TOWARDS A NUMERICAL SIMULATION OF AEROACOUSTICS IN A T-JUNCTION

George P. Mathew

Supervised by Dr. Craig Meskell

In partial fulfilment of the requirements of the Final Year Project in Mechanical Engineering for the degree of Baccalaureus in Arte Ingeniaria (BAI)

Department of Mechanical and Manufacturing Engineering

Trinity College Dublin

2012

i

Abstract T-Junction geometry is a very generic flow configuration in industry and one of

the greatest problems recognised with this type of geometry is the generation of

vibration due to the acoustic excitation of the flow. This is caused by a flow-acoustic

coupling in a sound field and is brought about at certain flow rates and T-Junction

geometries.

A review of past and present literature on the subject is presented in order to

understand the context of acoustic resonance in T-Junctions and pipes in general. The

factors that affect the future of CFD and whether it is feasible enough to be considered a

stand-alone alternative to experimentation are explored.

The acoustic resonance is numerically modelled using a commercial CFD

package to determine the feasibility of CFD for this complex flow. In order to achieve

reasonable accuracy, it was essential to determine the most suitable turbulence model

and mesh. A series of tests are carried out in order to assess this.

Initial tests revealed that the SST κ-ω model was the clear winner for modelling

the turbulent flow in this geometry and a meshing scheme using inflation layers

produced the most computationally efficient results. Successful modelling of the vortex

shedding in a T-Junction was eventually achieved by artificially fluctuating the velocity

at the inlets using a sine wave function.

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For Sneha

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Table of Contents

Abstract ............................................................................................................................. i

List of Figures ................................................................................................................. vii

List of Tables ................................................................................................................... ix

Abbreviations ................................................................................................................... x

Nomenclature ................................................................................................................... x

1 Introduction .............................................................................................................. 1

1.1 Project Background .................................................................................................. 2

1.2 T-Junction Geometry ............................................................................................... 4

2 Literature Review ...................................................................................................... 6

3 Computational Fluid Dynamics (CFD) ................................................................. 15

3.1 CFD Strategy ........................................................................................................... 15

3.2 Software ................................................................................................................... 15

3.3 Turbulence Modelling ........................................................................................... 16

3.3.1 Introduction ............................................................................................... 16

3.3.2 κ-ε Two-Equation Model ......................................................................... 17

3.3.3 SST κ-ω Two-Equation Model ................................................................. 19

3.3.4 Reynolds Stress Model (RSM).................................................................. 20

3.3.5 Large Eddy Simulation (LES) ................................................................... 21

3.4 Turbulence Model Testing .................................................................................... 23

4 Method .................................................................................................................... 27

4.1 Geometry ................................................................................................................. 27

4.2 Meshing ................................................................................................................... 29

4.3 Problem Setup ......................................................................................................... 32

4.4 Analysis and Post-Processing ............................................................................... 34

5 Results ..................................................................................................................... 35

5.1 Mesh Influence ....................................................................................................... 35

5.2 Effect of Convergence Criteria ............................................................................. 38

5.3 T-Junction with Chamfered Inlets ....................................................................... 40

5.4 Rectangular Duct T-Junction (Steady) ................................................................ 41

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5.5 Rectangular Duct T-Junction (Unsteady) ............................................................ 44

5.6 Rectangular Duct T-Junction (Unsteady with UDF) ......................................... 46

6 Conclusion .............................................................................................................. 50

List of References ........................................................................................................... 52

Appendix ........................................................................................................................ 57

T-Junction Geometries ..................................................................................................... 57

Unsteady flow UDF .......................................................................................................... 60

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List of Figures

Figure 1.1 Gas and Oil pipelines in Ireland and the UK [2] ................................................. 2

Figure 1.2 Different flow possibilities in a T-junction ........................................................... 3

Figure 1.3 Turbine bypass piping system. (1) Steam turbine isolation valve; (2) Bypass isolation valve; (3) Bypass control valve; (4) T-junction [1] ................................................. 4

Figure 1.4 (b) T-Junction without a transition zone (c) T-Junction with asymmetric transition zone [1] ...................................................................................................................... 5

Figure 2.1 Rectangular duct T-Junction test setup [1] ........................................................... 7

Figure 2.2 Flow visualisation arrangement [1] ....................................................................... 7

Figure 2.3 Typical responses for Case A T-Junctions with and without a transition zone [1] ................................................................................................................................................. 8

Figure 2.4 Responses for the four Case A T-Junction geometries [1] ................................. 8

Figure 2.5 Flow-induced acoustic resonance at the safety relief valve (SRV) side branch [7] ................................................................................................................................................. 9

Figure 2.6 The possible outcomes of Howe's equation [10] ................................................ 10

Figure 2.7 LES/SI method [14] ............................................................................................... 12

Figure 2.8 Experimental results (solid blue lines) vs. numerical results (dashed red lines) [14] ............................................................................................................................................. 12

Figure 2.9 3D mesh structure in the plane of symmetry and cross-section of the T-Junction used in [16] ................................................................................................................ 13

Figure 2.10 Contours of Mach number at the plane of symmetry [16] ............................. 13

Figure 2.11 Computations per kilowatt-hour over time [20] ............................................. 14

Figure 3.1 Visualisation of the continuous and discrete domain [21] ............................... 15

Figure 3.2 ANSYS Workbench workflow .............................................................................. 16

Figure 3.3 Residual plots of various turbulence models with a convergence criterion of 1 x 10-3 ........................................................................................................................................... 25

Figure 3.4 Velocity contours of three different turbulence models ................................... 26

Figure 4.1 Primary branch acoustic mode [1] ...................................................................... 27

Figure 4.2 (left) 3D flow in a pipe of cylindrical cross section [43] (right) 2D flow in a pipe of cylindrical cross section at the central axis but is the same for a rectangular duct [44] ............................................................................................................................................. 28

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Figure 4.3 (left) Dimensions of T-Junction with cylindrical geometry [1] (right) Table displaying the dimensions and relationships between L and D [1] .................................... 28

Figure 4.4 T-Junction geometry created using ANSYS DesignModeler ............................ 29

Figure 4.5 Two different meshing approaches ...................................................................... 31

Figure 5.1 Bad meshing and resulting y+ values .................................................................... 35

Figure 5.2 A more ideal mesh and resulting y+ values .......................................................... 36

Figure 5.3 Difference in velocity contours between the bad mesh (left) and the better mesh (right) ............................................................................................................................... 37

Figure 5.4 Results of a simulation between convergence criteria of 1 x 10-3 (left) 1 x 10-6 (right).......................................................................................................................................... 38

Figure 5.5 Residual plot for convergence criteria of 1 x 10-6 ............................................... 39

Figure 5.6 Vorticity contours of a T-Junction with chamfer (left) and without (right) .. 40

Figure 5.7 3D velocity profile of the flow in a rectangular duct .......................................... 41

Figure 5.8 Four cases of rectangular geometry T-Junctions based on the template on the left [1] ......................................................................................................................................... 41

Figure 5.9 Velocity swirling vector contour map of the flow in a T-Junction .................. 42

Figure 5.10 Vorticity contours of the flow development in the branch pipe (flow from right to left) ................................................................................................................................ 43

Figure 5.11 Vorticity contours of the four cases. Vortex formation is strongest in the red regions ........................................................................................................................................ 43

Figure 5.12 Normalised acoustic pressure as a function of reduced velocity for the four test cases [1] ............................................................................................................................... 44

Figure 5.13 Velocity plots of the four cases at various locations of the T-Junction ......... 45

Figure 5.14 Oscillation of branch velocity with time............................................................ 47

Figure 5.15 Instantaneous flow structure images and sketches for a complete cycle of Mode A oscillation (VR = 1.65 and f = 118 Hz) [1] ............................................................... 47

Figure 5.16 Velocity plots of the four cases at various locations of the T-Junction (with UDF) ........................................................................................................................................... 48

Figure 5.17 Combined vector and streamline map of regions of recirculation (blue regions) ....................................................................................................................................... 48

Figure 5.18 Vorticity (500 – 1000 s-1) (left) and dynamic pressure (500 – 1000 Pa) (right) contour maps at different phase angles for Case 1 ............................................................... 49

Figure 6.1 Three different modelling approaches ................................................................. 51

Figure 6.2 RANS vs. LES .......................................................................................................... 51

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Figure 0.1 Geometry #1 ........................................................................................................... 57

Figure 0.2 Geometry #2 ........................................................................................................... 58

Figure 0.3 Geometry #3 (Case 1) ............................................................................................ 59

Figure 0.4 Geometry #3 (Case 2) ............................................................................................ 59

Figure 0.5 Geometry #3 (Case 3) ............................................................................................ 59

Figure 0.6 Geometry #3 (Case 4) ............................................................................................ 59

List of Tables

Table 3.1 Iterations required for different turbulence models with a convergence criterion of 1 x 10-3 ................................................................................................................... 24

Table 4.1 Hydraulic diameters calculated for the cylindrical pipe and the rectangular duct ............................................................................................................................................. 33

Table 5.1 Section 5.1 pipe parameters (See Appendix) ....................................................... 35

Table 5.2 Section 5.2 pipe parameters.................................................................................... 38

Table 5.3 Section 5.3 pipe parameters.................................................................................... 40

Table 5.4 Section 5.4 pipe parameters (see Appendix) ........................................................ 41

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Abbreviations CFD Computational Fluid Dynamics DNS Direct Numerical Simulation LES Large Eddy Simulation RANS Reynolds-Averaged Navier-Stokes RSM Reynolds Stress Model SFS Sub-filter Scale SIMPLE Semi-Implicit Method For Pressure-Linked Equations SMG Standard Smagorinsky Model SST Shear Stress Transport UDF User-Defined Function URANS Unsteady Reynolds-Averaged Navier-Stokes

Nomenclature A Cross-sectional area D Diameter or height of the

transition zone and main pipe d Diameter or height of branch pipe dh Hydraulic diameter f Frequency (Hz) L Half-length of the transition zone ℓ Characteristic length LB Length of branch pipe LM Length of main pipe p Pressure Q Volume flow rate r Radius Re Reynolds number t Time

V Velocity VB Velocity of fluid in the branch pipe VM Velocity of fluid in the main pipe VR Reduced Velocity y+ Non-dimensional wall distance Δy First inflation layer height λ Wavelength μ Viscosity ν Kinematic viscosity ρ Density τ Time step size τ Time step ω Vorticity ωA Frequency (rad/s)

S.I. units used throughout.

