Additional Materials On The Accompanying Cd

Published Date: 02 Nov 2017

UNIVERSITY

Faculty of Engineering and Computing

Department of Aerospace, Electronic and Electrical Engineering

Aerospace Systems Engineering

303CDE – Individual Project – 1213AAA

"Turbulent Jets: Mesh influence on the accuracy of simulations"

Author: Chris van Schalkwyk

SID: 201082

Supervisor: Dr Humberto Jesus Medina

Submitted in partial fulfilment of the requirements for the Degree of Bachelor of Engineering Honours in Aerospace Systems Engineering

Academic Year: 2012/13

Declaration of Originality

This project is all my own work and has not been copied in part or in whole from any other source except where duly acknowledged. As such, all use of previously published work (from books, journals, magazines, internet, etc) has been acknowledged within the main report to an entry in the References list.

I agree that an electronic copy of this report may be stored and used for the purposes of plagiarism prevention and detection.

I understand that cheating and plagiarism constitute a breach of University Regulations and will be dealt with accordingly.

Signed: Date:

Office Stamp

Copyright

The copyright of this project and report belongs to Coventry University

Abstract

[Write here a summary of the project and its product or findings. It is a simple summary of the findings in the research paper, more like a sales pitch towards to readers of this research paper. It is aimed to be concise, between 250 and 500 words.

I would like to see a brief of, the aim of the project. What is the aim and objectives in a small paragraph? It should also be mentioned what the Re number in question is and also what the main aim is. For example, Humberto had to compare steady jets with pulsed jets. In my paper I will be comparing URANS, RANS, LES as well as comparing the solvers used, discretisation techniques, and mainly grid independence for LES.

Turbulence values:

U = free stream velocity = 0.1377

I = turbulent length scale = 0.16*(Re) ^ (-1/8) =5.639%

k = turbulent energy = 3/2*(U*I) ^2 = 8.2735e-5

Epsilon = turbulent dissipation rate = Cmu^ (3/4)*k^ (3/2)*l^-1 = 5.79e-6

Omega = specific turbulent dissipation rate = Cmu^ (-1/4)*({SQRT (8.2735e-5)}/

{0.02135}) = ({rho*k}/ {mu})*(mu_t/mu) ^-1 = 7.7783e-1

The question arises: how fine does the mesh need to be in the LES region? And, how do we, after having made an LES (assuming that there are no experimental data with which to compare), verify that the resolution was good enough?

The first measure is probably to compare the modelled turbulence and stresses with the resolved ones. The smaller the ratio, the better the resolution.

Another, similar way is to compare the resolved turbulent kinetic energy to the modelled one.

The energy spectra are commonly computed to find out whether they exhibit a −5/3 range and if they do the flow is considered to be well resolved.

Another measure of the resolution may be to look at the two-point correlations to identify, for example, the ratio of the integral length scale to the cell size.

A less common approach is to compare the SGS (i.e. modelled) dissipation due to fluctuating resolved strain-rates to that due to resolved or time-averaged strain-rates. This can be verified or disproved by making energy spectra of the SGS dissipation to find the wavenumbers at which the SGS dissipation does in fact take place.

Since this process takes place in the viscous-dominating near-wall region, the required grid resolution must be expressed in inner variables, i.e. viscous units. In LES, the required grid resolution is ∆x+ ≃ 100, y + ≃ 1 (wall-adjacent cell centres) and ∆z + ≃ 30 where x, y, z denote the stream wise, wall-normal and span wise directions, respectively.

[Lars Davidson, Int. J. of Heat and Fluid Flow, Vol. 30(5), pp. 1016-1025, 2009]

Table of Contents

Additional Materials on the Accompanying CD

Project Submission

Meshes

Coarse (URANS)

Coarse (RANS, LES)

Dense (LES)

Simulations

URANS

RANS

LES

Matlab script file for post processing

List of Figures

[To be populated on completion of paper]

Nomenclature

Latin Letters

d m inlet impinge diameter

H m plate-to-nozzle spacing

N u = hd/kf - Nusselt number

p m nozzle-to-nozzle spacing

P e = U d/α - Peclet number

P r = ν/α - Prandtl number

r m radial distance measured from the inlet impinge axis

k - Turbulent kinematic energy

Re = Ue d/ν - inlet impinge Reynolds number

Sc = ν/Dim - Schmidt number

Sh - Sherwood number

St = f d/Ue - Strouhal number

Stc = d/Ue - Strouhal number for which Reduction in entrainment

Reynolds stresses is expected

t s time

Greek Letters

m boundary layer thickness

m wave length

m2 /s kinematic viscosity

kg/ms dynamic viscosity

kg/m3 water density

m inlet impinge width

- Kolmogorov length scale

- scalar diffusivity

- mean Batchelor length scale

Co-Ordinate System

[I need to change this to my current mesh – y axis must be the vertical and not x. So i justneed to swop the x and y axis of this graph. Include H(m) where the left arrow is that link the confinement plate to the impingement plate. Denote the inlet and outlet in basic temanology.]

Acknowledgements

[This is an optional section, used to acknowledge the support or contribution of your family, friends, colleagues, university staff (usually including the supervisor), your client and any other external sources of help. Usually about 500 words long. ]

Chapter 1

Introduction

Impinging Jets has been a study for many researchers over the counting years, due to the complexity around obtaining useful results, the variables in question are of great concern. Jets, as commonly named, discharge fluid from a nozzle of specific dimensions and generate a pre-calculated fluid flow characteristic. Namely denoted by Navier Stoke Equations, which is detailed in 'Chapter 2 - Literature Review'. Impinging Jets have the denotation of a normalized jet by which the exiting fluid from the nozzle penetrates a 'plate', known and denoted as the Impingement Plate. This is more greatly known as the rapid deceleration of fluid by an object, which in turn disturbes the fluid flow, alters the heat dissipation as well as fluid characteristics. The creation of an impingement plate does not have to be characterized by a flat plate perpendicular to the exit fluid flow of the nozzle, how it is seen in this dissertation. However, when the exiting fluid build up is interupted, an impinging 'plate' is created. The evolution of the nozzle is user defined, in the case at hand, a free-jet has been selected as to be more appropiate and for simplicity reasoning. A free-jet is denoted/define as a jet that discharges fluid from a nozzle, irrispective of the nozzle's geometry.

