computational fluid dynamics (cfd) mec60703/mec4513

Computational Fluid Dynamics (CFD) MEC60703/MEC4513

1

Computational Fluid Dynamic (CFD)

MEC60703/MEC4513

Assignment I & II

NAME ID SIGNATURE

Adam Effendi Ashaari

0322851

Due Date: 23/10/2019 at 3.00 pm

Date of Submission: 23/10/2019 at 3.00 pm

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CONTENTS

1. INTRODUCTION 3

2. CFD ANALYSIS

2.1 INTRODUCTION 4

2.2 TASKS 5

a. TASK 1: GEOMETRY AND MESHING 5

b. TASK 2: MULTIPLE SOLVERS 17

c. TASK 3: TURBULENCE MODELS 23

3. CONCLUSION 37

4. REFERENCES 38


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1 INTRODUCTION

Computational fluid dynamics (CFD) plays an important role in the analysis and simulation of

fluid flow in various engineering applications. It helps to solve the Navier-Stokes equations which are

three dimensional and time dependent using pragmatic computing effort. Software like ANSYS Fluent

is able to carry this out with the usage of various turbulence models and solvers which allow the

avoidance of direct computer simulations that require an immense amount of computing effort.

Turbulence is a very complex phenomenon to solve analytically due to being three-dimensional,

nonlinear and dependent on time. By using computer simulations, engineers are able to design and

simulate part designs to better understand its behaviour when interacting with a flowing fluid. This

helps to save high amounts of money and resources as well as avoid the unnecessary need to build

physical yet inaccurate prototypes.

An area of intense CFD application is the flow analysis of pipes. Various industries like the energy,

construction, manufacturing and aerospace industries require large networks of pipes to provide

continuous fluid transport be it products or raw materials. To ensure continuous transport, energy is

given into the pipe system via electricity that power pumps. This indirectly affects the amount of natural

resources like coal or natural gas that is used up to transport fluids through pipe systems. Hence, it is

important that energy wastage due to frictional energy loss between the fluid and pipe wall as well as

internal fluid friction is reduced. CFD plays an important role in ensuring pipe components are designed

to provide optimal and efficient flow conditions for fluids. An important component of numerous pipe

systems is the T-junction pipe. It consists of a single main pipe with another pipe connected

perpendicularly to it. It is mainly used to diverge the flow of fluid from a single main pipe to a few

branching pipes or converge the fluid flow of several pipes into a single main pipe.

In order to simulate an accurate flow of fluid in a T-junction, suitable settings must be prepared in

the CFD software like ANSYS. This includes settings the most suitable number of mesh elements, mesh

element size, the type of solver, turbulence models and fluid parameters. Therefore, the objective of

this assignment is to develop a simulation model of an internal flow inside a T-junction pipe through a

step-by-step process from model meshing to turbulence model selection. The suitability of various

software settings is evaluated by comparing the results of various mesh settings, CFD solvers and

turbulence models in the form of contours and graphs of selected parameters in the T-junction pipe.


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2 CFD ANALYSIS

2.1 INTRODUCTION

Figure 1.1: T-junction geometry and the location of the inlets and outlets.

Based on Figure 1.1, the T-junction pipe in this analysis had a pipe diameter, d = 0.1 m and

horizontal length, L = 1 m. The location of the vertical inlet is 0.25 from the inlet of the horizontal

section and has a length of 0.5 m. The T-junction pipe has a constant cross section. The fluid flowing

in the T-junction is liquid water which flows through both inlet at a velocity of 1 m/s. The fluid flows

through the outlet which opens to the ambient atmosphere with pressure of 101325 Pa (1 atm). The fluid

has a constant velocity as it flows through the T-junction pipe. The material of fluid is water-liquid with

density = 998.2 kg/m3 and coefficient of viscosity μ = 1.003 x 10-3 kg/ms.

Figure 1.2: The T-junction model in Solidworks

Figure 1.3: The T-junction model drawing and dimensions in millimetres.


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To begin the task, the T-junction was drawn and modelled in Solidworks 2019 with the necessary

dimensions as shown in Figure 1.2 and 1.3. Then, the file was exported as an IGES file and imported

in an ANSYS 2019 workbench to begin meshing. In the ANSYS workbench, meshing was carried out

before any flow analysis was done. The mesh element size and number of mesh elements needed to be

determined so that the flow analysis can be carried out accurately and within a reasonable amount of

time. This meant that the mesh optimization needed to be carried out. To begin mesh optimization, a

grid independence test (GIT) and mesh refinement needed to be done. The GIT was carried out by over

a finite number of mesh elements and calculating the total pressure, total velocity and z-velocity flowing

through the T-junction outlet for each iteration. Increasing the number of elements meant that the mesh

element size needed to decrease. Hence, the manipulated variable in the GIT was the mesh element size.

2.2 TASKS

TASK 1: GEOMETRY AND MESHING

Figure 2.1.1: The Grid Independence Test (GIT) workbench

Figure 2.1.2: The results generated from the GIT workbench.


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Based on Figure 2.1.1, the GIT was set up in ANSYS Fluent with T-junction pipe IGES file.

The input parameter is the mesh element size and the out parameters were the mesh nodes, mesh

elements, orthogonal mesh quality, total pressure, total velocity and z-velocity. The calculations were

carried out on a range of mesh element sizes ranging from 0.05 m to 0.006 m as shown in Figure 2.1.2.

Any element size below 0.006 m could not generate results for total pressure, total velocity and z-

velocity due to reaching the resolution limit of the software. This was due to the ANSYS software used

being an educational version. The results of the GIT are tabulated in Table 2.1 and a set of graphs are

plotted in Figures 2.1.3, 2.1.4, 2.1.5 and 2.1.6.