The symbols shown above are valid throughout unless specified otherwise.

1

1 Introduction The basis of this study was motivated by the works of Ziada et al. [1] where the

flow in T-Junctions of various geometry and flow configuration was investigated. This

study focuses on modelling the unsteady flow in a T-Junction with a symmetric

transition zone using computational methods in order to gauge the feasibility of

computational fluid dynamics (CFD) compared to the actual experiment. A certain

element of trial-and-error is involved in CFD which raises the question as to whether a

solution can be considered as accurate as an actual experiment. Is CFD is a feasible

alternative for accurately representing the flow in a T-Junction? What are the difficulties

involved and the downfalls of CFD? Does CFD take into account effects such as acoustic

vibration? These are some of the questions that the study hopes to answer.

All T-Junction geometry created for simulations follow the dimensions of the

geometry of the pipes and ducts described in [1]. The flow is to be simulated using an

unsteady Reynolds-Averaged Navier-Stokes (URANS) model in ANSYS Fluent, a

commercial CFD package. This is explored in greater depth in the section on turbulence

modelling. A URANS model is used due to its relatively low computational cost and

reasonable accuracy compared to direct numerical simulation (DNS) or models such as

the Reynolds stress model (RSM).

The remainder of this section will briefly discuss the background of the project,

the geometry of the T-junction and some of the physics behind the flow. The following

chapter will review the relevant literature in this field. This includes the work has been

done and the work that is being done in relation to the coupling of acoustic resonance in

fluid flows in a T-Junction. Later, an introduction into turbulence modelling is

presented and it concludes upon the model most suited for this simulation. Once all the

relevant information has been detailed, the methodology required to carry out the

simulations can be described and subsequently, the results can be analysed.

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1.1 Project Background

Pipe networks are present in houses, sewage treatment, power plants, chemical

plants, oil rigs, cities and even across countries (Figure 1.1). These networks generally

transport liquids and gases over a range of distances. Pipes work in conjunction with

various other components and flow configurations to form a pipe network. Some of

these include elbows, bends, T-junctions, valves, pumps, turbines and compressors.

Figure 1.1 Gas and Oil pipelines in Ireland and the UK [2]

Pipe networks are prevalent in any major industry and therefore, many of the

components and flow configurations are widespread. One such flow configuration is a

T-junction. This is a generic design that allows two flows to converge into one flow or

one flow to diverge into two. Depending on the direction of fluid flow, a few different

possibilities are possible and are shown in Figure 1.2.

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Figure 1.2 Different flow possibilities in a T-junction

In many such T-junctions, one issue that arises time and time again is the

generation of acoustic resonance. This is caused when the sound field generated is

coupled with the unsteady separated fluid flow. Resonances cause high noise levels and

excessive vibration that can damage pipe networks by overstressing its components as

well as cause significant efficiency losses which results in fatigue failure over time. This

is a well-documented problem and is often encountered in high-pressure piping systems

transporting steam or natural gas.

This resonance is caused by unstable flow structures such as free shear layers, jets

and wakes. Small vorticity disturbances at the regions of flow separation rapidly grow

into vortex-like structures as they travel downstream with the flow [1]. Vorticity is a

vector field and is defined as the curl of the velocity as shown below:

V×∇=ω

Where ω is the vorticity, ∇ is the gradient and V is the velocity. When these vortex-like

structures travel downstream within a sound field, acoustic energy is either absorbed or

produced. In the case of the T-junction, the excessive creation of acoustic energy causes

resonance.

Howe’s work on the dissipation of acoustic energy [2] describes the dissipation

of acoustic energy between unsteady vorticity and acoustic fields by the following

integral over the fluid volume ∀ :

∏ ∫ ∀×⋅−= duV )(ωρ

where ω is the vorticity, V is the flow velocity, ρ is the density of the fluid and u is the

acoustic particle velocity of the sound field. The integral of the instantaneous acoustic

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power Π over a complete acoustic cycle may result in either positive or negative acoustic

energy. Acoustic resonances will only be excited by separated flows if the integral of the

above equation is positive over a complete acoustic cycle. [1]. Various studies have used

Howe’s equations to understand and explain the flow-sound interactions in various flow

configurations eg. [3].

A study conducted by Ziada et al. [1] was based on a specific vibration problem

involving a steam turbine bypass piping system in a power plant. It was observed that

fluid flowing through the valves at certain flow rates induced strong vibrations in the

bypass piping downstream of the valves (highlighted in red in Figure 1.3). The

mechanism of this failure was found to be the unsteady coupling of vortex shedding at

the mouth of the valve along with the side branch acoustic resonance.

Figure 1.3 Turbine bypass piping system. (1) Steam turbine isolation valve; (2) Bypass isolation valve; (3)

Bypass control valve; (4) T-junction [1]

1.2 T-Junction Geometry

As mentioned before, T-Junction geometry is a very generic flow configuration.

Although many variations exist, the basic design is the same – two pipes combining into

one. The basic geometry of the T-junction used in the majority of the simulations in this

project is shown in Figure 1.2 (left).

Here on, the following terminology will be used in regards to the T-junction.

The pipes entering (or upstream of) the T-Junction will be called the branch pipes. The

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pipe exiting (or downstream of) the T-Junction will be called the main pipe and the

expanded portion of the T-junction will be called the expansion zone or the transition

zone. This terminology is exactly the same terminology used by Ziada et al. [1, 3] in their

experiments.

Alternative T-Junction geometry have also made an appearance in their

experiments [1, 3] with a range of dimensions and flow directions. One such example is

a T-Junction without a transition zone where the branch pipes and the main pipe are of

the same diameter. This is the simplest possible T-Junction geometry as there is no

transition zone. Another variation of the T-Junction is where the transition zone is

asymmetric about the centreline of the main pipe (Figure 1.4).

Figure 1.4 (b) T-Junction without a transition zone (c) T-Junction with asymmetric transition zone [1]

The flow in each geometry is different and the addition of a transition zone

affects the flow and sometimes induces resonance. Furthermore, symmetry or

asymmetry of the transition zone influences the flow in the main pipe and the behaviour

of the resonance previously mentioned.

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2 Literature Review This basis for this project branched out from the work of Ziada et al. [1, 3] which

investigated the flow-acoustic coupling between separated flow in a T-Junction at the

lowest acoustic mode. These works focused on physically modelling the flow in a T-

Junction. The premise was that acoustic resonances are excited due to the coupling

between an acoustic mode and an unsteady flow. Acoustic resonances can lead to

excessive noise and dangerous levels of vibration. This vibration is caused as a result of

small vorticity perturbations growing rapidly into vortex-like structures at the flow-

separation region. The convection of these vortex-like structures within a sound field

results in the production or absorption of acoustic energy.

This is a well-recognised and documented phenomenon in industry especially

with gases and steam in high-pressure piping systems. Chen and Florjancic [4]

identified the source of failure in many safety relief valves to be flow-induced vibration.

The mechanism of this vibration was found to be the unstable coupling of acoustic

resonance and vortex shedding. The U.S. Nuclear Regulatory Commission reported the

failure of a steam dryer in Quad Cities nuclear power plant where flow-induced

vibration caused damage to components and supports for the main steam and feedwater

lines [5].

The present study in [1], although instigated by a specific vibration problem of a

steam turbine bypass piping system, addresses a generic T-Junction flow configuration

found commonly in industry. It was found that at certain flow rates, strong vibrations

were experienced by the piping. The experimentation in this study was carried out on T-

Junctions with an expansion zone so that the diameters of the two inlet pipes can be

matched. Two separate types of pipes were used. The first was a cylindrical pipe (Case

A) and T-Junction whereas in the second case a rectangular duct (Case B) was used. The

majority of the data obtained was from the rectangular duct case as flow visualisation

was only conducted in this case. Case A mainly focused on obtaining data using

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commercial piping and geometries such as using PVC piping and chamfered T-Junction

inlets. Figure 2.1 and Figure 2.2 shows the experimental setup for Case B. Case B was

used for flow visualisation because it allowed for the capture of reasonably clear images

due to the two-dimensional nature of the T-Junction (flat side walls).

