ENGR 401 ASSIGNMENT 2 2021
MATTHEW DURRANT
44626345
Part 1:
Question 1:
Approximating the flow as 2-Dimensional and axisymmetric ignores the effects of gravity, meaning
that the flow in the simulation will go on indefinitely, slowing spreading outwards, asymptotically
reducing to zero axial and radial velocity. Whereas, in reality the flow will be pulled down to the
ground by gravity and therefore unsymmetrical along the x-axis, and will not be able to spread out as
much in the x and y-directions before it reaches the ground. Therefore, the simulation will overpredict
the distance in the axial and radial directions the air from the sneeze and any particles it contains can
travel compared to reality. The simulation will also ignore the effects of external variables such as
wind, and the fact the air flow from a sneeze will not be symmetrical.
Question 2:
A rectangular computational domain was used initially with a height of 0.3 [m] and a length 1 [m] to
allow the use of rectangular mesh. Using rectangular mesh guarantees high mesh orthogonal quality
and low skewness. However, the downside of a rectangular computational domain in this particular
case is that it is less computationally efficient since additional computational resources are used
computing the flow in the top left corner, where the velocity gradient of the jet is very low. It was
thought that the improved mesh quality was a worthwhile trade-off for slightly increased
computational intensity and runtime.
Mesh biasing was used with a bias factor of ten, biased in the direction of flow on each edge of the
domain. This means that the largest cell length on each face is ten times larger than the smallest.
Biasing was used to resize the cells such that the cells in areas with large velocity gradients would be
smaller and the cells in areas with low velocity gradients larger, allowing the solution to capture the
large velocity gradients adjacent to the x-axis. A biasing factor of 10 was used to ensure that the cells
furthest from the x-axis did not have extreme aspects ratios, but that the cells adjacent to the x-axis
were small enough to capture large velocity gradients.
The mesh was refined by increasing the number of cell divisions per face and applying a face mesh
across the entire domain. Applying the face mesh kept the cells rectangular across the domain,
improving the mesh orthogonal quality as opposed to using triangular cells, which would result in
lower quality mesh. Refining the mesh, allowed mesh convergence to be observed as the mesh was
refined, until the solution became mesh independent. Refinements of the mesh with 10, 100, 200 and
300 cell divisions on each face may respectively be seen in Figures 1, 2 and 3 and 4.
Figure 1: Mesh of the computational domain with 10 divisions on each face
Figure 2: Mesh of the computational domain with 100 divisions on each face. Note: the biasing of small cells to the bottom
left where the velocity gradient of the jet is largest)
Figure 3: Mesh of the computational domain with 200 divisions on each face. Note: the biasing of small cells to the bottom
left where the velocity gradient of the jet is largest)
Figure 4: Mesh of the computational domain with 300 divisions on each face. Note: the biasing of small cells to the bottom
left where the velocity gradient of the jet is largest)
Question 3:
The initial laminar jet velocity was calculated from the Reynolds number for the laminar case (100)
using an assumed value for the kinematic viscosity of air at 35°C (the approximate temperature of
human exhalation) with the characteristic length taken as the full diameter of the jet slit (5.6 mm):
𝑅𝑒 =
𝑢𝐿
𝜈
Rearranged to:
𝑢=
𝑅𝑒𝜈 100 × 1.655 × 10−5
=
= 0.296 𝑚𝑠 −1
𝐿
5.6 × 10−3
Where: u is the initial jet laminar velocity, Re is the Reynolds number for the laminar case, L is the
width of the jet slit and ν is the kinematic viscosity of air at 35°C.
The centreline (axial) velocity was calculated for each of the mesh refinements. It can be seen from
Figure 5 that the velocity profile converges with approximately 200 nodes, as there is very little
change in the velocity profile between the solutions for 200 and 400 divisions on each face.
Therefore, it is fair to say that mesh convergence is achieved and the solution is mesh independent in
the laminar jet case with 200 cell divisions on each face.
Figure 5: Centreline (axial) velocity for the refinements of the mesh in the laminar case
In all mesh cases, the velocity begins at the calculated initial jet velocity and exponentially decays
along the x-axis, as expected.
Question 4:
The laminar streamwise (u) and spanwise (v) velocity components were plotted for 1, 5, 10, 20, 50
and 100 slit diameters for both the calculated ANSYS fluent laminar and analytical solutions in
Figures 6, 7, 8 and 9.
