................................................................................................................................................. 9

Viscosity Effects ................................................................................................................................................. 9

Turbulence Models ............................................................................................................................................ 9

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Summary This report summarises and discusses the results obtained from the CFD simulation of Laminar flow in tube

bundles using STAR CCM+. Symmetries and periodicity in boundary conditions were used to reduce the

problem to a simple ‘staggered tube bundle configuration’. The flow was computed as laminar for three

meshes. The first one consists of block structured quadrilateral cells with a resolution of 540 cells in total

with 72 points around one cylinder. The second mesh is an unstructured mesh made of quadrilateral cells

(PAVE). The total number of cells is 433 with 64 points around one cylinder. The third mesh (TETRA) is

made of triangular elements, with a total number of 490 cells with 64 points around one cylinder. The

following analyses have been conducted in this study:

1. Qualitative Analysis: The relative pressure and X-velocity contours obtained from simulation of the

three meshes were compared and discussed.

2. Quantitative Analysis: The Bulk Reynolds Number, Recirculation Length and the Pressure Drop

Coefficients obtained from the simulations have also been compared. Additionally, the X-velocity

along the inlet and the Y=0 axis have also been compared to the results obtained from a very fine

mesh setup to validate the accuracies of prediction.

3. Further Analysis: The viscosity was increased by twice and by four times and successively by 8 times

to increase the Reynolds Number. The Bulk Reynolds Number, Recirculation Length and the Pressure

Drop Coefficients were compared. Furthermore, the Reynolds Number was increased by subsequently

halving the viscosity to enforce turbulence. Different Turbulent models (Realizable 𝑘 − 𝜀 with

Enhanced Wall Y+ Treatment and 𝑘 − 𝜔 𝑆𝑆𝑇 with Enhanced Wall Y+ Treatment) were experimented

and the results were compared against the ‘Laminar’ model.

It was observed that the TETRA mesh showed the best agreement with the FINE mesh results, and that

changes in Viscosity had a significant impact on the recirculation length. Kp and KpxRe became insensitive

to further changes in Reynolds Number and viscosity respectively after a certain point. Finally, despite of

taking longer than the 𝑘 − 𝜀 model, the k-ω SST and Laminar Models showed good agreement with each

other giving similar values for both the Reynolds Number and Pressure Drop Coefficient. The 𝑘 − 𝜀 model

however gave more conservative numbers for velocity and pressure fields, and hence Reynolds number.

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Results and Discussions

QUALITATIVE ANALYSIS

Relative Pressure

Figure 1: Relative Pressure Contours :( In order) Block, PAVE, TETRA and FINE

Figure 1 shows the relative pressure contours obtained from the simulation of the four different meshes

simulated. Qualitatively speaking, good results for pressure field was obtained for most of the region in the

domain except for X=0 and the near the stagnation points. It was inferred that the TETRA mesh provides

the closest resemblance to the high resolution FINE mesh both in terms of the quality of plots and the

magnitude of relative pressure fields. The BLOCK grid was found to be the most inaccurate, especially in

the regions experiencing transverse gradients i.e. near the boundaries layers, shear layers and wakes. This

has been discussed in the ‘Validation’ section in more details. So, the maximum error for this case was

found to be about 58.8% as compared to the highly refined mesh, and this was found predominantly near

the sharp corners.

X-VELOCITY:

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Figure 2: X-Velocity Contours :( In order) Block, PAVE, TETRA and FINE

Figure 2 shows the X velocity contours for all the meshes. Qualitatively speaking, all the meshes showed

good values for the X-velocity across the flow domain, with high velocity flows in the centre of the domain,

away from the cylinders. Negative values indicate flow reversal caused due to vortex shedding in the

wake of the cylinder. The length at which the X-velocity changes in to positive is termed as the circulation

length, which have been compared for all meshes in the following section.