Figure 2.1 Rectangular duct T-Junction test setup [1]

Figure 2.2 Flow visualisation arrangement [1]

The results of the study revealed that in Case A, the onset of acoustic resonance

was noticeable at a VM between 25 m/s and 45 m/s and a significantly stronger

resonance was observed at VM = 65 m/s at ~118 Hz. But perhaps what was most

interesting was that acoustic resonance was only present in T-Junctions with a transition

zone as opposed to T-Junctions without a transition zone (Figure 2.3. The data for the

T-Junction without a transition zone comes from a previous study from Ziada et al. [3].

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The results of Case B also show a similar trend as in Figure 2.3 (Figure 2.4) where little

to no resonance was observed when the transition zone was omitted.

Figure 2.3 Typical responses for Case A T-Junctions with and without a transition zone [1]

Figure 2.4 Responses for the four Case A T-Junction geometries [1]

The main finding here was that the flow excitation is caused by the instability of

the shear layers separating from the step expansions at the inlets of the T-Junction. This

is also mentioned in a previous study by the same author [3]. Experiments by Michalke

[6] have shown that shear layers are unstable relative to disturbances within a certain

9

frequency range. It was also found that introducing geometrical asymmetry in the

transition zone produced little effect on the intensity of the acoustic resonance. The

study provides a guideline range of velocities that are outside the resonance range for a

range of L/D ratios. The previous study by Ziada et al. [3] was a precursor to the current

study as it laid the foundations for most of the experimentation. The T-Junctions tested

in this case did not have an expansion zone and the results were similar to those in case

4 for the rectangular duct in [1]. In the context of this project, the works of Ziada et al.

[1, 3] have been the most influential.

Okuyama et al. [7] investigated the acoustic resonance at the entrances to one or

two side branches. The context here was similar to the case of the Quad Cities Unit 2

Nuclear Power Plant [5]. The geometry of the pipes used here can be compared to a T-

Junction in some respects, but different to [1] in terms of flow directions and

dimensional proportions. Three pipe configurations were used: a single branch pipe

(Figure 2.5), coaxial branch pipe and the final case where the branch pipes were in

tandem.

Figure 2.5 Flow-induced acoustic resonance at the safety relief valve (SRV) side branch [7]

It was found that as the ratio of the cross-sectional area of the main pipes and the

branch pipes decreased, the amplitude of resonance increased. This was because the

radiation loss from the branch pipe to the main pipe increased at higher values of the

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cross-sectional ratio. This is one of the reasons why the vibration due to acoustic

resonance was relatively much stronger in [1, 3] .

Karlsson’s [8] doctoral thesis provides rigorous insight into the aeroacoustics of

duct branches in the context of silencers in cars. It focuses heavily on the acoustics and

it concludes upon the practical importance of the work when it comes to designing

complex flow duct networks. The study was motivated by the general concern of global

warming and the interest of developing new silencer concepts that can sufficiently

reduce exhaust noise in an internal combustion engine as well as being as compact as

possible and having minimal energy losses. The research focused on minimising the

flow losses in flow expansions (such as an expansion zone in a T-Junction), contractions

and reversals.

Howe’s equation [2] (see section 1.1) is mentioned in [8] to determine whether

acoustic energy is produced or absorbed. If the equation is positive, sound is generated;

if the equation is negative, sound is dissipated (Figure 2.6). Acoustic resonances will

only be generated by separated flows if the integral of Howe’s equation is positive over a

complete cycle. Howe’s equation has been used by numerous authors [9-11] to describe

the effects of flow-sound interaction mechanisms for various flow configurations such

as deep cavities exposed to grazing flows.

Figure 2.6 The possible outcomes of Howe's equation [10]

Karlsson’s work [8] was also relevant to this project because it investigated the

use of steady simulations using CFD to interpret the aeroacoustic phenomena. He

mentions that “steady-state CFD simulations are a computationally effective tool for

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rapid design cycles” and this is indeed true as CFD cuts down on expensive and time

consuming experimentation. The contour maps generated from the simulations are

surprisingly similar to the results obtained in the project in terms of the flow pattern

and behaviour. However, it is interesting to note that the κ-ε model is used here. The κ-ε

model models free stream flows with a reasonable amount of accuracy but it’s quite a

bad option to model flows at walls and boundary layers. The SST (Shear Stress

Transport) κ-ω model is a much better option in comparison as it combines the

strengths of the κ-ω model at the near-wall and boundary layers but avoids

oversensitivity in the free stream because as it switches to κ-ε formulation here. The

section 3 on computational fluid dynamics explores the strengths and weaknesses of

different turbulence models.

The above research is of practical important in the design of complex flow duct

networks as it provides as formalism for including flow-acoustic interaction effects in

linear multiports with the goal of measuring the amplification and reducing the incident

sound and also predicting non-linear phenomena such as whistling. This can be related

to this project as vortex shedding at the inlet of the T-Junction (which can be considered

a sharp-edged orifice) is very closely related to whistling. Other works such as [12] and

[13], both by Karlsson and Abom investigated aeroacoustics in T-Junctions using a new

method that they proposed and developed a quasi-steady model to describe the acoustic

scattering properties in a T-Junction in the low Strouhal number limit respectively. [13]

is applicable in the design of automotive intake and exhaust systems and the basis for

this is from Karlsson’s doctoral thesis [8].

Whistling of pipes and orifices and experimentally and numerically investigated

in [14]. The numerical approach involved using Large Eddy Simulation (LES) with an

acoustic signal analysis. LES is a very computationally expensive but a highly accurate

turbulence model and is explored in greater depth in section 3. The LES simulation was

coupled with a System Identification (SI) method for the acoustic signal analysis. This

was termed the LES/SI method and is shown in Figure 2.7. The solver used in this study

was AVBP developed by CERFACS [15]. This study was interesting because it involved

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numerical simulation as opposed to most other research in the area relying on

experimentation. The numerical results are then compared with the experimental

results and it shows how a numerical simulation can almost be as accurate as actual

experimentation (Figure 2.8) which once again reinforces Karlsson’s conclusion

regarding CFD simulations [8]. Another important note in [14] is regarding the y+

values used to model the flow. Low y+ values are used in the simulation as it was

important to generate a mesh that was not too coarse. Section 5.1 of this thesis on the

influence of meshing on the solution provides more insight regarding y+ values and the

accuracy of the solution.

Figure 2.7 LES/SI method [14]

Figure 2.8 Experimental results (solid blue lines) vs. numerical results (dashed red lines) [14]

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A study by Pérez-Garcia et al. [16] investigated the internal compressive flow in

T-Junctions in much greater detail from a numerical point of view than the studies

previously mentioned here. The study compared a range of turbulence models in order

to determine which was best suited for the problem at hand. The methodology used

closely resembles the methodology used in this project. Similarities include the choice of

turbulence models and the mesh generated (Figure 2.9) (see section 4.2 for the mesh

used in this project). One of the conclusions of [16] was that the SST κ-ω model agreed

best with the experimental and reference data. A similar conclusion on the choice of

turbulence models is reached in this project as well. A contour map of Mach numbers at

the plane of symmetry obtained in [16] was almost exactly the same as the contour maps

of velocity obtained in this project (section 5.2).

Figure 2.9 3D mesh structure in the plane of symmetry and cross-section of the T-Junction used in [16]

Figure 2.10 Contours of Mach number at the plane of symmetry [16]

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Numerous other studies have been conducted in the field for flow-acoustic

coupling and vibration in pipes. Examples of this include the works of Tonon et al. [17,

18]. However, the push towards numerical simulations of flows is a relatively recent

development. Pérez-Garcia et al. [16] mention that the computational time required for

each simulation was about a week on a Compaq HPC160 16 1 GHz processor. As

computational technologies advance, the computational time for simulations reduces

and numerical simulations become a much more feasible option. Moore’s law, in

practical terms states that the performance of a personal computer doubles every 18

months . This trend has held steadfast for decades. However, a better correlation

between electrical efficiency of computation and time has been developed by Koomey et

al. [20]. It states that

“The electrical efficiency of computation has doubled roughly every year and a

half for more than six decades, a pace of change comparable to that for

computer performance and electrical efficiency in the microprocessor era.”

Figure 2.11 Computations per kilowatt-hour over time [20]

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3 Computational Fluid Dynamics (CFD)

3.1 CFD Strategy

The basic idea behind CFD is to replace the continuous problem domain with a

discrete domain using a grid. In a continuous domain, each flow variable is defined at

every point in the domain whereas in the discrete domain, each flow variable is defined

only at grid points. The values at other locations other than the grid are approximated

by interpolating the values at grid points. This can be visualised by the comparison of

pressure p in the continuous and discrete time domain as shown in Figure 3.1.

Figure 3.1 Visualisation of the continuous and discrete domain [21]

The governing partial differential equations and boundary conditions are

defined in terms of the continuous variables p, V etc. These can be approximated in the

discrete domain in terms of the discrete variables pi, Vi etc. The discrete system is a large

set of coupled, algebraic equations in the discrete variables. Solving and setting up this

system requires a very large number of repetitive calculations which can easily be solved

by a computer. The size of the grid or mesh determines the accuracy of the solution and

the computational power required.