Figure 6: Spanwise Velocity Component v Numerical Solution
Figure 7: Spanwise Velocity Component v Analytical Solution
Figure 8: Streamwise Velocity Component u Analytical Solution
Figure 9: Streamwise Velocity Component u Numerical Solution
It can be seen from Figures 6 and 7 that the spanwise analytical profiles have a similar shape as their
numerical counterparts initially, but that the analytical solution predicts the spanwise velocities
becoming constant outside of the jet. A possible explanation for this disparity is the assumption of the
analytical solution that the pressure gradient outside of the jet is zero, and therefore that the span wise
velocity outside the jet is the same as at the edge of the jet. This is the likely cause of the analytical
spanwise velocity profiles becoming constant flat lines outside of the jet in Figure 7. However, the
numerical solutions suggest that there is a minor pressure gradient outside of the jet, with the pressure
gradient only reaching zero at the boundaries. This can be seen in Figure 6 as the velocity profiles are
still changing at 0.15 [m] but do eventually converge to zero at the boundary, where a zero-gauge
pressure boundary has been prescribed.
Question 5:
The momentum flux was computed for the laminar jet at the jet entrance and 1,5,10,20,50 and 100 jet
entrance diameters from the inlet. In Table 1 and Figure 10, it can be seen that the momentum flux
increases after the inlet with and decays from ten entrance lengths onwards. The increase in the
momentum flux diameters after the entrance is likely due to entrainment effects from the sharp entry
of the jet into the flow.
Table 1: Flow momentum flux at multiples of the jet slit diameter (s = 5.6 mm) from the entrance
Number of jet
diameters from slit
0s (inlet)
1s
5s
10s
20s
50s
100s
x-direction
Momentum Flux
[N/m2]
6.3457e-4
6.4159e-4
6.4124e -4
6.4145e-4
6.4062e-4
6.3708e-4
6.3486e-4
The momentum flux should be theoretically greatest at the inlet because the flow is at its maximum
velocity. As the jet is slowed with distance by viscosity, the flow loses momentum and the momentum
flux should decrease. Viscosity acts to slow down the flow as fluid lamina (layers) move over each
other, dissipating the kinetic energy of the flow through friction. A portion of the x-direction
momentum of the flow is also dissipated as the flow spreads out in the y-direction, further reducing it
with distance.
Figure 10: Momentum flux of the jet with respect to distance from the jet slit
Question 6:
To observe whether the flow developed to become self-similar, the velocity profiles of the jet at 1, 5,
10, 20, 50, and 100 entry lengths (s) were normalised in axial velocity by the peak axial velocity
u(x,0) at each increment of entry length, and in distance by the y-axis half-length which is defined as:
u(x,y) = u(x,0)/2, the distance along the y-axis at each increment of entry length where the axial
velocity component is half the peak value at that increment. The normalised velocity profiles were
plotted together in Figure 11.
Figure 11: Laminar jet normalised velocity profiles at various at 1,5,10,20,50 and 100 jet slit diameters (s)
Figure 11 shows that the velocity profiles become increasingly self-similar as the flow develops with
distance from the jet. It can be seen that the velocity profiles at 1, 5, 10 and 20, slit diameters (s) are
different from the others due to entry effects from the jet, whereas the remaining profiles are
increasingly lay on top of each other and become virtually indistinguishable. This matches the results
of (Mossad, Ravinesh, 2015) who identified the self-similarity region occurring after 20 jet entrance
diameters.
Part 2:
For the turbulent jet, a velocity value of 27.04 m/s was used to match the velocity used by (Mossad
and Ravinesh, 2015) for the same Reynolds number (104) and allow a direct comparison between
velocity profiles.
The kinematic viscosity was calculated from the known Reynolds number, slit diameter and velocity:
𝑅𝑒 =
𝑢𝐿
𝜈
Rearranged for the kinematic viscosity:
𝜈=
𝑢𝐿 27.04 × 5.6 × 10−3
𝑚2
−5
=
=
1.5142
×
10
[
]
𝑅𝑒
104
𝑠
Mesh convergence was checked and found to occur for meshes with 200 face divisions or more on
each face of the computational domain for the turbulent jet.
Question 7:
Experimental and reference simulation data from (Mossad, Ravinesh, 2015) was plotted alongside the
ANSYS fluent simulation results with the standard k-epsilon model for the jet with a Reynolds
number of 104 in Figure 12.
Figure 12: standard k-epsilon ANSYS fluent simulated profile with reference data from (Mossad, Ravinesh,2015) for the
turbulent jet
It can be seen from Figure 12 that the standard k-epsilon results correctly matches the experimental
data. This shows that the standard k-epsilon model can be successfully used to model turbulent jet
flow for moderate Reynolds numbers. The agreement with the experimental data shows that the kepsilon model used in this study is particularly effective at capturing the entrance effects of the jet
compared to the other refence numerical models. The difference between the k-epsilon profiles is
likely due to different types of mesh used in the two simulations. The reference values were calculated
using tetrahedral mesh, whereas this study used high orthogonal quality (>0.99) square mesh.