QUANTITATIVE ANALYSES

Validation

Figure 3: Validation Results

Figure 3 shows the plots of X-velocity against position for all the simulations to provide some numeric

insight into the accuracy of the results. The ‘Black’ line represents the results obtained from the FINE mesh

solution. The TETRA mesh (green) shows best agreement with the FINE mesh solution for the X-velocity

values on the axis X=0. The BLOCK mesh (blue) under-predicted the X-Velocity while the PAVE mesh over-

predicted it from X=0.5 to X=0.55. Additionally, all the coarse meshes over-predicted the X-velocity field

slightly at the start: a clear consequence of the incorrect pressure values in the region as discussed earlier.

Finer resolutions near the sharper corners or Corrective Boundary Wall Function may be adapted to

mitigate this issue. Otherwise, the values from all meshes were quite ‘reasonable’, given the

computational benefits.

On the other hand, on the Y=0 axis, none of the coarse meshes showed good agreement with the FINE

mesh solution from X=0.5 to X=0.85. Clearly, none of the COARSE meshes are fine enough to capture the

effects due to Vortex Shedding to a desired level of accuracy. Additionally, all the COARSE meshes under-

predicted and over-predicted the X-velocity field slightly at the start and from X=1.1 to X=1.4 respectively.

Overall, TETRA shows the best agreement with the FINE mesh results. Despite of inaccuracies on the Y=0

axis, it has been further been demonstrated in the later section that the TETRA mesh gave the minimum

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error for the some of the key parameters of interest. Furthermore, it could also be established that the

following coarse meshes are still Grid- Dependant, and that the results will continue to change as finer

meshes are chosen. Since BLOCK structured grids are predominately aligned in one direction, they

evidently fail to capture the effects due to transverse gradients such as the boundaries layers, shear

layers and wakes. Even though it has higher number of cells as compared to the PAVE (second) mesh, the

latter has finer unstructured grids near the boundary wall that can capture the effects due to transient

gradient. Despite of having lesser cells, TETRA provides better values than BLOCK due to presence of

unstructured cells and possibly due to the small changes in cell Aspect ratios and volumes.

But, the key trade-off here is the accuracy of the solution vs the computation time. Despite the

inaccuracies in the results, all the COARSE meshes provide great ease of ‘meshing’ and reduction in

computation time. Hence, coarser grids may be good choices where, at least for low Reynolds Number

Laminar flows where high accuracy is not a concern.

Recirculation Length Recirculation length is the length in the wake of any bluff body where the velocity changes direction from

positive to negative when plotted on the wake centreline i.e. Y=0 plane in this case. This was obtained

from Figure 3. The FINE mesh gave a recirculation length of 0.36 m which is almost exactly matched by

the TETRA mesh. BLOCK and PAVE meshes over-predict the recirculation lengths as 0.365 and 0.37

respectively.

Reynolds Number Reynolds number is defined as the ratio of inertial forces to viscous forces in the fluid. Hence, it indicates

whether the flow is laminar or turbulent. Table 1 shows the comparison between the Reynolds Number

obtained from the all the cases. The FINE mesh gave a Reynolds Number of 172.18. TERTA mesh showed

good agreement with the FINE results with a minimal error of 1.38%. PAVE was the only grid to over-

predict the Reynolds Number. Clearly, the flow is ‘laminar’- hence the viscous forces are dominant in this

case. Changes in viscosity affect the solutions considerably. This has been discussed in the next section.

GRIDS Bulk Velocity

Reynolds Number

Error

BLOCK 0.00254 163.35 5.13%

PAVE 0.00272 175.37 -1.85%

TETRA 0.00264 169.80 1.38%

FINE 0.00267 172.18 Table 1: Reynolds Number for all the meshes

Pressure Drop Coefficient

Pressure drop coefficient is a non-dimensional term defined as follows:

𝐾𝑝 = ∆𝑃/0.5𝜌𝑣2

Where∆𝑃, 𝜌 𝑎𝑛𝑑 𝑣 are the pressure drops, density and bulk velocity across the flow domain respectively.