3.2 Software

This project exclusively involved the use of CFD software for all simulations.

The software used was ANSYS, a commercial engineering simulation package. The

ANSYS Workbench suite offers a wide range of engineering simulation solution sets

that cover almost every aspect of engineering simulation required by a design process.

16

The geometry of the T-Junction was created in DesignModeler and an

appropriate mesh was created using the Meshing application which was the next step of

the workflow. Once these steps were completed, the fluid dynamics application in

ANSYS could be launched to run simulations. After the simulations have converged, the

results can be analysed using Fluent or analysed and presented using CFD-Post. Shown

below is a typical ANSYS Workbench fluid flow analysis workflow.

Of the four main steps in the workflow, the setup and analysis using Fluent was

the most important part. For this, a good understanding of turbulence modelling is

required. Fluent offers many turbulence models, each with its strengths and weaknesses.

3.3 Turbulence Modelling

3.3.1 Introduction

Computational Fluid Dynamics (CFD) is a vast field of study involving

numerical analysis in the field of a fluid’s flow phenomena. The development of new

CFD technologies dependent on the development of computational technologies and

our understanding of ordinary and partial differential equations [22].

Flows can be simulated using Direct Numerical Simulation (DNS). However, the

computational requirement for simulating complex flows in realistic conditions requires

an unrealistic amount of computational power. This is because DNS tries to capture all

eddy sizes, down to the smallest turbulence scales. Therefore, solving a problem

successfully is very much dependent on the physical models applied which is why it is

important to have good and appropriate turbulence models to base problems on.

Figure 3.2 ANSYS Workbench workflow

• Geometry – DesignModeler

• Mesh – Meshing

• Setup – Fluent

• Results – CFD-Post

17

In CFD, turbulence appears to be dominant over all other flow phenomena

when present. Properly modelling the turbulence in a given problem greatly increases

the quality of simulations [22]. There are many turbulence models available today, each

with its set of assumptions, advantages, disadvantages, mesh implications and

computational requirements.

Given the time constraints of this project, it was necessary to find a turbulence

model that was both reasonably accurate and computationally inexpensive. A basic

understanding of some of the commercially used turbulence models provided by Fluent

was essential for the advancement of this project.

3.3.2 κ-ε Two-Equation Model

The κ-ε two-equation turbulence model is the most common type of turbulence

model available today. More correctly, it is a family of turbulence models. Most

engineering problems involving fluid mechanics use this model and it is also very active

in research as new, refined two-equation models are still being developed. Two-equation

models include two extra transport equations that represent the turbulent properties of

the fluid flow and this enables two-equation models to account for history effects such

as convection and diffusion of turbulent energy .

One of the transported variables is usually κ which represents turbulent kinetic

energy. The second transported variable depends on the actual model itself. In the case

of the κ-ε model, the second transported model ε, represents the turbulent dissipation.

This model is derived by assuming the flow is fully turbulent and the effects of

molecular viscosity are negligible [24]. For most general purpose simulations, this

model offers a good compromise in terms of accuracy and robustness.

All two-equation models are based on the Boussinesq eddy viscosity assumption

. It assumes that the momentum transfer caused by the turbulent eddies can be

modelled with an eddy viscosity. One of the problems with this hypothesis is that it is a

huge simplification which allows one to think of the effect of turbulence in the mean

flow the same way as molecular viscosity affects a laminar flow. The other weakness of

18

the Boussinesq hypothesis is that while it is true in simple flows like straight boundary

layers such as boundary layers and wakes, it is not valid in more complex flows such as

those involving accelerations and decelerations. Therefore, problems arise when two-

equation models try to predict strongly rotating flows and other flows where curvature

effects are significant. Some other flows where the κ-ε model may not be suitable are:

• Flows with boundary layer separation

• Flows with sudden changes in the mean strain rate

• Flows in rotation fluids

However, the κ-ε model has been shown to be useful for free-shear flows with

relatively small pressure gradients. It is also less stable than models such as the SST κ-ω

model. Furthermore, where the κ-ε model sacrifices a certain amount of accuracy, it

makes up for it in lower computational requirements.

The standard κ-ε model and the realizable κ-ε model are suitable for coarse

meshes where the wall-cell y+ values are typically 30 and above. The realizable model

usually gives better results than the standard model. The standard two-layer κ-ε model

and the realizable two-layer κ-ε model offer the greatest flexibility in regards to the

mesh. Compared to other κ-ε models, they produce the least inaccuracies for

intermediate meshes (1 < y+ < 30) . The y+ value is the non-dimensional wall distance for

a bounded flow and can be defined as follows:

νyuy ⋅

≡+ *

where u+ is the friction velocity at the nearest wall, y is the distance to the nearest wall

and ν is the kinematic viscosity of the fluid . y+ is simply referred to as ‘y plus’ and is

commonly used in boundary layer theory.

The two major shortcomings of the κ-ε model are that it over-predicts the shear

stress in adverse pressure gradient flows because the length scale is too large and that it

requires near-wall modification [27]. Despite this, when an uncertainty arises as to

which turbulence model is to be used in a certain scenario, the realizable κ-ε model is a

reasonable choice.

19

3.3.3 SST κ-ω Two-Equation Model

The κ-ω two-equation model is another widely used turbulence model. The SST

κ-ω model was ultimately used as the primary turbulence model for this project for

reasons outlined in section 3.4.

The Shear Stress Transport (SST) κ-ω model, first introduced by Dr. Florian R.

Menter in 1994 [28] is a two-equation eddy-viscosity model. The development of the

SST κ-ω model was the need for the accurate prediction of aeronautic flows with strong

adverse pressure gradients and separation. The available turbulence models at the time

had consistently failed to compute these flows.

Once again, the SST κ-ω model is a two-equation model and this means that it

includes two extra transport equations to represent the turbulent properties of the flow.

The SST model is part of a family of κ-ω models which are another family of commonly

used turbulence models. Just the same as the two-equation κ-ε model, this allows the

model to account for history effects such as convection and diffusion of turbulent

energy. As before κ represents the turbulent kinetic energy and in this case, ω represents

the specific dissipation. The greater complexity of the model when compared to

standard models can be attributed to the fact that it is necessary to compute the distance

from the wall [29].

The SST model has become the accepted two-equation model in industry for

flow separation. The SST model combines the best of both the κ-ε model and the κ-ω

model and it is the most reliable model for fluids with flow separation [30]. The use of

κ-ω formulation in the inner parts of the boundary layer makes the model directly

usable all the way down to the wall through the viscous sub-layer. The SST formulation

switches to a κ-ε behaviour in the free stream and therefore avoids the common κ-ω

problem that the model is too sensitive to the inlet free-stream turbulence properties

[31]. The κ-ω model is substantially more accurate than the κ-ε model in the near wall

layers, and has therefore been successful for flows with moderate adverse pressure

gradients. Despite this, the κ-ω model model’s sensitivity in the free stream has largely

20

prevented the ω-equation from replacing the ε-equation as the standard scale-equation

in turbulence modelling [29]. This makes the SST model very attractive for simulations

in this project because of the need to model near wall as well as free stream flows.

Once again the Boussinesq hypothesis is the main assumption in the SST κ-ω

model because it is a two-equation model. It assumes that the Reynolds stress tensor, τij,

is proportional to the trace-less mean strain rate tensor, S*ij.

3.3.4 Reynolds Stress Model (RSM)

The RSM is a much higher level, elaborate turbulence model. It is the most

sophisticated model presented here so far. This modelling approach has its roots in the

work of B.E. Launder in 1975 [32]. In RSM, the isotropic eddy viscosity assumption has

been discarded and it closes the Reynolds-averaged Navier-Stokes (RANS) equations by

solving transport equations for the Reynolds stresses, together with an equation for the

dissipation rate. This means that five additional transport equations are required in two

dimensional flows and seven addition transport equations must be solved in three

dimensional fluid flow equations [33]. The directional effects of the Reynolds stress

fields are accounted for by the Reynolds stress transport equation. The fact that RSM

accounts for the effects of streamline swirl, curvature, rotation and rapid changes in

strain rate in a more exact manner than one-equation and two-equation models, it has a

greater potential to give accurate predictions for complex flows [34]. It is physically the

most complete turbulence model available. It accounts for history effects, transport, and

anisotropy of turbulent stresses.

Although RSM gives better results than the SST κ-ω model or the standard κ-ε

model, it is not without its limitations. Some problems arise such as:

• Greater difficulty in setting correct boundary layer conditions

• Difficulty in achieving convergence

• Discretization has to be done carefully

• Requires greater computational power than the SST κ-ω model and the standard

κ-ε model because of the seven additional equations to be solved in three

21

dimensions and five additional equations to be solved in two dimensions.

However, Large Eddy Simulation (LES) requires greater computational power

than all of these. Obviously this greater computational power requirement is due

to RSM including more physics than a two-equation turbulence model.