Question 8:
The potential core of the flow, is defined as the region of jet where the velocity remains effectively
unchanged from the original entrance value. It can be seen from Figure 9 that the there is a small ‘flat
plateau’ of the velocity profiles within approximately 0.04 [m] of the jet entrance for both the
reference values and the results of the k-epsilon and k-omega simulations. This corresponds to
approximately 4 -7 jet entrance diameters (s = 5.6 mm) from the jet entrance and contains the range
predicted by (Mossad, Ravinesh, 2015) of 4 -6 jet diameters.
Question 9:
To check the standard k-epsilon results, the simulation was recomputed using the k-omega turbulence
model. It can be seen in Figure 13 that the k-omega results do not conform to the experimental data as
well as the standard k-epsilon results but still agree relatively well, more so than the k-epsilon and
realisable k-epsilon results of (Mossad, Ravinesh, 2015). The k-omega model underpredicts the jet
velocity at every point. From this, it can be understood that the k-epsilon model is more effective at
modelling turbulent jet flow than the k-omega model for moderate Reynolds numbers. The inferiority
of the k-omega model compared to the standard k-epsilon model in this case is expected. The komega model is more accurate for modelling wall effects rather than free stream conditions, as in the
free stream both the turbulent kinetic energy and turbulence frequency unphysically tend to zero.
Whereas, the k-epsilon model does not suffer from this and can robustly model free stream
turbulence.
Figure 13: k-omega and standard k-epsilon ANSYS fluent simulated data with reference data from (Mossad, Ravinesh,2015)
for the turbulent jet
Question 10:
(Mossad, Ravinesh, 2015) describe the velocity profile of an air jet at a moderate turbulent Reynolds
of number (104). It can be seen in Figure 13 that their numerical results do not match their
experimental data.
Possible improvements in future similar studies could include:




Using rectangular mesh, or mesh with high (>0.90) orthogonal quality. This study used
rectangular mesh with extremely high orthogonal quality (>0.99) and found very close
agreement with the experimental results.
Using mesh biasing factors to bias the smallest cell size where the velocity gradients are
greatest. In this study, mesh biasing with a factor of ten was used in the positive flow
directions to ensure the mesh would accurately capture the velocity changes associated with
the jet centreline.
Using pressure outlet boundaries at the upper surface of the computational domain rather than
solid walls. The former was used in this study.
Adjusting the constants used in the turbulence models to empirically compensate for the
unknown factor(s) creating the disparity between the experimental and computational results.
Empirically adjusting for the unknown factors could allow the unknown factors affecting the
results to be deduced.
Part 3:
Question 11:
The computational domain was modified to take into the account effects of gravity making the flow
unsymmetrical. The new domain was constructed as a rectangle 2 metres high and 10 metres long.
The additional length of the domain was to consider the potential horizontal travel of the particles
after leaving the jet. The jet was placed 1.51 metres above the ground to take into to account the
height of the average person’s lips above the ground, as the average height of a person in New
Zealand in 1.71 metres (Statistics New Zealand 2018 Census), and the lips are approximately 20 cm
lower than the top of the head. The domain was extended 0.49 metres above the jet to prevent the
upper surface of the computational domain from constraining the paths of particles.
Question 12:
The trajectories of water particles injected into the air jet were plotted with and without turbulent
dispersion in Figures 14 and 15 respectively. The distributions of the particle exit positions against the
right boundary were plotted in Figures 16 and 17.
Figure 14: trajectories of 1µm diameter water particles without turbulent dispersion
Figure 15: trajectories of 1µm diameter water particles with turbulent dispersion
The difference between Figures 14 and 15 is clear, turbulent dispersion has resulted in a far larger
spread of the particles over the far-right boundary compared to the laminar distribution. The larger
spread of the turbulent particles is the result of turbulence in the jet dispersing the particles as the flow
moves to the right. The particle paths for both jets are initially relatively similar within approximately
the first metre of the domain from the jet. This has potentially useful implications for the modelling of
particle dispersion at short ranges as it shows that turbulent dispersion can be approximated
reasonably well at short ranges (< 1m) by a laminar particle distribution.
Figure 16: Distribution of particle exit positions against right boundary without turbulent dispersion
Figure 17: Distribution of particle exit positions against right boundary with turbulent dispersion (‘Number of tries’ =10)
It can be seen in Figures 16 and 17 that the turbulent jet possesses a much broader particle distribution
over the right boundary compared to the non-turbulent jet. This intuitively makes sense, as one would
expect turbulent dispersion to result in a greater spread of particles at any point in the jet. Although
the turbulent has a somewhat lopsided distribution with peaks at approximately 0.70 and 1.30, this
should be considered in the context of the statistical nature of turbulent dispersion. In the turbulent
simulation 10 “tries” were performed to produce a more accurate distribution of particles. The number
of tries effectively represents the sample size for the turbulent distribution. If the number of tries were
increased, a more representative distribution of the particles would be seen.