The three most widespread numerical simulation methods to predict turbulance is namely, Reynolds Averaged Navier Stokes (RANS), Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES), whereas RANS is the common method used in industry, benefitting from being the less resourcefull whilst DNS being the least common method used due to the high resource demand. RANS only resolves for the mean flow, which in turn averages out the turbulance fluctuations. Although used for over three decades, RANS is constantly under development and due for improvement. If the requirement is to resovle the turbulent fluctuations, LES and DNS are the preferred numerical simulation methods. LES' ground base principle is to resolve the energy carrying eddies, large eddies, while modelling the smaller eddies. LES is a step up from RANS, but a step down from DNS on computational time and requirement. The accuracy for LES is greatly grid dependant, thus user dependant, for wall bounded flows, like those experienced in Impinging Jets, a fine near wall grid is imperative. The previous has a direct relationship towards cost and resource availability, thus an appropiate meshing solution is vital, due to the sole fact that coarse LES meshs will not provide accurate predictions.

Impinging jets have a simple configuration but yet challenging geometry when the meshing aspects are taken into account. Due to the nature of fluid exiting the nozzle and impinging a surface, a basic symmetry or 'wedge' approach is not all that easy. The significance of this will be touched on in 'Chapter 4 - Methodology' and 'Chapter 5 - Meshing Guidelines'. The approach taken has been a simple, but yet effective manner by generating a simple, small Hexahedral mesh of the specific geometry chosen, at hand the fictive geometry is a 360 degree gemoetry. The basics of meshing and the simplicity of understanding how to generate a suitable mesh is still a wide field of study for CFD Engineers, esspecially those interested in LES and DNS. This is due to the sole fact that with RANS simulations one can increase the mesh size and the results will be directly propotunal. LES and DNS have a sligtly scewed approach, the mesh size can be increased to a certain point at where a plateu is reached. It is then to be noted that the benefit seen in RANS by having a directly propotunate relationship with the accuracy of results to grid size is not evident with LES and DNS. One will only increase resource requirements and incure greater expences and times on the simulation, which is not idealic towards many.

Literture studies provide some form of guidance onto meshing but none of them have directly stipulated on how to approach a problem from first base. The case at hand concentrates on transient stages (tending towards laminar flow) of flow, Re<4200, the main reason for this is due to computational resource, the greater the Reynolds Number, the more computational time requirement and resource requirement. Above the previous mentioned, a concentration of fluid vortecies is to be also to be briefly commented on in the boundry layer of the impinging plate, due to the geometry being ficitive.

Turbulent flows withing LES has to be understood prior to the completion of the disseration, thus saying it is vital to distiguish the variiance of smll-scale and large scale turbulance motion in the models at hand. In the high Reynolds number regieme, a larger seperation is prone with lengthscales, whereas the the geometry has a great influence on the large-scale motions, whilst the small-scales are vitually independant from the geometry. The Turbulent mixing, mainly found in the boundry layer, and transport is greatly controlled by the large eddies, large scale motions. The small-scale motions gradually decrease in size as the Reynolds number increases, most related to an exponential increase in Re, and a decrease in small-scale motions. Two main research areas of interest to small lengthscales, would be the energy cascade and the Kolmogorov hypothesis which will be touched on in 'Chapter 3 - Turbulence Lengthscales'.

Aims and Objectives

Aims

The aims are:

To provide a rough meshing guide to the average OpenFoam user. Ideally aimed for Impinging Jet cases, but the guide may be adapted accordingly.

To expand the current field knowledge of meshing with LES and RANS, using an Impinging Jet example.

Objectives

The objectives are

To validate the fictive geometry with previous research journals and papers published.

To identify key points and considerations when modelling and meshing, using Salome Meca v6.6.0.

To present a collaboration of data, graphical illusions, directly relating towards x/d and U. Proving the results obtained has a direct relationship towards grid dependency in LES and not that much of RANS.

To present a comparison of URANS, RANS and LES with coarse and dense mesh's.

To present a comparison of the results obtained from a coarse and dense mesh in LES solved using LES.

To present a comparison of results obtained using different two-equations models for RANS. Namely 'K-Epsilon' and 'K-OmegaSST'.

To present the direct relationship of Turbulence Length scales and grid scales, using a basic quality criteria method set out.

Dissertation overview

This section details the dissertation layout accompanied with a brief desciption of each chapter found in the pages to follow. The reader should not use this section of the dissertation as a 'deep-dive', but rather use it as a guideline for direction towards the correct chapter and section required

Chapter 1 – Introduction

This chapter keys out the relevance of the dissertation as well as the motivation behind this work, it clearly identifies the aims and objectives of this paper.

Chapter 2 - Literature Review

Most known literature on the subject of meshing and impinging jets, where relevant to the dissertation, notably including literature on free-jets, URANS, RANS, LES, Turbulence lengthscales, grid resolution for LES and also guidelines for designing a grid.

Chapter 3 - Turbulence Lengthscales

This Chapter highlights the importance of hand calculations for meshing, those specifically relating to Kolmogorov hypothesis, Taylor's hypothesis, turbulence scales, Turbulence spectrum, and lengthscales. This is a deep dive into lengthscales, combined with the relation to grid dependency for LES.

Chapter 4 – Methodology

Here the methods are justified, as well as the values chosen for comparison. The method for obtaining the results can be found here, with the added reference to the specific annex which will specifically detail the method of creating the geometry as well as the mesh. Pre- and Post-Processing will be touched on, but a deep dive on post-processing can be found to the end of the chapter. The software and hardware used will also be detailed for the user's benefit, if the models were to be recreated for validation purposes.

Chapter 5 - Meshing Guidelines

This chapter details a brief meshing procedure for obtaining useful LES results with Salome Meca v6.6.0, including the use of Turbulence Calculations, Length Scale comparisons.

Chapter 6 - Results and Discussion

The main body for the obtained results are to be found here, with the benefit of a RANS/LES comparison, LES – coarse/dense comparison, as well as an adaptive geometry creation for the investigation on sectioned meshing with RANS.

Chapter 7 - Project Management

This chapter details the time scales set out for the project, combinded with the resource availability and how the created Gantt Chart has evolved over time. A quality management review can also be found, as this benefits the user to see how projects can either be complicated, over engineered and/or blown out of perspective.

Chapter 8 - Critical Appraisal

A dispassionate and detailed discussion and analysis of the work and its outcomes, both positive and negative. The section will demonstrate the knowledge and expertise that you have gained from your project.

Chapter 9 – Conclusions

The findings from the work is summerized in this chapter as well as the future work to be considered relating to this topic.

Chapter 10 - Student Reflections

A reflective and critical appraisal of my personal performance, problems encounter along the way and a brief mention of how they were approached, resolved and what could have been done better or differently.

Appendices

Useful information to the reader can be found within the appendices, those including meshing illustrations, geometry design process, etc.