Because of the computational complexity of the RSM, they have not been widely

used for engineering applications. It would seem that the RSM would have the best

chance of emerging as the “ultimate” turbulence model because it is not restricted by the

Boussinesq hypothesis and because the closure contains the greatest number of model

PDEs and constants of all the models considered.

3.3.5 Large Eddy Simulation (LES)

As discussed above, the κ-ε model simply attempts to model the turbulence by

performing time or space averaging. Under certain conditions, this method can be very

accurate, but is not suitable for transient flows, because the averaging process wipes out

most of the important characteristics of a time dependent solution. On the other hand,

DNS attempts to solve all time and spatial scales. Although that would provide a very

accurate result, it is computationally unrealistic. A compromise between these two

methods is LES. It was initially proposed in 1963 by Joseph Smagorinsky [35] to

simulate atmospheric currents. It was originally implemented in the 1970s to study

weather.

LES is generally used in flows with medium to large Reynolds numbers where

the length scale is in meters to kilometres. RANS is suitable for all Reynolds numbers

and DNS is only suitable for small to medium flows because of its much greater

computational cost. Examples of applications of LES are in simulating geophysical

turbulence in pollution layers, deep convection, convection in the sun etc.

The technique of LES has emerged as a viable alternative to the RANS approach

in order to remedy the scale-complexity problem inherent to high Reynolds number

turbulent flows. In LES, the motion is separated into small and large scale and equations

are solved for the latter. The principal operation in LES is low pass-filtering [36]. This

22

operation when applied to the Navier-Stokes equations eliminates small scales of the

solution. In large eddies, most energy and fluxes are explicitly calculated. In small

eddies, little energy and fluxes are parameterized – SFS (sub-filter scale) model. LES is

supposed to be insensitive to the SFS model [37]. This reduces the computational cost of

the simulation. The governing equations are thus transformed, and the solution is a

filtered velocity field. LES resolves large scales of the flow field solution allowing better

fidelity than alternative approaches such as RANS methods. It also models the smallest

scales of the solution rather than resolving them as DNS does [38]. Because of this, it

makes the computational cost for practical engineering systems with complex geometry

or flow configurations attainable using supercomputers.

In LES, the standard Smagorinsky model (SMG) is still widely used due to its

algorithmic and numerical simplicity. However, the major drawback associated with

this model is that the optimal model parameter is flow-dependent and ad hoc

modifications of this parameter are required near solid surfaces [39]. The most

compelling case for LES can be made for momentum, heat and mass transfer in free

shear flows at high Reynolds numbers. There is good reason to expect LES to be

successful, primarily because both the quantities of interest and the rate controlling

process are determined by the resolved large scales.

In a study by Y. Cheng et al. [39] in the context of turbulent flow over a matrix

of cubes, a detailed comparison between the RANS (κ-ε model) and LES was carried out.

Some of the conclusions that were drawn from the study were:

• The RANS results (more specifically, predictions provided by the standard κ-ε

model) were found to be considerably different from the LES predictions and the

experimental measurements. The size of the recirculation zone was

overestimated in the RANS simulation. This creates a severe underestimation of

the mean streamwise velocity component.

• The complex features (vortex shedding, large separation zones etc.) of the fully

developed flow within and above an array of cubes are reproduced better with

LES calculations, albeit at the disadvantage of much greater computation times.

23

In the present study, the computational cost associated with LES is about 100

times greater than that incurred with the κ-ε RANS model.

The setup and use of turbulent flow simulations requires a profound knowledge

of fluid mechanics, numerical techniques and the application under consideration. The

susceptibility of LES to errors in modelling, in numerics, and in the treatment of

boundary conditions, can be quite large due to nonlinear accumulation of different

contributions over time, leading to an intricate and unpredictable situation[40]. One of

the main difficulties arising in the evaluation of errors in LES is the non-linear

accumulation of different error sources. The worst is the possible interaction between

SGS modelling errors and numerical errors.

In physical LES as described in [41], good numerical accuracy comes at a higher

computational price. With the numerical methods usually employed, halving the grid

spacing increases the computational cost by about a factor of 24 = 16.

In conclusion, LES seeks to solve large spatial scales (like DNS) while modelling

the smaller scales (κ-ε). Firstly, the larger scales carry the majority of the energy and

therefore are more important methodology is a hybrid between these two methods

which allows for the generation of useful solutions to transient flows, while still

maintaining computationally realistic problems [42].

3.4 Turbulence Model Testing

With a basic understanding of the various turbulence models available for use in

Fluent, the next goal of the project was to find a suitable turbulence model for the

problem at hand. For this, a simple T-Junction geometry was created and meshed.

Simulations were run for the cases of the κ-ε (standard, RNG and realizable), κ-ω

(standard and SST) and RSM models. A range of results were compiled in order to

compare the aforementioned. The first test completed was the time and computational

power required to complete the calculation.

Residuals in Fluent are a measure of how well a solution satisfies the discrete

form of each governing equation. Therefore, the smaller the convergence criterion, the

24

more accurate the result becomes. However, setting a steep convergence criterion also

means that the simulation takes much longer to run. For these simulations, the

convergence criteria were set at 1 x 10-3 as this was the default and anything higher

would be an unnecessary, time-consuming investment for this project. The residual

plots from six different simulations are shown below. The inlet velocities in all cases

were 30m/s.

(a) Standard κ-ε (b) Realizable κ-ε

(c) RNG κ-ε (d) Standard κ-ω

Model Iterations Standard κ-ε 99 Realizable κ-ε 104 RNG κ-ε 262 Standard κ-ω 235 SST κ-ω 296 RSM 1235

Table 3.1 Iterations required for different turbulence models with a convergence

criterion of 1 x 10-3

25

(e) SST κ-ω (f) RSM

Figure 3.3 Residual plots of various turbulence models with a convergence criterion of 1 x 10-3

The obtained results were as theory indicated. The standard κ-ε required the

least number of iterations for the solution to converge. The Reynolds stress model

simulation was at the other end of the scale with 1235 iterations. This is because there

are five additional transport equations that must be solved in a 2D flow. Given the time

constraints of this project, this was an unrealistic model to use.

Furthermore, the velocity contours of the κ-ε, κ-ω and RSM models were

compared to finalise the choice for the most suitable turbulence model. The choice was

immediately clear. The κ-ε model, while fast in obtaining results, produces results that

are quite inaccurate. This can be seen in Figure 3.4 (a) where the velocity at the central

axis of each branch pipe is in the region of ~40 m/s to ~48 m/s. This is simply not true

because the inlet velocity is specified at 30m/s and this value cannot be higher unless

influenced by another flow or geometry. The RSM produces results that are quite

accurate in terms of the velocity contours. The velocity profile on the main pipe is what

would normally be expected and the velocity in the branch pipes are around 30m/s.

However, as mentioned previously, the weakness of RSM is that it is very

computationally expensive. The SST κ-ω model was found to be an excellent

compromise between the two. The velocity profile, while not as accurate as the RSM, it

comes very close as can be seen in Figure 3.4 (b) & (c).

26

(a) Standard κ-ε (b) SST κ-ω

In conclusion, the SST κ-ω model was chosen because it incorporates the free

stream behaviour of the κ-ε model and the near wall behaviour of the κ-ω model. The

velocity profile obtained from the SST κ-ω model very closely matched the velocity

profile from the RSM simulation. Additionally, it was able to solve the problem in an

acceptable timeframe and computational requirement (296 iterations) whereas RSM

required nearly four times as many iterations.

(c) RSM

Figure 3.4 Velocity contours of three different turbulence models

27

4 Method The procedure for setting up and running each simulation was the same.

Different simulations sometimes shared the same geometry and meshing and therefore

a standard geometry and mesh could be created and transferred over to each simulation.

The workflow involved the following steps.

4.1 Geometry

The geometry was first created in DesignModeler, the geometry creation

application in ANSYS Workbench. DesignModeler is a parametric modelling

application. Parametric modelling allows for the use of parameters to define a geometry

or model. As a result, the geometry can be altered with ease as opposed to direct

modelling. Direct modelling is analogous to using constants in defining dimensions

whereas in parametric modelling, variables are used. While direct modelling is a

perfectly viable modelling approach, for complex parts or assemblies or parts that have

multiple dimensions and variations, parametric modelling is a better choice.

The geometry of the T-Junction was identical to the geometries described in the

cylindrical pipe and rectangular duct cases by Ziada et al. in [1]. The cylindrical duct 2D

cross-sectional geometry was only used for a few initial simulations as the behaviour of

the flow would be drastically different in a 3D scenario because the flow is cylindrical

(Figure 4.2). Therefore a 2D simulation would not be an accurate representation. On the

other hand, the cross-section of the rectangular duct could be considered the same in

the z-direction and so a 2D representation of the flow would

be reasonably accurate at the central axis of the pipe.

The most important dimension of the pipe setup was

the length of the branch pipe as this was determined by the

wavelength of the primary branch acoustic node as shown in

Figure 4.1.