The larger spread of the turbulent particles has significant implications for the modelling of the
transmission of airborne diseases. The distribution of the turbulently dispersed particles is
significantly more widely dispersed than that of the laminar particles and will therefore be more
effective at spreading infected droplets or aerosols will increasing distance compared to the laminar
distribution.
Question 13:
The distribution on the ground was calculated for particles of 1, 10, 100 and 1000 µm. An aerosol
particle is defined as a particle emitted by coughing or sneezing capable of being suspended in the
ambient atmosphere. Particles of up to 5 µm are classified as aerosols for SARS - nCov-2 (Covid-19)
(World Health Organization, 2020), although the aerosol size depends on the particular respiratory
disease. Therefore, only the modelled particles of 1µm would be classified as aerosols by the World
Health Organization definition, with the 10, 100 and 1000 µm sized particles being classified as
droplets.
Using the sample function in ANSYS, it was found that none of the particles of size 1 and 10 µm
struck the ground with laminar dispersion. Half of the 100 µm particles were found to strike the
ground, with the other half escaping through the far-right boundary. All of the 1000 µm particles were
found to fall shorter than the 100 µm particles. This makes physical sense, as it would expected that
the larger the particle, the heavier it would be and the less distance it would travel horizontally before
falling to the ground. Since the dispersion is laminar, the distributions are deterministic and it should
not be possible for a larger, heavier particle to travel further than a smaller, lighter particle.
Figure 18: Distribution of 100 µm particles across the ground with laminar dispersion
Figure 19: Distribution of 1000 µm particles across the ground with laminar dispersion
The particle tracks were recomputed using turbulent dispersion and plotted in Figures 20, 21 and 22.
All 1 µm particles were found to escape from the computational domain, therefore travelling more
than 10 metres. 97% of 10 µm particles were found to escape out of the right boundary, with the
remaining 3 striking in the last half-metre of the computational domain.
Figure 20: Distribution of 10 µm particles across the ground with turbulent dispersion
91% of 100 µm particles were found to escape the computational domain out of the right boundary.
Figure 21: Distribution of 100 µm particles across the ground with turbulent dispersion
Figure 22: Distribution of 1000 µm particles across the ground with turbulent dispersion
At first glance in Figures 21 and 22, the heavier 1000 µm particles appear to be travelling further on
average than the smaller diameter, lighter 100 µm particles. However, it must be considered that in
the case of the 1000 µm particles, all of the tracked particles are striking within the computational
domain. Whereas, in the case of the 100 µm particles, the vast majority (91%) are escaping from the
control volume, in other words travelling more than 10 metres. Therefore, on average the 100 µm
particles are actually travelling significantly further than the 1000 µm particles. Because of the
random nature of the turbulence dispersion of the particles, the distance travelled by the particles must
be considered statistically, where it is highly likely that the vast majority of larger particles will fall
shorter than the lighter particles, but it is still possible for a heavier particle to travel further than a
lighter particle.
To validate the numerically calculated particle trajectories, a literature review was performed to see if
other sources suggested similar travel distances for similarly sized particles. (Fennely, 2020) states
that aerosol particles of 5 µm can be carried 7 – 8m from the mouth by sneezing and coughing, with
larger droplets of 60 – 100 µm carried up to 6 m, which approximately matches the findings of this
study, which found 1µm particles carried 10 m and particles of 100 µm carried 5 to 9.5 m. Therefore,
the results of this study should be considered valid in the context of the parameters used.
Question 14:
The results of the particle deposition study show that the smallest particles in both the laminar and
turbulent cases were easily capable of travelling the full ten metre length of the computational
domain, with only the largest 1000 µm particles unable to travel the full length. Therefore, it can be
concluded from this study that aerosol particles projected from the mouth can easily spread airborne
particles in a lecture theatre. This has significant implications for the potential ease of transmission of
particles carrying airborne diseases within other indoor spaces. These particle deposition findings
support the use of public health measures such as the wearing of masks and social distancing to
prevent the spread of airborne and other communicable diseases in indoor spaces. Future possibilities
to make similar studies more representative could be to model the flow in three dimensions and use
data from anatomical studies to better model the geometry of human lips. The study could also be
made more representative by measuring the Reynolds number of a human sneeze and modelling the
jet flow at this Reynolds number. Using a larger computational domain will also improve the results
by capturing more the particles which were seen to travel more than 10 metres and escape the domain.