Table 2.1: Table of orthogonal quality, mesh nodes, mesh elements, total pressure, total velocity

and z-velocity as element size changes from 0.05 m to 0.006 m

Element

Size

(m)

Orthogonal Quality No. of

mesh

nodes

No. of

mesh

elements

Total

pressure

(Pa)

Total

Velocity

(m/s)

z-velocity

(m/s) Min Max Average

0.05 0.0343546

4

0.9669617

6

0.3215846

51

2074 3781 1577.536

3

1.689538

2

-

1.540173

6

0.04 0.0146754

29

0.9599131

25

0.3788290

23

2737 5322 1853.931

7

1.980696

8

-

1.969063

9

0.03 0.0049741

06

0.9584313

42

0.4540639

24

4192 8677 1924.694

1

1.990573

9

-

1.986453

6

0.02 0.0577904

73

0.9803453

25

0.6633193

75

7654 17733 1957.213

4

1.980620

2

-

1.979052

6

0.01 0.1227099

78

0.9899036

08

0.7761862

4

31361 95492 1890.911

1

1.924750

4

-

1.924372

6

0.009 0.1700578

87

0.9952909

55

0.7716746

92

41062 130480 1908.409

4

1.929137

2

-

1.928883

3

0.008 0.1749931

29

0.9901677

45

0.7729291

32

53909 179917 1910.535

5

1.931614

8

-

1.931299

4

0.007 0.1839556

82

0.9912140

88

0.7720854

61

76653 268600 1920.831

3

1.935514

5

-

1.935202

8

0.006 0.1665986

62

0.9922026

84

0.7735924

47

11002

8

410959 1887.729

4

1.919026 -1.918743


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Figure 2.1.3: The graph of number of elements against the average orthogonal mesh quality

Figure 2.1.4: The graph of number of elements against the total pressure at the tee junction

outlet

Figure 2.1.5: The graph of number of elements against the total velocity at the tee junction

outlet


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Figure 2.1.6: The graph of number of elements against the z-velocity at the tee junction outlet

Based on Figure 2.1.3, the average orthogonal mesh quality increased until it reaches an

approximate value of 0.99 when there is 179917 mesh elements or a mesh element size of 0.008. Based

on Figure 2.1.4, the total pressure at the tee junction outlet increased until it reaches an approximately

constant value of 1900 Pa when there is 130480 mesh elements or a mesh element size of 0.009. Based

on Figure 2.1.5, the total velocity at the tee junction outlet increased until it reaches an approximate

constant value of 1.92 m/s at 130480 mesh elements or a mesh element size of 0.009. Based on Figure

2.1.6, the z-velocity at the tee junction outlet decreased until it reaches an approximate constant value

of -1.92 m/s at 95492 mesh elements or a mesh element size of 0.01.

By observing the relationship between the number of mesh elements and the orthogonal mesh

quality, total pressure, total velocity and z-velocity, the mesh achieves grid independence when the

element size is set between 0.01 and 0.008. Hence, the optimal mesh element size chosen is 0.008 m.

To further refine the mesh, an inflation setting is given in the mesh settings whereby the area

surrounding the T-junction walls are segregated into 5 layers. This gives more mesh refinement along

the area where the wall boundary layer will form in the T-junction due to the effects of shear stress and

viscosity.

To better confirm the effect of mesh element size and number of mesh elements on the

calculations, contours and graphs of three types of mesh sizes are done. The coarse mesh has an element

size of 0.04 m, the intermediate mesh has an element size of 0.02 m and the optimal mesh has an element

size of 0.008 m.


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Figure 2.1.7: The ANSYS Workbench to carry out Task 1, Task 2 and Task 3.

To carry out Task 1, Task 2 and Task 3, a single ANSYS workbench is set up as shown in

Figure 2.1.7. Each task uses the same IGES file of the T-junction pipe. By completing Task 1, the

optimum mesh is determined and used in Task 2. In Task 2, the optimum solver is determined. Using

the optimal mesh and solver from Task 1 and Task 2 respectively, the optimal turbulence model is

chosen in Task 3. Since each subsequent task carries forward the settings of the preceding task, a single

ANSYS workbench integrating all three tasks is created.

When carrying out Task 1, ANSYS Fluent was set to use the pressure-based solver and k-

epsilon turbulence model. Other settings included Parallel processing option which allowed the usage

of multiple processes to allow the calculations to be done in a shorter time frame by splitting the mesh

and assigning it into several computer processes or nodes. A Series processing option only allowed a

single process to run the entire calculation which causes the processing time to increase. Moreover, the

COUPLED method is used to allow for pseudo-transient and high order term relaxation settings to be

switched on which allows the case being calculated to be stabilized and converge in a shorter timeframe.


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Figure 2.2.1: Residual plot of coarse mesh with element size of 0.04 m

Figure 2.2.2: Residual plot of intermediate mesh with element size of 0.02 m


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Figure 2.2.3: Residual plot of optimal mesh with element size of 0.008 m

Based on Figure 2.2.1, the coarse mesh with element size of 0.04 m required 176 iterations

which took 0.029 seconds per iteration and 5.189 seconds in total wall-clock time to complete. Based

on Figure 2.2.2, the intermediate mesh with element size of 0.02 m required 110 iterations which took

0.070 seconds per iteration and 7.650 seconds in total wall-clock time to complete. Based on Figure

2.2.3, the optimal mesh with element size of 0.008 m required 94 iterations which took 0.617 seconds

per iteration and 58.00 seconds in total wall-clock time to complete. Hence, based on the time taken for

residuals to converge, the optimal mesh requires the least amount of time followed by the intermediate

and coarse meshes.

Figure 2.3.1: Pressure contour for coarse mesh with element size of 0.04 m


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Figure 2.3.2: Pressure contour for intermediate mesh with element size of 0.02 m

Figure 2.3.3: Pressure contour for optimal mesh with element size of 0.008 m

Figure 2.3.4: Graph of pressure along the longitudinal line in the tee junction for coarse (0.04

m), intermediate (0.02 m) and optimal (0.008 m) meshes.


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Based on Figure 2.3.1, 2.3.2 and 2.3.3, the distribution of total pressure bands becomes more

balanced as the element size decreases. In Figure 2.3.1, the total pressure bands are mainly between

3.238e+03 and 2.646e+03 at both inlets as well as between 2.801e+02 and -3.115e+02 at the outlet of

the T-junction pipe. Values are mainly at the extreme ends of the pressure range. The lowest pressure

value is found at a very small region of the pipe neck. The intermediate values of pressure become more

apparent as the element size increases as shown in Figure 2.3.3.

Based on Figure 2.3.4, the graph of element size 0.04 m is rough and jagged. It becomes

smoother at element size of 0.02 m and it is the smoothest at 0.008 m. Moreover, at 0.1 m from the

horizontal inlet of the T-junction pipe, the lowest value of the pressure graph is reached by the 0.008 m

element size.