Chapter 2

Literature Review

Introduction

[emphasis should be placed on the topics touched in this section and what the main concern is, meaning the sole reason for writing this research paper. In this case our main focus would be the effect of using RANS and LES for impinging jets simulations, and how the grid dependancy effects the results obtained. But furthermore, how does this relate to a client, is LES actually that beneficial or is it not that beneficial?]

Numerical Analysis of Turbulence

The Navier-Stokes Equations

Navier-Stokes equations consist of two parts, namely the conservation of mass (continuity) and the conservation of momentum equations. The Navier-Stokes equations are used to solve fluid flow, the NS equations for incompressible Newtonian fluid may be written as:

(2.1)

(2.2)

It may be noted that in the equation 2.2, represents the fluid velocity at a given point in the flow domain,is the fluid density,is the pressure andis the given kinematic viscosity.Is given to take the form of:

(2.3)

Fluid can be denoted in two means, commonly known as laminar or turbulent, the decidiing factor which would relate to the predominante forces, either viscous or inertial.

Viscous forces:

(2.4)

Inertial forces:

(2.5)

In the year 1883, Osborne Reynolds, had derived a non-dimensional number used to quantitatively define the transition point of fluid flow, from laminar to turbulent flow. This is more commonly known as, Reynolds number and takes the form of:

(2.6)

Reynolds number is described to be the fluid velocity,, multiplied by the characteristic length of the flow domain,, devided by the fluid kinematic viscosity,. The Reynolds number has a great significance in fluid flow characteristics, for the sole reason that when the Reynolds number is low, lower than the given transition limit, the viscous stresses dampen out smaller perturbations, which in turn restrains the flow form turning turbulent. The adverse effect happens when the Re moves past the transitional limit, and the fluid flow is said to be turbulent, whereas when the viscous stresses become insufficient relative to the inertial perturbation then the fluid flow is known to be fully turbulent.

Reynolds Averaged Navier Stokes

The Reynolds Averaged Navier Stokes (RANS) computational method utlizes a time averaging of the main fluid flow field that the contribution of the fluctuating fluid flow components are grouped into what is known as the 'Reynolds stresses'. These stresses have to be modelled directly within the computational domain with the use of a turbulance model, which averages the Reynolds stresses to forfil the Navier-Stokes equations.

The RANS methods is a great favorite within many companies and universities, the reason is due to its cost effectiveness and yet simple computational method, the computational resource is ten fold less that of LES and many times less resourcefull than DES or DNS. The effective results obtained are weighted up against the resource required, thus results may not yield great or to the extent of effective results, but for the less resource comutational simulations, RANS is the better turbulance method to utilize. It is greatly beneficial to those running LES, DES and DNS simulations who would appreciate to validate journals and/or get the more dense mesh refinement done. The negative aspects of RANS is that the dimensions of the flow field is limited to the dimensions of the geometry.

Large Eddy Simulation

Large Eddy Simulations, also incorporate the base of the Navier-Stoke equations, as they are used to compute the larger eddies, known as the main flow field, whilst the smaller eddies are modelled. The extraction of the small-scale motions, commonly known as the smaller eddies from the main flow field is done by means of spatial filtering. LES is yet a step up from the previously mentioned RANS turbulance model, due to the fact of being able to solve more eddies and as mentioned models the smaller eddies whilst solving the larger eddies. The method of defining what portion is to be modelled by LES is utilized by the grid density or by means of the filter width. Though saying the above, the LES results obtained is not directily proportunal to the grid density, their is a point in which the grid has been refined 'too much' and the small eddies are 'too small' to be solved by LES, the utilization of DES and/or DNS has to be considered for excessive grid refinement.

With LES, the equations are solved by applying a low-filter to the continuity aswell as the Navier-Stokes eqautions, this low-filter seperates the small and large scales, a one-dimensional filtering operation for the velocity field is defined as:

(2.7)

In equation 2.7,is known as the filter width, andis denoted as the spatial co-ordinates, whileis the filter function.

Later seen, the use of length-scales assist in determining the applicable grid density in the streamwise (x), wall normal (y) and spanwise (z) directions. It is noted that the two-point correlation method is the considered approach for such determinations.

Subgrid-Scale Models

OneEqEddy

This one equation eddy model utilizes a balanced model equation to simulate the behaviour of k, turbulent kinematic energy. When referring to the openfoam guide and when looking in 'src/LESmodels/compressible/oneEqEddy/oneEqEddy.H' the following equations may be found for the SGS model:

(2.8)

and

where:

When the Smagorinsky SGS model equations is reviewed, it is noted that they are both of similar contex although the onEqEddy benefits from an additonal transport euqation for the use of the SGS Turbulent kinetic energy, k. This can be seen as:

(2.9)

(2.10)

The above term is the base of the oneEqEddy SGS model, breaking it down and describing the terms. On the left, as noted in the first term, this is the change in turbulent kinetic energy over time. The second convection and the third diffusion. Eugene de Villiers [2006] has noted that the right hand side of the above SGS model is:

"the decay of turbulance form the resolved scales to the sub-grid scales via the energy cascade".

The last term in equation 2.8 is the turbulent dissipation,

Explicit filtering in LES

Explicit filtering is a method introduced and modeled to insure that the frequency content modelled for all terms in the known equations are the same, the filtering method also removes unwanted high frequencies which are incorrectly interpreted by the the discrete grid. The common explicit filter equation may be seen in the form of:

(2.11)

The Sub-grid Scale term is denoted by:

(2.12)

In equation 2.11, the over bar represents the explicit filter and the tilde function refers to the implicit filtering. Lund and Kaltenbach [1995] have devised a simple and easy method of explicit filtering, this is to filter the velocity field upon the completion of each time step, this was known as the spectral cutoff filter.

One time step for and explicit time integrated method for the resolved velocity field can be seen as:

(2.13)

in equation 2.13, n is denoted as time levels andas the timestep,is known to be the change in the velocity field at one given timestep. When filtering the resolved velocity field in equation 2.13, we derive the following formula:

(2.13)

Extensive research has been done by Lund [1997], Gullbrand [2001], Amiri et al. [2005], Carati et al. [2001] and many more authors.

Impinging Jets (Single)

Impinging Jets are widely uesd in industry for vaious process, that including heating and cooling effects. The impinging jet is a simple inlet face with a downwards force of fluid, usually gravity induced, and nears the impinging plate/surface where the fluid impinges the surface and spreads out radially in the case at hand. The use of heating and cooling is due to the fact that jet impingement generates high levels of heat-transfer coefficient near the stagnation point, this is termed to be the point at where the flow field of the local velocity is equal to zero. Impinging jets may also be used for anti-icing systems, as can be seen in Figure 2.1 below, the anti icing system on aircraft wings could assit in the generation of great lift and lower drag, this is due to the fact that ice accretation on aircraft wings could cause a decrease in lift by circa 30% and similary increase the drag by circa 40%.