Figure 4.1 Primary branch acoustic mode [1]

28

Figure 4.2 (left) 3D flow in a pipe of cylindrical cross section [43] (right) 2D flow in a pipe of cylindrical

cross section at the central axis but is the same for a rectangular duct [44]

Figure 4.4 shows one of the first geometries created for this project. This

geometry was created using the geometry described for the cylindrical pipe case in [1] as

shown in Figure 4.3. The half-length of this specific transition zone was determined

using the relationship:

DL ⋅= 1

where L is the half-length of the transition zone and D is the diameter of the transition

zone and the main pipe. The length of the branch pipe, LB was 1.42 m. These figures are

obtained from a table from [1] that details the dimensions of eight cylindrical pipe cases

(Figure 4.3(right)). The 45° chamfer shown in the dotted circle in Figure 4.3(left) was

ignored so that the flow separation and vortex formation could be better visualised. The

diameter of the branch pipe, d was 0.076 m and D was 0.102m. The length of the

transition zone 2L was calculated to be 0.204 m. The length of the main pipe was 1.5LB.

This is derived from the fact that the wavelength of the primary acoustic node λ, was

equal to 2LB and the length of the main pipe LM was equal to 3/4λ.

Figure 4.3 (left) Dimensions of T-Junction with cylindrical geometry [1] (right) Table displaying the

dimensions and relationships between L and D [1]

29

Figure 4.4 T-Junction geometry created using ANSYS DesignModeler

4.2 Meshing

Perhaps the part of the workflow that influences the accuracy of the solution

most is the mesh. The mesh represents the continuous domain as a discrete domain. In

order to analyse fluid flows, the flow domain is split into smaller subdomains. The

governing equations are then discretized and solved inside each of these subdomains.

The subdomains are generally called cells or elements and a collection of these is called a

mesh or grid. The subdomains are made of basic geometry such as hexahedra and

tetrahedra in 3D and quadrilaterals and triangles in 2D.

The mesh density has a direct correlation to the accuracy of the solution. The

greater the mesh density, the more accurate the solution becomes. However, greater

mesh densities result in more computationally expensive simulations. Therefore, the

ideal mesh must be dense enough to obtain a reasonably accurate solution but not

30

excessively dense that a simulation requires an unreasonable amount of time. Meshes

can be optimised so that the density is greater in areas of interest or areas of important

flow structures. This allows for efficient use of computational resources.

Mesh creation in ANSYS is achieved using the Meshing application. This is the

second step of the workflow. Initially, the geometry shown in Figure 4.4 was meshed

using a quadrilateral based mapped face meshing method. Mapped face meshing creates

a very uniform mesh of either quadrilaterals or triangles as specified. In order to achieve

greater mesh complexity, the edge sizing tool was used in conjunction with mapped face

meshing. This gave greater control over the mesh as it allowed for the manual division

of cells at each region or wall.

The other meshing scheme that was used involved the use of inflation layers.

This is a much better meshing approach as inflation layers are useful for greater

resolution at the boundary layers of the flow. This means that the mesh density is much

greater at the boundary layer and every layer after the first layer grows in height by a

growth rate that can be specified. The first layer height Δy is calculated as follows:

−

+ ⋅⋅∆⋅=∆ 1413

Re80yLy , where µ

ρ dV ⋅⋅=Re

This equation for Δy is from the ANSYS 11 documentation [45]. Δy+ is the desired y+

value which is 1 for κ-ω turbulence models . ρ is the density of the fluid, V is the

velocity, d is the diameter of the pipe and μ is the viscosity. Using this, the first layer

height was calculated to be 2x10-4 m. However, the first layer height was set at 1x10-4m as

this was to capture the boundary layer with a good level of accuracy. For this case, a

growth rate of 1.1 was used, i.e. each layer was 10% taller than the layer before it. The

mesh density was lower in other parts of the flow such as the central region of the T-

Junction and around the central axis of the branch and main pipes. In this region,

triangles were used rather than the quadrilateral scheme in the inflation layers.

The results for both methods were expected to be different in that the

method using inflation layers should result in a much more accurate solution. This is

detailed in the results section. Figure 4.5 shows the two different meshing approaches.

31

(a) Mapped face meshing with edge sizing (quadrilaterals)

(b) Meshing using inflation layers (quadrilaterals and triangles)

(c) Inflation layers at the edge of the geometry magnified.

Figure 4.5 Two different meshing approaches

32

4.3 Problem Setup

The problem can now be set up using Fluent. This is a very important step as it is

crucial that the inputs are correct. A CFD solution is only as strong as its inputs. Initially

setting up a problem in Fluent is a long process. However, for subsequent problems, the

setup file may be duplicated and worked upon. This means that the settings for all the

simulations will be the same. Caution must be exercised using this method because a

wrong setting during the initial setup will carry on to duplicated files.

The most important step in the problem setup was indicating the time domain –

steady or unsteady (transient). Initially, the Fluent solver was set to obtain a steady-state

solution. This can be set in the menu on the left under the ‘General’ heading. Steady-

state refers to the flow system when the fluid properties do not change with respect to

time. This solution isn’t exactly valid for the case of a T-Junction as the flow is turbulent

and turbulent flows are unsteady by definition. However, a turbulent flow can be

statistically stationary [38]. The goal of the steady simulation was to obtain a flow field

that resembled an actual flow field in a T-Junction. Once this was achieved, the goal of

the simulation shifted to generating an unsteady or transient solution. A transient

solution models the flow at every time step and it becomes possible to visualise the flow

as it reaches as it reaches a steady-state from a time of t = 0s. The time step can be

specified by the user. Obviously, this type of simulation would require significantly

greater time investment and computational power. The following steps were then taken

to set up the problem:

1. The turbulence model was specified from the list of models as SST κ-ω.

2. The fluid in question was natural gas (methane). The Fluent database

contains a large selection of materials to choose from. Methane was selected

from this list with the properties: Density, ρ = 0.6679 kg/m3; Viscosity μ =

1.087x10-5 kg/ms.

3. If a UDF (User-defined function) is created, it should be added to the

problem setup under the ‘Define’ menu.

33

4. Next, the cell zone conditions were set. This step assigned the material

selected in step 2 to the geometry. Since air was the default material, this was

changed to methane.

5. The boundary conditions for the problem can now be set. The main user

inputs here are the inlet velocity and the turbulence specification. For all the

simulations in the project, the velocity was input as components and the

turbulence was specified by the turbulent intensity and the hydraulic

diameter. Only the x-velocity is entered as the flow is considered horizontal

at the inlets. The turbulence intensity was set at the default value of 10%. The

values of hydraulic diameter were calculated depending on the cross

sectional profile of the pipe or duct. The hydraulic diameter was calculated

using the following equations from :

rr

rdh ⋅=⋅⋅⋅⋅

= 22

4 2

ππ

(Circular Pipe) where r is the radius of the pipe.

)(2

)(24

baba

babadh +

⋅⋅=

+⋅⋅

= (Rectangular Duct) where a and b are the height and

width of the duct. All dimensions in m.

Cylindrical Pipe Rectangular Duct dh, branch dh, main dh, branch dh, main

0.076 0.102 0.07 0.090254 Table 4.1 Hydraulic diameters calculated for the cylindrical pipe and the rectangular duct

6. The pressure-velocity coupling scheme was set to SIMPLE (Semi-implicit

method for pressure-linked equations) and the spatial discretization method

for the momentum, turbulent kinetic energy, and the specific dissipation rate

were set to ‘Second Order Upwind’. Upwind schemes are a class of numerical

discretisation methods for solving hyperbolic partial differential equations.

The origin of upwind methods can be credited to the work of Courant,

Isaacson and Rees [48]. The order of discretization refers to the convective

terms of the equations. Second order discretization generally yields better

accuracy while the convergence is more robust in first order discretization.

34

7. In the monitors tab, the convergence criteria for the residuals of the solution

can be set for the values of continuity, x-velocity, y-velocity, κ and ω. As

mentioned before, residuals are a measure of how well a solution satisfies the

discrete form of each governing equation. Some tutorials suggest a steep

convergence criterion such as 1x10-6 . Initially for the simulations, the

convergence criteria were set at 1x10-6. However, it was soon discovered that

such a high degree of accuracy was not absolutely required and was an

unnecessary time investment. This is demonstrated in the results section.

The monitors tab also allows the user to create surface and volume monitors.

This enables the creation of very useful custom plots that retrieve data from a

desired location in the flow.

8. The solution must now be initialized. Since the method of solving a problem

in Fluent is an iterative process, it requires an initial set of values to begin the

calculation. These values can be obtained from one or all of the parts of the

flow as specified in the drop down menu. The initial values were computed

‘all zones’ for all simulations.

9. For a transient simulation, Fluent gives the user the option of recording

animations of the solution once the solution has been initialized. This is an

extremely useful tool in the analysis of the results.

10. Finally, the calculation can be run. The number of iterations of the solution

to be calculated for and a few other plotting options can be specified here.

4.4 Analysis and Post-Processing

The results of the simulation can be analysed either within Fluent or CFD-Post.

Fluent offers a range of tools such as contour maps, vector plots, graphs, reports and

animations that can be generated in order to analyse data. CFD-Post enables the

creation of presentation-quality images and graphs.

35

5 Results

5.1 Mesh Influence

The difference in mesh density and composition can

greatly affect the solution. Two different meshes were

presented earlier. Steady simulations were run on each of the

meshes and the differences were quite evident. The y+ values

at the walls of the branch pipe, main pipe and T-Junction

were smaller by an order of magnitude of two in the case of

the mesh shown in Figure 5.2

.