Figure 2.4.1: Velocity contour for coarse mesh with element size of 0.04 m

Figure 2.4.2: Velocity contour for intermediate mesh with element size of 0.02 m


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Figure 2.4.3: Velocity contour for optimal mesh with element size of 0.008 m

Figure 2.4.4: Graph of velocity along the longitudinal line in the tee junction for coarse (0.04 m),

intermediate (0.02 m) and optimal (0.008 m) meshes.

Based on Figure 2.4.1, 2.4.2 and 2.4.3, the bands on the velocity contours become smoother

as the element size decreases. Based on Figure 2.4.1, the bands have rough edges and not clearly

defined areas. Based on Figure 2.4.2, the bands start to smoothen out but are still rough at the 2.843 –

2.559 m/s band. Based on Figure 2.4.3, all the bands in the contour have smoothen out and show very

clearly defined areas of different fluid velocities.

Based on Figure 2.4.4, the graph becomes smoother as the element size decreases. The

highest peak value was achieved by element size of 0.008 m followed by 0.02 m and 0.04 m. Element

size of 0.008 m showed the smoothest graph especially after it reached its peak value. Element size of

0.04 m had a rough and jagged graph before and after its peak value.


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Figure 2.5.1: z-velocity contour for coarse mesh with element size of 0.04 m

Figure 2.5.2: z-velocity contour for intermediate mesh with element size of 0.02 m

Figure 2.5.3: z-velocity contour for optimal mesh with element size of 0.008 m


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Figure 2.5.4: Graph of z-velocity along the longitudinal line in the tee junction for coarse (0.04

m), intermediate (0.02 m) and optimal (0.008 m) meshes.

Based on Figure 2.5.1, 2.5.2 and 2.5.3, the bands of the z-velocity contour becomes smoother

as the element size decreases. The bands in Figure 2.5.1 has very rough edges but it becomes smoother

in Figure 2.5.2. The bands of the contour becomes smooth and well-defined in Figure 2.5.3 which has

the optimal mesh element size. Furthermore, the region with the highest z-velocity becomes more well-

defined and distinguishable at the neck of the T-junction as the element size reduces from 0.04 m to

0.008 m.

Based on Figure 2.5.4, the graph also becomes smoother as the mesh element size decreases.

The peak value is also greatest at the mesh element size of 0.008 m, followed by 0.02 m and 0.04 m.

The graph of mesh element size 0.008 m has the smoothest graph whereas the graph of mesh element

size 0.04 m has a very jagged and rough pattern.

TASK 1 – CONCLUSION

Based on the residual plots, graphs and contours of three different meshes, the results show that

the mesh with element size of 0.008 m possesses the smoothest contours and graphs for total pressure,

total velocity and z-velocity. Element sizes of 0.04 m has the most jagged graphs and contours whereas

mesh element size of 0.02 m still has rough sections in the contours and graphs. Therefore, to obtain

accurate calculations, an element size of 0.008 m is chosen for Tasks 2 and 3.


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TASK 2 – MULTIPLE SOLVERS

To carry out Task 2, the mesh with element size of 0.008 m. Settings were kept constant from

the previous task except that two versions of the same calculations were made; one with the pressure-

based solver and another with the density-based solver. The output parameters used to generate graphs

and contours were still total pressure, total velocity and z-velocity. The residual values are also lowered

from 10-3 to 10-4 to increase the accuracy of the flow analysis results.

Figure 3.1.1: Residuals for pressure-based solver

Figure 3.1.2: Residuals for density-based solver


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Based on Figure 3.1.1, using the pressure-based solver, 163 iterations were needed for the

residuals to reach convergence which correlated to 99.254 seconds of total wall clock time and 0.609

seconds average wall clock time per iteration. Based on Figure 3.1.2, using the density-based solver,

177 iterations were needed for the residuals to reach convergence which correlated to 109.984 seconds

of total wall clock time and 0.621 seconds average wall clock time per iteration. This meant that the

pressure-based solver required less iterations and time to complete the flow analysis than the density-

based solver.

Figure 3.2.1: Pressure Contour for Pressure-based Solver

Figure 3.2.2: Pressure Contour for Density-based Solver


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Figure 3.2.3: Graph of pressure along the longitudinal line in the tee junction for pressure-

based and density-based solvers

Based on Figure 3.2.1 and 3.2.2, the total pressure contour for pressure-based and density-based

contour have smooth and well defined bands. However, the bands at the neck of the T-junction pipe is

wider and more balanced in the pressure-based solver than the density-based solver. The region of

lowest value of total pressure is also slightly larger for the pressure-based solver than the density-based

solver. Based on Figure 3.2.3, the graphs of the pressure-based and density-based solver is very similar

in terms of the graph shape and range of values on the graph.

Figure 3.3.1: Velocity contour for pressure-based solver


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Figure 3.3.2: Velocity contour for density-based solver

Figure 3.3.3: Graph of velocity along the longitudinal line in the tee junction for pressure-based

and density-based solvers

Based on Figure 3.3.1 and 3.3.2, the velocity contours of the pressure-based and density-based

solvers are very similar in that they both have smooth bands and well-defined regions. However, the

2.585 – 2.298 m/s, 0 – 2.872e-01 m/s and 2.872e-01 – 5.745e-01 m/s bands in the pressure-based solver

contour are slightly larger than that of the density-based solver contour. This pressure-based solver

contour shows a more balanced distribution of contour bands than that of the density-based solver

contour. Based on Figure 3.3.3, the graph generated for both pressure-based and density-based solvers

are very similar in shape but slightly different in the values especially in the middle section of the T-

junction pipe as both graphs reach their peak values. The pressure-based solver graph has slightly higher

values when compared to that of the density-based solver as it reaches its peak value. However, the

pressure-based solver graph has a slightly lower peak value than the density-based solver graph.


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Figure 3.4.1: z-velocity contour for pressure-based solver

Figure 3.4.2: z-velocity contour for density-based solver

Figure 3.4.3: Graph of z-velocity along the longitudinal line in the tee junction for pressure-

based and density-based solvers


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Based on Figure 3.4.1 and 3.4.2, the z-velocity contours of both pressure-based and density-

based solvers are similar in terms of the location of bands in the T-junction. However, the pressure-

based solver contour has a slightly larger band regions at the neck of the T-junction than the density-

based solver contour. This includes the two highest value bands, 6.901e-01 – 3.355e-01 m/s and -

1.910e-02 – -3.737e-01 m/s as well as the lowest value band, -2.501- 2.856 m/s. This shows that the

pressure-based solver has a more balanced distribution of contour bands than the density-based solver.