Figure 2.1 Iminging Jet basics, taken from 'Kito et. al. Heat Transfer- AsianA description...

Research, 37(8), 2008'

The stagnation point is located directly below the impinging jet, and as can be seen in Figure 2.2 below, this is the point when maximum heat transfer coefficient occurs, similarly it is where the highest Nussalt number will be found. This decreases with an increase in r, which is the distance from the centre of the Impinging Jet surface, in the case at hand, the further radially away from the centre.

Figure 2.2 Stagnation PointA description...

An excellent detailed description is given by Yule [1978], where the most importnant characteristics have been categorized of the exit flow from an impinging jet, Figure 2.3 is an extract of his work.

A description...

Figure 2.3 Impinging Jet/Tube exit flow characteristics, formation of eddies - "Physical structure of transitional jet." (Yule, 1978).

As noted by Yule [1978] Figure 2.3 the natural instability of the laminar shear layer produces vortex concentrations, in the formation of rings, the further downstream, away from the nozzle, the vortex 'rings' travel they increase in size. Yule also proved that thefluctuations increased as the distance from the nozzle inscreased, this closley related to the growth of wave deformations which resulted in a decrease of circumferential cross-correlations. Small scale motions are generated as the vortex rings have developed over time and distance from the nozzle, where as the core deformations have developed past larger than the known critical size. Yule noted that the known entangled vortex rings, vortices may be visible in the turbulent region downstream where they are now developed into large scale eddies.

Key Parameters

Nozzle – Impinge Inlet

The nozzle of choice, is one of a larger length, specifically the nozzle aspect ratio, nozzle length to nozzle diameter, is, the reason for this is due to the sole reason for exciting the flow and increasing the heat transfer coefficient. Garimella [1996] experimented upon the relationship between heat transfer and nozzle aspect ratio. Similarly to that mentioned above, Garimella proved that with

high heat transfer coefficients were recorded, whenthen the heat transfer coefficient dropped drastically and adversly, with the heat transfer coefficient had a slight tendancy to increase. Garimella also proved that the increase of the ratio of the, plate-nozzle-spacing and nozzle diameterhad a less distingtive effect on the nozzle aspect ratio. It is also to be noted that the Nussalt number,is directly proprortunal to the nozzle diameter, which inturn relates to an increase in turbulance intensity .

Non-Dimensional Distances

Non-dimensional distances relating to the nozzle are as follows:

Nozzle-Plate Seperation:

Plate-to-Plate distance:

Nozzle-to-Plate distance

Radial distance from the stagnation point:

Nozzle length to nozzle diameter:

The nozzle-plate seperation, has a distinctive effect on the jet flow when relating towards impinging jets. Cooper [1993] has studied a vast range of nozzle-plate seperation cases, in all if not most of his cases it is found that the discharge height, has a linear relationship between discharge height, stagnation turbulance and Nussalt number. Cooper's cases are based around high Re numbers, but this does shed light on the effect of .

Impingement Surface

As noted in Figure 2.1, the target area is classed and referred to as the Impinging surface/plate. The impinging jet is yet another characterisation of the jet, specifically as it determines the behaviour of the flow exiting the nozzle. Mostafa [2007] had done some extensive research into the differences of impingment on flat-, concave-, and convex-surfaces. Aydore [1997] has also added great research towards the topic, it was stated that the effect of the impingement surface may be visable as early as whereas it was noted that the formation of vortices became visable at around

Reynolds Number

As noted the Reynolds number is a fluid flow definition, it characterizes the fluid flow to be either laminar or turbulent, the jet characteristics are greatly influenced by the Reynolds number. Laminar flow is usualy that with a and adversly turbulent flow is flow with a .

Grid size / resolution

Thier are many published ways in determining the grid density, the most common and preferred method by many reseachers is the two-point correlation method. This is detailed in a later stage of the dissertation. Many found the energy spectra to be a unrealistic and non accurate measure of LES resolution, coupled with the ratio of the resolved turbulent kinetic energy to the total one. Assumed from journal articles and released papers, for LES, the recommended required grid resolutions are as follows:

Streamwise:

Wall-normal:

spanwise:

Other than the boundary layer thickness which decreases with an increase in Reynlds number, the viscous length unit increases as the Reynolds number increases and this drastically increases simulation times at higher Reynolds numbers. Davidson [2009] mentioned that their are a few methods in determining the resolution for LES simulations. The first being to compare the modelled turbulance and stresses with the resolved ones, the smaller the ratio, the better the resolution. The second way, similar to the first, is to compare the resolved kinetic energy with the modelled ones. It is said that the energy spectra is commonly computed to compare it with a range, if it does then the flow will be considered to be well resolved. As mentioend, the preferred method is to utilize the two-point correltation. Davidson identified this to be an example of the ratio of the integral length scale to the cell size. Another less common method detailed by Davidson is to compare the Sub-grid Scale dissipation due to the fluctuating resolved strain-rates to that due to the resolved/time-averaged strain-rates. It is also to be noted that the Sub-grid Scale dissipation taes place at notably the highest wavenumbers.

Chapman [1979] had two distinctive methods for resolving the grid resolution for LES and namely DNS, as this is at interest to compare LES towards. Reasoning for the above is their is no uses in comparing a turbulance method with one that is less powerful, but rather one that is more powerful, and then to draw a conclusion on how close one is to the more powerful model. The two methods discussed and researched by Chapman [1979] is that for both wall modelled and wall resolved motions. The method will be detailed in Chapter 3.

Add more things on meshing

Chapter 3

Methodology

[ Too many students waste valuable words talking about the "waterfall model" when in fact they used a prototyping or iterative/incremental approach. What really interests us isn’t the theory of the process model you used, but the reasons for choosing it – can you justify it? ]

Introduction

[This is just a small introduction into the methodology, what it is all about and what wil be touched on, for example the calculations for

Governing Equations

[still to populate]

Turbulence Lengthscales

Introduction

Once the flow regime has been broken down into two distinctive parts, namely, small and large lengthscales, it is easier to analyse the flow and apply the given calculation methods for grid refinement. Firstly, it is vital to understand the importance of the mathematical relations and as noted below this will be detailed. Resolving for the large-scale motions is of utmost importance in LES, with the combination of modelling the small-scale motions, the turbulence. This is more evident in high Reynolds numbers, but is seen in most flow ranges.