Table 2 shows the pressure drop coefficients obtained from all the meshes. Though slightly over-predicted,

the TETRA showed the best agreement with the FINE mesh value of 2.44 with a minimal error of 2.82%.

Hence, the mesh TETRA was chosen for further analyses.

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GRIDS Pressure Drop Coefficient

%Error

BLOCK 2.64 -11.10%

PAVE 2.29 3.61%

TETRA 2.44 -2.82%

FINE 2.37 Table 2: Pressure Drop Coefficients for all meshes

EFFECTS DUE TO CHANGES IN VISCOSITY The viscosity was systematically increased by multiplying with 2, 4 and 8. It was also halved for the

following analysis, and three important comparisons were made which have been discussed in this section.

Table 3 summarises the results obtained.

Increment x8 x4 x2 x1

Dynamic Viscosity (Pa.s)

1.46E-04 7.32E-05 3.66E-05 1.83E-05

Bulk Velocity (m/s)

0.000715 0.001344 0.002117 0.002636

Re 5.759056 21.63608 68.17281 169.8048

Kp 33.16034 9.397759 3.786337 2.44119

KpxRe 190.9723 203.3307 258.1252 414.5258 Table 3: Effects due to changes in viscosity

Recirculation Length As mentioned earlier, Recirculation length is where the X velocity changes its sign from negative to

positive. As it can be seen from Figure 4 below, the previous analysis gave a recirculation length of 0.36 m.

As the viscosity was doubled, the Recirculation length drastically reduced to 0.05m. When it was

increased further by 4 or 8 times, the Recirculation length appeared to nullify. This is because the viscous

forces are so dominant that there is no flow separation from the cylinder. On the other hand, when the

viscosity was halved, Recirculation length occupied the complete domain. This is because to the

dominance of the inertial forces. The flow separates from the cylinders giving rise to large unsteady vortex

shedding that continues till the end of the domain.

Hence, it can be inferred that viscosity has a significant impact on the recirculation length. Increase in

viscosity results in a smaller recirculation length and vice versa.

Figure 4: X-velocity on Y=0 axis

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Non Dimensional Analysis The variation of the non-dimensional parameters Kp, Reynolds Number and their mutual product with

changing viscosity have been analysed in the section. Figure 5 (a) shows how Kp varied with the Reynolds

Number. Kp dropped sharply as Reynolds Number was increased up to about 70, after which it became

relatively insensitive to further increment in Reynolds Number. Figure 5 (b) shows the variation of KpxRe

with viscosity. Similar to Kp, KpxRe dropped sharply as viscosity was increased up to about 8E-5, after

which it became relatively insensitive to further increment in viscosity.

Figure 5: (a) Kp vs Reynolds Number (b) KpxRe vs Dynamic Viscosity

TURBULENCE MODELLING The analysis was further extended to study how changing the turbulence models affects the results from

the TETRA mesh. The viscosity was halved to 9.16 E-6 Pa.s in order to increase turbulence. Three

turbulence models were then tested: Laminar, Realizable 𝒌 − 𝜺 with Enhanced Wall Y+ Treatment and

𝒌 − 𝝎 𝑺𝑺𝑻 with Enhanced Wall Y+ Treatment.

Convergence Figure 4 shows the plots of residuals for all the three cases. A convergence criterion of 10^-6 was chosen

for the residuals. The Laminar model showed slow convergence with 5000 iterations. The 𝑘 − 𝜀 model with

Enhanced Wall Y+ Treatment achieved a much faster convergence (in 1200 steps) as compared to 𝑘 −

𝜔 𝑆𝑆𝑇 with Enhanced Wall Y+ Treatment (5000 iterations).

Figure 6: (in order) Laminar, k-ε with Enhanced Wall Y+ Treatment and k-ω SST with Enhanced Wall Y+ Treatment.