Based on Figure 3.4.3, the pressure-based and density-based solvers generated graphs with

similar shapes. However, the pressure-based solver has slightly lower values than that of the density-

based solver when it approaches its peak value. The pressure-based solver also has a slightly higher

peak value than the density-based solver.


Based on the residual plots, graphs and contours of the pressure-based and density-based

solvers, the pressure-based solver provides a more optimal solution to carry out the flow analysis of the

T-junction pipe than the density-based solver. This is because the pressure-based solver requires less

time and iterations to reach convergence than the density-based solver. The contours for total pressure,

total velocity and z-velocity also show a more balanced distribution of contour bands for the pressure-

based solver than the density-based solver. The graphs of both solvers are very similar in pattern to

deduce any discernible differences except for slight differences in values at each point along the

longitudinal line of the T-junction pipe.

From a theoretical perspective, the pressure-based solver is more appropriate for the T-junction

and its input parameters. The density-based solver needs a higher computational cost and time due to

using a coupled approach where the set of equations for continuity, momentum and energy are solved

in a single system simultaneously [1]. However, the pressure-based solver uses the segregated approach

which decouples the equations which allows individual equations for momentum, continuity and energy

sequentially [2]. From a fluid mechanics point of view, the pressure-based solver is also shown to be

more suitable for the given T-junction and input parameters. A density-based solver requires

calculations to be done based on the acoustic timescale. In the case given, the Mach number is nearly

zero which means that the timestep on the timescale is very small. This leads to more time and

computational cost needed for the density-based solver. However, the pressure-based solver solves the

equations on a flow timecscale and ignores the acoustic timescale features. This is more suitable for

incompressible flow which is the case given in the T-junction pipe [3].

In conclusion, the pressure-based solver is chosen as the most suitable solver for the T-junction

pipe and its given input parameters.


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TASK 3 – TURBULENCE MODELS

To carry out Task 3, the exact same settings were used since the beginning of the flow analysis. Element

size of 0.008 m from Task 1 and pressure-based solver from Task 2 was used in Task 3. Four turbulence

models were used and compared to determine the most suitable turbulence model for the T-junction

and its input parameters.

Before comparing the contours and graphs of the T-junction simulations, the physical meaning of

turbulence as well as the various approaches to computational fluid dynamics is elaborated. This is to

provide clarification and supporting evidence for the justification of the turbulence model chosen in the

last section of Task 3 labelled Task 3 – Conclusion.

Meaning of Turbulence

Turbulence is a fluid motion that is three-dimensional, time dependent, rotational, intermittent,

highly disordered, diffusive and dissipative. Turbulence is primarily defined by the transfer of energy

to lower spatial scales across a continuous wave-number spectrum [4]. The main mechanisms are eddies

and high-vorticity regions. Using the analogy of molecular viscosity, turbulence is described as the eddy

viscosity which is the local fluid property whereby the mixing length is akin to the molecular mean-free

path derived from the kinetic theory of gases. Eddies can also be described as bundles of vortex elements

stretched by mean flow and have a preferred direction similar to the mean flow while having random

directions to one another. This leads to the breakdown of large eddies to smaller-sized eddies in the

form of an energy cascade. In this mechanism, energy is transferred from large eddies to adjacent

smaller eddies until the energy is dissipated as internal thermal energy of the smallest scale eddies [5].

The usage of eddy viscosity is the critical concept in various Reynolds-Averaged Navier-Stokes

(RANS) models and it is described in the Boussinesq eddy viscosity hypothesis. In this hypothesis,

momentum transfer due to eddies are modelled with an eddy viscosity and it is analogous to how

momentum motion in gases can be described by molecular viscosity [5]. The hypothesis states that the

Reynolds-stress tensor is derived from the effective viscosity formulation which is

𝜏𝑖𝑗 −2

3𝑘𝛿𝑖𝑗 − 2𝑣𝑡𝑆𝑖𝑗

where 𝑘 = 𝑢𝑖′𝑢𝑗

′̅̅ ̅̅ ̅̅ which is the turbulent kinetic energy, 𝛿𝑖𝑗 is the Kronecker delta and 𝑣𝑡 is the turbulent

kinematic viscosity. The Boussinesq hypothesis is elaborated in greater detail in the section about

RANS models where they play a central role [6].

In turbulence, Reynolds numbers affects the difference between the largest and smallest eddies

or between the low-frequency and high-frequency fluctuations. This understanding leads to high-

Reynolds numbers turbulent motion being estimated in three levels of motions which are mean, large-


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scale and small-scale motions. Furthermore, the effect of viscosity is only at the viscous sublayer and

does not affect the turbulence as the density fluctuations are comparatively small to the mean fluid

density [4].

Due to its non-linear and difficult precise definition, the capability to model this phenomenon

is able to save tremendous amounts of money by avoiding the construction and testing of physical

prototypes. This allows complex designs to be refined and its parameters to be optimized. ANSYS

Fluent is able to model fluid turbulence within practical computer resources by solving the Navier-

Stokes equations which are three dimensional and time dependent. The Navier Stokes equations are the

application of Newton’s 2nd law of motion on fluid elements which describes the linear momentum

conservation [5]. The Navier-Stokes equations along with the continuity and energy equation are able

to describe fluid characteristics in the following equations:

Continuity Equation:

𝜕𝜌

𝜕𝑡+ ∇ ∙ (𝜌�̅�) = 0 (1)

Navier-Stokes Equation:

𝜕𝜌𝑢

𝜕𝑡+ ∇ ∙ 𝜌𝑢𝑉 = −

𝜕𝑝

𝜕𝑥+

𝜕

𝜕𝑥(𝜆∇ ∙ 𝑉 + 2𝜇

𝜕𝑢

𝜕𝑥) +

𝜕

𝜕𝑦(𝜇 (

𝜕𝑣

𝜕𝑥+

𝜕𝑢

𝜕𝑦)) +

𝜕

𝜕𝑦(𝜇 (

𝜕𝑣

𝜕𝑥+

𝜕𝑢

𝜕𝑦)) +

𝜕

𝜕𝑧(𝜇 (

𝜕𝑢

𝜕𝑧+

𝜕𝑤

𝜕𝑥)) + 𝜌𝑓𝑥,𝑏𝑜𝑑𝑦

𝜕𝜌𝑢

𝜕𝑡+ ∇ ∙ 𝜌𝑢𝑉 = −

𝜕𝑝

𝜕𝑥+ ∇ ∙ (𝜇∇u) + 𝑆𝑀,𝑥 (2)

where

∇ ∙ (𝜇∇u) =𝜕

𝜕𝑥(𝜇

𝜕𝑢

𝜕𝑥) +

𝜕

𝜕𝑦(𝜇

𝜕𝑢

𝜕𝑦) +

𝜕

𝜕𝑧(𝜇

𝜕𝑢

𝜕𝑧)

𝑆𝑀,𝑥 = 𝜕

𝜕𝑥(𝜇

𝜕𝑢

𝜕𝑥) +

𝜕

𝜕𝑦(𝜇

𝜕𝑣

𝜕𝑥) +

𝜕

𝜕𝑧(𝜇

𝜕𝑤

𝜕𝑥) +

𝜕

𝜕𝑥(𝜆∇ ∙ 𝑉) + 𝜌𝑓𝑥,𝑏𝑜𝑑𝑦

Energy Equation:

𝜕𝜌𝑒

𝜕𝑡+ ∇ ∙ (𝜌𝑒�̅�) = −𝑝∇ ∙ �̅� + ∇ ∙ (𝑘∇T) + Φ + 𝑆𝑖 (3)

where

Φ = 𝜇 [2 (𝜕𝑢

𝜕𝑥)

2+ 2 (

𝜕𝑣

𝜕𝑦)

2+ 2 (

𝜕𝑤

𝜕𝑧)

2+ (

𝜕𝑢

𝜕𝑦+

𝜕𝑣

𝜕𝑥)

2+ (

𝜕𝑢

𝜕𝑧+

𝜕𝑤

𝜕𝑥)

2+ (

𝜕𝑣

𝜕𝑧+

𝜕𝑤

𝜕𝑦)

2]

𝑆𝑖 = [𝜌𝑓 ∙ �̅� − �̅�]


25

Due to the common terms in the continuity, Navier-Stokes and energy equations, a general

transport equation is formulated in a generic form:

𝜕𝜌𝜙

𝜕𝑡+ ∇ ∙ (𝜌𝑒�̅�) = ∇ ∙ (Γ∇𝜙) + Φ + 𝑆𝑖 (4)

where the first term describes the rate of change of ϕ in the fluid element, the second term is the net rate

of flow due to convection of ϕ out of the fluid element, the third term is the rate of increase due to

diffusion of ϕ in the fluid element and the fourth term refers the any sources or sinks of ϕ.

There are several approaches to model turbulence in a CFD software using the combination of

Navier-Stokes, continuity and energy equations. These include direct numerical simulation (DNS),

Reynolds-Averaged Navier Stokes (RANS) models and Large Eddy Simulation (LES) models.

Direct Numerical Simulation

This approach to computational fluid dynamics requires the numerical solution of the full three-

dimensional, time-dependent Navier Stokes equations in the absence of a turbulence model. This

solution is done for the entire spectrum of scales. It is useful to study turbulence mechanisms and

improve turbulence models. However, it requires very high computing resources in terms of storage

and speed. Therefore, it is very overwhelmingly costly and not a practical approach on conventional

computers. The usage of DNS is limited to flows with low to moderate Reynolds numbers with the

highest to date being 1200. In spite of this, the accuracy of the solutions are much greater than the other

methods. To overcome the resource-intensive approach of DNS, turbulence models are introduced [4-

5].

Reynolds-Averaged Navier Stokes (RANS) models

Reynolds-Averaged Navier Stokes (RANS) models refer to the solutions that use ensemble-

averaged or time-averaged Navier-Stokes equations. This means that by averaging the flow quantities

over a specific time period, the flow quantities can be described as a sum of mean and fluctuating parts

in a procedure called Reynolds decomposition. This can be described as

𝑢 (𝑡) = �̅� + 𝑢′(𝑡)

where the average mean value of a flow quantitiy is �̅� and the fluctuating component is 𝑢′(𝑡).

RANS Navier-Stokes equations are written as

ρ (𝜕𝑢𝑖̅̅ ̅

𝜕𝑡+ 𝑢𝑘̅̅ ̅

𝜕𝑢𝑖̅̅ ̅

𝜕𝑥𝑘) = −

𝜕�̅�

𝜕𝑥+

𝜕

𝜕𝑥𝑗(𝜇

𝜕𝑢𝑖̅̅ ̅

𝜕𝑥𝑗) +

𝜕𝑅𝑖𝑗

𝜕𝑥𝑗


26

where the Reynolds stress tensor is 𝑅𝑖𝑗 = 𝜌𝑢𝑖′𝑢𝑗

′̅̅ ̅̅ ̅̅ = 𝜇𝑇 (𝜕𝑢𝑖̅̅ ̅

𝜕𝑥𝑗+

𝜕𝑢𝑗̅̅ ̅

𝜕𝑥𝑖) −

2

3𝜇𝑇

𝜕𝑢𝑘̅̅ ̅̅

𝜕𝑥𝑘𝛿𝑖𝑗 −

2

3𝜌𝑘𝛿𝑖𝑗 . The term

𝜇𝑇 is the eddy or turbulent viscosity term as hypothesised in the Boussinesq hypothesis and is based on

the turbulence time or velocity scale and a length scale [5].

RANS models with linear eddy viscosity are divided into several categories including zero-equation

models, half equation models, one-model equations and two-equation models.

Large Eddy Simulation (LES)

The LES approach uses a spatially averaged Navier-Stokes equations which solves the various scales

of flow differently. Large-scale turbulence uses a full analytical solution whereas small-scale flow are

modelled. This causes the LES approach to be more accurate than the LES approach and produces very

reliable solutions for high Reynolds numbers turbulence. Despite being less costly than the DNS

approach, it still requires higher computing resources than the RANS approach.

In this study, four RANS turbulence models are used as it is the most common approach used in

industrial applications and are readily available in ANSYS Fluent. These models include the Spalart-

Allmaras turbulence model, k-ε turbulence model and a k-ω variant called SST turbulence model [5].