Kolmogorov hypothesis

Richardson [1922] has denoted the large eddies, with a size to be comparable to the overall flow region L, to break up and have instability, thus transferring their energy to the smaller eddies, this process circulates and smaller eddies transfer energy to smaller eddies. This process continues until the Reynolds number,, is of a magnitude to be stable. When stable the molecular viscosity has reached a suitable effectivness in dissipating kinetic energy. Kolmogorov [1941] has added great value to the work by Richardson, in identifying what is known as the Kolmogorov Scales. The small-scale depends only at the rate at which energy is supplied to it from the mean vlow and the kinamatic viscosity of the fluid in question. Noted from Tennekes and Lumley [1972], which is what the universal equilibrium theory of small turbulance is based on by Kolmogorov, the rate of dissipation form large-scale motions. Kolmogorov's hypothesis for local isotropy states the following: motions is equal in magnitude to the rate of energy supply to the small-scale

"at sufficiently high Reynolds number, the small-scale turbulent motions are statistically isotropic" [Pope, 2000].

This states that the statistics for the small-scale motions are universal in most high-Reynolds flows, Re>4500. Vital to point out that the lengthscale,, with the direct relationship towards Kolmogorov's Hypothesis for isotropy, this can be simplified into writing, . Noted by Kolmogorov, the first simularity hypothesis for small-scale motions and high-Reynolds numbers states:

"in every turbulent flow at sufficiently high-Reynolds number, the statistics of the small-scale motions have a universal form that is uniquely determined by the viscosity, v, and the rate of energy dissipation, 'epsilon'" [Pope, 2000].

After the above hypothesis, the Kolmogorov microscales are derived, giving the following relationships:

Kolmogorov microscales of length:

(3.1)

Kolmogorov microscales of velocity:

(3.2)

Kolmogorov microscales of time:

(3.3)

Kolmogorov has another hypothesis, from the coninuation of the first, it states:

"in every turbulent flow at sufficiently high Reynolds number, the statistics of the motions of scale 'l' in the range have a universal form that is uniquely determined by , independant of "

This denotes that introducing the lengthscales, splits what is known as the universal equllibrium range, into two unique subranges, namely the inertial subranges and the dissipations subrange. The hypothesis may be written as,.

Universal equilibrium range

(3.4)

Dissipation Range

(3.5)

Inertial subranges

(3.6)

Energy-containing range

(3.7)

Taylor’s hypothesis

Taylor's hypothesis [Taylor, 1938] is the approximation of spatial correlations by mundane approximations, the importance is great for the emperical solution of spatial correlations, which in turn would require the incorporation of the two-point correlation for . One technique discussed in the Taylor's hypothesis, is known as the 'flying hot-wire' approach which simply involves a moving, single wired probe. This moves rapidly through the turbulent field with a constant velocity along a line parallel to the direction 'x' with the unit vector set to . It can be noted that if the probe is at position at then:

The time is at location:

(3.8)

The velocity in question,

(3.9)

From the above, the mundane autocovariance can be obtained from the measured velocityis:

(3.10)

where is the probes distance travelled, measured with time, in seconds.

For stationary flows, whereas the turbulance intensity is small when compared to the mean velocity in the given direction, , a single stationary probe shall be utilized. The 'flying hot-wire' approach can therefor be applied with . As seen by, Lumley [1965], when relating to grid turbulance, is quite accurate when facilitating high order corrections in free shear flows, yet, Tong and Warhaft [1995], proved that under experimental data the free shear flows had failed.

The two-point correlation

The two-point correlation is one of the simplest and proven to be one of the accurate measurements in determining grid resolution. This can be referred to the spatial structure of any random field, a simple second order formula can be seen below: ;;

(3.11)

For turbulent fields, equation 3.11 can be rearranged as follows:

(3.12)

Equation 3.12, the correlation function may be defined as the effect of one point in the field on another point in the same field in question. This directly relates to the relationship between adjoining velocity fluctuations if referred and linked to turbulence.

The two-point correlation formula found in 3.12 can be rearranged when considering homogeneous isotropic turbulence, as this is expressed with two scalar functions:

(3.13)

Functions,andare known as longitudinal and transverse autocorrections. Introducing the co-ordinate system, with directions,, unit vector, and the relationyields the following:

(3.14)

(3.15)

and

The continuity equation implies:

(3.16)

Therefore, equation 3.12 equates to:

(3.17)

Equation 3.17 implies that during isotropic turbulence,is completly determined by. The two lengthscales that are of great importance would be the integral lengthscales and the Taylor microscales. See Chapter 3.5.2 and 3.5.3.

Turbulence Scales

Turbulent energy lengthscales

Once a simple RANS case has been run, it is possible to obtain the energy carrying eddies more commonly known to be the turbulent energy lengthscales:

(3.18)

Equation 3.18 is denoted as an estimated lengthscale from the RANS model simulated, the subscript "ERANS" is denoted for a RANS model. To get a true indication of the integral scale, Kang et al [2003], noted that when the constant 'A' in equation 3.18 is taken to harmony one may achieve this due to the sole fact that the turbulent energy lengthscale defines the aize for the large eddies, carrying energy.

Following from equation 3.18, the turbulance Reynolds number is:

(3.19)

From equation 3.19, we can deduce the following:

(3.20)

(3.21)

Combining equation 3.20 and 3.21, the following findings may be concluded:

(3.22)

Integral lengthscales

From Chapter 3.4, we can define the integral scale, one example would be if we utilize the same co-ordinate system with directionand unit vector , then the Integral scale,is define as:

(3.23)

The longitudinal integral scale:

(3.24)

The tranverse integral scale:

(3.25)

when considering equation 3.17, 3.18, 3.20, .

Taylor microscales

Second to that of the Integral lengthscale, the Taylor microscalehas just such a great importance, it is defined as:

The longitudinal Taylor scale:

(3.26)

The tranverse integral scale:

(3.27)

Considering equation 3.17, the Taylor microscale can be derived as:

(3.28)

when referring to equation 3.28, then equation 3.26 and equation 3.27 can be related as:

(3.29)

Batchelor length scale

The average batchelor length scale,is notably a legthscale of the smallest scalar motions:

3.30)

which is derived by deviding the average Kolmogorov length scaleby the square root of the Schemidit number, denoted as:

(3.31)

Grid Size / Resolution / Cost

Comment on the Chapman [1979] method – from: [6]

H. Choi, P. Moin, 2011. Center for Turbulence Research Annual Research Briefs 2011. Grid-point requirements for large eddy simulation: Chapman’s estimates revisited, pp.31-36.