Figure 5.1 Bad meshing and resulting y+ values

Geometry # 1 L 0.102 m LB 0.482 m D 0.102 m d 0.076 m Chamfer No VB 30 m/s

Table 5.1 Section 5.1 pipe parameters (See Appendix)

36

Figure 5.2 A more ideal mesh and resulting y+ values

Although there were more nodes in the case of the better mesh (64,338 elements

vs. 18,223 elements), the bad mesh required a greater amount of iterations (254 vs. 205

for the better mesh) for the solution to converge. However, the time required per

iteration was greater in the case of the better mesh. This leads to the conclusion that the

number of iterations required are not an indication of how complex the mesh is, but

rather a characteristic of the quality of the mesh. An extremely dense mesh is extremely

wasteful and although the calculation may finish in fewer iterations, the time required

per iteration would be much higher. Therefore, it is necessary to generate a mesh which

has enough complexity where required (such as near-wall and boundary layers) but also

reduced density in areas where the flow does not have to be modelled at such a fine

level.

37

A velocity contour map reveals drastic differences between the two meshes. It is

evident that near-wall and boundary layer fluid behaviour is badly modelled in the case

of the bad mesh. The contours are much more realistic in the contour map on the right.

Notice the areas of separated areas high velocities (indicated by the orange to red areas)

on the contour map on the large mesh. It makes more sense that the areas would be

connected and the resulting velocity would be greater at the entrance to the main pipe as

a result of continuity. The continuity equation states that the flow rate into a system is

equal to the flow rate out of a system so the reduction in area would mean an increase in

velocity from the following equations.

∑ ⋅= iiAQ υ and outin QQ =

where Q is the volume flow rate, A is the area of the cross section of the flow and V is the

velocity of the fluid. Figure 5.3 clearly shows that the mesh used on the left is a very poor

choice when it comes to modelling near-wall behaviour. Therefore, it is logical to say

that better results can be achieved with a denser mesh near the walls of the T-Junction.

From my experience, I believe that the time invested in generating a well-tailored mesh

is rewarding when it comes to the solution generated. In this case, the mesh used on the

right required a lot more time to create than the one on the left.

Figure 5.3 Difference in velocity contours between the bad mesh (left) and the better mesh (right)

38

Table 5.2 Section 5.2 pipe parameters

5.2 Effect of Convergence Criteria

Previously in the section on problem setup, it was

mentioned that a convergence criteria of 1 x 10-6 was highly

unnecessary. It was noticed that reducing the convergence

criteria greatly increased the accuracy of the flow

downstream of the T-Junction. As, the region beyond the T-

Junction was not of great importance in this project for

analysis, it was decided that a loss of accuracy was

acceptable.

Figure 5.4 Results of a simulation between convergence criteria of 1 x 10-3 (left) 1 x 10-6 (right)

According to Figure 5.4 above, it is clear that the flows upstream and at the T-

Junction are almost identical in both cases. The difference is that the simulation with

convergence criteria of 1 x 10-6, the flow downstream of the T-Junction is more

accurately modelled, but at an unjustifiable computational cost as the region beyond the

T-Junction is not being analysed in this project. Although there is a marginal increase in

accuracy in the flow coming into the junction, it is offset by the computational cost.

Therefore, it was decided that convergence criteria of 1 x 10-3 would be used throughout

Geometry # 1 L 0.102 m LB 0.482 m D 0.102 m d 0.076 m Chamfer No VB 30 m/s

39

the project. The solution had converged after 205 iterations for the 1 x 10-3 case whereas

the 1 x 10-6 required 1621 iterations. Figure 5.5 shows this and a few conclusions can be

drawn from this plot. Firstly, I believe that if time is not an issue, opting for a lower

convergence criteria is a good choice. Therefore, for a particular geometry, with the

same mesh, a pilot simulation with unreasonably low convergence criteria should be

run. From the residual plot of this, the point at which all the residuals reduce with a

steady slope (after ~920 iterations in this case) is a good indicator as to what

convergence criteria must be used. Following simulations can be run with the

convergence criteria corresponding to ~920 iterations. According to the plot below,

5x10-5 is a good convergence criteria provided time is not an issue. The difference

between 5x10-5 and 1 x 10-6 is small compared to the difference in iterations between the

two points. From other simulations conducted, the plot continues with the same slope

indefinitely.

Figure 5.5 Residual plot for convergence criteria of 1 x 10-6

It is important to note that for these simulations, although LB = 1.42 m, it didn’t

necessarily have to be so. Ignoring the effects of acoustics, LB only needed to be long

enough so that the fluid at the inlet would not have a significant influence on the fluid

flow in the T-Junction as would be the case if LB was too short.

40

5.3 T-Junction with Chamfered Inlets

These were a series of test simulations run to

investigate the flow in a T-Junction with chamfered inlets.

Geometries #2 was used (Table 5-3). T-Junctions with

chamfered inlets are featured in [1] where pipes of circular

cross section are used. Once again, convergence criteria of 1

x 10-3 were used.

The major difference in this case as compared to a T-

Junction without chamfered inlets is the reduction of vorticity. The simplest definition

of vorticity is the tendency of an element in a fluid to turn. It is also known as the curl of

the velocity vector. In the context of fluid dynamics, vorticity can be related to the

amount of circulation in a fluid flow [50].

Figure 5.6 Vorticity contours of a T-Junction with chamfer (left) and without (right)

The contours were set at a range from 0-1000 s-1. Therefore, it is clear from

Figure 5.6 that the maximum vorticity is a greater value in the T-Junction on the right. It

is important to minimise the amount of vorticity because it excites acoustic resonance.

However, for the purposes of simulations, T-Junctions without chamfered inlets will be

used because it is easier to generate oscillation. This is also one of the reasons why a

majority of the experimentation in [1] was carried out in non-chamfered T-Junctions.

Geometry # 2 L 0.102 m LB 0.482 m D 0.102 m d 0.076 m Chamfer Yes VB 30 m/s

Table 5.3 Section 5.3 pipe parameters

41

5.4 Rectangular Duct T-Junction (Steady)

From this point onwards, the geometry

of the T-Junction used was the same as in the

case of the rectangular duct in [1] as extensive

experimentation has been carried out in with

this setup. Therefore, data from these tests can

be used for the simulation. Four different

rectangular cross-section T-Junction

geometries of the same basic template and

dimensions were used (Figure 5.8).

Figure 5.8 Four cases of rectangular geometry T-Junctions based on the template on the left [1]

Table 5.4 Section 5.4 pipe parameters (see Appendix)

Geometry # 3 (Case 1) 3 (Case 2) 3 (Case 3) 3 (Case 4) L (m) 0.22225 0.22225 0.22225 0.22225 LB (m) 1.22 1.22 1.22 1.22 D (m) 0.127 0.0985 0.0985 0.07 d (m) 0.07 0.07 0.07 0.07 Chamfer No No No N/A VB (m/s) 43.27 43.27 43.27 43.27 dh, branch (m) 0.07 0.07 0.07 0.07 dh, main (m) 0.090254 0.0818 0.0818 0.07

Figure 5.7 3D velocity profile of the flow in a rectangular duct

42

The geometry in this series of simulations followed those described in [1]. The

T-Junction had a length (2L) that was 3.5 times the width (see Figure 5.8). The length of

each branch pipe was 1.22 m. VB was calculated using the following equation from [1]:

smVfVV B

BR /27.43=⇒

⋅=

Where VR is the reduced velocity (1.65), f is the frequency of the lowest acoustic mode of

the branches (118 Hz) and ℓ is the characteristic length which was equal to the half-

length of the transition zone (0.22225 m). In this configuration, VB was about 9% [1]

lower than VM. VM can be calculated using the continuity equation as was found to be

9.3% lower than the calculated VB.

( ) ( )

smVV

VAVAQQ

M

M

MMBB

MB

/7.47))127.007.0(()27.4307.0(2

22

2

=⋅×=⋅

⋅=⋅=⋅

The first set of simulations were carried out for the steady-state solution.

Vorticity contour plots obtained from the simulations of the four cases correspond to

the flow pattern and vortex formation shown in Figure 5.8. A velocity swirling vector

contour map clearly reveals the locations of these vortices (orange to red being the most

severe). It also shows an almost fully developed flow at the entrance of the T-Junction

Figure 5.9 Velocity swirling vector contour map of the flow in a T-Junction

43

Figure 5.10 Vorticity contours of the flow development in the branch pipe (flow from right to left)

(a) Case 1 (b) Case 2

(c) Case 3 (d) Case 4

Figure 5.11 Vorticity contours of the four cases. Vortex formation is strongest in the red regions

The geometry that is most prone to vortex shedding and interaction is Case 1.

The transition zone means that strong vortices are allowed to develop in the upper half

of the T-Junction especially around the central axis as shown in Figure 5.11 (Case 1).

44

Although Case 2 and Case 3 are identical in terms of cross section, the placement of the

transition zone has a major influence on the flow. Vortex formation is extremely mild in

Case 2 because there is no significant vortex formation taking place in the upper portion

of the T-Junction. The vortex being formed in the lower portion, though not ideal, is

acceptable because it is contained in the corner and not allowed to interact with other

vortices by the flow. However, in Case 4 we see the least amount of vortex formation.