Spalart-Allmaras turbulence model

The Spalart-Allmaras turbulence model is a one-equation model which has been designed for

flows in aerospace applications such as wings and airfoils. It is also used for supersonic or transonic

flows over airfoils and boundary-layer flows. It requires a low amount of computational resource [6].

The turbulence model involves the usage of a transport equation with a modified eddy viscosity in the

form of 𝜇𝑇 = 𝑓(�̃�). This means that the transport equation comes in the following form:

𝜕𝜌�̃�

𝜕𝑡+ ∇ ∙ (𝜌�̃�𝑈) = ∇ ∙ [(𝜇 + 𝜌�̃�)∇�̃� + 𝐶𝑏2𝜌

𝜕�̃�

𝜕𝑥𝑘

𝜕�̃�

𝜕𝑥𝑘] + 𝐶𝑏1𝜌�̃�Ω̃ − 𝐶𝑤1𝜌 (

�̃�

𝜅𝑦) 𝑓𝑤

where 𝜕𝜌�̃�

𝜕𝑡 is the rate of change of �̃� , ∇ ∙ (𝜌�̃�𝑈)is the convective transport of �̃� , (𝜇 + 𝜌�̃�)∇�̃� is the

transport of �̃� by diffusion, 𝐶𝑏2𝜌𝜕�̃�

𝜕𝑥𝑘

𝜕�̃�

𝜕𝑥𝑘 is the transport of �̃� by turbulent diffusion, 𝐶𝑏1𝜌�̃�Ω̃ is the rate

of �̃� production and 𝐶𝑤1𝜌 (�̃�

𝜅𝑦) 𝑓𝑤is the rate of �̃� dissipation [5].

The Spalart-Allmaras turbulence has an advantage where it can be used to gain accurate results

even with coarse meshes. It also provides accurate results for boundary layers that contain significant

pressure gradients. However, a disadvantage of the Spalart-Allmaras turbulence model include the

creation of severe diffusion in three-dimensional vertical flow regions. It is unable to predict the decay

of homogenous and isotropic turbulence [6].

k-ε turbulence model


27

The k-ε turbulence model is a common two-equation model which uses two additional transport

equations in conjunction with the mean-flow Navier-Stokes equations. The first equation calculates the

turbulence kinetic energy (k) and the second equation calculates the dissipation rate of turbulence

kinetic energy (ε). The eddy viscosity is calculated in the form of 𝜇𝑇 = 𝑓 (𝜌𝑘2

𝜀). The transport equations

for k and ε are written as:

𝜕𝜌𝑘

𝜕𝑡+ ∇ ∙ (𝜌𝑘𝑈) = ∇ ∙ (

𝜇𝑡

𝜎𝑘∇𝑘) + 2𝜇𝑡𝑆𝑖𝑗 − ρε

𝜕𝜌𝜀

𝜕𝑡+ ∇ ∙ (𝜌휀𝑈) = ∇ ∙ (

𝜇𝑡

𝜎𝜀∇휀) + 𝐶1𝜀

𝜀

𝑘2𝜇𝑡𝑆𝑖𝑗 ∙ 𝑆𝑖𝑗 − 𝐶2𝜀ρ

ε2

𝑘

where the first term is the rate of change of k or 휀, the second term is the transport of k or 휀 convection,

the third term is the transport of k or 휀 by diffusion, the fourth term is the production rate of k or 휀 and

the fifth term is destruction rate of k or 휀 [5].

This model assumes that the flow is fully turbulent and molecular viscosity has no effect on the

flow. The model is able to provide accurate results for low-Reynolds numbers and the model equation

given is derived from Launder and Sharma in 1974. However, this means that the k-ε model is unable

to accurately determine the flow characteristics near wall-bounded regions. Regions near walls require

high-order derivatives. Moreover, this model is unable to accurately predict flows with great pressure

gradients and strains due to streamline curvature, skewing and rotation [7-8]. Therefore, the standard k-

ε model requires modification to provide more accurate results in regions near walls where there is

separated flows.

A modification of the k-ε model is the Renormalization Group (RNG) k-ε model which contains

a modified 𝐶2𝜀 term. This model provides more accurate results of the recirculation length found in

separating flows. However, this version of k-ε model does not predict flows that contain acceleration

[4].

k- ω turbulence model

Another commonly used two equation model is the k-ω turbulence model which also uses two

extra transport equations in conjunction with the mean-flow Navier-Stokes equations. The first equation

is the turbulent kinetic energy, k which is similar to the k term used in the k-ε model and the second

equation is the specific dissipation, ω which is the ratio of ε to k. This means that the eddy viscosity is

calculated as 𝜇𝑇 = 𝑓 (𝜌𝑘

ω). The k- ω model is written as:

𝜕𝜌𝑘

𝜕𝑡+ ∇ ∙ (𝜌𝑘𝑉) = 𝜏𝑖𝑗

𝜕𝑢𝑖̅̅ ̅

𝜕𝑥𝑗+

𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝑘)

𝜕𝑘

𝜕𝑥𝑗] − 𝜌𝛽∗𝑓𝛽𝑘ω

𝜕𝜌ω

𝜕𝑡+ ∇ ∙ (𝜌𝑘𝑉) = α

𝜔

𝑘𝜏𝑖𝑗

𝜕𝑢𝑖̅̅ ̅

𝜕𝑥𝑗+

𝜕

𝜕𝑥𝑗[(𝜇 +

𝜇𝑡

𝜎𝜔)

𝜕𝜔

𝜕𝑥𝑗] − 𝜌𝛽𝑓𝛽ω2


28

where the first term is the rate of change of k or ω, the second term is the transport of k or ω convection,

the third term is the transport of k or ω by diffusion, the fourth term is the production rate of k or ω and

the fifth term is destruction rate of k or ω [5].

This model is used in the turbo-machinary and aerospace applications It also incorporates

modifications to account for the effects of low Reynolds numbers, compressibility and shear flow

spreading [9].