X – axis

Y – axis

Z - axis

All Mesh's

0.21625

0.4575

0.21625

Table 3.1 Overall domain bounding box

Coarse Mesh

Dense Mesh

Quater Mesh

Total number of Points

-

1576436

918636

Total number of cells

-

1550500

894250

Total number of faces

-

4677225

2706850

Max. Cell Volume ()

-

3.71403e-07

3.71403e-07

Min. Cell Volume ()

-

1.60938e-11

1.68887e-11

Total Volume ()

-

0.0226135

0.00591078

Max. Mesh non-orthogonality

-

38.6556

38.6556

Ave. Mesh non-orthogonality

-

3.66653

4.83695

Max skewness

-

0.845657

0.845657

Max aspect ratio

-

117.224

117.224

Time Step ()

-

1

1

Average time ()

-

30000

30000

No. iterations during averaging

-

30000

30000

CPU effort (hours)

-

23

15

Table 3.2 RANS Simulation Mesh Characterisitics

Full Mesh

(Dense)

Full Mesh

(Coarse)

Quater Mesh (Coarse)

Total number of Points

15509416

1537176

927526

Total number of cells

15395625

1507500

900000

Total number of faces

46300050

4551900

2727150

Max. Cell Volume ()

6.05259e-08

7.80024e-07

7.80024e-07

Min. Cell Volume ()

6.55289e-12

2.21113e-11

2.21113e-11

Total Volume ()

0.0226165

0.0226108

0.00591007

Max. Mesh non-orthogonality

39.5309

38.1871

38.1871

Ave. Mesh non-orthogonality

2.58279

4.22951

5.48457

Max skewness

0.651528

0.992249

0.992249

Max aspect ratio

66.5041

49.8885

49.8885

Time Step ()

1e-05

1e-05

1e-05

Average time ()

1

1

1

No. iterations during averaging

100000

100000

100000

Approx. CPU effort (hours)

528

355

Table 3.3 LES Simulation Mesh Characteristics

As noted from the above tables, the LES take a great deal longer to simulate that that of the RANS turbulence models. This has proven to add limitations on the dissertation; computational resource has been to a limit as per the degree classification studying at this point in time. The RANS simulations have successfully run, with the added benefit of simulating various two equation models, namely those listed in table 3.3 below. It is also to be noted that due to the resource limit, many of the larger simulations could not be attempted; this will reflect in the Chapter 7 under Future Work as the cases have been set up and are ready to run. A partial simulation has been attempted for the LES meshes. The residuals plot of the cases indicates good mesh definition and good simulation denotation. By this engineering assumption, it is safe to run the mesh's on a cluster till the cases have converged or till the specified average time. When cost is a concern, as it is to many companies, the question is normally asked, does one need such a dense/fine grid/mesh? Is the results so important that the companies has to lay out 'x' times more on the budget just for some extra resolved and modelled eddies? Most will revert to a simple, no. Thus saying the Coarse_Full mesh is takes c. 528 hours to compute, which is 1.49 times more computational time. In this instance it would seem feasible for a company to cost up both simulations, or rather the Coarse_Full as one yields much better results, due to the non-symmetry plane applied when meshing the Quarter model, which does not represent a true flow pattern on the impingement surface.

RANS

Full Mesh

RANS

Quarter Mesh

LES - Dense

Full Mesh

LES - Coarse

Full Mesh

LES - Coarse

Quarter Mesh

KOmegaSST

NoWallFunctions

Yes

Yes

-

-

-

KOmegaSST

WallFunctions

Yes

Yes

-

-

-

KE

WallFunctions

Yes

Yes

-

-

-

KOmegaMedina

NoWallFunctions

Yes

No

-

-

-

OneEqEddy

-

-

Resource Limit

Resource Limit

Resource Limit

Smagorinsky

-

-

Resource Limit

Resource Limit

Resource Limit

Table 3.4 Simulations Attempted

Boundary Conditions

URANS

RANS

Two equation [K-e model, k-w model, SST]

LES

Data Analysis

Chapter 4

Results and Discussion

Introduction

As best compiled as possible, below is a compilation of results ranging from turbulence kinetic energy, to that of the axial/radial r.m.s fluctuating velocity relative to the radial distance measured from the center axis of the downstream flow. It is to be noted that the time dedicated to this thesis has not allowed for the compilation of data that was wanted from the start, but rather a solid base has been set for future and further studies. In the best interest to the reader(s), please refer to the Nomenclature for reference to specific letters denoted in the chapter ahead.

Please note that for simplicity all the graphical illusions have been placed after one another, this is due to an ease of reading for the user. All Figures in chapter 4 will follow after one another at the end of the chapter.

The following chapter serves multiple purposes:

it serves as a validation to the experimental results obtained from [1] which was the study of Steady and Pulsating Impinging Jets, quoted from the thesis states: "The effects of the Reynolds number, the nozzle-to-plate spacing and the Strouhal number were investigated systematically". The validation below proves that the open-source CFD work compiled has been on par with the experimental results obtained for Steady Jets as well as adding useful information for further researchers.

it examines the comparison of RANS, using various Reynolds Averaged Simulation (RAS) turbulence models for the computation of the specific geometries in question.

it examines the effect of applying wall functions and not applying wall functions to the specified RAS turbulence models mentioned above in (2).

it compares the mesh resolution of RANS and LES with the added benefit of visualizations for graphical understandings to the reader(s).

it acts as a base line for grid manipulation with LES and RANS simulations, whilst delivering a engineering comment on each method.

Lastly, it also acts as a base line for LES meshing, whereas the two-point correlations have been the preferred method as previously stated, which as seen in the later part of the document, provides a strong base line for further research to be attempted at ease. The effect of the meshing method has drastically improved the results, this can be seen in Figure 'x' where a coarse mesh has been compared with a dense mesh, and a Full mesh has been compared with a Quarter mesh. The study of Pulsating Jets is yet to be simulated, and will be reflected in Chapter 7, Further Work

The below results have been compiled for the benefit of the user/reader(s), they are not to be taken as engineering standards to lay comparisons to, but rather research based, to gain knowledge from and add benefit to.

Reynolds number choice and why?

As future work will be carried out on the heat transfer within LES on impinging jets it was vital to select a Reynolds number for this dissertation that could add benefit in the near future. With close relation to that said by Jamabunathan [1992] [2], Huang [1994] [3], and Garimella [2001] [4] it is said that the heat transfer increases in the stagnation point as the Reynolds number increases following the relationship, set out by Jamabunathan [1992, . As set out by Dr Medina [2009], there is great debate on the expression, within the research conducted. Thus it is safe to assume a Reynolds number bordering the laminar and turbulent region, namely the transitional flow phase. This is where the flow is neither laminar nor turbulent. It is said that the transitional Reynolds number is less than the average turbulence flow, which starts at and higher than laminar flow,. For the flow to be classed as turbulent, it has to pass through the stagnation point and transitional phase, as illustrated in Figure 'x1' below.