Therefore, perhaps it would not be completely wrong to assume that a T-Junction

without a transition zone is the best type of T-Junction. This is best reinforced by the

fact that these results correspond with the data obtained in [1] as shown below.

Figure 5.12 Normalised acoustic pressure as a function of reduced velocity for the four test cases [1]

5.5 Rectangular Duct T-Junction (Unsteady)

A steady simulation of the flow simply isn’t enough to visualise the vortex

formation and interaction in the T-Junction. It does not show the initial development of

the flow and the flow characteristics once the flow has settled down. In order to visualise

this and periodic vortex formation as observed in [1], a transient or unsteady simulation

was set up in Fluent. The main difference in this case is specifying a time step and the

number of time steps in the solution calculation. In an unsteady simulation, a steady-

state solution is calculated at every time step and compiled together. The T-Junction

45

parameters for these simulations are exactly the same as shown in Table 5-3. A time step

of 0.000847 s is calculated from the frequency of the lowest acoustic mode (f = 118 Hz)

mentioned in section 5.4.

101×

=f

τ

where τ is the time step. f is multiplied by 10 so that a tenth of each period can be

visualised, otherwise only the beginning of every period can be captured. The results

obtained from these simulations were disappointing for the fact that the flow was

completely symmetric. Periodic vortex shedding was expected but the velocity in each

simulation, after reaching a steady state, simply “flat lined”. Vorticity contours in this

case, after reaching a steady state would resemble images such as those shown in Figure

5.11.

(a) Case 1 (b) Case 2

(c) Case 3 (d) Case 4 Figure 5.13 Velocity plots of the four cases at various locations of the T-Junction

Many other simulations were performed, some with slightly different inlet

velocities. In all cases, there was no periodic vortex shedding at the T-Junction. One of

the reasons for this was that since the flow was contained in a small pipe, any oscillation

in the flow would be contained by the walls. Also, since the flow is being sucked down

the duct, the effect of vortices would be negated. This is unlike external flows across a

46

cylinder or a flat plate where periodic vortex shedding is easily induced and can be

simulated with ease. The slightest difference in flow symmetry creates a snowballing

effect and this brings about periodic vortex shedding. The other reason that an

oscillation could not be induced was that unlike the real world where there can be

imperfections and asymmetries, Fluent models both branch pipes as completely

identical and perfect. This means that the flow is exactly the same on both sides if the

parameters at both inlets are the same.

5.6 Rectangular Duct T-Junction (Unsteady with UDF)

Since trying to simulate periodic vortex shedding at the T-Junction was not

possible, it was decided that a UDF (User-Defined Function) would be used. The Fluent

6.3 UDF manual describes a UDF as a function created by the user that can be

dynamically loaded with the Fluent solver. A UDF can be used to define custom

boundary conditions, material properties and source terms for the flow. UDFs are

written in the C programming language. Although knowledge of C programming was

useful, it was not completely essential. There was a wealth of information in the Fluent

manual and on the internet which enabled the creation of a UDF that would artificially

introduce oscillation in the flow (see Appendix for this UDF, modified from an example

UDF in [52]).

The idea was to generate a flow with a fluctuating velocity. A sine wave function

was therefore implemented in the UDF. The following function was used:

)( tSinaVV AInletB ⋅⋅+= ω

where ωA was the frequency in rad/s (ωA = 118 x 2π = 741.42 rad/s), a was the

amplification factor and t was the current time of the simulation. This increases in

increments of time step. The amplification factor simply increases the amplitude of the

oscillation in flow velocity. The effect of this UDF on the inlet velocity at the branch

pipe is shown in Figure 5.14. The velocity oscillated around 43.27 m/s at both inlets with

an amplitude of ± 2.86 m/s (this is controlled by a). Figure 5.15 shows the results

47

obtained by Ziada et al. [1]. The effect of using a UDF is immediately evident (Figure

5.16). The sampling point in Case 3 was placed in a region of almost no oscillation.

Figure 5.14 Oscillation of branch velocity with time

Figure 5.15 Instantaneous flow structure images and sketches for a complete cycle of Mode A oscillation

(VR = 1.65 and f = 118 Hz) [1]

0

0.005

0.01

0.015

0.02

0.025

-60-40-200204060

Flow

Tim

e (s

)

Velocity at branch pipe inlets relative to the central axis of the T-Junction (m/s)

Left InletRight Inlet

48

(a) Case 1 (b) Case 2

(c) Case 3 (d) Case 4

Figure 5.16 Velocity plots of the four cases at various locations of the T-Junction (with UDF)

Comparing Figure 5.18 to

Figure 5.15 it was evident that the

simulation was quite successful at

modelling the periodic vortex

shedding observed in [1]. The

regions bordered by red contours in

the vorticity contour maps in Figure

5.18 represent areas with the

strongest vortex shedding. The

dynamic pressure contours also

fluctuate at the specified frequency.

The lower, upper and central

regions of the T-Junction and the walls at the entrance of the main pipe show regions of

recirculation (Figure 5.17).

Figure 5.17 Combined vector and streamline map of regions of recirculation (blue regions)

49

(a) 0°

(b) 90°

(c) 180°

(d) 270°

Figure 5.18 Vorticity (500 – 1000 s-1) (left) and dynamic pressure (500 – 1000 Pa) (right) contour maps at different phase angles for Case 1

50

6 Conclusion This project initially set out with the goal of computationally simulating the fluid

flow in a T-Junction as investigated by Ziada et al. [1]. During the course of the project

however, it became a question as to whether CFD is actually viable for a flow

configuration such as this given the complexity of the physics involved.

The most practical model for numerical simulation of flow in T-Junctions was

found to be the SST κ-ω model as it produced results that were almost as good as the

RSM while only requiring a reasonable amount of computational power. It was also

better suited for this problem as it took on the traits of the κ-ε model’s reduced

sensitivity at the free stream and the κ-ω model’s accuracy at near wall and boundary

layers. The effects of meshing on the solution are also presented which shows the

importance of inflation layers for simulating the flow in a very efficient manner.

This study has shown that while it is possible to replicate the flow in a T-

Junction using CFD, the solution greatly depends on the amount of information that

can be conveyed to the software. Simulating external flows are easier in comparison as

the onset of vortex shedding is brought about by the slightest instability. The flow in a

T-Junction is inherently stable but at certain flow rates, acoustic resonance is generated

and this could not be modelled by Fluent using a simple simulation. Therefore, a UDF

was used to generate an oscillation in the flow which would initiate periodic vortex

shedding.

Using a UDF proved to be a viable method of imitating the flow. The effect of

flow-acoustic coupling was artificially simulated using a sin wave function which

created a periodic oscillation in the inlet velocities. The results of the simulations were

surprisingly close to the results from experiments [1]. Despite this, the fact remains that

this modelling was only possible because the flow characteristics were known. This is

not ideal when creating optimal T-Junction geometry as the user may not be aware of

the flow-acoustic coupling phenomenon. Therefore, the results of CFD alone cannot be

51

considered the definitive authority in this problem, at least from this study. However, if

the effects of flow-acoustic coupling are understood, then Fluent becomes a much more

potent tool especially when used in conjunction with actual experimentation.

Furthermore, as computational technologies advance the computational effort required

for simulations decrease. “Koomey’s Law” [20] describes how the computational

efficiency doubles roughly every 1.5 years. Considering these, CFD has a very promising

future.

Further study in this area should focus on modelling the acoustic aspect of the

flow. The end result i.e. the unsteady flow in this study was modelled by artificially

stimulating vortex shedding as RANS produces inherently stable results. Then perhaps,

an LES is a better modelling approach. LES offers a much better representation of

turbulent flow structures (Figure 6.1, Figure 6.2). The reason for using a RANS model is

its relatively low computational cost. However, when this is not an issue, LES is a

superior option. Obviously, if one desires greater accuracy still, DNS can be used but the

computational power required is simply unrealistic.

(a) DNS (unsteady) (b) RANS (steady/unsteady) (c) LES (unsteady)

Figure 6.1 Three different modelling approaches

Figure 6.2 RANS vs. LES

52

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57

Appendix

T-Junction Geometries

All dimensions in m.

Figure 0.1 Geometry #1

58

Figure 0.2 Geometry #2

59

Figure 0.3 Geometry #3 (Case 1)

Figure 0.4 Geometry #3 (Case 2)

Figure 0.5 Geometry #3 (Case 3)

Figure 0.6 Geometry #3 (Case 4)

60

Unsteady flow UDF

#include "udf.h"

DEFINE_PROFILE (unsteady_velocity_L, thread, position)

{

face_t f;

real t = CURRENT_TIME;

begin_f_loop (f, thread)

{

F_PROFILE (f, thread, position) = 43.27 + 3*sin(741.42*t);

}

end_f_loop (f, thread)

}

DEFINE_PROFILE (unsteady_velocity_R, thread, position)

{

face_t f;

real t = CURRENT_TIME;

begin_f_loop (f, thread)

{

F_PROFILE (f, thread, position) = -43.27 + 3*sin(741.42*t);

}

end_f_loop (f, thread)

}