Shear Stress Transport (SST) turbulence model

The SST model is a turbulence model that has four extra transport equations with the mean

flow Navier-Stokes equations. It combines the advantages of the k-ω and k-ε models which are the

robustness of the k-ω model near walls and the accuracy of the k-ε model far from walls. The turbulent

viscosity is modified to include the transport of turbulent shear stress. It is able to provide accurate

results within boundary layers with high pressure gradients. Applications include turbomachinery

blades, zero pressure gradient and wind turbines [10]. The turbulent viscosity is determined using:

𝜇𝑇 =𝑎1𝑘

max (𝑎1𝜔,𝑆𝐹2)

The k and ω are calculated using:

𝜕

𝜕𝑥𝑗(𝑈𝑗𝑘) =

𝜕

𝜕𝑥𝑗[(𝑣 + 𝜎𝑘𝑣𝑡)

𝜕𝑘

𝜕𝑥𝑗] + 𝑃�̃� − 𝛽∗𝑘ω

𝜕

𝜕𝑥𝑗(𝑈𝑗𝑘) =

𝜕

𝜕𝑥𝑗[(𝑣 + 𝜎𝜔𝑣𝑡)

𝜕𝜔

𝜕𝑥𝑗] + 𝛼𝑆2 − 𝛽ω2 + 2(1 − 𝐹1)𝜎𝜔2

1

𝜔

𝜕𝑘

𝜕𝑥𝑗

𝜕𝜔

𝜕𝑥𝑗

The SST model has a blending function, 𝐹1 which is zero when far away from walls and becomes one

within the boundary layer. This allows the model to switch between the k-ω model near walls and k-ε

model further from the wall. An advantage of the SST Model is that it combines the advantages of both

k-ε and k-ω models. However, the disadvantage is the higher cost of computing resources when

compared to the previous turbulence models [11].


29

ANALYSIS OF GRAPHS AND CONTOURS

Figure 4.1.1: Residual plot of Laminar turbulence model

Figure 4.1.2: Residual plot of k-espilon turbulence model


30

Figure 4.1.3: Residual plot of Spalart-Allmaras turbulence model

Figure 4.1.4: Residual plot of SST turbulence model


31

Based on Figure 4.1.1, using the laminar turbulence model, the residuals did not converge even

after 2000 iterations which took a total wall clock time of 788.709 seconds at 0.394 seconds of average

wall clock time per iteration. Based on Figure 4.1.2, using the k-epsilon turbulence model, 163 iterations

were needed for the residuals to reach convergence which correlated to 99.254 seconds of total wall

clock time and 0.609 seconds average wall clock time per iteration. Based on Figure 4.1.3, using the

Spalart-Allmaras turbulence model, 371 iterations were needed for the residuals to reach convergence

which correlated to 193.110 seconds of total wall clock time and 0.521 seconds average wall clock time

per iteration. Based on Figure 4.1.4, using the SST turbulence model, 336 iterations were needed for

the residuals to reach convergence which correlated to 268.763 seconds of total wall clock time and

0.800 seconds average wall clock time per iteration.

Figure 4.2.1: (Clockwise, from top left to bottom left) Pressure contour for the 1. laminar, 2.

Spalart-Allmaras, 3. k-epsilon and 4. SST turbulence models


32

Figure 4.2.2: Graph of pressure along the longitudinal line in the tee junction for laminar,

Spalart-Allmaras, k-epsilon and SST turbulence models

Based on Figure 4.2.1, the total pressure contour of the laminar turbulence model has rough

contour bands on the neck of the T-junction. The total pressure contour of the k-epsilon, Spalart-

Allmaras and SST turbulence models have similar distribution of contour bands. These three turbulence

models have smooth contour bands and only have slight variations in the size of the lowest value contour

band at the edge joining the vertical and horizontal sections of the T-junction pipe.

Based on Figure 4.2.2, the graphs generated from the k-epsilon, Spalart-Allmaras and SST

models are similar to each other whereas the graph generated from the laminar model is rough and

deviates from the other graphs especially at the points before it reaches the peak value. Excluding the

laminar turbulence model, the k-epsilon turbulence model achieves the lowest peak value, followed by

the SST and Spalart-Allmaras turbulence models.


33

Figure 4.3.1: (Clockwise, from top left to bottom left) Velocity contour for the 1. laminar, 2.


Figure 4.3.2: Graph of velocity along the longitudinal line in the tee junction for laminar,



34

Based on Figure 4.3.1, the velocity contour of the laminar turbulence model has an uneven and

rough contour band distribution of low values on the lower half of the horizontal pipe. There is also a

relatively large region of the highest value contour band on the top half of the horizontal pipe. The

Spalart-Allmaras turbulence model generated a contour quite similar to that of the SST turbulence

model but with slightly different sized regions for the highest and lowest value contour band. These

regions are also not as smooth as the contour bands found on the rest of the T-junction pipe. The k-

epsilon turbulence model has similar distribution of contour bands as the Spalart-Allmaras and SST

turbulence models but with much smoother contour bands at the neck of the T-junction.

Based on Figure 4.3.2, the velocity graphs plotted from the k-epsilon, SST and Spalart-

Allmaras turbulence models are similar to each other whereas the laminar turbulence model deviates

greatly before reaching the peak value of the graphs. The k-epsilon turbulence model reaches the

greatest peak value followed by the Spalart-Allmaras and SST turbulence models. All the graphs have

very similar values after they reached their peak value.

Figure 4.4.1: (Clockwise, from top left to bottom left) z-velocity contour for the 1. laminar, 2.



35

Figure 4.4.2: Graph of z-velocity along the longitudinal line in the tee junction for laminar,


Based on Figure 4.4.1, the z-velocity contour of the laminar turbulence model has rough contour

band edges on the lower half of the horizontal pipe section and a relatively large region of the lowest

value contour band on the top half of the horizontal pipe section. The Spalart-Allmaras and SST

turbulence models have very similar contour band distributions with the exception of a slightly larger

lowest value region in the SST model than the Spalart-Allmaras model. The k-epsilon turbulence model

has a z-velocity contour that is quite similar to the SST model with the difference being a wider region

of mid-value contour bands that cover the lower half of the horizontal pipe section.

Based on Figure 4.4.2, the z-velocity graph plotted from the k-epsilon, Spalart-Allmaras and

SST turbulence models are highly similar to each other whereas the graph plotted from the laminar

turbulence model is highly jagged and deviates greatly from the graphs of the other turbulence models

before reaching its peak value. The k-epsilon turbulence model reached the greatest peak value followed

by the SST and Spalart-Allmaras turbulence models. All the graphs are plotted on very similar points

after reaching the peak value.