Bulk Velocity, Reynolds Number and Kp Laminar K-E K-w

Bulk Velocity 0.00287589 0.003103 0.002896

Reynolds Number

370.45 399.65 373.04

Kp 2.05 1.76 2.02 Table 4: Bulk Velocity, Reynolds Number and Kp for all the models

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Table 4 shows the Bulk Velocity, Reynolds Number and Kp for all the models. Highest predictions for Bulk

velocity and hence, Reynolds Number were obtained from Realizable 𝒌 − 𝜺 model whereas the lowest

came from the 𝒌 − 𝝎 𝑺𝑺𝑻 model. The highest and lowest Reynolds numbers obtained were 399.65 and

370.45 respectively. Similarly, the highest value of Kp was obtained at 2.05 for the laminar flow whereas

the lowest was at 2.02 for the 𝒌 − 𝝎 𝑺𝑺𝑻 model. Since, the domain is not highly turbulent, it is expected

that the ‘Laminar’ model should still be a reasonable choice. The predictions from the 𝒌 − 𝝎 𝑺𝑺𝑻 model

shows good agreement with the predictions from the ‘Laminar’ models. Some of the other qualitative

results obtained have been provided below.

Pressure and Velocity Contours

Figure 7: (in order) Relative Pressure Contours for Laminar, k-ε with Enhanced Wall Y+ Treatment and k-ω SST with Enhanced Wall Y+ Treatment.

Figure 7 shows the Relative Pressure Contours for Laminar, k-ε with Enhanced Wall Y+ Treatment and k-ω

SST with Enhanced Wall Y+ Treatment. Whilst both Laminar and k-ω SST gave similar values for relative

pressures with a peak of 6.54E-6, the k-ε model seemed to overestimate it slightly in the regions of high

pressure with a peak of 8.54 E-6. However, the k-ε model predicted a lower pressure wake region

(stronger) wake as compared to the other two.

Figure 8 shows the contours of X-Velocity magnitude for Laminar, k-ε with Enhanced Wall Y+ Treatment

and k-ω SST with Enhanced Wall Y+ Treatment. Qualitatively speaking, Laminar and k-w models show good

agreement between the results obtained. However, the k-ε model predicts slightly faster flow across a

wider section of the free path in the domain.

Figure 8: (in order) contours of X-Velocity magnitude for Laminar, k-ε with Enhanced Wall Y+ Treatment and k-ω SST with Enhanced Wall Y+ Treatment.

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Conclusion The following conclusions can be drawn from this activity:

Accuracy The coarser grids examined may be good choices, at least for low Reynolds Number Laminar flows

where a high degree of accuracy is not of concern. The trade-off is computation cost vs accuracy.

The TETRA mesh showed the best agreement with the FINE mesh results. It also gave the least error

for the key parameters of interest such as Reynolds Number, Recirculation length and the Pressure

Drop Coefficient with a minimal error around 2%. BLOCK structured grids also gave good results but

since they are predominately aligned in one direction, they evidently failed to be accurate in the

regions of transverse gradients such as the boundaries layers, shear layers and wakes.

Viscosity Effects Changes in Viscosity has a significant impact on the recirculation length. Increase in viscosity results

in a smaller recirculation length and vice versa. As the viscosity was doubled, the Recirculation

length drastically reduced to 0.05m, and seemed to nullify with further increments in viscosity as

the strong viscous forces resisted the flow separation from the cylinder. On the other hand, when

the viscosity was halved, Recirculation length occupied the complete domain.

Kp dropped sharply as Reynolds Number was increased up to about 70, after which it became

relatively insensitive to further increment in Reynolds Number.

KpxRe dropped sharply as viscosity was increased up to about 8E-5 Pa.s, after which it became

relatively insensitive to further increment in viscosity.

Turbulence Models The k-ω SST and Laminar Models took about 5000 iterations to converge whereas the k-ε model

was much faster taking only 1200 iterations to converge.

For a dynamic viscosity of 9.16 E-6 Pa.s, the k-ε model gave the highest and lowest value for

Reynolds Number (400) and Pressure Drop Coefficient (1.76) respectively.

The k-ω SST and Laminar Models showed good agreement with each other giving similar values for

both the Reynolds Number and Pressure Drop Coefficient.