Figure 'x1' Laminar separation bubble (Lock, 2007)

The above caption is taken from Lock [2007], whereas Mayle [1991] also denoted a volume of slow recirculating fluid in between the points of separation and reattachment a 'Laminar separation bubble', otherwise known as a reattachment bubble. The Reynolds number is thus vital for the experiment at hand, due to the sole facts for; when a laminar boundary layer battles/fights to overcome the viscous effects and adverse pressure gradients, the laminar flow then separates and transition is known to occur in the shear flow near the surface. This then may lead to the reattachment of the molecules from the fluid to the surface, and thus forming a Laminar Separation Bubble (LSB) (Mayle, 1991). When closely studied, the Laminar Separation Bubble involves all stages for natural transition, those including reverse flow, circulating flow, etc.

RAS Turbulence models and why?

It has been aimed to concentrate on three different RAS turbulence models, namely: K-Epsilon model, K-OmegaSST, kklMedina. The reason for choice is due to the different two-equations in each of the models used. The model kklMedina has not been validated and proven for impinging jets to date, especially with the specific geometry arrangement, thus it will be a good stage to partly validate the model to that of the K-Epsilon and K-OmegaSST with and without wall functions. Notably, kklMedina is a model without wall functions, thus comparable to the K-OmegaSST without wall functions used in this dissertation. Below is a concise description of the standard K-Epsilon and K-OmegaSST model. From the literature review it is noted that the K-OmegaSST will produce better results without wall functions and will yield better overall results to that of the K-Epsilon model. From the literature review, it is also noted that the K-Epsilon model is and should only be used for turbulence models, but it is a great starting point with regards to mesh refinement for a cost saving entity. The sole reason is that one is able to compute a low cost simulation and still gain partial results, which in turn would yield as a building block for LES meshing and simulation case set up. Not mentioning its benefit in further research towards other RAS turbulence models, it may assist in shortening the simulation period of DES and/or DNS simulations.

Standard k-É› Model

This is the baseline of models, whereas it only solves for the kinetic energy (k) and the turbulent dissipation (ɛ), the rate of dissipation for velocity fluctuations. This turbulence model derives eddy viscosity from one single turbulence length scale, whereas when looked at closely the calculated turbulent diffusion is only that of which occurs at a specific scale. This is not ideal, as in reality all/most scales of motions contribute to the turbulent diffusion. This model also uses what is known as, the gradient diffusion hypothesis, which it uses to relate the Reynolds stresses to the mean velocity gradients as well as the turbulent viscosity. One great weakness of this model is it lack of sensitivity to adverse pressure gradients and its shortcoming is numerical stiffness when equations are integrated through the viscous sub-layer which are treated with dampening functions which have coherent stability issues [F.R. Menter [1993, 1994]. Despite some drawbacks, this mode has some valuable points, it is a robust model, it is very easy to implement, it’s not computational demanding meaning it is cost effective. It is also suitable for initial computations, as mentioned before, it’s used for ‘initial screening’ for the cases in this dissertation and is used as a baseline for future research.

k-ω SST model

This model, the k-ω Shear Stress Transport (SST) model is a modified k-ω whereas it combines the known Wilcox k-ω [2006] and the standard k-ω model. The Wilcox model is known to use the Boussinesq assumption; Wilcox used this in most/all of his models. The Wilcox k-ω model is used for near walls and the standard k-ω model for far walls, the SST model is fused using a blended function whereas the eddy viscosity formulation is modified to account for the transport effect of the principle turbulent shear stress (F.R. Menter [1993, 1994]]. The k-ω SST model has one greater drawback, namely that of which the turbulent viscosity is limited, it requires mesh resolution near the wall whereas the k-ɛ model does not. This model is more widely used for separated flows. On the other hand, the k-ω SST model is great in determining the amount of flow separation under opposing pressure gradients, whereas it accounts for the transport of the shear stresses. As mentioned by F.R. Menter, this model is highly recommended for high accuracy in the boundary layer simulations.

Flow characterization

As noted from Figure 4.1, 4.2 and the left graphs on Figure 4.3 and Figure 4.4 the exit flow type for this dissertation is fully developed, the base of this is due to the study done by Dr H.J Medina [2009] where it was mentioned to be a critical area of research for future engineering, due to the beneficial aid of cooling and heating effects on surfaces. To be noted, the study of heat has not been conducted in this dissertation, but heat has been included in the simulations for future study (Heat flow has been modeled and accounted for in most/all RAS turbulence models, as well as some LES simulation cases set up and ready to be run). Figure 4.1 (axisymmetric) has a detailed explanation within the graphs itself. As noted, axisymmetric has been achieved in the computational domain, as only one Reynolds number has been modeled, there may be no comparison to others Reynolds numbers, but the comparison of RAS models may be seen. It is to be noted that all RAS turbulence models used reveal a comparable exit flow contour. This is due to the Boundary Conditions set up in each individual case, special attention has been paid towards this, and as noted, the k-ɛ model has thus been converted to accommodate laminar flow. This is a great achievement, whereas predicted the k-ɛ model would not have the same contours as the adaptive Wilcox k-ω model, namely the k-ω SST. Figure 4.3 compares the Full 3D geometry to that of the Quarter 3D geometry, as seen at x/d = 2.5 and x/d 2.85, the exit flow has been disturbed and the Quarter 3D is not comparable to that of the Full 3D model in this case. In the current Quarter 3D mesh, the sides of the mesh has been denoted to be a SymmetryPlane, this is where the problem occurs when referring to the exit flow. As seen, the k-ω SST without wall functions is the most comparable model to that within the Full 3D case. As seen in Figure 4.2 is a detailed comparison of the k-ω SST model where the Full 3D and Quarter 3D is compared. It is also to be noted that x/d = 1, at the nozzle exit, a momentary velocity plateau is reached at around r/d = 0.5 for the Quarter 3D mesh. This is transparent over all x/d values for the Quarter 3D mesh. It is also to be noted from Figure 4.5 that the Quarter 3D mesh has some prone axial velocity spikes, yet again, in and around r/d = 0.5. When x/d = 2.5 is analyzed, it is safe to say that this has no comparison at all and should be neglected for comparison purposes. The nozzle exit, x/d = 1 does show to have some relationship, but yet again the sharpness of the Quarter 3D mesh is prone is all x/d values. One may assume that a Quarter 3D may be modelled solely for illustration and guidance purposes, one may model a Quarter 3D mesh, using the same BC’s as this dissertation for solely an initial set-up simulation and not one to use for results comparison.