Based on the residual plots, contours and graphs generated from four different turbulence

models, the most suitable turbulence model for the flow analysis of the T-junction pipe and its input

parameters is the k-epsilon turbulence model. The reason being the k-epsilon model requires the least

amount of iterations and time followed by the SST and Spalart-Allmaras turbulence models. In fact, the

k-epsilon only needs half the number of iterations and time to reach convergence when compared to the

second fastest turbulence model, the SST model. Furthermore, the k-epsilon turbulence model shows


36

smoother contour bands than the SST and Spalart-Allmaras models especially at the neck of the T-

junction pipe which is the most crucial part of the pipe. The k-epsilon turbulence model also shows a

more balanced distribution of contour bands in the pipe. From the graphs, it can be seen that the k-

epsilon model reaches the greatest peak values when compared to the SST and Spalart-Allmaras

turbulence models despite showing similar peaks and patterns.

From a theoretical perspective, the laminar turbulence model is not suitable for the given T-

junction and its input parameters. This is because the flow in the T-junction pipe has a Reynolds number

of 9.95 × 104 and is therefore turbulent flow [5]. This rules out the laminar turbulence model and also

explains the reason why the residual plot does not achieve convergence even after 2000 iterations. The

Spalart-Allmaras model is a one-equation model was originally designed for wall-bounded flows and

low-Reynolds number model for aerospace applications. An example is transonic flow over airfoils. It

possess smaller near-wall gradients which makes it less sensitive to errors for non-layered meshes near

walls. It also is not effective at rapidly calculating areas where the fluid flow abruptly changes due to

changes in length scale [6]. Hence, it might use the least amount of computing effort when compared

to the other turbulence models but due to the two dimensional flow geometry of the T-junction, it may

encounter errors when passing the pipe neck.

The SST turbulence model is a combination of the k-epsilon and k-omega models whereby k-

epsilon is effective at free-stream, far field flow and k-omega is effective at near-wall regions. Therefore,

it also requires the highest number of equations compared to the other turbulence models. It can be used

to analyse viscous-affected regions without extra modification but may lead to over-prediction and is

sensitive to inlet boundary conditions [9-10]. The k-epsilon turbulence model is the most widely-used

and proven turbulence model that is used for a wide range of applications like planar surfaces,

recirculation, streamlined curvature and complex geometries. It also does not require high computing

effort and analytically accounts for the fluid velocity in the viscous sublayer near the wall [7-8]. Hence,

through careful consideration of each turbulence model, the most suitable turbulence model is k-epsilon.


37

3 CONCLUSION

From Task 1, the results show that as the mesh element size decreases, so does the number of

elements. Hence, the mesh reaches grid independence where the values for total pressure, total velocity,

z-velocity and orthogonal mesh quality reach an approximately constant value at a specific mesh

element size or number. A fine but optimal mesh element size provides smoother graphs and contours.

In this investigation, the optimal mesh size is 0.008 m. Further mesh quality improvement is done with

an inflation setting along the walls of the T-junction pipe. From Task 2, the pressure-based solver is

most suitable for the T-junction pipe and its input parameters due to requiring less iterations, less

computing time and theoretically being more suitable for low Mach and incompressible flow. From

Task 3, the k-epsilon turbulence model is shown to be the most suitable turbulence model for the T-

junction and its input parameters. This is due to it requiring the least amount of iterations and computing

time as well as generating contours with the smoothest and most well-defined contour bands. In

conclusion, the flow analysis of the given T-junction can be done with the optimal balance of calculation

accuracy and computing efficiency by using a 0.008 m mesh element size, a pressure-based solver and

a k-epsilon turbulence model.


38

4 REFERENCES

[1] "ANSYS FLUENT 12.0 Theory Guide - 18.1.2 Density-Based Solver", Afs.enea.it, 2009. [Online].

Available: http://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node362.htm. [Accessed:

19- Oct- 2019].

[2] "ANSYS FLUENT 12.0 Theory Guide - 18.1.1 Pressure-Based Solver", Afs.enea.it, 2009. [Online].

Available: http://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node361.htm. [Accessed:

19- Oct- 2019].

[3] A. Miettinen and T. Siikonen, "Application of pressure‐ and density‐based methods for different

flow speeds", International Journal for Numerical Methods in Fluids, vol. 79, pp. 243-267, 2015.

[4] C. Argyropoulos and N. Markatos, "Recent advances on the numerical modelling of turbulent

flows", Applied Mathematical Modelling, vol. 39, no. 2, pp. 693-732, 2015.

[5] "Modelling Turbulent Flows: Introductory FLUENT Training", Fluentusers.com, 2006. [Online].

Available:

http://www.southampton.ac.uk/~nwb/lectures/GoodPracticeCFD/Articles/Turbulence_Notes_Flue

nt-v6.3.06.pdf. [Accessed: 19- Oct- 2019].

[6] P. Raje, "Spalart-Allmaras turbulence model for Compressible flows", Department of Aerospace

Engineering, Indian Institute of Technology, Bombay, 2015.

[7] S. El-Behery and M. Hamed, "A comparative study of turbulence models performance for separating

flow in a planar asymmetric diffuser", Computers and Fluids, vol. 44, pp. 248-257, 2011.

[8] X. Loyseau, P. Verdin and L. Brown, "Scale-up and turbulence modelling in pipes", Journal of

Petroleum Science and Engineering, vol. 162, pp. 1-11, 2018.

[9] S. Salehi, M. Raisee and M. Cervantes, "Computation of Developing Turbulent Flow through a

Straight Asymmetric Diffuser with Moderate Adverse Pressure Gradient", Journal of Applied Fluid

Mechanics, vol. 10, no. 4, pp. 1029-1043, 2017.

[10] M. Abd Halim, N. Nik Mohd, M. Mohd Nasir and M. Dahalan, "The Evaluation of k-ε and k-ω

Turbulence Models in Modelling Flows and Performance of S-shaped Diffuser", International

Journal of Automotive and Mechanical Engineering, vol. 15, no. 2, pp. 5161-5177, 2018.

[11] J. Fürst, P. Straka, J. Příhoda and D. Šimurda, "Comparison of several models of the

laminar/turbulent transition", EDP Sciences, vol. 45, 2013.

computational fluid dynamics (cfd) mec60703/mec4513

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