Graph Comments

Figure 4.1-4.5

Figures 4.1/2/3/4/5 details the radial profiles for both mean axial and radial velocity components at different axial, x/d, locations. Where x/d=2.85 defines the effect of the mean velocity near the impinging plate/wall. The mean axial component, U, shows no significant differences, this is mirrored for x/d=1 and x/d=0.5. Where x/d=1 is at the nozzle exit flow, it may be noted that the k-ω SST without wall functions model has some slight variations in x/d=1 and x/d=2.85. This is due to the model being developed around near wall effectiveness.

Figure 4.4 details the comparison of the mean axial and radial velocity components for x/d=0.5 ; x/d=2.5 ; x/d=2.85. It can be noted that with relation to the experimental data supplied by Dr H.J Medina [2009], the k-ω SST with wall functions has similar relations besides the mean radial velocity component for x/d=0.5. It is also to be noted that Figure 4.4c/g/d/h details the significance of using a model without wall functions. It is concluded that the k-ω SST without wall functions has a slightly stronger accuracy than the experimental data provided, which is greatly understandable due to this being a theoretical simulation. The mean radial component near the wall (Figure 4.4h) details a greater stability margin for the exit flow, this effect is mirrored in the mean axial velocity component (Figure 4.4d). The flow around the stagnation point (r/d=0.5, for Re=4000) around x/d=2.85 moves slightly onwards using the k-ω SST with wall functions model, notably the maximum axial velocity. Whereas the maximum radial velocity is the same for x/d=2.85 and x/d=2.5, but the kkl-Medina model shows the that the maximum velocity has shifted onwards from the stagnation point. Detailed in from Figure 4.4e/f/g/h the kkl-Medina model used did not provide great accuracy on the mean radial velocity components for x/d=2.5 and x/d=0.5. Though the model did prove to be within comparison limits to that of the near wall region, x/d=2.85, as well as the nozzle exit flow 2.5, some abnormal excitation is distinctive in x/d=2.5 and x/d=0.5. In Figure 4.4h the kkl-Medina shows the lowest mean axial velocity for x/d=2.85.

The models all exhibit a very high local radial velocity component, which is followed by a sudden/rapid decrease in velocity, as detailed in x/d=0.5 ; x/d=2.5 and x/d=2.85. In turn this means that the initial exit velocity is not that great and does not exhibit great fluid moving forces. From previous research, it is noted that higher Reynolds number flows, have a more gradual drop in velocity, indicating a greater capability to force the fluid in the radial direction. As seen in Figure 4.5d, there is no recirculation (negative values of radial velocity, V), but in Figures 4.5a/b, x/d=0.5 and x/d=1 (which is near the nozzle exit) it can be concluded that there is distinctive recirculation near the centreline. The kkl-Medina shows the greatest values of recirculation near the nozzle exit, x/d=1. When comparing the radial velocities of the Full 3D model to that of the Quarter 3D model, it can be seen that the Quarter 3D model does not transpose the Full 3D model as per the recirculation activity prone in x/d=0.5 and x/d=1. Due to the ineffectiveness of the kkl-Medina model in previous studies, it was not attempted on the Quarter 3D model in this dissertation. The reason for this is due to the comparison made with the experimental data, the mean radial velocity, theoretically and practically has to have a rapid decrease after reaching its maximum, this is not the case with the kkl-Medina model.

Conclusion

Errors and uncertainty in CFD

Versteeg H.K., Malalasekera W. (2007). An Introduction to Computational Fluid Dynamics. England: Pearson Education Limited. 1-517.

Referances:

Keppens R.. (2008). Solaire postgraduate school, Copenhagen 3-7 November 2008.Paraview: A (novice) user perspective. 1 (1), 1-28.

Sdcsd

Figure 4.1: Axial Radial Velocity Components for various r/d values. Comparing the Full Geometry to the Quarter Geometry. Sampled at Re=4000, H/d=3

Figure 4.2: Axial and Radial Velocity Components for various r/d values. Compromised of the Full geometry, comparing various RAS turbulence models with and without Wall Functions to experimental data, Dr H.J. Medina [2009]. Sampled at Re=4000, H/d=3

Figure 4.3: Axial Velocity Components for various r/d values. Comparing the Full and Quarter geometry, plus various RAS turbulence models with and without Wall Functions. Sampled at Re=4000, H/d=3

Figure 4.4: Axial Skewness Factor for various r/d values. Comparing various RAS turbulence models with and without Wall Functions. Sampled at Re=4000, H/d=3

Figure 4.5: Axial Skewness Factor for various r/d values. Comparing various RAS turbulence models with and without Wall Functions, for Full and Quarter Geometry. Sampled at Re=4000, H/d=3

Figure 4.6: Centre-line Axial Velocity decay for various r/d values. Comparing various RAS turbulence models with and without Wall Functions, for Full and Quarter Geometry. Compared to experimental data, Dr H.J. Medina [2009]. Sampled at Re=4000, H/d=3

Figure 4.7: Centre-line Skewness Factor for various r/d values. Comparing various RAS turbulence models with and without Wall Functions, for Full and Quarter Geometry. Sampled at Re=4000, H/d=3

Figure 4.8: Turbulent Kinetic Energy for various r/d values. Comparing various RAS turbulence models with and without Wall Functions, for Full and Quarter Geometry. Sampled at Re=4000, H/d=3

ssssssssssss

Chapter 5

Project Management

[ The subsections shown below are only one possible structure for this section covering the conduct of the project. ]

Project Schedule

[ This could include the work breakdown structure, Gantt chart, and comments about how well you managed to keep to the original plan, or what adjustments were necessary. ]

Quality Management

[ Standards adopted, techniques used to review progress and evaluate outcomes, etc. ]

Chapter 6

Critical Appraisal

[ A dispassionate and detailed discussion and analysis of the work and its outcomes, both positive and negative. The section will demonstrate the knowledge and expertise that you have gained from your project.]

Chapter 7

Conclusions

[Optional introduction ]

Achievements

[Comment on what you have achieved in terms of product or other results, with reference to the original project objectives. ]

Future Work

[ Outline possible enhancements or extensions to the product, or further work needed to address outstanding issues, etc. ]

Chapter 8

Student Reflections

[ A reflective and critical appraisal of your personal performance, problems encountered and how they were resolved, lessons learnt, what could have been done better or differently, etc. ]